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1, , SETS, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , 2, , Number of Questions, , JEE MAIN, BITSAT, , 1, , 0, , 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , % Weightage, , JEE Main, BITSAT, , 2, 2, , Critical Concepts, , Rating of Difficulty, , CUS (chapter utility score), out of 10, , Types of Sets, Subsets,, Power Set, Operations on, Sets, Venn Diagram,, Cardinal Number of a Set,, Number of Elements in, Different Sets, , 3/5, , 5.5

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SETS, , 3

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EBD_7762, 4, , MATHEMATICS, , Topic 1: Sets, Cardinal Number of Sets, Types of Sets,, Disjoint Sets, Complementary Sets, Subset, Power Set, 1., , 2., 3., 4., , 5., , 6., , 7., , 8., 9., , The set of intelligent students in a class is :, (a) a null set, (b) a singleton set, (c) a finite set, (d) not a well defined collection, The number of the proper subset of {a, b, c} is:, (a) 3, (b) 8, (c) 6, (d) 7, Which one is different from the others ?, (i) empty set (ii) void set (iii) zero set (iv) null set :, (a) (i), (b) (ii), (c) (iii), (d) (iv), If A = {x, y} then the power set of A is :, (a) {xx, yy}, (b) {f, x, y}, (c) {f, { x} , {2y}}, (d) {f,{x} ,{y} ,{x, y}}, Consider the following sets., I. A = {1, 2, 3}, II. B = {x Î R : x2 – 2x + 1 = 0}, III. C = {1, 2, 2, 3}, IV. D = {x Î R : x3 – 6x2 + 11x – 6 = 0}, Which of the following are equal?, (a) A = B = C, (b) A = C = D, (c) A = B = D, (d) B = C = D, If X = {1, 2, 3, …, 10} and ‘a’ represents any element of X,, then the set containing all the elements satisfy a + 2 = 6, a Î, X is, (a) {4}, (b) {3}, (c) {2}, (d) {5}, Two finite sets have m and n elements. The total number of, subsets of the first set is 56 more than the total number of, subsets of the second set. The values of m and n are:, (a) 7, 6, (b) 6, 3, (c) 5, 1, (d) 8, 7, The set {x : x is an even prime number} can be written as, (a) {2}, (b) {2, 4}, (c) {2,14} (d) {2, 4, 14}, Given the sets, , A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}. Which of the, following may be considered as universal set for all the, three sets A, B and C?, (a) {0, 1, 2, 3, 4, 5, 6}, (b) f, (c) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, (d) {1, 2, 3, 4, 5, 6, 7, 8}, 10. Which of the following collections are sets ?, (a) The collection of all the days of a week, (b) A collection of 11 best hockey player of India., (c) The collection of all rich person of Delhi, (d) A collection of most dangerous animals of India., , 11. Consider the following sets., A = {0},, B = {x : x > 15 and x < 5},, C = {x : x – 5 = 0},, D= {x : x2 = 25},, E= {x : x is an integral positive root of the equation, x2 – 2x – 15 = 0}, Choose the pair of equal sets, (a) A and B, (b) C and D, (c) C and E, (d) B and C, 12. If a set is denoted as B = f, then the number of element in, B is, (a) 3, (b) 2, (c) 1, (d) 0, 13. Let X = {1, 2, 3, 4, 5}. Then, the number of elements in X, are, (a) 3, (b) 2, (c) 1, (d) 5, 14. Let A = {(1, 2), (3, 4), 5}, then which of the following, is incorrect?, (a) {3, 4} Ï A as (3, 4) is an element of A, (b) {5}, {(3, 4)} are subsets of A but not elements of, A, (c) {1, 2}, {5} are subsets of A, (d) {(1, 2), (3, 4), 5} are subset of A, 15. If f denotes the empty set, then which one of the following, is correct ?, (a), , fÎf, , (c) {f} Î {f}, , (b) f Î{f}, (d) 0 Î f, , 16. Which one of the following is an infinite set ?, (a) The set of human beings on the earth, (b) The set of water drops in a glass of water, (c) The set of trees in a forest, (d) The set of all primes, 17. The set A = {x : x Î R, x2 = 16 and 2x = 6} equals, (a) f, (b) {14, 3, 4} (c) {3}, (d) {4}, 18. A = {x : x ¹ x} represents, (a) {x}, (b) {1}, (c) { }, (d) {0}, 19. Which of the following is a null set ?, (a) {0}, (b) {x : x > 0 or x < 0}, (c) {x : x 2 = 4 or x = 3}, (d) {x : x 2 + 1 = 0, x Î R}, 20. Let A = {1, 3, 5} and B = {x : x is an odd natural number, less than 6}. Then, which of the following are true?, I. A Ì B, II. B Ì A, III. A = B, IV. A Ï B, (a) I and II are true, (b) I and III are true, (c) I, II and III are true, (d) I, II and IV are true

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SETS, , 5, , 21. If X = {1, 2, 3}, then the number of proper subsets is, (a) 5, (b) 6, (c) 7, (d) 8, 22. The number of non-empty subsets of the set {1, 2, 3, 4} is, 3 ´ a. The value of ‘a’ is, (a) 3, (b) 4, (c) 5, (d) 6, 23. If A = {a, {b}}, then P(A) equals., (a) {f, {a}, {{b}}, {a, {b}}}, (b) {f, {a}}, (c) {{a}, {b}, f}, 24. The set builder form of given set A = {3, 6, 9, 12} and, B = {1, 4, 9, ....., 100} is, (a) A = {x : x = 3n, n Î N and 1 £ n £ 5},, B = {x : x = n2, n Î N and 1 £ n £ 10}, (b) A = {x : x = 3n, n Î N and 1 £ n £ 4},, B = {x : x = n2, n Î N and 1 £ n £ 10}, (c) A = {x : x = 3n, n Î N and 1 £ n £ 4},, B = {x : x = n2, n Î N and 1 < n < 10}, (d) None of these, 25. Which of the following sets is a finite set?, (a) A = {x : x Î Z and x2 – 5x + 6 = 0}, (b) B = {x : x Î Z and x2 is even}, (c) D = {x : x Î Z and x > –10}, (d) All of these, 26. Which of the following is a singleton set?, (a) {x : |x| = 5, x Î N}, (b) {x : |x| = 6, x Î Z}, (c) {x : x2 + 2x + 1 = 0, x Î N}, (d) {x : x2 = 7, x Î N}, 27. Which of the following is not a null set?, (a) Set of odd natural numbers divisible by 2, (b) Set of even prime numbers, (c) {x : x is a natural number, x < 5 and x > 7}, (d) {y : y is a point common to any two parallel lines}, 28. If A = {x : x = n2, n = 1, 2, 3}, then number of proper, subsets is, (a) 3, (b) 8, (c) 7, (d) 4, 29. Which of the following has only one subset?, (a) { }, (b) {4}, (c) {4, 5} (d) {0}, 30. Which of the following statement is FALSE, , {, , 2, , }, , Reason : The total number of proper subsets of a set, containing n elements is 2n – 1., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., , ì1 2 3 4 5 6ü, 33. The set í , , , , , ý in the set-builder form is, î2 3 4 5 6 7þ, , ì, ü, n, , where n Î N and 1 < n < 6ý, (a) í x : x =, n +1, î, þ, ì, ü, n, , where n Î N and 1 £ n < 6ý, (b) í x : x =, n +1, î, þ, , 34., , 35., , 36., , 37., , (a) If A = x : x = 4, x Î N , B = {-2} then A ¹ B, (b) If A = {x : | x | < 2, x Î I}, B = {-1,1} , then A = B, (c) If {1, 2, 3, 4, 5}, B = {2, 1, 3, 3, 4, 4, 5}, then A = B, , {, , 2, , }, , (d) If A = x : x - 5x + 7 = 0, x Î R and, B = f, then A = B, 31. Which of the following is/ are true?, I. If A is a subset of the universal set U, then its, complement A¢ is also a subset of U., II. If U = {1, 2, 3, ....., 10} and A = {1, 3, 5, 7, 9}, then, ( A ¢ )¢ = A., (a) Only I is true, (b) Only II is true, (c) Both I and II are true (d) None of these, 32. Assertion : The subsets of the set {1, {2}} are, { }, {1}, {{2}} and {1, {2}}., , 38., , 39., 40., , ì, ü, n, , where n Î N and 1 £ n £ 6ý, (c) í x : x =, n, +, 1, î, þ, (d) None of the above, The set {x : x is a positive integer less than 6 and 3x– 1, is an even number} in roster form is, (a) {1, 2, 3, 4, 5}, (b) {1, 2, 3, 4, 5, 6}, (c) {2, 4, 6}, (d) {1, 3, 5}, If B = {x : x is a student presently studying in both classes, X and XI}. Then, the number of elements in set B are, (a) finite, (b) infinite, (c) zero, (d) None of these, Consider:, X = Set of all students in your school., Y = Set of all students in your class., Then, which of the following is true?, (a) Every element of Y is also an element of X, (b) Every element of X is also an element of Y, (c) Every element of Y is not an element of X, (d) Every element of X is not an element of Y, If A Ì B and A ¹ B, then, (a) A is called a proper subset of B, (b) A is called a super set of B, (c) A is not a subset of B, (d) B is a subset of A, Which of the following is correct?, I. Number of subsets of a set A having n elements is, equal to 2n., II. The power set of a set A contains 128 elements then, number of elements in set A is 7., (a) Only I is true, (b) Only II is true, (c) Both I and II are true, (d) Both I and II are false, If A = f, then the number of elements in P(A) is, (a) 3, (b) 2, (c) 1, (d) 0, The set of real numbers {x : a < x < b} is called, (a) open interval, (b) closed interval, (c) semi-open interval, (d) semi-closed interval

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EBD_7762, 6, , MATHEMATICS, , 41. Which of the following is true?, (a) a Î {{a}, b}, (b) {b, c} Ì {a, {b, c}}, (c) {a, b} Ì {a, {b, c}} (d) None of these, 42. The interval [a, b) is represented on the number line as, (a), (b), a, a, b, b, (c), (d), a, a, b, b, 43. The interval represented by, , 44., 45., 46., , 47., , a, b, (a) (a, b) (b) [a, b], (c) [a, b) (d) (a, b], The number of elements in P[P(P(f))] is, (a) 2, (b) 3, (c) 4, (d) 5, The cardinality of the set P{P[P(f)]} is, (a) 0, (b) 1, (c) 2, (d) 4, Let A, B, C be three sets. If A Î B and B Ì C, then, (a) A Ì C, (b) A Ë C, (c) A Î C, (d) A Ï C, The number of elements in the set, {(a, b) : 2a2 + 3b2 = 35, a, b Î Z}, where Z is the set of all, integers, is, (a) 2, (b) 4, (c) 8, (d) 12, Topic 2: Venn-Diagram, Operation on Sets., , 48., , 49., , If the sets A and B are given by A = {1, 2, 3, 4},, B = {2, 4, 6, 8, 10} and the universal set, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then, (a), , (A È B)' = {5, 7, 9}, , (b), , (A Ç B)' = {1, 3, 5, 6, 7}, , (c), , ( A Ç B )' = {1, 3, 5, 6, 7, 8}, , (d) None of these, If A = {1, 2, 3, 4}, B = {2, 3, 5, 6} and C = {3, 4, 6, 7}, then, (a) A – (B Ç C) = {1, 3, 4}, , (b) A – (B Ç C) = {1, 2, 4}, (c) A – (B È C) = {2, 3}, (d) A – (B È C) = {f}, 50. If the sets A and B are as follows :, A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, then, (a) A – B = {1, 2}, (b) B – A = {5}, (c) [(A – B) – (B – A)] Ç A = {1, 2}, (d) [(A – B) – (B – A)] È A = {3, 4}, 51. Consider the following relations:, I. A – B = A – (A Ç B), II. A = (A Ç B) È (A – B), III. A – (B È C) = (A – B) È (A – C), Which of these is/are correct?, (a) Both I and III, (b) Only II, (c) Both II and III, (d) Both I and II, , 52. What does the shaded portion of the Venn diagram given, below represent?, , P, , Q, , R, (a), (b), , (P Ç Q) Ç (P Ç R), , (c), , ((P È Q) - R) Ç ((P Ç R) - Q), , (d), , ((P Ç Q) È R) Ç ((P È Q) - R), , ((P Ç Q) - R) È ((P Ç R) - Q), , 53. Let A = {x : x Î R, x < 1}; B = {x : x Î R, x - 1 ³ 1} and, A È B = R - D, then the set D is, , (a) {x :1 < x £ 2}, , (b) {x :1 £ x < 2}, , {x :1 £ x £ 2}, , (d) None of these, , (c), , 54. If A È B ¹ f, then n ( A È B) = ?, (a) n( A) + n( B) - n( A Ç B), (b) n( A) - n( B) + n( A Ç B), (c) n( A) - n ( B ) - n( A Ç B ), (d) n( A) + n ( B ) + n( A Ç B ), 55. Which of the following properties are associative law ?, (a) A È B = B È A, , 56., , (b), , AÈC = C È A, , (c), , AÈ D = D È A, , (d), , ( A È B) È C = A È ( B È C ), , Let V = {a, e, i, o, u} and B = {a, i, k, u}. Value of, V – B and B – V are respectively, (a) {e, o} and {k}, (b) {e} and {k}, (c) {o} and {k}, (d) {e, o} and {k, i}, , 57. Let A = {a, b}, B = {a, b, c}. What is A È B ?, (a) {a, b}, (b) {a, c}, (c) {a, b, c}, (d) {b, c}, 58. Let A = {x : x is a multiple of 3} and, B = { x : x is a multiple of 5}. Then A Ç B is given by:, (a) {15, 30, 45,...}, (b) {3, 6, 9,...}, (c) {15, 10, 15, 20...}, (d) {5, 10, 20,...}, 59. Statement - I : Let A = {a, b} and B = {a, b, c}. Then,, A Ë B., Statement - II : If A Ì B, then A È B = B., (a) Statement I is true, (b) Statement II is true, (c) Both are true, (d) Both are false

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SETS, , 7, , 60. What does the shaded region represent in the figure given, below ?, P, , Q, , B, , 1, , (b), , 7, , 3, 2, , 6, , 4, , 9, , 5, , 1, , (c), R, , (a) (P È Q) – (P Ç Q), (b) P Ç (Q Ç R), (c) (P Ç Q) Ç (P Ç R), (d) (P Ç Q) È (P Ç R), 61. The shaded region in the given figure is, , 2, 8, 10, , B, 4, 6, , 9, , 1, , B, 4, 6, 8, 10, , A, , 3, , 2, , (d), 9, , 5, 7, , 65. Most of the relationships between sets can be represented, by means of diagrams which are known as, (a) rectangles, (b) circles, (c) Venn diagrams, (d) triangles, 66. Which of the following represent the union of two sets, A and B?, , A, , È, , È, , C, , B, , (a) A Ç (B È C), (b) A È (B Ç C), (c) A Ç (B – C), (d) A– (B È C), 62. The shaded region in the given figure is, A, B, , C, , (a) B Ç (A È C), (b) B È (A Ç C), (c) B Ç (A – C), (d) B – (A È C), 63. If A = {x : x is a multiple of 3} and, B = {x : x is a multiple of 5}, then A – B is equal to, (b), , AÇB, , (c) A Ç B, (d) A Ç B, 64. If U = {1, 2, 3, 4, ....., 10} is the universal set of A, B and, A = {2, 4, 6, 8, 10}, B = {4, 6} are subsets of U, then given, sets can be represented by Venn diagram as, 1, (a), , 3, 5, , A, , 7, , 2 4 6, 8 10, , B, 4, 6, , 9, , (a), , A, , B, , (b), , A, , AÈ B, , (c), , B, AÈ B, , È, , È, A, , B, AÈ B, , (a) A Ç B, , 5, , 8 10, , A, , 3, 7, , A, , (d), , A, , B, AÈ B, , 67. Which of the following are correct?, I. A – B = A – (A Ç B)., II. A = (A Ç B) È (A – B)., III. A – (B È C) = (A – B) È (A – C)., (a) I and II, (b) II and III, (c) I, II and III, (d) None of these, 68. Assertion : For any two sets A and B, A – B Ì B ¢., Reason : If A be any set, then A Ç A¢ = f., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 69. If A is the set of the divisors of the number 15, B is the set of, prime numbers smaller than 10 and C is the set of even, numbers smaller than 9, then (A È C) Ç B is the set, (a) {1, 3, 5}, (b) {1, 2, 3}, (c) {2, 3, 5}, (d) {2, 5}, 70. Let X and Y be two non-empty sets such that, X Ç A = Y Ç A = f and X È A = Y È A for some nonempty set A. Then

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EBD_7762, 8, , MATHEMATICS, , (a) X is a proper subset of Y, (b) Y is a proper subset of X, (c) X = Y, (d) X and Y are disjoint sets, 71. Let X = {Ram, Geeta, Akbar} be the set of students of, Class XI, who are in school hockey team and Y = {Geeta,, David, Ashok} be the set of students from Class XI, who, are in the school football team. Then, X Ç Y is, (a) {Ram, Geeta}, (b) {Ram}, (c) {Geeta}, (d) None of these, 72. Which of the following represent A – B?, (a) {x : x Î A and x Î B}, (b) {x : x Î A and x Ï B}, (c) {x : x Î A or x Î B}, (d) {x : x Î A or x Ï B}, 73. The shaded region in the given figure is, , È, , A, , of all right-angled triangles. What do the sets P Ç Q and, R – P represents respectively ?, (a) The set of isosceles triangles; the set of nonisosceles right angled triangles, (b) The set of isosceles triangles; the set of right angled, triangles, (c) The set of equilateral triangles; the set of right angled, triangles, (d) The set of isosceles triangles; the set of equilateral, triangles, 79. If aN = {ax : x Î N} and bN Ç cN = dN, where b, c Î N are, relatively prime, then, (a) d = bc, (b) c = bd, (c) b = cd, (d) None of these, 80. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 5},, B = {2, 4, 6, 7} and C = {2, 3, 4, 8}, then which of the, following is true?, (a) ( B È C)¢ = {1, 5, 9, 10}, , B, , C, (a) A Ç (B È C), (b) A È (B Ç C), (c) A Ç (B – C), (d) A – (B È C), 74. If A and B are non-empty subsets of a set, then, (A – B) È (B – A) equals to, (a) (A Ç B) È (A È B) (b) (A È B) – (A – B), (c) (A È B) – (A Ç B) (d) (A È B) – B, 75. Let A, B, C are three non-empty sets. If A Ì B and B Ì C,, then which of the following is true?, (a) B – A = C – B, (b) A Ç B Ç C = B, (c) A È B = B Ç C, (d) A È B È C = A, 76. In the Venn diagram, the shaded portion represents, , È, , 81., 82., 83., , 84., 85., , A, , (b) (C – A )¢ = {1, 2, 3, 5, 6, 7, 9, 10}, (c) Both (a) and (b), (d) None of these, If A and B are any two sets, then A È (A Ç B) is equal to, (a) A, (b) B, (c) Ac, (d) Bc, The smallest set A such that A È {1, 2} = {1, 2, 3, 5, 9} is, (a) {2, 3, 5}, (b) {3, 5, 9}, (c) {1, 2, 5, 9}, (d) None of these, Consider the following relations :, I. A = (A Ç B) È (A – B), II. A – B = A – (A Ç B), III. A – (B È C) = (A – B) È (A – C), Which of these is correct?, (a) I and III, (b) I and II, (c) Only II, (d) II and III, If A and B are non-empty sets, then P(A) È P(B) is equal to, (a) P(A È B), (b) P(A Ç B), (c) P(A) = P(B), (d) None of these, Let U be the set of all boys and girls in school. G be the, set of all girls in the school. B be the set of all boys in, the school and S be the set of all students in the school, who take swimming. Some but not all students in the, school take swimming., U, , (a) complement of set A (b) universal set, (c) set A, (d) None of these, 77. Let A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8,12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, and D = {5, 10, 15, 20}, Which of the following is incorrect?, I. A – B = {4, 8, 16, 20}, II. (C – B) Ç (D – B) = f, III. B – C ¹ B – D, (a) Only I & II, (b) Only II & III, (c) Only III & I, (d) None of these, 78. Let S = the set of all triangles, P = the set of all isosceles, triangles, Q = the set of all equilateral triangles, R = the set, , (a), , B, , G, , U, , (b), , S, , B, , S, , G, , U, , (c), , S, B, , G, , (d), , None of these, , 86. If A and B are two sets, then (A – B) È (B – A) È (A Ç B), is equal to, (a) Only A, (b) A È B, ¢, (c) ( A È B), (d) None of these

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SETS, , 87. If aN = {ax : x Î N} , then the set 3N Ç 7N is, (a) 21 N, (b) 10 N, (c) 4 N, (d) None, 88. If A = {x Î R : 0 < x < 3} and B = {x Î R :1 £ x £ 5} then, A D B is, (a) {x Î R : 0 < x < 1}, (b) {x Î R : 3 £ x £ 5}, (c) {x Î R : 0 < x < 1 or 3 £ x £ 5} (d) f, 89. If A and B are sets, then A Ç (B – A) is, (a) f, (b) A, (c) B, (d) None of these, 90. If n(A) = 3, n(B) = 6 and A Í B. Then, the number of, elements in A È B is equal to, (a) 3, (b) 9, (c) 6, (d) None of these, 91. If A = {x : x is a multiple of 4} and B = {x : x is a multiple, of 6}, then A Ç B consists of all multiples of, (a) 16, (b) 12, (c) 8, (d) 4, 92. Each student in a class of 40, studies at least one of the, subjects English, Mathematics and Economics. 16 study, English, 22 Economics and 26 Mathematics, 5 study English, and Economics, 14 Mathematics and Economics and 2 study, all the three subjects. The number of students who study, English and Mathematics but not Economics is, (a) 7, (b) 5, (c) 10, (d) 4, 93. A survey of 500 television viewers produced the following, information, 285 watch football, 195 watch hockey, 115 watch, basket-ball, 45 watch football and basket ball, 70 watch, football and hockey, 50 watch hockey and basket ball, 50 do, not watch any of the three games. The number of viewers,, who watch exactly one of the three games are, (a) 325, (b) 310, (c) 405, (d) 372, 94. Out of 800 boys in a school, 224 played cricket, 240 played, hockey and 336 played basketball. Of the total 64 played, both basketball and hockey, 80 played cricket and basketball, and 40 played cricket and hockey, 24 played all the, three games. The number of boys who did not play any, game is :, (a) 128, (b) 216, (c) 240, (d) 160, 95. Let V = {a, e, i, o, u}, V – B = {e, o} and B – V = {k}., Then, the set B is, (a) {a, i, u}, (b) {a, e, k, u}, (c) {a, i, k, u}, (d) {a, e, i, k, u}, 96. In a statistical investigation of 1003 families of Calcutta, it, was found that 63 families has neither a radio nor a T.V, 794, families has a radio and 187 has T.V. The number of families, in that group having both a radio and a T.V is, (a) 36, (b) 41, (c) 32, (d) None of these, Topic 3: Algebraic Operations on Sets, Demorgan’s Law, Number of Elements in Different Sets., 97. If A and B are finite sets, then which one of the following is, the correct equation?, (a) n (A – B) = n (A) – n (B), (b) n (A – B) = n (B – A), (c) n (A – B) = n (A) – n (A Ç B), (d) n (A – B) = n (B) – n (A Ç B), [n (A) denotes the number of elements in A], , 9, , 98. In a group of 52 persons, 16 drink tea but not coffee, while, 33 drink tea. How many persons drink coffee but not tea ?, (a) 17, (b) 36, (c) 23, (d) 19, 99. If S and T are two sets such that S has 21 elements, T has 32, elements, and S Ç T has 11 elements, then number of, elements S È T has, (a) 42, (b) 50, (c) 48, (d) None of these, 100. Let n (U) = 700, n (A) = 200, n (B) = 300, n (A Ç B) = 100, then, n (A' Ç B') is equal to, (a) 400, (b) 600, (c) 300, (d) None of these, 101. There are 600 student in a school. If 400 of them can speak, Telugu, 300 can speak Hindi, then the number of students, who can speak both Telugu and Hindi is:, (a) 100, (b) 200, (c) 300, (d) 400, 102. In a group of 500 students, there are 475 students who can, speak Hindi and 200 can speak Bengali. What is the number, of students who can speak Hindi only ?, (a) 275, (b) 300, (c) 325, (d) 350, 103. In a survey of 400 students in a school, 100 were listed, as taking apple juice, 150 as taking orange juice and, 75 were listed as taking both apple as well as orange juice., Then, which of the following is/are true?, I. 150 students were taking at least one juice., II. 225 students were taking neither apple juice nor, orange juice., (a) Only I is true, (b) Only II is true, (c) Both I and II are true (d) None of these, 104. If n(A) = 8 and n (A Ç B) = 2, then n[(A Ç B)¢ Ç A] is equal to, (a) 8, (b) 6, (c) 4, (d) 2, 105. A market research group conducted a survey of, 2000 consumers and reported that 1720 consumers like, product P1 and 1450 consumers like product P2. What is, the least number that must have liked both the products?, (a) 1150 (b) 2000, (c) 1170, (d) 2500, 106. Out of 500 car owners investigated, 400 owned car A and, 200 owned car B, 50 owned both A and B cars. Then, (a) 100 owned car A only, (b) 150 owned car B only, (c) 150 owned exactly one car, (d) The given data is incorrect, 107. In a city 20 percent of the population travels by car, 50, percent travels by bus and 10 percent travels by both car, and bus. Then persons travelling by car or bus is, (a) 80 percent, (b) 40 percent, (c) 60 percent, (d) 70 percent, 108. In a battle 70% of the combatants lost one eye, 80% an, ear, 75% an arm, 85% a leg, x% lost all the four limbs. The, minimum value of x is, (a) 10, (b) 12, (c) 15, (d) None of these, 109. Which of the following is correct?, I. n(S È T) is maximum when n (S Ç T) is least., II. If n(U) = 1000, n(S) = 720, n(T) = 450, then least value of, n(S Ç T) = 170., (a) Only I is true, (b) Only II is true, (c) Both I and II are true (d) Both I and II are false

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EBD_7762, 10, , MATHEMATICS, , 110. In a school, there are 20 teachers who teach Mathematics, or Physics of these, 12 teach Mathematics and 4 teach, both Maths and Physics. Then the number of teachers, teaching only Physics are, (a) 4, (b) 8, (c) 12, (d) 16, 111. In a town of 10000 families, it was found that 40% families, buy newspaper A, 20% families buy newspaper B and, 10% families buy newspaper C, 5% buy A and B, 3% buy, B and C and 4% buy A and C. If 2% families buy all of, three newspapers, then the number of families which buy, A only, is, (a) 4400 (b) 3300, (c) 2000, (d) 500, 112. A town has total population of 25,000 out of which 13,000, read “The Times of India” and 10,500 read “ The Hindustan, Times”. 2,500 read both papers. The percentage of, population who read neither of these newspapers is, (a) 16% (b) 18%, (c) 20%, (d) 25%, 113. If n(A) = 1000, n(B) = 500 and if n(A Ç B) ³ 1 and, n(A È B) = p, then, (a) 500 £ p £ 1000, (b) 1001 £ p £ 1498, (c) 1000 £ p £ 1498, (d) 1000 £ p £ 1499, 114. Consider the following statements., I., , 10, , If An is the set of first n prime numbers, then U An, n =2, , is equal to {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}, II. If A and B are two sets such that n (A È B) = 50,, n (A) = 28, n (B) = 32, then n (A Ç B) = 10., Which of these is correct?, (a) Only I is true, (b) Only II is true, (c) Both are true, (d) Both are false, 115. A class has 175 students. The following data shows the, number of students opting one or more subjects., Maths–100, Physics–70, Chemistry–40, Maths and, Physics–30, Maths and Chemistry–28, Physics and, Chemistry–23, Maths, Physics and Chemistry–18., How many have offered Maths alone?, (a) 35, (b) 48, (c) 60, (d) 22, , 116. A set A has 3 elements and another set B has 6 elements., Then, (a) 3 £ n (A È B) £ 6, (b) 3 £ n (A È B) £ 9, (c) 6 £ n (A È B) £ 9, (d) 0 £ n (A È B) £ 9, 117. 60 employees in an office were asked about their, preference for tea and coffee. It was observed that for every, 3 people who prefer tea, there are 2 who prefer coffee., For every 6 people who prefer tea, there are 2 who drink, both of tea and coffee. The number of people who drink, both is the same as those who drink neither., How many people drink both tea and coffee?, (a) 10, (b) 12, (c) 14, (d) 16, 118. In a certain town 25% families own a phone and 15% own a car, 65% own neither a phone nor a car. 2000 families own both a car, and a phone. Consider the following statements in this regard, (1) 10% families own both a car and a phone, (2) 35% families own either a car or a phone., (3) 40,000 families live in the town., Which one of the statements are correct?, (a) 1 and 2 (b) 1 and 3 (c) 2 and 3 (d) 1, 2 & 3, 119. A market research group conducted a survey of 1000, consumers and reported that 720 consumers liked product, A and 450 consumers liked product B. What is the least, number that must have liked both products ?, (a) 170, (b) 280, (c) 220, (d) None, 120. Given n(U) = 20, n(A) = 12, n(B) = 9, n(A Ç B) = 4, where U, is the universal set, A and B are subsets of U, then, n((A È B)c) =, (a) 17, (b) 9, (c) 11, (d) 3, , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, 1., , Suppose A1, A2, ......, A30 are thirty sets each having 5, elements and B1, B2, ......, Bn are n sets each with 3 elements., 30, , n, , i =1, , j =1, , 3., 4., , Let U Aj = U Bj = S and each element of S belongs to, , 2., , exactly 10 of the Ai's and exactly 9 of the Bj's. Then n is, equal to, (a) 15, (b) 3, (c) 45, (d) 35, Two finite sets have m and n elements. The number of subsets, of the first set is 112 more than that of the second set. The, values of m and n are, respectively,, (a) 4, 7, (b) 7, 4, (c) 4, 4, (d) 7, 7, , 5., , The set (A Ç B')' È (B Ç C) is equal to, (a) A' È B È C, (b) A' È B, (c) A' È C', (d) A' Ç B, Let F1 be the set of parallelograms, F2 the set of rectangles,, F3 the set of rhombuses, F4 the set of squares and F5 the set, of trapeziums in a plane. Then F1 may be equal to, (a) F2 Ç F3, (b) F3 Ç F4, (c) F2 È F5, (d) F2 È F3 È F4 È F1, Let S = set of points inside the square, T = set of points, inside the triangle and C = set of points inside the circle. If, the triangle and circle intersect each other and are contained, in a square. Then,, (a) S Ç T Ç C = f, (b) S È T È C = C, (c) S È T È C = S, (d) S È T = S Ç C

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SETS, , 11, , 6., , If R be the set of points inside a rectangle of sides a and, b (a, b > 1) with two sides along the positive direction of, X-axis and Y-axis. Then,, (a) R = {(x, y) : 0 £ x £ a, 0 £ y £ b}, (b) R = {(x, y) : 0 £ x < a, 0 £ y £ b}, (c) R = {(x, y) : 0 £ x £ a, 0 < y < b}, (d) R = {(x, y) : 0 < x < a, 0 < y < b}, 7., In a town of 840 persons, 450 persons read Hindi, 300 read, English and 200 read both. Then, the number of persons, who read neither, is, (a) 210, (b) 290, (c) 180, (d) 260, 8., If X = {8n – 7n – 1 : n Î N} and Y = {49(n – 1): n Î N}, then, (a) X ÌY (b) YÌ X, (c) X = Y, (d) X Ç Y = f, 9., A survey shows that 63% of the people watch a news, channel whereas 76% watch another channel. If x% of the, people watch both channel, then, (a) x = 35, (b) x = 63, (c) 39 £ x £ 63, (d) x = 39, 10. If sets A and B are defined as :, 1, , x ¹ 0, x Î R }, x, B = {(x, y) : y = – x, x Î R}, then, (a) A Ç B = A, (b) A Ç B = B, f, Ç, (c) A, B=, (d) A È B = A, 11. If A and B are two sets, then A Ç (A È B) equals to, (a) A, (b) B, (c) f, (d) A Ç B, 12. If A = {1, 3, 5, 7, 9, 11, 13, 15, 17}, B = {2, 4, ...., 18} and N the, set of natural numbers is the universal set, then, (A' È (A È B) Ç B') is, (a) f, (b) N, (c) A, (d) B, 13. If S = {x | x is a positive multiple of 3 less than 100} and P =, {x | x is a prime number less than 20}. Then n(S) + n(P) is, equal to, (a) 34, (b) 31, (c) 33, (d) 41, , A = {(x, y ) : y =, , 14. If X and Y are two sets and X' denotes the complement of, X, then X Ç (X È Y )' is equal to:, (a) X, , (b) Y, , (c), , f, , (d) X Ç Y, , Past Year MCQs, 15. Let X ={1,2,3,4,5}. The number of different ordered pairs, (Y,Z) that can formed such that Y Í X , Z Í X and Y Ç Z is, empty is :, [JEE MAIN 2012, A], (a) 52, (b) 35, (c) 25, (d) 53, 16. The set (A \ B) È (B \ A) is equal to, [BITSAT 2014, C], (a) [ A \ ( A Ç B)] Ç [ B \ ( A Ç B)], (b) ( A È B) \ ( A Ç B), (c) A \ ( A Ç B ), (d) A Ç B \ A È B, 17. Two finite sets have m and n elements. The number of subsets, of the first set is 112 more than that of the second set. The, values of m and n respectively are,, [BITSAT 2016, C], (a) 4, 7 (b) 7, 4, (c) 4, 4, (d) 7, 7, 18. Let A, B, C be finite sets. Suppose that n (A) = 10, n (B) = 15,, n (C) = 20, n (AÇB) = 8 and n (BÇC) = 9. Then the, possible value of n (AÈBÈC) is, [BITSAT 2017, S], (a) 26, (b) 27, (c) 28, (d) Can be 26 or 27 or 28, 19. Let S = {x Î R : x ³ 0 and, , 2 | x - 3 | + x ( x - 6) + 6 = 0 . Then S :, [JEE MAIN 2018, A], (a) contains exactly one element., (b) contains exactly two elements., (c) contains exactly four elements., (d) is an empty set., , Exercise 3 : Try If You Can, 1., , 2., 3., , In a market research project, 20% opted for 'Nirma' detergent, whereas 60% opted for 'Surf blue' detergent. The remaining, individuals were not certain. If the difference between those, who opted for 'Surf blue' and those who were uncertain was, 720, how many respondents were covered in the survey?, (a) 1100, (b) 1150, (c) 1800, (d) None of these, If A = {x : x2 = 1} and B = {x : x4 = 1}, then A D B is equal to, (a) {i, –i}, (b) {–1, 1}, (c) {–1, 1, i, –i} (d), None of these, Let A = {q : sin(q) = tan(q)} and B = {q : cos(q) = 1} be two, sets. Then :, (a) A = B, (b) A Ë B, (c), , BË A, , (d), , A Ì B and B - A ¹ f, , 4., , 5., , 6., , Let A, B, C be finite sets. Suppose that n (A) = 11, n (B) = 16,, n (C) = 21, n (AÇB) = 9 and n (BÇC) = 10. Then the, possible value of n (AÈBÈC) is, (a) 27, (b) 28, (c) 29, (d) Any of the three values 26, 27, 28 is possible, A survey shows that 61%, 46% and 29% of the people, watched “3 idiots”, “Rajneeti” and “Avatar” respectively., 25% people watched exactly two of the three movies and, 3% watched none. What percentage of people watched all, the three movies?, (a) 39% (b) 11%, (c) 14%, (d) 7%, Which is the simplified representation of, (A' Ç B' Ç C) È (B Ç C) È (A Ç C) where A, B, C are subsets, of set X?, (a) A, (b) B, (c) C, (d) X Ç (A È B È C)

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EBD_7762, 12, , MATHEMATICS, , 7., , In a class of 60 students, 23 play Hockey 15 Play Basketball and 20 play cricket. 7 play Hockey and Basket-ball, 5, play cricket and Basket-ball, 4 play Hockey and Cricket, and 15 students do not play any of these games. Then, (a) 4 play Hockey, Basket-ball and Cricket, (b) 20 play Hockey but not Cricket, (c) 1 plays Hockey and Cricket but not Basket-ball, (d) All above are correct, 8., A survey of 500 television viewers produced the following, information, 285 watch football, 195 watch hockey, 115 watch, basket-ball, 45 watch football and basket ball, 70 watch, football and hockey, 50 watch hockey and basket ball, 50 do, not watch any of the three games. The number of viewers,, who watch exactly one of the three games are, (a) 325, (b) 310, (c) 405, (d) 372, 9., If n(A) = 4 and n(B) = 7, then the difference between, maximum and minimum value of n(A È B) is, (a) 1, (b) 2, (c) 3, (d) 4, 10. A dinner party is to be fixed for a group of 100 persons. In, this party, 50 persons do not prefer fish, 60 prefer chicken, and 10 do not prefer either chicken or fish. The number of, persons who prefer both fish and chicken is., (a) 20, (b) 22, (c) 25, (d) None of these, 11. Let N be the set of non-negative integers, I the set of, integers, Np the set of non-positive integers, E the set of, even integers and P the set of prime numbers. Then, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, , (d), (d), (c), (d), (b), (a), (b), (a), (c), (a), (c), (d), , 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, , (d), (c), (b), (d), (a), (c), (d), (c), (c), (c), (a), (b), , 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, , (a), (a), (b), (c), (a), (b), (c), (b), (c), (a), (c), (a), , 1, 2, , (c), (b), , 3, 4, , (b), (d), , 5, 6, , (c), (d), , 1, 2, , (c), (a), , 3, 4, , (b), (d), , 5, 6, , (d), (c), , (b) N Ç Np = f, (d) N D Np = I – {0}, , (a) I – N = Np, (c) E Ç P = f, 12., , Let P = {q : sin q – cos q = 2 cos q} and, , Q = {q : sin q + cos q = 2 sin q} be two sets. Then, (a) P Ì Q and Q – P ¹ Æ (b) Q Ë P, (c) P Ë Q, (d) P = Q, 13. If U = {x : x5 – 6x4 + 11x3 – 6x2 = 0},, A = {x : x2 – 5x + 6 = 0) and, B = {x : x2 – 3x + 2 = 0}, then n (A Ç B)¢, is equal to:, (a) 2, (b) 3, (c) 4, (d) 5, 14. From 50 students taking examination in Mathematics,, Physics and Chemistry, each of the students has passed, in at least one of the subject, 37 passed Mathematics,, 24 Physics and 43 Chemistry. Atmost 19 passed, Mathematics and Physics, atmost 29 Mathematics and, Chemistry and atmost 20 Physics and Chemistry. Then,, the largest numbers that could have passed all three, examinations, are, (a) 12, (b) 14, (c) 15, (d) 16, 15. Let A, B and C be finite sets such that A Ç B Ç C = f and, each one of the sets A D B, B D C and C D A has 100, elements. The number of elements in A È B È C is, (a) 250, (b) 200, (c) 150, (d) 300, , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (d) 73 (d), 37 (a) 49 (b), 61, (d) 74 (c), 38 (c) 50 (a), 62, (b) 75 (c), 39 (c) 51 (d), 63, (d) 76 (a), 40 (a) 52 (b), 64, (c) 77 (a), 41 (d) 53 (b), 65, (c) 78 (a), 42 (b) 54 (a), 66, (a) 79 (a), 43 (b) 55 (d), 67, (b) 80 (c), 44 (c) 56 (a), 68, (c) 81 (a), 45 (d) 57 (c), 69, (c) 82 (b), 46 (b) 58 (a), 70, (c) 83 (b), 47 (c) 59 (b), 71, (b) 84 (d), 48 (a) 60 (d), 72, Exercise 2 : Exemplar & Past Year MCQs, (b), (c), (a) 13 (d), 7, 9, 11, (a), (c), (b), 8, 10, 12, 14 (c), Exercise 3 : Try If You Can, (c), (d), (d) 13 (b), 7, 9, 11, (a) 10 (a), (d) 14 (b), 8, 12, , 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, , (b), (b), (a), (c), (a), (c), (b), (b), (a), (d), (c), (b), , 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, , (c), (d), (a), (c), (a), (b), (b), (b), (c), (d), (c), (a), , 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, , (c), (b), (b), (a), (d), (c), (c), (c), (b), (c), (a), (d), , 15, 16, , (b), (b), , 17, 18, , (b), (d), , 19, , (b), , 15, , (c)

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2, , RELATIONS AND, FUNCTIONS-1, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010 - 2018), , Number of Questions, , 2, , JEE MAIN, BITSAT, 1, , 0, 2010 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), , Exam, , JEE Main, BITSAT, , % Weightage, , Critical Concepts, , Rating of, Difficulty, , CUS (chapter utility score), out of 10, , 2, 4, , Cartesian Product of Two Sets, Relation-domain,, codomain and range of a relation, Function-domain ,, codomain and range of a function, Solving Rational, Inequalities (Wavy Curve Method) Different types of, Common Functions and their graphs, Transdental, Functions, Graphing new Function, Piecewise, Functions., , 3.5/5, , 6.7

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RELATIONS AND FUNCTIONS-1, , 15

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EBD_7762, 16, , MATHEMATICS, , Exercise 1 : Topic-wise MCQs, Topic 1 : Cartesian Product of Sets, Relations, Domain,, Co-domain and Range of a Relation., 1., 2., , 3., , 4., 5., , 6., , 7., 8., 9., , If A × B = { (5, 5), (5, 6), (5, 7), (8, 6), (8, 7), (8, 5)}, then the, value A is, (a) {5}, (b) {8}, (c) {5, 8} (d) {5, 6, 7, 8}, The relation R defined on the set of natural numbers as, {(a, b) : a differs from b by 3}is given, (a) {(1, 4), (2, 5), (3, 6),.....}, (b) {(4, 1), (5, 2), (6, 3),.....}, (c) {(1, 3), (2, 6), (3, 9),.....}, (d) None of these, The Cartesian product of two sets P and Q, i.e., P × Q = f, if, (a) either P or Q is the null set, (b) neither P nor Q is the null set, (c) Both (a) and (b), (d) None of the above, A relation is represented by, (a) Roster method, (b) Set-builder method, (c) Both (a) and (b), (d) None of these, Consider the following statements :, I. If n (A) = p and n (B) = q, then n (A × B) = pq, II. A × f = f, III. In general, A × B ¹ B × A, Which of the above statements are true ?, (a) Only I, (b) Only II, (c) Only III, (d) All of the above, Which of the following is/ are not true?, (a) If P = {m, n} and Q = {n, m}, then, P × Q = {(m, n), (n, m)}., (b) If A and B are non-empty sets, then A × B is a, non-empty set of ordered pairs (x, y), such that, x Î A and y Î B., (c) If A = {1, 2} and B = {3, 4}, then A × (B Ç f) = f., (d) All of the above, If (4x +3, y) = (3x + 5, – 2), then the sum of the values of x and, y is, (a) 0, (b) 2, (c) –2, (d) 1, If (x + 3, 4 – y) = (1, 7), then the value of 4 + y is, (a) 3, (b) 4, (c) 5, (d) 1, Let A = {1, 2, 3, 4, 6}. If R is the relation on A defined, by {(a, b) : a, b Î A, b is exactly divisible by a}., Assertion : The relation R in Roster form is {(6, 3), (6, 2),, (4, 2)}., Reason : The domain and range of R is {1, 2, 3, 4, 6}., (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., , 10. If A is the set of even natural numbers less than 8 and B is, the set of prime numbers less than 7, then the number of, relations from A to B is, (a) 29, (b) 92, (c) 32, (d) 29 – 1, 11. Let A = {1, 2}, B = {3, 4}. Then, number of subsets of, A × B is, (a) 4, (b) 8, (c) 18, (d) 16, 12. Let A = {x, y, z} and B = {a, b, c, d}. Then, which one of, the following is not a relation from A to B?, (a) {(x, a), (x, c)}, (b) {(y, c), (y, d)}, (c) {(z, a), (z, d)}, (d) {(z, b), (y, b), (a, d)}, 13. Let R be the relation on Z defined by, R = {(a, b) : a, b Î Z, a – b is an integer}. Then, (a) domain of R is {2, 3, 4, 5, .....}, (b) range of R is Z, (c) Both (a) and (b), (d) None of the above, 14. If A = {2, 3, 4, 5} and B = {3, 6, 7, 10}. R is a relation, defined by R = {(a, b) : a is relatively prime to b, a Î A, and b Î B}, then domain of R is, (a) {2, 3, 5}, (b) {3, 5}, (c) {2, 3, 4}, (d) {2, 3, 4, 5}, 15. The number of elements in the domain of relation, R = {(x, y) : x2 + y2 = 16, x, y Î Z} is, (a) 1, (b) 2, (c) 3, (d) 4, 16. If A = {1, 2}, B = {1, 3}, then (A × B) È (B × A) is equal to, (a) {(1, 3), (2, 3), (3, 1), (3, 2), (1, 1), (2, 1), (1, 2)}, (b) {(1, 3), (3, 1), (3, 2), (2, 3)}, (c) {(1, 3), (2, 3), (3, 1), (3, 2), (1, 1)}, (d) None of these, 17. Let R be a relation from N to N defined by, R = {(a, b) : a, b Î N and a = b2}. Then, which of the, following is/ are true?, I. (a, a) Î R for all a Î N., II. (a, b) Î R implies (b, a) Î R., III. (a, b) Î R, (b, c) Î R implies (a, c) Î R., (a) I and II are true, (b) II and III are true, (c) All are true, (d) None of these, 18. Consider the following statements., Let A = {1, 2, 3, 4} and B = {5, 7, 9}, I. A ´ B = B ´ A, II. n (A ´ B) = n (B ´ A), Choose the correct option., (a) Statement-I is true., (b) Statement-II is true., (c) Both are true., (d) Both are false., 19. Consider the following statements., I. Let A and B are non-empty sets such that A Í B. Then,, A ´ C Í B ´ C., II. For any two sets A and B, A ´ B = B ´ A, Choose the correct option., (a) Only I is true., (b) Only II is true., (c) Both are true., (d) Both are false., 20. The number of elements in the set {(x, y) : 2x2 + 3y2 = 35,, x, y Î Z}, where Z is the set of all integers,, (a) 8, (b) 2, (c) 4, (d) 6

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RELATIONS AND FUNCTIONS-1, , 17, , 21. If the set A has 3 elements and the set B = {3, 4}, then the, number of elements in A ´ B is, (a) 6, (b) 9, (c) 8, (d) 2, 22. If A, B and C are three sets, then, (a) A × (B Ç C) = (A × B) Ç (A × C), (b) A × ( B ¢ È C ¢ )¢ = (A × B) Ç (A × C), 23., , 24., , 25., , 26., , (c) Both (a) and (b), (d) None of the above, If A = {8, 9, 10} and B = {1, 2, 3, 4, 5}, then the number, of elements in A × A × B are, (a) 15, (b) 30, (c) 45, (d) 75, If A = {1, 2, 4}, B = {2, 4, 5}, C = {2, 5}, then, (A – C) × (B – C) is equal to, (a) {(1, 4)}, (b) {(1, 4), (4, 4)}, (c) {(4, 1), (4, 4)}, (d) {(1, 2), (2, 5)}, Let set X = {a, b, c} and Y = f. The number of ordered, pairs in X × Y are, (a) 0, (b) 1, (c) 2, (d) 3, The figure given below shows a relation R between the sets, A and B., B, ·5, A, , ·3, , 9·, , ·2, , 4·, 25 ·, , ·1, · –2, · –3, · –5, , Then which of the following is correct?, I. The relation R in set-builder form is {(x,y) : x is the, square of y, x Î A, y Î B}, II. The domain of the relation R is {4, 9, 25}, III. The range of the relation R is {–5, –3, –2, 2, 3, 5}, (a) Only I and II are true. (b) Only II and III are true., (c) I, II and III are true, (d) Neither I, II nor III are true., 27. Consider the following statements., I. If (a, 1), (b, 2) and (c, 1) are in A ´ B and n(A) = 3,, n (B) = 2, then A = {a, b, c} and B = {1, 2}, II. If A = {1, 2} and B = {3, 4}, then A ´ (B Ç f) is equal to, A ´ B., Choose the correct option., (a) Only I is true, (b) Only II is true, (c) Both are true, (d) Neither I nor II is true, 28. If A, B and C are any three sets, then A × (B È C) is, equal to, (a) (A × B) È (A × C), (b) (A È B) × (A È C), (c) (A × B) Ç (A × C), (d) None of these, , 29. If the set A has p elements, B has q elements, then the, number of elements in A × B is, (a) p + q, (b) p + q + 1, (c) pq, (d) p2, 30. Let A = {x, y, z} and B = {a, b, c, d}. Which one of the, following is not a relation from A to B?, (a) {(x, a), (x, c)}, (b) {(y, c), (y, d)}, (c) {(z, a), (z, d)}, (d) {(z, b), (y, b), (a, d)}, 31. A relation R is defined in the set Z of integers as follows, (x, y) Î R iff x2 + y2 = 9. Which of the following is false?, (a) R = {(0, 3), (0, –3), (3, 0), (–3, 0)}, (b) Domain of R = {–3, 0, 3}, (c) Range of R = {–3, 0, 3}, (d) None of these, 32. The domain and range of the relation R given by, 6, ; where x, y Î N and x < 6} is, x, (a) {1, 2, 3}, {7, 5}, (b) {1, 2}, {7, 5}, (c) {2, 3}, {5}, (d) None of these, 33. Consider the following statements., I. If the set A has 3 elements and set B = {3, 4, 5}, then the, number of elements in A ´ B = 9., II. The domain of the relation R defined by, R = {(x, x + 5) : x Î (0, 1, 2, 3, 4, 5)} is {5, 6, 7, 8, 9, 10}., Choose the correct option., (a) Only I is true., (b) Only II is true., (c) Both I and II are true. (d) Both I and II are false., 34. Consider the following statements., I. The relation R = {(x, x3) : x is a prime number less than, 10 } in Roster form is {(3, 27), (5, 125),, (7, 343)}, II. The range of the relation, R = {(x + 2, x + 4) : x Î N, x < 8} is {1, 2, 3, 4, 5, 6, 7}., Choose the correct option., (a) Only I is true, (b) Only II is true, (c) Both are true, (d) Both are false, 35. Consider the following statements, I. Let n(A) = m and n(B) = n. Then the total number of, non-empty relations that can be defined from A to B is, 2mn – 1, II. If A = {1, 2, 3}, B = {3, 8}, then (A È B) ´ (A Ç B) is, equal to {(1, 3), (2, 3), (3, 3), (8, 3)}., , R = {(x, y) : y = x +, , y ö, æx, III. If ç - 1, + 1÷ = (2, 1), then the values of x and y, 9 ø, è2, , respectively are 6 and 0., Choose the correct option., (a) Only I and II are false., (b) Only II and III are true., (c) Only I and III are true., (d) All the three statements are true

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EBD_7762, 18, , 36., , 37., , 38., , 39., , 40., , 41., , 42., , 43., 44., , 45., , 46., , MATHEMATICS, , If n (X) = 5 and n (Y) = 7, then the number of relations on, X ´ Y is 25m. The value of ‘m’ is, (a) 5, (b) 7, (c) 6, (d) 8, If A = {a, b, c}, B = {b, c, d} and C = {a, d, c}, then, (A – B) × (B Ç C) =, (a) {(a, c), (a, d)}, (b) {(a, b), (c, d)}, (c) {(c, a), (a, d)}, (d) {(a, c), (a, d), (b, d)}, If P = {a, b, c} and Q = {r}, then, (a) P × Q = Q × P, (b) P × Q ¹ Q × P, (c) P × Q Ì Q × P, (d) None of these, Let n(A) = 8 and n(B) = p. Then, the total number of, non-empty relations that can be defined from A to B is, (a) 8p, (b) np – 1 (c) 8p – 1 (d) 28p – 1, If A = {a, b}, B = {c, d}, C = {d, e}, then {(a, c),, (a, d), (a, e), (b, c), (b, d), (b, e)} is equal to, (a) A Ç (B È C), (b) A È (B Ç C), (c) A × (B È C), (d) A × (B Ç C), Suppose that the number of elements in set A is p, the, number of elements in set B is q and the number of, elements in A × B is 7. Then p2 + q2 =, (a) 42, (b) 49, (c) 50, (d) 51, The cartesian product of A × A has 9 elements, two of, which are (–1, 0) and (0, 1), the remaining elements of, A × A is given by, (a) {(–1, 1), (0, 0), (–1, –1), (1, –1), (0, –1)}, (b) {(–1, –1), (0, 0), (–1, 1), (1, –1), (1, 0), (1, 1), (0, –1)}, (c) {(1, 0), (0, –1), (0, 0), (–1, –1), (1, –1), (1, 1)}, (d) None of these, Let A = {1, 2, 3}. The total number of distinct relations, that can be defined over A, is, (a) 29, (b) 6, (c) 8, (d) 26, The relation R defined on the set A = {1, 2, 3, 4, 5} by R, = {(x, y) : |x2 – y2 | < 16} is given by, (a) {(1, 1), (2, 1), (3, 1), (4, 1), (2, 3)}, (b) {(2, 2), (3, 2), (4, 2), (2, 4)}, (c) {(3, 3), (4, 3), (5, 4), (3, 4)}, (d) None of these, The relation R defined on set A = {x : |x| < 3, x Î I} by, R = {(x, y) : y = |x|} is, (a) {(–2, 2), (–1, 1), (0, 0), (1, 1), (2, 2)}, (b) {(–2, –2), (–2, 2), (–1, 1), (0, 0), (1, –2), (1, 2),, (2, –1), (2, –2)}, (c) {(0, 0), (1, 1), (2, 2)}, (d) None of these, Let X = {1, 2, 3}. The total number of distinct relations that, can be defined over X is 2n. The value of ‘n’ is, (a) 9, (b) 6, (c) 8, (d) 2, Topic 2 : Functions, Domain, Codomain and, Range of a Function., , 47., , Which of the following relation is a function ?, (a) {(a, b) (b, e) (c, e) (b, x)}, (b) {(a, d) (a, m) (b, e) (a, b)}, (c) {(a, d) (b, e) (c, d) (e, x)}, (d) {(a, d) (b, m) (b, y) (d, x)}, , 48. The domain of f (x) =, , ù1 é, ú ,1ê, ûú 2 ëê, (c) [1, ¥ [, (a), , 49., , f (x) =, , 1, 2x ,1, , , 1, x 2 is:, , (b) [ – 1, ¥ [, (d) None of these, , ( x + 1) ( x - 3), is a real valued function in the domain, ( x - 2), , (a), , (-¥, - 1] È [3, ¥), , (b), , (-¥, - 1] È (2, 3], , (c), , [-1, 2) È [3, ¥], , (d) None of these, , 50. The domain of the function x 2 - 5x + 6 + 2 x + 8 - x 2 is, (a) [2, 3], (b) [–2, 4], (c) [–2, 2] È [3, 4], (d) [–2, 1] È [2, 4], 51. There are three relations R1, R2 and R3 such that, R1 = {(2, 1), (3, 1), (4, 2)},, R2 = {(2, 2), (2, 4), (3, 3), (4, 4)} and, R3 = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}, Then,, (a) R1 and R2 are functions, (b) R2 and R3 are functions, (c) R1 and R3 are functions, (d) Only R1 is a function, 52. The value of the function f ( x ) =, , x 2 - 3x + 2, x2 + x - 6, , lies in the, , interval, (a), , ì1 ü, ( -¥, ¥) – í , 1ý, î5 þ, , (b), , (c), , (-¥, ¥) - {1}, , (d) None of these, , 53. The range of the function f (x) =, (a) (– ¥, 3], , (-¥, ¥), , x2 - x +1, , x2 + x +1, (b) (–¥, ¥), , where x Î R, is, , é1 ù, êë 3 , 3úû, 54. Let N be the set of natural numbers and the relation R be, defined such that {R = (x, y) : y = 2x, x, y Î N}. Then,, (a) R is a function, (b) R is not a function, (c) domain, range and co-domain is N, (d) None of the above, 55. The domain for which the functions f(x) = 2x2 – 1 and, g(x) = 1 – 3x is equal, i.e. f(x) = g(x), is, (c) [3, ¥), , (d), , (a) {0, 2}, , (b), , ì1, ü, í , – 2ý, î2, þ, , ì 1 ü, ì1 ü, (d) í , 2ý, í – , 2ý, î 2 þ, î2 þ, 56. The domain of the function, (c), , f (x) =, , x - 1 - x 2 is

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RELATIONS AND FUNCTIONS-1, , (a), , 19, , 1 ù é 1 ù, é, ,1ú (b) [–1, 1], ê -1, úÈê, 2û ë 2 û, ë, , 1ù é 1, ö, æ, çè -¥, - ú È ê , + ¥÷ø (d), 2û ë 2, 57. The domain of the function, , (c), , f (x) =, , 1, x 12 - x 9 + x 4 - x + 1, , 65. The domain of the function f(x) =, , is given by, , (-¥, - 1), , (b), , (1, ¥), , (c), , (-1, 1), , (d), , (-¥, ¥), , 58. The range of the function f (x) =, (a), , æ, 11 ù, ç -¥, 3 ú, è, û, , (b), , 3x 2 - 4 x + 5 is, , æ, 11 ö, ç -¥, 5 ÷, è, ø, , é 11 ö, æ 11 ö, , ¥÷, , ¥÷, ê, (d) ç, è 5 ø, ë 3 ø, Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and, f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Then,, (a) f is a relation from A to B, (b) f is a function from A to B, (c) Both (a) and (b), (d) None of these, Consider the following statements., I. Relation R = {(2, 0), (4, 8), (2, 1), (3, 6)} is not a function., II. If first element of each ordered pair is different with, other, then the given relation is a function., Choose the correct option., (a) Only I is true., (b) Only II is true., (c) Both I and II are true. (d) Neither I nor II is true., Consider the following statements., I. If X = {p, q, r, s} and Y = {1,2, 3, 4, 5}, then, {(p, 1), (q, 1), (r, 3), (s, 4)} is a function., II. Let A = {1, 2, 3, 4, 6}. If R is the relation on A defined by, {(a, b) : a, b Î A, b is exactly divisible by a}., The relation R in Roster form is {(6, 3), (6, 2), (4, 2)}, Choose the correct option., (a) Only I is false., (b) Only II is false., (c) Both I and II are false. (d) Neither I nor II is false., Find the domain of the function, , (c), , 59., , 60., , 61., , 62., , f (x) =, , æ, 2, 1, 2 x –1ö, –, –, ç 2, ÷, è x – x + 1 x + 1 x 3 + 1ø, , (a) (– ¥, 2] – {–1}, (c) ] –1, 2], , (b) (–¥, 2), (d) None of these, , 63. The domain of the function f(x) =, , 1, 9 – x2, , x 2 – 5x + 4, , is, , (a) R, (b) R – {1, 4}, (c) R – {1}, (d) (1, 4), 66. Which of the following relation is NOT a function ?, , é 1 ù, ,1ú, ê, ë 2 û, , (a), , x 2 + 3x + 5, , is, , (a) –3 £ x £ 3, (b) –3 < x < 3, (c) –9 £ x £ 9, (d) –9 < x < 9, 64. Let f = {(1, 1), (2, 3), (0, –1), (–1, –3)} be a linear function, from Z into Z, then f(x) =, (a) 2x – 1 (b) 2 x (c) 2x + 1, (d) –2x + 1, , (a) f = {(x, x) | x Î R}, , (b) g = {(x, 3) | x Î R}, , 1, (c) h = { (n, ) | n Î I}, n, , (d) t = {(n, n2) | n Î N}, , Topic 3 : Value of a Function, Some Functions and, their Graphs (i.e., Identity Function, Constant Function,, Polynomial Function, Rational Function, Modulus Function,, Signum Function, Greatest Integer Function),, Algebra of Real Functions., 67. If f(x) =, , x, f (a), , then, is equal to:, x ,1, f (a ∗ 1), , é ,a ù, (d) f ê, ú, êë a ,1úû, 68. If g = {(1, 1), (2, 3), (3, 5), (4, 7)} is a function described by, the formula, g (x) = ax + b then what values should be, assigned to a and b?, (a) a = 1, b= 1, (b) a = 2, b = – 1, (c) a = 1, b = – 2, (d) a = – 2, b = – 1, 69. If f : R ® R is defined by f(x) = 3x + | x |, then, f(2x) – f (– x) – 6x =, (a) f(x), (b) 2f(x), (c) – f(x), (d) f(– x), 70. Find the range of f (x) = sgn(x2 – 2x + 3)., (a) {1, –1}, (b) {1}, (c) {–1}, (d) None of these, (a) f(a2), , æ1 ö, (b) f çç ÷÷÷, çè a ø, , (c) f(– a), , 71. The Domain of the function f ( x) =, , x, 1- | x |, , (a), , (-¥, - 1) È [0, 1), , (b), , (-¥, - 1) È (0, ¥), , (c), , (0, ¥), , (d), , None of these, , 72. If f ( x) = x3 -, , 1, , , then f (x) + f æç 1 ö÷ is equal to, è xø, x, 3, , 1, (b) 2, (c) 0, (d) 1, x3, 73. If f (x + 1) = x2 – 3x + 2, then f (x) is equal to:, (a) x2 – 5x – 6, (b) x2 + 5x – 6, 2, (c) x + 5x + 6, (d) x2 – 5x + 6, (a) 2 x3, , 74. If f (x) =, , (a) x, , 1– x, 1– xö, , then f æç, is equal to:, 1+ x, è 1 + x ÷ø, , (b), , 1– x, 1+ x, , (c), , 1+ x, 1– x, , (d) 1/x

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EBD_7762, 20, , 75., , MATHEMATICS, , Let f (x) = [x], where [x] denotes the greatest integer less, , than or equal to x. If a = 20112 + 2012, then the value of, f (a) is equal to, (a) 2010 (b) 2011, (c) 2012, (d) 2013, 76. If P = {x Î R : f (x ) = 0} and, , Y, , 78., , –3, –2, –3 –2 –1, X¢, , Q ={x Î R : g ( x ) = 0} then P È Q is, (a), (b), (c), 77., , {x Î R, {x Î R, , {xÎ R : (f (x)), , 2, , + ( g ( x ))2 = 0, , 6–, 4–, 2–, , (a), , X¢, , –6 –4 –2 O 2 4 6, –2 –, –4 –, –6 –, , X, , – –3, , }, , –6 –4 –2 O, –4 –, –8 –, –12 –, , æ1ö, g (x) = 3x + 5, then f ç ÷ × g(14) is, è2ø, , 1336, 5, , (b), , 1363, 4, , (d) 1608, , ì x, 0 £ x £ 1, ï, 80. Let f1(x) = í1, x >1, ï0, otherwise, î, , 12 –, 8–, 4–, , X¢, , Which of the following options identify the above graph?, (a) Modulus function, (b) Greatest integer function, (c) Signum function, (d) None of these, 79. If f and g are real functions defined by f(x) = x2 + 7 and, , (c) 1251, , Y, , (b), , Y¢, f(x) = [x], , (a), , Y¢, , X, , – –2, , (d) None of these, The graph of the function f : R ® R and f(x) = x2,, x Î R, is, Y, , 1 2 3 4 5, , O, – –1, , : f (x ) + g ( x ) = 0}, , : f ( x ) g ( x) = 0}, , –1, , 2 4 6, , X, , f2(x) = f1(–x) for all x, f3(x) = – f2(x) for all x, f4(x) = f3(–x) for all x, Which of the following is necessarily true?, , Y¢, , (a) f4 (x) = f1(x) for all x, (b) f1 (x) = – f3(– x) for all x, , Y, , (c), , X¢, , 8–, –, 4–, –, , (c) f2 (– x) = f4(x) for all x, , –4 –2 – 0 2 4, –4 –, –, –8 –, , X, , Y¢, , (d) X¢, , Y, 20 –, 16 –, 12 –, 8–, 4–, –4 –2 –4 – 0, –8 –, –12 –, –16 –, –20 –, , Y¢, , 2 4, , X, , (d) f1 (x) + f3(x) = 0 for all x, 81. If f : R ® R is defined by f(x) = 2x + | x |, then, f(2x) + f(–x) –f(x) =, (a) 2x, (b) 2 | x |, (c) –2x, (d) –2 | x |, 82. Let f(x) = 1 + x, g(x) = x2 + x + 1, then (f + g) (x) at, x = 0 is, (a) 2, (b) 5, (c) 6, (d) 9, x, 3, 83. If f(x) = a , then [f(p)] is equal to, (a) f (3p) (b) 3f (p), (c) 6f (p), (d) 2f (p), 84. Consider the following statements., I. Let f = {(1, 1), (2, 3), (0, –1), (–1, –3)} be a linear function, from Z to Z. Then, f(x) is 2x –1., II., , 1, æ1ö, 3, If f(x) = x - 3 , then f(x) + f ç ÷ is equal to 0., x, è xø

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RELATIONS AND FUNCTIONS-1, , 21, , Choose the correct option., (a) Only I is true., (b) Only II is true., (c) Both are true., (d) Both are false., 85. The domain of the function f (x) =, (a) (– ¥, 2], (c) (–¥, 2), , 1, is :, | x – 2 | – ( x – 2), , (2, ¥), (d) [2, ¥), , (b), , 86. The domain of f ( x ) = log(|| x - 2 | -2 | -1) is, , (a) {–3}, (c) R – {3}, 95. If f(x) =, , (-¥, - 1) È (1, 3) È (5, ¥), , (c), , (5, ¥), , (d) None of these, 87. If f (x) = 4x– x2, x Î R, then f (b + 1) – f (b – 1) is equal to, m (2 – b). The value of ‘m’ is, (a) 2, (b) 3, (c) 4, (d) 5, 88. If f(y) = 2y2 + by + c and f(0) = 3 and f(2) = 1, then the value, of f(1) is, (a) 0, (b) 1, (c) 2, (d) 3, 1, , x ¹ 2 and g(x) = (x – 2)2, then, 89. Assertion : If f(x) =, x–2, , 1 + ( x – 2), , x ¹ 2., x–2, Reason : If f and g are two functions, then their sum is, defined by (f + g) (x) = f(x) + g(x) " x Î D1 Ç D2, where, D1 and D2 are domains of f and g, respectively., (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., 90. Which of the following statements is incorrect, (a) x sgn x = |x|, (b) |x| sgn x = x, (c) x (sgn x) (sgn x) = x, (d) |x| (sgn x)3 = |x|, 3, , (f + g) (x) =, , | x |2 -5 | x | +6 +, , 91. f(x) =, , in, (a) [–4, –3], (c) [–2, 2], 92., , f (x) =, , 8 + 2 | x | - | x |2 is real for all x, (b) [–3, –2], (d) [3, 4], , x(x - p) x(x - q), +, , p ¹ q. What is the value of, q-p, p -q, , f (p) + f (q) ?, (a) f (p – q), (b) f (p + q), (c) f (p (p + q)), (d) f (q (p – q)), 93. If f (x) = x and g (x) = | x |, then (f + g) (x) is equal to, (b) 2x for all x Î R, (a) 0 for all x Î R, (c), , ì 2x,for x ³ 0, í, î 0, for x < 0, , (d), , ì 0, for x ³ 0, í, î2x, for x < 0, , x +3, , is, , (b) R – {–3}, (d) R, , 2x + 2-x, , then f(x + y). f(x – y) =, 2, , (a), , 1, [f(2x) + f (2y)], 2, , (b), , 1, [f (2x ) + f (2y)], 4, , (c), , 1, [f (2x ) - f (2 y)], 2, , (d), , 1, [f (2x ) - f (2 y)], 4, , (a) R – (1, 3), (b), , x +3, , 94. The domain of the function f(x) =, , 96. The domain of the function f ( x) = 3, , x, 1- | x |, , (a) (–¥ –1) È (–1, 1) È (1, ¥) (b) (–¥ –1], (c) [0, ¥), (d) None of these, 97. The domain and range of the real function f defined by, f(x) = |x – 1| is, (a) R, [0, ¥), (b) R, (–¥, 0), (c) R, R, (d) (–¥, 0), R, 98. The domain of the function f defined by, 1, , f(x) =, , x– x, , is, (b) R+, (d) {f}, , (a) R, (c) R–, , BEYOND NCERT, Topic 4 : Logarithmic Functions, Exponential Functions,, Even and Odd Functions, Periodic Functions., 99. If f : R ® R is a function satisfying the property, f(2x + 3) + f(2x + 7) = 2 , x Î R , then the period of f(x) is, (a) 2, (b) 4, (c) 5, (d) 10, 100. Domain of definition of the function, f ( x) =, , 3, 4 - x2, , + log10 ( x 3 - x) , is, , (a), , ( -1,0) È (1,2) È ( 2, ¥ ), , (b) (a, 2), , (c), , ( -1,0) È ( a,2), , (d) (1,2) È (2, ¥ ) ., , 101. If f (x) =, , 2 x + 2- x, , then f (x + y). f (x – y) is equal to, 2, , (a), , 1, [f (x + y) + f (x – y)], 2, , (b), , (c), , 1, [f (x + y) . f (x – y)], 2, , (d) None of these, , (, , 1, [f (2x) + f (2y)], 2, , ), , 102. The function f ( x) = log x + x 2 + 1 , is, (a), (b), (c), (d), , neither an even nor an odd function, an even function, an odd function, a periodic function

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EBD_7762, 22, , MATHEMATICS, , 103. Which of the following functions are periodic?, (a) f (x) = log x, x > 0, (b) f (x) = ex, x Î R, (c) f (x) = x – [x], x Î R, (d) f (x) = x + [x], x Î R, , æ 1+ xö, 104. The function f (x) = log ç, satisfies the equation, è 1 - x ÷ø, (a) f (x + 2) – 2f (x + 1) + f (x) = 0, (b) f (x + 1) + f (x) = f (x (x + 1)), (c) f (x1) · f (x2) = f (x1 + x2), , 107. Which of the following is wrong?, (a) Every constant function is an even function., (b) A constant function may be odd function also., (c) Every constant function is an odd as well as an even, function., (d) Every constant function is a periodic function, , f (– a), is equal to, f (b), (a) f (a + b), (b) f (a – b), (c) f (– a + b), (d) f (– a – b), 109. The domain of definition of the function, 1, y=, + x + 2 is, log10 (1 - x), 108. If f (x) = e – x, then, , æ x1 + x2 ö, (d) f (x1) + f (x2) = f çè 1 + x x ÷ø, 1 2, 105. If the real -valued function f ( x ) =, , a x -1, x n (a x + 1), , is even, then, , n equals, (a), , 1, 4, , (b), , 2, 3, , (c), , 106. The range of the function f (x) =, (a) (– ¥, ¥), (c) (– 1, 0], , (a), , 1, 3, , (d) 2, , e x – e| x|, x, , | x|, , e +e, , is, , (-3, - 2) excluding –2.5 (b) [0, 1] excluding 0.5, , (c) [–2, 1] excluding 0, 110. The domain of F (x) =, , log 2 ( x + 3), x2 + 3x + 2, , (a) R – {– 1, – 2}, (c) R – {– 1, – 2 – 3}, , (b) [0, 1), (d) (– 1, 1), , (d) None of these, is, , (b) (– 2, ¥), (d) (– 3 , ¥) – {– 1, – 2}, , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, 1., , 2., , 3., , 4., , 5., , Let n(A) = m, and n(B) = n. Then the total number of nonempty relations that can be defined from A to B is, (a) mn, (b) nm – 1, (c) mn – 1, (d) 2mn – 1, 2, If [x] – 5[x] + 6 = 0, where [.] denote the greatest integer, function, then, (a) x Î [3, 4], (b) x Î (2, 3], (c) x Î [2, 3], (d) x Î [2, 4), Range of f (x) =, , 1, is, 1– 2cos x, , 7., , (a), , (b), , (c), , é1 ö, (– ¥, –1] È ê , ¥÷, ë3 ø, , é 1 ù, (d) ê – ,1ú, ë 3 û, , é 1ù, ê –1, 3 ú, ë, û, , (b) f (xy) > f (x) . f ( y), (d) None of these, , Domain of a 2 – x 2 , (a > 0) is, (a) (–a, a), (b) [–a, a], (c) [0, a], (d) (–a, 0], , If f (x) = ax + b, where a and b are integers, f(–1) = –5 and, f(3) = 3, then a and b are equal to, (a) a = –3, b = –1, (b) a = 2, b = –3, (c) a = 0, b = 2, (d) a = 2, b = 3, The domain of the function f defined by, , f (x) =, , 8., , é1 ù, êë 3 ,1úû, , Let f (x) = 1 + x 2 , then, (a) f (xy) = f (x) . f (y), (c) f (xy) < f (x) . f (y), , 6., , 1, x 2 –1, , is equal to, , (a) (– ¥, – 1) È (1, 4], (b) (– ¥, – 1] È (1, 4], (c) (– ¥, – 1) È [1, 4], (d) (– ¥, – 1) È [1, 4), The domain and range of the real function f defined by, f (x) =, , 9., , 4– x +, , (a), (b), (c), (d), The, , 4– x, is given by, x–4, , Domain = R, Range = {– 1, 1}, Domain = R – {1}, Range = R, Domain = R – {4}, Range = {– 1}, Domain = R – {– 1}, Range = {– 1, 1}, domain and range of real function of defined by, , f (x) = x –1 is given by, (a) Domain = (1, ¥), Range = (0, ¥), (b) Domain = [1, ¥), Range = (0, ¥), (c) Domain = (1, ¥), Range = [0, ¥), (d) Domain = [1, ¥), Range = [0, ¥)

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RELATIONS AND FUNCTIONS-1, , 10., , 23, , The domain of the function f given by f(x) =, , x2 + 2 x + 1, x2 – x – 6, , ., , (a) R – {3, – 2}, (b) R – {– 3, 2}, (c) R – [3, – 2], (d) R – (3, – 2), 11. The domain and range of the function f given by, f(x) = 2 – |x – 5| is, (a) Domain = R+, Range = (–¥, 1], (b) Domain = R, Range = (–¥, 2], (c) Domain = R, Range = (–¥, 2), (d) Domain = R+, Range = (–¥, 2], 12., The domain for which the functions defined by f (x) = 3x2 – 1, and g(x) = 3 + x are equal to, (a), , é 4ù, êë –1, 3 úû, , (b), , é 4ù, êë1, 3 úû, , (c), , 4ù, é, êë –1, – 3 úû, , (d), , 4ù, é, êë –2, – 3 úû, , Past Year MCQs, 1, , 13. The domain of the function f ( x) =, , x -x, , is, , [JEE MAIN 2011, C], (b) (– ¥ , 0), (d) (– ¥ , ¥ ), , (a) (0, ¥), (c) (– ¥, ¥ ) – {0}, , 14. The domain of the function, æ, 1 ö ö, æ, f ( x ) = log 2 ç – log1/ 2 ç1 + 1/ 4 ÷ – 1÷ is [BITSAT2014, S, BN], è, è, x ø ø, (a) (0, 1) (b) (0, 1] (c) [1, ¥) (d) (1, ¥), æ1 ö, ç ÷, ÷ = 3x , x ¹ 0 and, 15. If f(x) + 2f ç, [JEE MAIN 2016, A], èx ø, S = {x Î R : f(x) = f(–x)}; then S:, (a) contains exactly two elements., (b) contains more than two elements., (c) is an empty set., (d) contains exactly one element., x, 16. If x is real number, then 2, must lie between, x - 5x + 9, [BITSAT 2017, A], 1, 1, (a), and 1, (b) –1 and, 11, 11, 1, (c) –11 and 1, (d) –, and 1, 11, 17. The domain of the function f ( x ) = x 2 - [ x ] 2 , where [x], denotes the greatest integer less than or equal to x, is, [BITSAT 2018, A], (a) (0, ¥), (b) (-¥ , 0), (c) (-¥, ¥), (d) None of these, , Exercise 3 : Try If You Can, 1., , æ 1 ö, If af (x + 1) + bf ç, ÷ = x , x ¹ –1, a ¹ b, then f (2) is equal to, è x +1 ø, (a), , (c), 2., , (, , 2a + b, , 2 a2 - b 2, , (b), , ), , a + 2b, , 4., , a, 2, , a -b, , 2, , (d) None of these, , a 2 - b2, , 5., , If f : R ® R satisfies f ( x + y ) = f ( x) + f ( y ) , for all x,, n, , 6., , y Î R and f(1) = 7, then S f (r ) is, r=1, , (a), (c), 3., , 7 n (n + 1), 2, , (b), , 7 (n + 1), 2, , (d), , f (x) =, , 7n, 2, , 7n + (n + 1) ., , [ x]3 - 4 [ x], , the greatest integer function) is, , 7., , (where [.] represents, , (0.625) 4 – 3 x – (1.6) x ( x + 8), , (a) [–3, 2] (b) [1, 4], (c) [2, 5], The set of all x satisfying the inequality, , 1, 2x - 1, - 3, ³0, x - x +1 x +1 x +1, (a) (–¥, 2], (b) [1, 2], (c) (–¥, –1) È (–1, 2], (d) (2, ¥], 2, , 2, , The domain of the function, f(x) = loge {sgn(9 – x2)} +, , (a) [–2, 1) È [2, 3), (b) [–4, 1) È [2, 3), (c) [4, 1) È [2, 3), (d) [2, 1) È [2, 3), If x and y satisfy the equations, max (| x + y |, | x - y |) = 1 an d | y |= x - [ x ] , then the, number of ordered pairs (x, y) is, (a) 0, (b) 4, (c) 8, (d) infinite, If f (1) = 1 and f (n + 1) = 2 f (n) + 1, if n ³ 1 , then f (n) is., (a) 2n+1, (b) 2n, (c) 2n - 1, (d) 2n-1 - 1, Find the domain of, , -, , (d) [–4, –1]

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EBD_7762, 24, , MATHEMATICS, 2 æ1– xö, = 64 x "x ÎR – {–1, 0,1} then f (x) is, If [ f ( x)] . f ç, è 1 + x ÷ø, , 8., , 1/ 3, , (a), , é æ1+ xö ù, 4 êx2 ç, ÷ú, ë è1– xø û, , (c), , é æ 1+ xö ù, 4 êx ç, ÷ú, ë è1– xø û, , 1/ 3, , (b), , é æ1– xö ù, 4 êx2 ç, ÷ú, ë è1+ xø û, , (d), , é æ1– xö ù, 4 êx ç, ÷ú, ë è 1+ xø û, , 1/ 3, , 1/ 3, , | x|, , | x|, , 9., , Let f ( x ) = e{e sgn x} and g ( x) = e[e sgn x ] , x Î R where {}, and [] denotes the fractional and integral part functions, respectively. Also h (x) = log (f (x)) + log (g (x)), then for real, x, h (x) is, (a) An odd function, (b) An even function, (c) Neither an odd nor an even function, (d) Both odd as well as even function., , 10., , If f : ¡ ® ¡ & g: ¡ ® ¡ be two given functions, then, 2 min {f (x) – g(x), 0} equals, (a) f (x) + g(x) – |g(x) – f (x)|, (b) f (x) + g(x) + |g(x) – f (x)|, (c) f (x) – g(x) + |g(x) – f (x)|, (d) f (x) – g(x) – |g(x) – f (x)|, e–x, , where [x] is the greatest, 1 + [ x], 1+ x 2, integer less than or equal to x. Then,, (a) D( f + g) = R –[– 2, 0), (b) D( f + g) = R – [– 1, 0), , 11., , Let f (x) =, , x, , and g(x) =, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, , (c), (b), (a), (c), (a), (a), (a), (d), (d), (a), (d), , 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, , (d), (d), (d), (c), (a), (d), (b), (a), (a), (a), (c), , 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, , (c), (b), (a), (c), (a), (a), (c), (d), (d), (a), (a), , 1, 2, , (d), (d), , 3, 4, , (c), (c), , 5, 6, , (b), (b), , 1, 2, , (a), (a), , 3, 4, , (a), (d), , 5, 6, , (c), (d), , 1ö, æ, (c) R( f ) Ç R(g) = ç – 2, ÷, è, 2ø, (d) None of these, x, and ‘a’ be a real number. If x0 = a,, 1- x, x1 = f (x0), x2 = f (x1), x3 = f (x2)....... If x2009 = 1,, then the value of a is, , 12. Let f (x) =, , 2009, 1, 1, (c), (d), 2010, 2009, 2010, 13. Define relations R1 and R2 on set A = [2, 3, 5, 7, 10] as xR1y if, x(y – 1) and xR2y if x + y = 10, then the relation R given by R, = R1 Ç R2 is, (a) { }, (b) {(3, 7)}, (c) {(3, 7), (5, 5)}, (d) None of these, 14. Let [x] represent the greatest integer less than or equal to x., , (a) 0, , (b), , é 2, ù é 2 ù, If ê n + l ú = ê n + 1 ú + 2, where l, n Î N, then l can, ë, û ë, û, assume, (a) (2n + 4) different values, (b) (2n + 5) different values, (c) (2n + 3) different values, (d) (2n + 6) different values, 15. The domain of two definition of the function f (x) is given by, the equation 2x + 2y = 2 is, (a) 0 < x £ 1, (b) 0 £ x £ 1, (c) – ¥ < x £ 0, (d) – ¥ < x < 1, , ANSWER KEYS, EXERCISE-1 : TOPIC-WISE MCQs, (a), (d) 67 (a), 34 (d) 45, 56, (a), (d) 68 (b), 35 (d) 46, 57, (c), (c) 69 (a), 36 (b) 47, 58, (a) 48, (a), (a) 70 (b), 37, 59, (c), (c) 71 (a), 38 (b) 49, 60, (c), (b) 72 (c), 39 (d) 50, 61, (c) 51, (c), (a), 40, 62, 73 (d), (c) 52, (b), (b) 74 (a), 41, 63, (d), (a) 75 (b), 42 (b) 53, 64, (a) 54, (a), (b) 76 (b), 43, 65, (b), (c) 77 (d), 44 (d) 55, 66, EXERCISE-2 : Exemplar & Past Year MCQs, (a), (d), (b) 13 (b), 7, 9, 11, (c) 10, (a), (a) 14 (a), 8, 12, EXERCISE-3 : Try If You Can, (c), (a), (b) 13 (b), 7, 9, 11, (a) 10, (d), (d) 14 (b), 8, 12, , 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, , (b), (b), (b), (b), (a), (a), (c), (c), (b), (c), (a), , 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, , (a), (d), (d), (b), (c), (b), (a), (a), (a), (d), (b), , 15, 16, , (a), (d), , 17, , (d), , 15, , (d), , 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, , (a), (b), (c), (c), (d), (c), (c), (c), (d), (c), (d)

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3, , TRIGONOMETRIC, FUNCTIONS, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , 5, JEE MAIN, BITSAT, , Number of Questions, , 4, , 3, , 2, , 1, , 0, , 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 6, 9, , Critical Concepts, , Relation between degree and radian,, Trigonometric Functions: Domain, & Range, Sum & Difference of, two angles, Conditional Trigonometric, Identities, Trigonometric equations &, their Solutions, Properties of TriangleSine Rule, Cosine Rule, etc., , Rating of, Difficulty Level, , CUS, (Chapter Utility Score), Out of 10, , 4.5/5, , 8.2

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TRIGONOMETRIC FUNCTIONS, , 27

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EBD_7762, 28, , MATHEMATICS, , Exercise 1 : Topic-wise MCQs, Topic 1 : Circular System, Trigonometric Ratios,, , (a), , Domain and Range of Trigonometric Functions., 1., , 2., 3., , 4., , 5., , 6., , 7., , 8., , p [ x], is, 2, (a) {0, 1}, (b) {– 1, 1}, (c) {–1, 0, 1}, (d) [–1, 1], A circular wire of radius 7 cm is cut and bent again into, an arc of a circle of radius 12 cm. The angle subtended, by the arc at the centre is, (a) 50°, (b) 210°, (c) 100°, (d) 60°, A circular wire of radius 3 cm is cut and bent so as to lie, along the circumference of a hoop whose radius is 48 cm., The angle in degrees which is subtended at the centre of, hoop is, (a) 21.5°, (b) 23.5°, (c) 22.5° (d) 24.5°, The radius of the circle in which a central angle of 60°, The range of f (x) = cos, , 22 ö, æ, intercepts an arc of length 37.4 cm is ç Use p =, ÷, è, 7 ø, 9., , (a) 37.5 cm (b) 32.8 cm (c) 35.7 cm (d) 34.5 cm, The degree measure of the angle subtended at the centre, of a circle of radius 100 cm by an arc of length, 22 ù, é, 22 cm as shown in figure, is ê Use p = ú, 7û, ë, (a), , 12° 30¢, , (b) 12° 36¢, (c), , 11° 36¢, , m, 22 cm, 0c, 10 q, O 100 cm, , (d) 11° 12¢, 10., , Domain of the function f (x) =, , 1, - 1 , is, sin x, , πö, , (b), , U [(2n - 1) π, 2nπ], , (d) None of these, , U (2 nπ, (2n + 1) π], , n ÎI, , nÎI, , The value of tan2 q sec2q (cot2q – cos2q) is, 1, (a) 0, (b) 1, (c) –1, (d), 2, The large hand of a clock is 42 cm long. How much, distance does its extremity move in 20 minutes?, (a) 88 cm (b) 80 cm, (c) 75 cm (d) 77 cm, The angle in radian through which a pendulum swings, and its length is 75 cm and tip describes an arc of length, 21 cm, is, 7, 6, 8, 3, (a), (b), (c), (d), 25, 25, 25, 25, The length of an arc of a circle of radius 3 cm, if the angle, subtended at the centre is 30° is (p = 3.14), (a) 1.50 cm (b) 1.35 cm (c) 1.57 cm (d) 1.20 cm, , æ, , U çè 2nπ, 2nπ + 2 ÷ø, , (c), , nÎI, , 11., , The range of the function f (x) =, , 1, is, 2 - cos3x, , (a), , ( -2, ¥ ), , (b), , [ -2,3], , (c), , æ1 ö, ç 3 ,2÷, è, ø, , (d), , æ1 ö, ç 2 ,1÷, è, ø, , 12. I : cos a + cos b + cos g = 0, II : sin a + sin b + sin g = 0, , 3, If cos (b – g) + cos (g – a) + cos (a – b) = - , then, 2, (a) I is false and II is true (b) I and II both are true, (c) I and II both are false (d) I is true and II is false, -4, 13. If tan q =, , then sin q is, 3, -4, 4, -4 4, or, (a), but not, (b), 5, 5, 5, 5, 4, 4, (c), but not (d) None of these, 5, 5, 24, 14. If sin q =, and 0° < q < 90° then what is the value of, 25, æqö, sin ç ÷ ?, è2ø, 3, 4, 12, 7, (a), (b), (c), (d), 5, 5, 25, 25, 15. Assertion : Area of unit circle is p unit2., , Reason : Radian measure of 40° 20¢ is equal to, , 2 p, 540, , radian., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 16. Consider the statements given below:, I. sin x is positive in first and second quadrants., II. cosec x is negative in third and fourth quadrants., III. tan x and cot x are negative in second and fourth, quadrants., IV. cos x and sec x are positive in first and fourth quadrants., Choose the correct option.

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TRIGONOMETRIC FUNCTIONS, , 29, , (a) All are correct, (b) Only I and IV are correct, (c) Only III and IV are correct, (d) None is correct, 17. Which pairs of function is identical ?, (a), , f ( x) = x2 , g ( x) = x, , (b) f (x) = sin2x + cos2x ; g (x) = 1, x, , g (x) = 1, x, (d) None of these, , (c) f (x) =, , 18. Radian measure of 40° 20 ¢ is equal to, 120 p, 121 p, (a), radian, (b), radian, 504, 540, 121 p, radian, (d) None of these, (c), 3, 1, 19. If x = sec q + tan q, then x +, =, x, (a) 1, (b) 2 sec q (c) 2, (d) 2 tan q, 20. Which among the following is/ are true?, I. The values of cosec x repeat after an interval of 2p., II. The values of sec x repeat after an interval of 2p., III. The values of cot x repeat after an interval of p., (a) I is true, (b) II is true, (c) III is true, (d) All are true, 21. The domain of ; f(x) =, , -2 6, and x lies in III quadrant, then the value, 5, 1, of cot x is, . Value of m is, m 6, (a) 1, (b) 2, (c) 3, (d) 5, 27. Consider the following statements., I. cot x decreases from 0 to –¥ in first quadrant and, increases from 0 to ¥ in third quadrant., II. sec x increases from –¥ to –1 in second quadrant, and decreases from ¥ to 1 in fourth quadrant., III. cosec x increases from 1 to ¥ in second quadrant, and decreases from –1 to –¥ in fourth quadrant., Choose the correct option., (a) I is incorrect, (b) II is incorrect, (c) III is incorrect, (d) IV is incorrect, , 26. If sin x =, , æ cosec q + cot q ö, 1, . Value of m is, ç, ÷ is equal to, m, è sec q - tan q ø, (a) 2, 29. Let f ( x ) =, , cos(sin x ) + log x { x} ; {.} denote, , (c), , (a), , [1, p), , (b), , ( 0, 2p ) - [1, p), , (d) (0, 1), , sin x, 2, , 1 + tan x, , -, , cos x, 1 + cot 2 x, , 30., , then range of f(x) is, , (a) [–1, 0], (b) [0, 1], (c) [–1, 1], (d) None of these, 23. If sin q + cos q = 1, then sin q cos q =, (a) 0, , (b) 1, , (c) 2, , (d), , 31., , 1, 2, , -p, -p, cos A cos B 1, < A< 0, < B < 0, then, =, = ,, 2, 2, 3, 4, 5, value of 2 sin A + 4 sin B is – a. The value of ‘a’ is, (a) 4, (b) 2, (c) 3, (d) 0, , 24. If, , 25. If tan q =, , (c) 5, , (d) 6, , 1, æ p xö, - tan ç ÷ , -1 < x < 1, è 2ø, 2, , é1 ù, êë 2 ,1úû, , ö, é1, (b) ê , -1 ÷, ë2, ø, , é 1, ù, é 1 ö, (d) ê - , -1ú, ê - 2 ,1÷, ë 2, û, ë, ø, The domain of the function, x, f ( x) =, is, sin(ln x) – cos(ln x ), (a) (e2np, e(3n + 1/2)p), (b) (e(2n + 1/4)p, e (2n + 5/4)p), (c) (e2n + 1/4)p, e(3n – 3/4)p), (d) None of these, Find the distance from the eye at which a coin of a, diameter 1 cm be placed so as to hide the full moon, it is, being given that the diameter of the moon subtends an, angle of 31¢ at the eye of the observer., (a) 110 cm, (b) 108 cm, (c) 110.9 cm, (d) 112 cm, A wheel rotates making 20 revolutions per second. If the, radius of the wheel is 35 cm, what linear distance does a, (c), , æ pö, çè 0, ÷ø - {1}, 2, , 22. Let f(x) =, , (b) 4, , and g ( x ) = 3 + 4 x - 4 x 2 , then dom (f + g) is given by, , the fractional part, is, (a), , -3, 3p, and p < q <, , then the value of, 5, 2, , 28. If cos q =, , æ cosec2 q – sec 2 q ö, , then ç, ÷ is equal to, ç cosec2 q + sec 2 q ÷, 7, è, ø, , 1, , m, . The value of m is, m +1, (a) 1, (b) 2, (c) 3, , (d) 4, , 32., , 22 ö, æ, point of its rim travel in three minutes? ç Take p =, ÷, è, 7 ø, (a) 7.92 km, , (b) 7.70 km, , (c) 7.80 km, , (d) 7.85 km

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EBD_7762, 30, , MATHEMATICS, , 33. The minute hand of a watch is 1.5 cm long. How far does, its tip move in 40 minutes? (Use p = 3.14), (a) 2.68 cm, , (b) 6.28 cm, , (c) 6.82 cm, , (d) 7.42 cm, , 34. If the arcs of the same lengths in two circles subtend, angles 65° and 110° at the centre, the ratio of their radii, is, (a) 12 : 13 (b) 22 : 31, 35., , If tan A + cot A = 4, then tan4 A + cot4 A is equal to, (a) 110, , 36., , (c) 22 : 13 (d) 21 : 13, , If, , (b) 191, , (c) 80, , (d) 194, , sin A, cos A, = m and, = n, then the value of tan B;, sin B, cos B, , n2 < 1 < m2, is, (a) n2, , (c), 37., , (b), , n2, , (m – 1), 2, , 1– n, , ±, , 2, , m2 – 1, , (d) m2, , If tan q + sec q = p, then what is the value of sec q ?, (a), , p2 + 1, , (b), , p2, , p2 + 1, p, , Topic 2 : Trigonometric Ratios of compound Angles,, Trigonometric Ratios of sum and Difference of Angles,, Trigonometric Ratios of Multiple and Sub-Multiple Angles,, Formula for Lowering the Degree of Trigonometric Functions, 41. Value of cot 5° cot 10° ..... cot 85° is, (a) 0, (b) –1, (c) 1, (d) 2, 42. Value of sin 10° + sin 20° + sin 30° +......+sin360° is, (a) 1, , (c), 38., , 39., , p +1, 2p, , p +1, 2p, , If 12 cot q – 31 cosec q + 32 = 0, then the value of, sin q is, (a), , 3, or 1, 5, , (b), , 2, –2, or, 3, 3, , (c), , 4, 3, or, 5, 4, , (d), , ±, , If 5 tan q = 4, then, , (b) 1, , 1, 2, , 5 sin q – 3 cos q, =, 5 sin q + 2 cos q, , (c), , 1, 6, , (d) 6, , The range of f (x) = cos x – sin x is, (a) [– 1, 1], (b) [– 1, – 1], (c), , [ - 2, 2], , (d), , 1, 2, , 1, 1, and tan B = , then value of A + B is, 2, 3, p, p, p, (a) p, (b), (c), (d), 6, 2, 4, 44. If sin 2q + sin 2f = 1/2, cos 2q + cos 2f = 3/2, then value of, cos2 (q – f) is, 5, 3, 5, 3, (b), (c) (d), 8, 8, 8, 5, 45. Consider the statements given below:, I. 2 cos x . cos y = cos(x + y) – cos(x – y)., II. –2 sin x . sin y = cos(x + y) – cos(x – y)., III. 2 sin x . cos y = sin(x + y) – sin(x – y)., IV. 2 cos x . sin y = sin(x + y) + sin(x – y)., Choose the correct statements., (a) I is correct, (b) II is correct, (c) Both I and II are correct, (d) III is correct, 46. The value of cosec (–1410)° is equal to, , (a), , (b), , 1, 2, , 47. The value of sin 765° is, , 2, , (a) 0, 40., , (d), , (c) 2, , 43. If tan A =, , (a) 1, 2, , (b) 0, , (a) 2, , (c) 2, , (d) None of these, , 1, , (b) 3, , . Value of n is, n, (c) 4, (d) 0, , cos ( p + x ) × cos ( – x ), = cot2 x, æp, ö, sin ( p – x ) × cos ç + x ÷, è2, ø, Reason : cos (p + q) = –cos q and cos (–q) = cos q., Also, sin (p – q) = sin q and sin (–q) = –sin q., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 49. The value of tan 20° + 2 tan 50° – tan 70° is equal to, (a) 1, (b) 0, (c) tan 50°, (d) None of these, , 48. Assertion :, , 50. If a and b lies between 0 and, , (d) [– 2, – 2], sin (a – b) =, (a), , 55, 56, , p, 12, and if cos (a + b) =, and, 2, 13, , 3, , then value of sin 2a is, 5, 13, 56, (b), (c) 0, (d), 58, 65

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TRIGONOMETRIC FUNCTIONS, , 31, , 51. Value of 3 cosec 20° – sec 20° is, (a) 3, , (b), , 52. Value of 2cos, , (a), , -, , 1, 2, , 3, 2, , 62. If sin x + cos x =, , (c) 1, , (d) 4, , p, 9p, 3p, 5p, cos, + cos + cos, is, 13, 13, 13, 13, , (b) 0, , (c) 1, , (d), , 63., , 3, 2, , 53. Value of sin 47° + sin 61° – sin 11° – sin 25° is, (a) cos 7º (b) sin 7º, (c) sin 61º (d) –sin 25º, 2sin a, 1 - cos a + sin a, 54. If y =, , then value of, is, 1 + cos a + sin a, 1 + sin a, y, 3, y, (a), (b) y, (c) 2y, (d), 3, 2, 3, p, -12, 3p, 55. If sin A = , 0 < A < and cos B =, ,p<B<, , then, 5, 2, 13, 2, value of sin (A – B) is, 13, 15, 13, 16, (a) (b) (c) (d) 82, 65, 75, 65, 56. Value of tan15°. tan45° tan75° is, (a) 0, , (b) 1, , 57. Value of, , pö æ, 3p ö, æ, çè 1 + cos ÷ø çè1 + cos ÷ø, 8, 8, , 59., , 60., , 61., , 5p ö, æ, çè1 + cos ÷ø, 8, , (d) –1, , 7p ö, æ, çè1 + cos ÷ø is, 8, , 1, 2, 5, 3, (b), (c), (d), 8, 3, 8, 4, If A + B = 45°, then (cot A – 1) (cot B – 1) is equal to, 1, (a) 1, (b), (c) –1, (d) 2, 2, 3, If sin A =, and A is in first quadrant, then the values, 5, of sin 2A, cos 2A and tan 2A are, 24 7 24, 1 7 1, ,, ,, ,, ,, (a), (b), 25 25 7, 25 25 7, 24 1 24, 1 24 1, ,, ,, ,, ,, (c), (d), 25 25 7, 25 25 24, 1, ,, The value of tan(a + b), given that cot a =, 2, –5, æp, ö, 3p ö, , b Î ç , p ÷ is, a Î æç p,, and sec b =, ÷, è2, ø, 3, è, 2 ø, 1, 2, 5, 3, (a), (b), (c), (d), 11, 11, 11, 11, 4, 12 3p, If cos A = , cos B = ,, < A, B < 2 p , the value of the, 5, 13 2, cos (A + B) is, 30, 33, 65, 65, (a), (b), (c), (d), 33, 65, 65, 30, , (a), , 58., , (c), , 3, 2, , 7, 25, 24, (c), (d), 25, 7, 7, cot 54° tan 20°, +, The value of, is, tan 36° cot 70°, (a) 2, (b) 3, (c) 1, (d) 0, The value of cosec (–1410)° is equal to, , (a), , 64., , 25, 17, , 1, , then tan 2x is, 5, , (a) 1, , (b), , (b) 2, , (c), , 1, 2, , (d) None of these, , 19p, is n . Value of ‘n’ is, 3, (b) 2, (c) 3, (d) 5, , 65. The value of tan, (a) 1, , 3, - 11p ö, 66. The value of sin æç, ÷ is m . Value of ‘m’ is, è 3 ø, (a) 1, (b) 2, (c) 3, (d) 5, 67. If none of the angles x, y and (x + y) is a multiple of p, then, , (a) cot (x + y) =, , cot x × cot y – 1, cot y + cot x, , cot x × cot y + 1, cot y – cot x, (c) (a) and (b) are true, (d) (a) and (b) are not true, 68. cos (A + B). cos ( A – B) is given by:, (a) cos2 A – cos2B, (b) cos( A2 – B2), 2, 2, (c) cos A – sin B, (d) sin2A – cos2B, æ 5p ö, 69. What is the value of sin ç ÷ ?, è 12 ø, , (b) cot (x – y) =, , (a), , 3 +1, 2, , (b), , 6+ 2, 4, , 3+ 2, 6 +1, (d), 4, 2, 1, 1, 70. If x + = 2 cos q , then x 3 +, is:, x, x3, (c), , (a), , 1, cos 3q, 2, , (b) 2 cos 3q, , 1, cos 3q, 3, If 1 + cot q = cosec q, then the general value of q is, p, p, (a) np +, (b) 2np –, 2, 2, p, p, (c) 2np +, (d) 2np ±, 2, 2, , (c) cos 3q, , 71., , (d), , 72. The value of sin, (a), , 3, 2, , 31p, is, 3, , (b) –, , 3, 2, , (c) –, , 1, 2, , (d), , 1, 2

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EBD_7762, 32, , MATHEMATICS, , æ –15p ö, 73. The value of cot ç, is, è 4 ÷ø, , 75., , 76., , 77., , (c) – 2 sin x, sin x – sin 3x, , (d) None of these, , p, 5p, p, p, sec, – 4 sin, cot, is equal to, 6, 4, 6, 3, , (b) 1, , (c) 3, , (d) 4, , 78. cot x cot 2x – cot 2x cot 3x – cot 3x cot x is equal to, (a) 0, (b) 1, (c) 2, (d) 3, tan 70° – tan 20°, =, 79. The value of, tan 50°, (a) 1, (b) 2, (c) 3, (d) 0, , 81., , 82., 83., , 1, 3, –, =, sin 10° cos 10°, (a) 0, (b) 1, (c) 2, (d) 4, 1 + cos 2x + cos 4x + cos 6x =, (a) 2 cos x cos 2x cos 3x, (b) 4 sin x cos 2x cos 3x, (c) 4 cos x cos 2x cos 3x, (d) None of these, cosec A – 2 cot 2A cos A =, (a) 2 sin A, (b) sec A, (c) 2 cos A cot A, (d) None of these, If cos x + cos y + cos a = 0 and sin x + sin y + sin a = 0,, æx+ yö, then cot ç, ÷=, è 2 ø, , æx+ yö, (d) sin ç, ÷, è 2 ø, If angle q is divided into two parts such that the tangent, of one part is K times the tangent to other and f is their, difference, then sin q is equal to, , (a) sin a, 84., , (a), , (d), , (b) cos a, , K +1, q, sin, K –1, 2, , (c) cot a, , (b), , K +1, f, sin, K –1, 2, , K –1, sin f, K +1, , 1 + sin A – cos A, =, 1 + sin A + cos A, , (a), , 86, , pö æ, pö, æ, The value of ç 1 + cos ÷ ç 1 + cos ÷, 6ø è, 3ø, è, 2p ö æ, 7p ö, æ, m, . Value of m is, ç 1 + cos, ÷ ç 1 + cos, ÷ is, 3 ø è, 6 ø, 16, è, (a) 1, (b) 2, (c) 3, (d) 8, The value of, , (a) 2, , 85., , (d) None of these, , is equal to, sin 2 x – cos 2 x, (a) sin 2x (b) cos 2x, (c) tan 2x, , 3 sin, , 80., , K +1, sin f, K –1, , –1, , (b) 1, (c) 3, (d) – 3, 3, æ 3p, ö, æ 3p, ö, cos ç, + x ÷ – cos ç, – x ÷ is equal to, 4, 4, è, ø, è, ø, (a), (b) –2 sin x, 2 sin x, (a), , 74., , (c), , sin, , A, 2, , (b) cos, , A, 2, , (c) tan, , A, A, (d) cot, 2, 2, , 1, é 3 cos 23° – sin 23°ù =, û, 4ë, , (a) cos 43° (b) cos 7°, 87. If sin 2q + sin 2f =, , (c) cos 53° (d) None of these, , 1, 3, and cos 2q + cos 2f = , then, 2, 2, , cos2 (q – f) =, (a), , 3, 8, , (b), , 5, 8, , (c), , 3, 4, , (d), , 5, 4, , Topic 3 : Solution of Trigonometric Equations, 88. If 0 < q < 360°, then solutions of cos q = – 1/2 are, (a) 120º, 360º, (b) 240º, 90º, (c) 60º, 270º, (d) 120º, 240º, 1, 89. If tan q = –, , then general solution of the equation is, 3, p, p, (a) 2np + , n Î I, (b) np + , n Î I, 6, 6, p, p, (c) 2np - , n Î I, (d) np - , n Î I, 6, 6, 90. If 2 tan 2 q = sec2q, then general value of q are, p, p, (b) np ± , n Î I, (a) np ± , n Î I, 6, 4, p, p, (c) 2np + , n Î I, (d) 2np ± , n Î I, 6, 4, 91. If sin 5x + sin 3x + sinx = 0 and 0 £ x £ p/2, then value of x is, p, p, p, p, (b), (c), (d), (a), 6, 3, 2, 4, 92. If sin 2x + cos x = 0, then which among the following, is/ are true?, I. cos x = 0, 1, II. sin x = –, 2, p, III. x = (2n + 1) , n Î Z, 2, 7p, , nÎZ, IV. x = np + (–1)n, 6, (a) I is true, (b) I and II are true, (c) I, II and III are true (d) All are true, 5p, p, p, + cosec, + 3 tan2, is equal to, 6, 6, 6, (a) 1, (b) 5, (c) 3, (d) 6, , 93. cot2

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TRIGONOMETRIC FUNCTIONS, , 94. Value of 2 sin2, , p, 7p, m, p, + cosec2, ·cos2, is, . The, 3, 6, m -1, 6, , value of ‘m’ is, (a) 3, (b) 2, 95. If, , 33, , (c) 4, , 102. If tan q + tan 2q +, , sin x, sec x, tan x, æ pö, ´, ´, = 9, where x Î ç 0, ÷ , then, cos x cosec x cot x, è 2ø, , the value of x is equal to, p, p, p, (b), (c), (d) p, 4, 3, 2, Find x from the equation:, cosec (90° + q) + x cos q cot (90° + q) = sin (90° + q)., (a) cot q (b) tan q, (c) –tan q (d) –cot q, Number of solutions of the equation tan x + sec x =, 2 cos x lying in the interval [0, p] is, (a) 0, (b) 1, (c) 2, (d) 3, If sin 3a = 4 sin a sin (x + a) sin (x – a), then x =, p, p, (a) np ±, (b) np ±, 6, 3, p, p, (c) np ±, (d) np ±, 4, 2, The general value of q satisfying the equation, , 96., , 97., , 98., , 99., , 2, tan q + tan æç p – q ö÷ - b ± b - 4ac = 2, is, 2a, è2, ø, , p, 4, p, (c) 2np ±, 4, , (a) np ±, , (b) np +, , p, 4, , pö, æ, 100. Assertion : If tan 2x = – cot ç x + ÷ , then, 3ø, è, 5p, , n Î Z., 6, Reason : tan x = tan y Þ x = np + y, where n Î Z., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., , x = np +, , (a) q = np + (–1)n + 1, , 3 tan q = 0 is, , p, , q = np; n Î I, 3, , (b) q = np; n Î I, (c) q =, , np, , nÎI, 2, , (d) q = np + (–1)n + 1, , p, , " nÎI, 18, p, (b) q = (6n + 1) , " n Î I, 9, p, (c) q = (3n + 1) , " n Î I, 9, p, (d) q = (3n + 1), 18, The most general value of q satisfying the equations, sin q = sin a and cos q = cos a is, (a) 2np + a, (b) 2np – a, (c) np + a, (d) np – a, If sec 4q – sec 2q = 2, then the general value of q is, p, p, (a) (2n + 1), (b) (2n + 1), 4, 10, p, np p, p, +, (c) np +, or, (d) (2n + 1), 2, 5, 10, 2, General solution of the equation tan q tan 2q = 1 is, given by, p, p, (a) (2n + 1) , n Î I, (b) np + , n Î I, 4, 6, p, p, (c) np – , n Î I, (d) np ± , n Î I, 6, 6, The most general value of q satisfying the equation, , 104., , 105., , cosq =, , (d) np + (–1)n, , 101. The general solution of sin2 q sec q +, , 103., , 106., , p, 4, , p, , q = np; n Î I, 2, , 3, then, , (a) q = (6n + 1), , (d) None of these, , (a), , 3 tan q tan 2q =, , (a), , 1, 2, , and tanq = – 1 is, , 2np - 7, , p, 4, , (b), , np -, , p, 4, , p, 7p, (d) 2np +, 2, 4, 2, 107. The solution of the equation cos q + sinq + 1 = 0, lies in the, interval, , (c), , np +, , (a), , æ p pö, çè - , ÷ø, 4 4, , æ p 3p ö, (b) çè , ÷ø, 4 4, , (c), , æ 3p 5p ö, çè , ÷ø, 4 4, , æ 5p 7p ö, (d) çè , ÷ø, 4 4, , 108. The number of values of x in the interval [0, 3p] satisfying, the equation 2sin2 x + 5 sin x – 3 = 0 is, (a) 4, (b) 6, (c) 1, (d) 2, æp, ö, 109. If cot q + cot ç + q ÷ = 2, then the general value of q is, è4, ø, p, p, (b) 2np ±, (a) 2np ±, 6, 3, p, p, (c) np ±, (d) np ±, 3, 6

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EBD_7762, 34, , MATHEMATICS, 2, , 110. If 2 cos x + 3 sin x – 3 = 0, 0 £ x £ 180°, then x =, (a) 30°, 90°, 150°, (b) 60°, 120°, 180°, (c) 0°, 30°, 150°, (d) 45°, 90°, 135°, 111. If cot q + tan q = 2 cosec q, the general value of q is, (a) np ±, (c) 2np ±, , p, 3, , p, 6, , (b) np ±, , p, 3, , (d) 2np ±, , p, 5, , (b), , p, 6, , 1ö p, æ, çn + ÷, 6ø 5, è, , 1ö p, 1ö p, æ, æ, (d) ç n + ÷, ç 2n ± ÷, 6ø 5, 3ø 5, è, è, 113. If cos 7q = cos q – sin 4q, then the general value of q is, (c), , (a), , np np p, ,, +, 4 3 18, , (c), , np np, n p, ,, + ( –1), 4, 3, 18, , 114. The solution of sin x = –, , (a), , 2np +, , p, 3, , (b) np +, , p, 4, , p, p, (d) (2n + 1), 6, 6, 119. Number of solutions of equation,, sin 5x cos 3x = sin 6x cos 2x, in the interval [0, p] is, (a) 4, (b) 5, (c) 3, (d) 2, 120. If tan (cot x) = cot (tan x), then, , (c) 2np -, , 112. If 3 tan 2q + 3 tan 3q + tan 2q tan 3q = 1, then the, general value of q is, (a) np ±, , 118. The solution of tan 2q tan q = 1 is, , (a) sin 2x =, , 2, (2n + 1) p, , (b) sin x =, , (c) sin 2x =, , 4, (2n + 1) p, , (d) None of these, , 2, , 4, (2n + 1) p, , , the smallest positive, , (b), , np np, n p, + ( –1), ,, 3 3, 18, , 121. If tan(A – B) = 1, sec(A + B) =, , (d), , np np, n p, ,, + ( –1), 6, 3, 18, , 25p, 19 p, 13p, 7p, (b), (c), (d), 24, 24, 24, 24, 122. The solution of the equation, [sin x + cos x]1 + sin 2x = 2, –p £ x £ p is, , 3, is, 2, , value of B is, (a), , (a), , 4p, , where n Î Z, (a) x = np + (–1), 3, n, , 3, , p, 2, , (b) p, , 2p, , where n Î Z, (b) x = np + (–1), 3, , p, 3p, (d), 4, 4, 123. The number of solutions of the given equation, , 3p, , where n Î Z, 3, (d) None of the these, , tan q + sec q = 3, where 0 £ q £ 2p is, (a) 0, (b) 1, (c) 2, (d) 3, 124. If n is any integer, then the general solution of the, , (c), , n, , (c) x = np + (–1)n, , 115. If 3 tan 2q + 3 tan 3q + tan 2q tan 3q = 1, then the, general value of q is, (a) np +, (c), , p, 5, , 1ö p, æ, ç 2n ± ÷, 6ø 5, è, , 116. If tan q –, , 2 sec q =, , (a) np + (–1)n, (c) np + (–1)n, , 1ö p, ÷, 6ø 5, , (b), , æ, çn +, è, , (d), , 1ö p, æ, çn + ÷, 3ø 5, è, , (b) np + (–1)n, , p p, +, 3 4, , (d) np + (–1)n, 2, , 117. The general solution of sin q sec q +, (a) q = np + (–1)n + 1, (b) q = np, n Î Z, (c) q = np + (–1)n + 1, np, , nÎZ, (d) q =, 2, , (b) x = np ±, , p p, +, 4 3, , 3 tan q = 0 is, , p, , q = np, n Î Z, 3, p, , nÎZ, 3, , p p, –, 3 4, , 125. If 4 sin 2 q + 2, , 2, , is, , p, 12, , p, 7p, or x = 2n p –, 12, 12, , (c) x = 2n p +, (d) x = np +, , 1, , p, 7p, or x = 2n p +, 12, 12, , (a) x = 2n p –, , 3, then the general value of q is, , p p, –, 4 3, , equation cos x – sin x =, , p, 7p, or x = np –, 12, 12, , (, , ), , 3 + 1 cos q = 4 +, , 3, then the general, , value of q is, (a) 2np ±, , (c) np ±, , p, 3, p, 3, , (b) 2np +, , (d) np –, , p, 4, p, 3

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TRIGONOMETRIC FUNCTIONS, , 35, , 126. The number of values of x in the interval [0, 3p] satisfying, the equation, 2 sin2 x + 5 sin x – 3 = 0 is, (a) 4, (b) 6, (c) 1, (d) 2, 127. If sin q + cos q = 1, then the general value of q is, (b) np + (–1)n, , (a) 2np, p, (c) 2np +, 2, , p p, –, 4 4, , p, (d) (2n – 1) +, 4, , 128. sin 6q + sin 4q + sin 2q = 0, then q =, (a), , (c), 129. If, , np, p, or np ±, 4, 3, , (b), , np, p, or 2np ±, 4, 6, , (d) None of these, , 3p, 4, , np, p, or np ±, 4, 6, , p, (c) 2np –, 4, , 130. If sec2 q =, , (a) 2np ±, , (b) 2np +, , 133. If D is the area and 2s the sum of three sides of a triangle,, then, (a), , D£, , (c), , D>, , s2, 3 3, s2, 3, , s, 2, , (b), , D£, , (d), , None of these, , 5, 13, 9, (b), (d) 2, (c), 4, 4, 4, 135. Each side of a square subtends an angle of 60° at the top of, tower h metres high standing in the centre of the square. If, a is the length of each side of the square, then, (a) 2a2 = h2, (b) 2h2 = a2, 2, 2, (c) 3a = 2h, (d) 2h2 = 3a2, , (a), , p, 4, , 136. Period of, , p, (d) 2np ±, 4, , (a) 2p, , sin q + sin 2q, is, cos q + cos 2q, , (b), , p, 5, , (c), , 2p, 3, , (d), , p, 3, , 9, , 4, , then the general value of q is, 3, , x sgn x, is, 137. f (x) = (sin x7) · e, (a) an even function, (b) an odd function, (c) neither even nor odd (d) None of these, , p, 6, , 138. (sec 2 q + cos ec 2 q) can never be less than, (a) 4, (b) 3, (c) 2, (d) None of these, , p, (c) 2np ±, 3, , p, 6, , (b) np ±, , 139. If p1 , p 2 , p 3 are respectively the perpendiculars from the, vertices of a triangle to the opposite sides, then, , p, (d) np ±, 3, , cos A cos B cos C, +, +, is equal to, p1, p2, p3, , 131. General solution of tan 5q = cot 2q is, (a) q =, , np p, +, 7 14, , (b) q =, , np p, +, 7, 5, , (c) q =, , np p, +, 7, 2, , (d) q =, , np p, +, 7, 3, , (a), , np, p, + (–1)n, 2, 4, , p, (c) q = np –, 3, , (b) q = np + (–1)n, p, (d) q = np –, 4, , 1, r, , (b), , 1, R, , R, (d) None of these, D, 140. A pole 50 m high stand on a building 250 m high. To an, observer at a height of 300 m the building and the pole, subtend equal angles. The distance of the observer from, the top of the pole is, , (c), , 132. Solution of the equation 3 tan(q – 15) = tan(q + 15) is, (a) q =, , Topic 4 : Trigonometric Functions as Periodic, Even, and Odd Functions, Conditional Trigonometric Identities,, Maximum and Minimum Values of Trigonometric Expressions,, Properties of Triangles, Heights and Distances., , 134. The ratio of the greatest value of 2 – cosx + sin 2x to its least, value is, , 2 sec q + tan q = 1, then the general value of q is, , (a) np +, , BEYOND NCERT, , p, 3, , (a) 20 6, (b) 25 6, (d) 15 6, (c) 25 5, 141. If sin q1 + sin q2 + sin q3 = 3 then cos q1 + cot q2 + cos q3 is, equal to, (a) 0, (b) 1, (c) 2, (d) 3

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EBD_7762, 36, , MATHEMATICS, , 142. The least values of (cos ec 2 q + sin 2 q) is, (a) 1, (b) 2, (c) 3, 143. In a D PQR, Ð R =, , (d) 4, , p, P, Q, . If tan and tan are the roots of the, 2, 2, 2, , equation a x 2 + bx + c = 0 (a ¹ 0), then, (a) a + b = c, (b) b + c = a, (c) c + a = b, (d) b = c, 144. If A + B + C = p and cos A + cos B + cos C = 1 + k, A, B, C, sin sin sin then k =, 2, 2, 2, (a) 2, (b) 4, (c) 8, (d) –8, , 147. Which of the following is a periodic function?, (a) x – [x] + sin x, (b) sin [x], (c) x sin x, (d) (x – [x]) cos (px), , (a), , (b) cot A + cot B + cot C = cot A cot B cot C, (c) cos 2 A + cos 2 B + cos 2C = 1 + 4sin A sin B sin C, (d) All three are correct, 149. The period of the function f (x) = cos 4x + tan 3x is, , 145. Period of sin q - 3 cos q is, (a), (c), , p, 4, p, , (b), , p, 2, , (d), , 2p, , 146. Given b = 2, c = 3 , ÐA = 30° , then inradius of DABC is :, (a), , 3 -1, 2, , (b), , (c), , 3 -1, 4, , (d) None of these, , p, then, 2, tan A tan B + tan B tan C + tan C tan A = 1, , 148. If A + B + C =, , (a), , p, 12, , (b), , (c), , p, 2, , (d) p, , 3, 150. Period of the function sin, , (a) 2 p, (c) 8 p, , 3 +1, 2, , p, 6, , x, x, + cos5, is :, 2, 5, (b) 10 p, (d) 5 p, , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, 1., 2., , If sin q + cosec q = 2, then sin 2 q + cosec 2 q is equal to, (a) 1, (b) 4, (c) 2, (d) None of these, If y = cos2x + sec2x, then, (a) y £ 2, (b) y £ 1, (c), , 3., , y³2, , 6., , (d) 1 < y < 2, , p, 6, , (b) p, , (c) sec q =, , 1, 2, , 1, 5, , (b) cos q = 1, (d) tan q = 20, , (d) Not defined, , The value of, , 1– tan 2 15°, 1 + tan 2 15°, , is, , (b), , 3, , 3, (d) 2, 2, The value of cos1° cos 2° cos 3° ... cos 179° is, , (c), 7., , 1, , (a), , (d), , (a) sin q = –, , 1, 2, , (a) 1, , p, 4, Which of the following is not correct?, , (c) 0, , The value of tan 1° tan 2° tan 3° ... tan 89° is, (a) 0, (b) 1, (c), , 1, 1, If tan q = and tan f = , then the value of q + f is, 2, 3, , (a), , 4., , 5., , (b) 0, , 2, , (c) 1, 8., , (d) – 1, , If tan q = 3 and q lies in IIIrd quadrant, then the value of, sin q is, (a), , 1, 10, , (b), , 2, 10, , (c), , –3, 10, , (d), , –5, 10

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TRIGONOMETRIC FUNCTIONS, , 9., , 37, , The value of tan 75° – cot 75° is equal to, (a), , (b), , 2 3, , 2+ 3, , (d) 1, 2– 3, Which of the following is correct?, (a) sin 1° > sin 1, (b) sin 1° < sin1, (c) sin 1° = sin 1, , 11., , (d) sin 1° =, , p, 3, , p, 4, p, (c) zero, (d), 2, 12., The minum value of 3 cos x + 4 sin x + 8, (a) 5, (b) 9, (c) 7, (d) 3, , 15., , 16., 17., , æp ö, æp ö, The value of cot ç + q÷ cot ç – q÷ is, è4 ø, è4 ø, , 1, 2, –, , 19., , 1, 2, , –4, and q lies in third quadrant, then the value of, 5, , If sin q =, cos, , q, is, 2, , (a), , 1, 5, , (c), , –, , (b), , 1, , 10, 1, , (d), , 5, , 1, , –, , 10, , 24. The number of solution of tan x + sec x = 2cos x in (0, 2 p ) is, (a) 2, (b) 3, (c) 0, (d) 1, 25., , The value of sin, sin, , p, p, 2p, 5p, is, + sin + sin, + sin, 18, 9, 18, 18, , 7p, 4p, + sin, 18, 9, , (b) 1, , p, 3p, p, p, cos + cos, (d) cos + sin, 6, 7, 9, 9, 26. If A lies in the second quadrant and 3 tan A + 4 = 0, the value, of 2 cot A – 5cos A + sin A is equal to, , (c), , (a), , -, , 53, 10, , (b), , 23, 10, , 37, 7, (d), 10, 10, The value of cos2 48° – sin2 12° is, , (c), 1, 2, , (d), , 1, 8, , 27., , 1, 1, , tan B = , then tan(2A + B) is equal to, 2, 3, (b) 2, (c) 3, (d) 4, , The value of sin, (a), , p, , then the value of (1 + tan a)(1 + tan b) is, 4, (b) 2, (d) Not defined, , (a) 1, (c) – 2, , (b) 1, , 18. If tan A =, (a) 1, , 1, 2, (d) – 1, , If a + b =, , (a), , (a) – 1, (b) 0, (c) 1, (d) Not defined, cos 2q × cos 2f + sin2(q – f) – sin2(q + f) is equal to, (a) sin 2(q + f), (b) cos 2(q + f), (c) sin 2(q – f), (d) cos 2(q – f), The value of cos12° + cos 84° + cos 156° + cos 132° is, , (c), , 22., , (c), , (b), , (c) 0, , 23., , The value of sin(45° + q) – cos(45° – q) is, (a) 2 cos q, (b) 2 sin q, (c) 1, (d) 0, , (a), , 1, (d) 2, 2, If sin q + cos q = 1, then the value of sin 2q is, , (b) 0, , (a) 1, , (b), , 13. The value of tan 3A - tan 2A - tan A is equal to, (a) tan 3A tan 2A tan A, (b) - tan 3A tan 2A tan A, (c) tan A tan 2A - tan 2A tan 3A - tan 3A tan A, (d) None of these, 14., , 21., , p, sin1, 18°, , 1, m, , then a + b is equal to :, If tan a =, and tan b =, 2, m, +1, m +1, , (a), , The value of sin50° – sin70° + sin10° is, (a) 1, , (c), 10., , 20., , p, 13p, sin, is, 10, 10, , (b) –, , 1, 2, , (c) –, , 28., 1, 4, , (d) 1, , (a), , 5 +1, 8, , (b), , (c), , 5 +1, 5, , (d), , 5 –1, 8, 5 +1, 2 2, , 1, 1, and tan b = , then cos 2a is equal to, 7, 3, (a) sin 2b, (b) sin 4b, (c) sin 3b, (d) cos 2b, , If tan a =

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EBD_7762, 38, , 29., , 30., , MATHEMATICS, , a, , then b cos 2q + a sin 2q is equal to, b, a, (a) a, (b) b, (c), (d) None of these, b1, If for real values of x, cos q = x + , then, x, (a) q is an acute angle, (b) q is right angle, (c) q is an obtuse angle, (d) No value of q is possible, , If tan q =, , Past Year MCQs, 31., , (, , ), , 1, sin k x + cosk x where x Î R and k ³ 1., k, Then f 4 ( x ) - f 6 ( x ) equals, [JEE MAIN 2014, C], , Let f k ( x ) =, , 1, 1, 1, 1, (b), (c), (d), 6, 3, 4, 12, A bird is sitting on the top of a vertical pole 20 m high and its, elevation from a point O on the ground is 45° . It flies off, horizontally straight away from the point O. After one, second, the elevation of the bird from O is reduced to 30° ., Then the speed (in m/s) of the bird is [JEE MAIN 2014, A], , (a), 32., , 33., , (a), , 20 2, , (c), , 40, , ), , 2 -1, , (d), , (, 40 (, , ), , 3 -1, , 20, , 3- 2, , (c), , ), , p 5p, p, (b), ,, 3 3, 3, p 5p, æ 3ö, , , cos –1 ç – ÷ (d) None of these, 3 3, è 2ø, 2æp, , ö, æp, ö, cos ç + q÷ - sin 2 ç - q÷ =, è6, ø, è6, ø, , (a), (c), , 1, cos 2q, 2, 1, - cos 2q, 2, , 37., , 38., , 1: 3 (b) 2 : 3, , æ 1 ö, ÷÷, 2np + (-1) n sin -1 çç, è 3ø, p, (d) np 4, 43. If tan (cot x) = cot (tan x), then sin 2x is equal to :, [BITSAT 2016, C], , (a), , 2, (2n + 1)p, , (b), , 4, (2n + 1)p, , (c), , 2, n (n + 1)p, , (d), , 4, n (n + 1)p, , cos q1 + cos q2 + cos q3 =, [BITSAT 2016, C], (a) 0, (b) 1, (c) 2, (d) 3, 45. If A and B are positive acute angles satisfying, , 1, (d), 2, , (c), , p, 4, p, 2np +, 4, , 3np -, , 44. If sin q1 + sin q2 + sin q3 = 3 , then, , (b) 0, , If 5 tan q = 4, then, , (a), , (b), , [BITSAT 2014, C], , 5sin q - 3cos q, = [BITSAT 2014, C], 5sin q + 2 cos q, (a) 0, (b) 1, (c) 1/6, (d) 6, 36. If the angles of elevation of the top of a tower from three, collinear points A, B and C, on a line leading to the foot of the, tower, are 30°, 45° and 60° respectively, then the ratio,, AB : BC, is :, [JEE MAIN 2015, A], , 35., , (a), , (c), , The solution of (2 cos x – 1) (3 + 2 cos x) = 0 in the interval, 0 £ x £ 2p is, [BITSAT 2014, C], (a), , 34., , (, , (b), , 39. In a DABC, the lengths of the two larger sides are 10 and, 9 units, respectively. If the angles are in AP, then the length, of the third side can be, [BITSAT 2015, A], (a) 5 ± 6, (b) 3 3, (c) 5, (d) None of these, 40. A man is walking towards a vertical pillar in a straight path,, at a uniform speed. At a certain point A on the path, he, observes that the angle of elevation of the top of the pillar is, 30°. After walking for 10 minutes from A in the same direction,, at a point B, he observes that the angle of elevation of the, top of the pillar is 60°. Then the time taken (in minutes) by, him, from B to reach the pillar, is:, [JEE MAIN 2016, A], (a) 20, (b) 5, (c) 6, (d) 10, 41. If 0 £ x < 2p, then the number of real values of x, which, satisfy the equation, cos x + cos 2x + cos 3x + cos 4x = 0 is: [JEE MAIN 2016, A], (a) 7, (b) 9, (c) 3, (d) 5, 42. The general solution of the equation, sin 2x + 2sin x +2 cos x+ 1 = 0 is, [BITSAT 2016, A], , 3 :1, , (d), , 3: 2, , cos A, sin A, = n,, = m, then the value of (m2 – n2) sin 2 B is, cos B, sin B, [BITSAT 2015, C], (a) 1 + n2 (b) 1 – n2 (c) n 2, (d) – n2, The period of tan 3q is, [BITSAT 2015, C], (a) p, (b) 3p/4, (c) p/2, (d) None of these, , 3 sin A 2 cos B, =, ,, sin B, cos A, Then the value of A + 2B is equal to : [BITSAT 2016, A], p, p, p, p, (a), (b), (d), (c), 6, 3, 2, 4, cos, A, cos, B, cos, C, 46. In a DABC, if, , and the side a = 2,, =, =, a, b, c, then area of the triangle is, [BITSAT 2016, A], 3 cos 2 A + 2 cos 2 B = 4 and, , If, , 3, (d) 3, 2, 2, 2, If 5(tan x – cos x) = 2cos 2x + 9, then the value of cos 4x is :, [JEE MAIN 2017, A], , (a) 1, 47., , (a), , -, , (b) 2, , 7, 9, , (b), , -, , (c), , 3, 5, , (c), , 1, 3, , (d), , 2, 9

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TRIGONOMETRIC FUNCTIONS, , 39, , 48. Let a vertical tower AB have its end A on the level ground., Let C be the mid-point of AB and P be a point on the ground, such that AP = 2AB. If ÐBPC = b, then tan b is equal to :, [JEE MAIN 2017, A], 4, 6, 1, (a), (b), (c), 9, 7, 4, 49. The number of roots of equation, , (d), , 2, 9, , æ, æp, ö, æp, ö 1ö, 8cos x × ç cos ç + x ÷ × cos ç - x ÷ - ÷ = 1 in [0, p] is kp,, 6, 6, è, ø, è, ø 2ø, è, then k is equal to :, [JEE MAIN 2018, S], 13, 8, 20, 2, (b), (c), (d), 9, 9, 9, 3, 52. PQR is a triangular park with PQ = PR = 200 m. A T.V. tower, stands at the mid-point of QR. If the angles of elevation of, the top of the tower at P, Q and R are respectively 45°, 30°, and 30°, then the height of the tower (in m) is :, [JEE MAIN 2018, A], , (a), , (b) 100 3, , (c) 50 2, , (d) 100, , 53. If msin q = n sin(q + 2a) then tan(q + a ) is, [BITSAT 2018, A], , cos x + cos 2 x + cos 3x = 0 is (0 £ x £ 2p), , [BITSAT 2017, A], (a) 4, (b) 5, (c) 6, (d) 8, 50. An observer on the top of a tree, finds the angle of, depression of a car moving towards the tree to be 30°. After, 3 minutes this angle becomes 60°. After how much more, time, the car will reach the tree?, [BITSAT 2017, A], (a) 4 min. (b) 4.5 m (c) 1.5 min (d)2 min., 51. If sum of all th e solution s of th e equation, , (a) 50, , (a), , m+n, tan a, m-n, , (b), , m+n, tan q, m-n, , (c), , m+n, cot a, m-n, , (d), , m+n, cot q, m-n, , 54. Number of solutions of equation sin 9q = sin q in the, interval [0, 2p] is, (a) 16, (c) 18, , [BITSAT 2018, A], (b) 17, (d) 15, , 55. A pole stands vertically inside a triangular park ABC. If the, angle of elevation of the top of the pole from each corner, of the park is same, then the foot of the pole is at the, [BITSAT 2018, C], (a) centroid, (b) circumcentre, (c) incentre, (d) orthocentre, 56. Let A, B, and C are the angles of a plain triangle and, A 1, B 2, C, = , tan = . Then tan, is equal to, 2 3, 2 3, 2, [BITSAT 2018, A], (a) 7/9, (b) 2/9, (c) 1/3, (d) 2/3, , tan, , Exercise 3 : Try If You Can, 1., , sin 2 q, , If 0 £ q £ p and 81, p, 6, (c) p, The ran ge, , (a), 2., , cos 2 q, , + 81, , = 30, then q is equal to :, , p, 2, (d) None of these, values of the expression, (b), , of, , 5., , The value of tan6 20° – 33 tan4 20° + 27 tan2 20° is :, (a) 2, (b) 3, (c) 4, (d) 5, , 6., , If exp{(sin 2 x + sin 4 x + sin 6 x + .....¥) log e 2} satisfies the, equation x 2 - 9x + 8 = 0, then the value of, 0<x<, , pö, æ, 5cos q + 3cos çq + ÷ + 1 is, 3ø, è, , 3., , 4., , (a), , (a) [–7, 7], , (b) [–6, 8], , (c) [– 8, 6], , (d) [ -3 3,13], , p, = ( p - q ) ( r - s ) , then the value of, 24, p + q + r + s is, (a) 6, (b) 7, (c) 8, (d) 9, If f (x) = [cos x] + [sin x + 1] = 0, (where [.] denotes, the greatest integer function), then value of x, satisfying f (x) = 0, where x Î [0, 2p], (a) x Î (p/2, p] È [3p/2, 2p] (b) x Î (0, p/2), (c) x Î [p/2, p] È [3p/2, 2p] (d) x Î [p, 2/ p], , p, is, 2, , 3 +1, , (b), , cos x, ,, cos x + sin x, , 3 -1, , 3 +1, 3 -1, (d), 2, 2, The number of solutions of the equation, , (c), 7., , If tan, , 2 sin x - 3, 8., , 2cos2 x - 3cos x + 1, , = 1 in [0, p] is, , (a) 2, (b) 3, (c) 4, (d) 5, A pole stands vertically inside a triangular park DABC . If the, angle of elevation of the top of the pole from each corner of, the park is same, then in DABC the foot of the pole is at the, (a) centroid, (b) circumcentre, (c) incentre, (d) orthocentre

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EBD_7762, 40, , MATHEMATICS, , If sin q = x +, , 9., , 1, (b) p < 0, (c) p > 0, (d) p ³ 1, 4, Is the equation (ab + ca + bc) sin q = 2(a 2 + b2 + c 2 ), possible for real values of a, b, c ?, (a) possible, (b) not possible, (c) insufficient data, (d) None of these, If the equation a sin3 x + (b – a) sin2 x + (c – b) sin x – c = 0 has, exactly three distinct solutions in [0, p], where a + b + c = 0,, then which of the following is not the possible value of cla?, , (a), 10., , 11., , b, is possible for some real x then, x, , p£, , 12., , 1, 2, 2, (a) 1, (b), (c), (d), 4, 7, 7, The number of real values of x such that, (2x + 2–x – 2cos x) (3x + p + 3–x–p + 2cos x) (5p– x + 5x – p – 2cos x) = 0 is:, (a) 1, (b) 2, (c) 3, (d) infinite, , 13., , A man observes when he has climbed up, , 1, of the length of, 3, an inclined ladder, placed against a wall, the angular, depression of an object on the floor is a and that after he, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, , (b), (a), (a), (c), (c), (b), (c), (c), (b), (b), (c), (b), (b), (c), (b), , 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, , (a), (b), (b), (b), (d), (d), (c), (a), (a), (c), (b), (a), (d), (c), (b), , 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, , (c), (a), (b), (c), (d), (b), (c), (c), (c), (c), (c), (b), (d), (a), (b), , 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, , 1, 2, 3, 4, 5, 6, , (c), (c), (d), (c), (b), (c), , 7, 8, 9, 10, 11, 12, , (b), (c), (a), (b), (b), (d), , 13, 14, 15, 16, 17, 18, , (a), (d), (c), (b), (c), (c), , 19, 20, 21, 22, 23, 24, , 1, 2, , (a), (b), , 3, 4, , (c), (a), , 5, 6, , (b), (c), , 7, 8, , has climbed the ladder fully, the depression is b . If the, inclination of the ladder to the floor is q, then cot q =, 3 cot b - cot a, 3 cot a - cot b, (b), 2, 2, cot b - cot a, cot a + cot b, (c), (d), 2, 2, 14. The most general values of q for which, , (a), , sin q - cos q = min{1, a 2 - 6a + 10}, a Î R are given by, n, (a) np + (-1), , p p, 4 4, , n, (b) np + (-1), , p p, +, 4 4, , p, (d) None of these, 4, 15. ABC is a triangular park with AB = AC = 100 m. A TV tower, stands at the mid-point of BC. The angles of elevation of, the top of the tower at A, B, C are 45°, 60°, 60° respectively., The height of the tower is, , (c) 2np +, , (a) 50 m, , (b), , 50 3m, , (c), , (d), , 50 (3 - 3 ) m, , 50 2 m, , ANSWER KEYS, Exercise-1 : Topic-wise MCQs, (c), (c), 61 (b) 76, 91 (c), (a), (b), 62 (d) 77, 92 (d), (a), (b), 63 (a), 78, 93 (d), (b), (b), 64 (b) 79, 94 (a), (d), (d), 65 (c), 80, 95 (b), (d), (c), 66 (b) 81, 96 (b), (b), (a), 67 (c), 82, 97 (c), (a), (c), 68 (c), 83, 98 (b), (b), (c), 69 (b) 84, 99 (b), (d), (c) 100 (a), 70 (b) 85, (b), (d) 101 (b), 71 (c), 86, (a), (b) 102 (c), 72 (a), 87, (d), (d) 103 (a), 73 (b) 88, (a), (d) 104 (c), 74 (c), 89, (b), (a) 105 (d), 75 (d) 90, Exercise-2 : Exemplar & Past Year MCQs, (c), (b), 25 (a), 31, 37 (b), (b), (b), 26 (b) 32, 38 (d), (c), 27 (a), 33 (b), 39 (a), (b), (a), 28 (b) 34, 40 (b), (c), (c), 29 (b) 35, 41 (a), (b), (c), 30 (d) 36, 42 (d), Exercise-3 : Try If You Can, (b), (a), (a), 9, 11, 13 (a), (b), (b), 10 (b) 12, 14 (b), , 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, , (d), (d), (a), (d), (a), (c), (b), (c), (a), (b), (d), (b), (d), (b), (c), , 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, , (d), (c), (c), (c), (a), (a), (b), (a), (c), (b), (a), (a), (a), (c), (b), , 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, , (c), (b), (c), (b), (b), (a), (b), (a), (b), (d), (a), (d), (d), (d), (b), , 43, 44, 45, 46, 47, 48, , (b), (a), (b), (d), (a), (d), , 49, 50, 51, 52, 53, 54, , (c), (c), (a), (d), (a), (b), , 55, 56, , (a), (a), , 15, , (b)

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4, , PRINCIPLE OF, MATHEMATICAL INDUCTION, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2002-2018), , Number of Questions, , 2, JEE MAIN, BITSAT, , 1, , 0, 2002, , 2004, , 2008, , 2009, , 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 0, 1, , Critical Concepts, , First principle of Mathematical, Induction, Second principle, of Mathematical Induction, , Rating of Difficulty, , CUS, (chapter utility, score) out of 10, , 3/5, , 5

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PRINCIPLE OF MATHEMATICAL INDUCTION, , 43

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EBD_7762, 44, , MATHEMATICS, , Exercise 1 : Topic-wise MCQs, 1., , 2., , Let P(n) be statement 2n < n!. Where n is a natural number,, then P(n) is true for:, (a) all n, (b) all n > 2, (c) all n > 3, (d) None of these, If P(n) = 2 + 4 + 6 + .....+ 2n, n ÎN, th en, Þ P (k + 1) = (k + 1)(k + 2) + 2 for all k ÎN. So we can, , For every positive integral value of n, 3n > n3 when, (a) n > 2 (b) n ³ 3, (c) n ³ 4, (d) n < 4, 10. If p is a prime number, then n p – n is divisible by p when n, is a, (a) Natural number greater than 1, (b) Irrational number, (c) Complex number, (d) Odd number, , conclude that P (n) = n(n + 1) + 2 for, , 11., , P (k ) = k (k + 1) + 2, , 3., , 4., , 5., , (a) all n Î N, (b) n > 1, (c) n > 2, (d) nothing can be said, Let T(k) be the statement 1 + 3 + 5 + .... + (2k – 1) = k2 +10, Which of the following is correct?, (a) T(1) is true, (b) T(k) is true Þ T(k + 1) is true, (c) T(n) is true for all n Î N, (d) All above are correct, If x > –1, then the statement (1 + x)n > 1 + nx is true for, (a) all n Î N, (b) all n > 1, (c) all n > 1 provided x ¹ 0 (d) None of these, For all n Î N,, 1+, , 1, 1, 1, +, + ..... +, 1+ 2 1+ 2+ 3, 1 + 2 + 3 + ..... + n, , is equal to, (a), , 3n, n +1, , (b), , n, n +1, , (c), , 2n, n –1, , (d), , 2n, n +1, , 7., , 10n + 3(4n + 2) + 5 is divisible by (n Î N), (a) 7, (b) 5, (c) 9, (d) 17, The statement P(n), “1 × 1! + 2 × 2! + 3 × 3! + ..... + n × n! = (n + 1)! – 1” is, (a) True for all n > 1, (b) Not true for any n, (c) True for all n Î N, (d) None of these, , 8., , Statement-I : 1 + 2 + 3 + ..... + n <, , 6., , 1, (2n + 1)2, n Î N., 8, Statement-II : n(n + 1) (n + 5) is a multiple of 3, n Î N., (a) Only Statement I is true, (b) Only Statement II is true, (c) Both Statements are true, (d) Both Statements are false, , 9., , Let S ( K ) = 1 + 3 + 5... + (2 K - 1) = 7 + K 2 , then which of, the following is true?, (a) Principle of mathematical induction can be used to, prove the formula, (b) S ( K ) Þ S ( K + 1), (c), , S (K ) Þ, / S ( K + 1), , (d) S (1) is correct, 12. Let P(n) : “2n < (1 × 2 × 3 × ... × n)”. Then the smallest, positive integer for which P(n) is true is, (a) 1, (b) 2, (c) 3, (d) 4, 13. A student was asked to prove a statement P(n) by induction., He proved that P(k + 1) is true whenever P(k) is true for, all k > 5 Î N and also that P (5) is true. On the basis of, this he could conclude that P(n) is true, (a) for all n Î N, (b) for all n > 5, (c) for all n ³ 5, (d) for all n < 5, 14. If P(n) : 2 + 4 + 6 +... + (2n), n Î N, then P(k) = k (k + 1) + 2, implies P (k + 1) = (k + 1) (k + 2) + 2 is true for all k Î N., So statement P(n) = n (n + 1) + 2 is true for:, (a) n ³ 1, (b) n ³ 2, (c) n ³ 3, (d) None of these, 15. Let P(n) be a statement and let P(n) Þ P (n + 1) for all, natural numbers n, then P(n) is true, (a) for all n, (b) for all n > 1, (c) for all n > m, m being a fixed positive integer, (d) nothing can be said, 16. The statement P(n) "1 × 1! + 2 × 2! + 3 × 3! + .... + n × n!, = (n + 1)! – 1" is, (a) true for all n >1, (b) not true for any n, (c) true for all n Î N, (d) None of these, 17. Assertion : For every natural number n ³ 2,, , 1, 1, , +, , 1, 2, , + ..... +, , 1, n, , > n, , Reason : For every natural number n ³ 2,, n (n + 1) < n + 1.

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PRINCIPLE OF MATHEMATICAL INDUCTION, , (a) Assertion is correct, Reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., n, , æ n + 1ö, 18. If n is a natural number, then ç, ³ n! is true when, è 2 ÷ø, , (a) n > 1, (b) n ³ 1, (c) n > 2, (d) n ³ 2, 19. Use principle of mathematical induction to find the value, of k, where (102n – 1 + 1) is divisible by k., (a) 11, (b) 12, (c) 13, (d) 9, 2, 2, 2, 20. For all n ³ 1, 1 + 2 + 3 + 42 + ..... + n2 =, (a), (c), , n (n + 1), , 2, , (d), , n (n + 1) (2n + 1), 6, , 2n < n, , (b), , n 2 > 2n, , (c) n 4 < 10n, (d) 23n > 7 n + 1, 22. P(n) : 2.7n + 3.5n – 5 is divisible by, (a) 24, " n Î N, (b) 21, " n Î N, (c) 35, " n Î N, (d) 50, " n Î N, 23. For all n ³ 1,, , (n + 1), , 1, 1, 1, 1, +, +, + ..... +, =, 1.2 2.3 3.4, n ( n + 1), , (b), , n, n +1, , 4n + 3, (d), 2n, n, n, n, 24. For every positive integer n, 7 – 3 is divisible by, (a) 7, (b) 3, (c) 4, (d) 5, 25. For natural number n, (n!)2 > nn, if, (a) n > 3, (b) n > 4, (c) n ³ 4, (d) n ³ 3, 26. Assertion : 11m + 2 + 122m + 1 is divisible by 133 for all, m Î N., Reason : xn – yn is divisible by x + y, " n Î N, x ¹ y.., (a) Assertion is correct, Reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., , (c), , Let a(n) = 1 +, , 1, 1 1 1, + + +…+ n, . Then, 2 3 4, (2 ) - 1, , (a) a(100) £ 100, (b) a(100) > 100, (c) a(200) £ 100, (d) a(200) < 100, 28. If P = n(n 2 – 1 2) (n 2 – 2 2) (n 2 – 3 2) ...... (n 2 – r 2),, n > r, n Î N then P is necessarily divisible by, (a) (2r + 2) !, (b) (2r + 4) !, (c) (2r + 1) !, (d) None of these, 29. By mathematical induction,, 1, 1, 1, +, + ..... +, is equal to, 1× 2 ×3 2 × 3× 4, n (n + 1)(n + 2), , (a), , (c), , n (n – 1) (2n + 1), , (a) n, , 27. For a positive integer n,, , (b) n(n + 1) (2n – 1), , 6, , 21. For each n Î N , the correct statement is, (a), , 45, , n (n + 1), , 4 ( n + 2 ) ( n + 3), n (n + 2), , 4 (n + 1) (n + 3), , (b), , n (n + 3), , 4 (n + 1) (n + 2), , (d) None of these, , 30. By using principle of mathematical induction for every, natural number, (ab)n =, (a) an bn, (b) an b, (c) abn, (d) 1, 31. If n Î N, then 11n + 2 + 122n + 1 is divisible by, (a) 113, (b) 123, (c) 133, (d) None of these, 32. For all n Î N, 41n – 14n is a multiple of, (a) 26, (b) 27, (c) 25, (d) None of these, 33. The remainder when 54n is divided by 13, is, (a) 1, (b) 8, (c) 9, (d) 10, 34. The greatest positive integer, which divides, n (n + 1) (n + 2) (n + 3) for all n ÎN, is, (a) 2, (b) 6, (c) 24, (d) 120, 35. Let P(n) : n2 + n + 1 is an even integer. If P(k) is assumed, true then P(k + 1) is true. Therefore P(n) is true., (a) for n > 1, (b) for all n Î N, (c) for n > 2, (d) None of these, 36. By the principle of induction " n Î N, 32n when divided, by 8, leaves remainder, (a) 2, (b) 3, (c) 7, (d) 1, 37. If n Î N , then the number (2 + 3 ) n + (2 - 3 ) n is, (a), (b), (c), (d), , an integer for all values of n, an integer if n is even, an integer if n is odd, always an irrational number

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EBD_7762, 46, , 38., , MATHEMATICS, , If m, n are any two odd positive integers with n < m, then, the largest positive integer which divides all the numbers, of the type m2 – n2 is, (a) 4, (b) 6, (c) 8, (d) 9, , 39. x ( x n -1 - na n -1 ) + a n (n - 1) is divisible by ( x - a ) 2 for, (a) n > 1, (b) n > 2, (c) all n Î N, (d) None of these, n, 40. For natural number n, 2 (n – 1)! < nn, if, (a) n < 2 (b) n > 2, (c) n ³ 2 (d) n > 3, 41. Principle of mathematical induction is used, (a) to prove any statement, (b) to prove results which are true for all real numbers, (c) to prove that statements which are formulated in terms, of n, where n is positive integer, (d) in deductive reasoning, 42. For all n Î N, 1.3 + 2.32 + 3.33 + ..... + n.3n is equal to, (a), , (b), , (c), , (d), , 43., , 44., , (2n + 1) 3n + 1 + 3, 4, , (2n – 1) 3n + 1 + 3, 4, , (2n + 1) 3n, , +3, , 4, , (2n – 1) 3n + 1 + 1, , 4, If n is a positive integer, then 52n + 2 – 24n – 25 is, divisible by, (a) 574, (b) 575, (c) 674, (d) 576, The greatest positive integer, which divides, (n + 1) (n + 2) (n + 3) ..... (n + r) for all n Î W, is, (a) r, (b) r!, (c) n + r, (d) (r + 1)!, , 45. If n Î N , then x 2 n -1 + y 2 n -1 is divisible by, (a) x + y, 46. If, , (b) x – y, , (c) x2 + y2, , (d) x2 + xy, , ( 2n )!, 4n, <, , then P(n) is true for, n + 1 (n!)2, , (a) n ³ 1, (c) n < 0, 47. For all n Î N,, , (b) n > 0, (d) n ³ 2, , æ, ( 2n + 1) ö, 3ö æ, 5öæ, 7ö, æ, ÷, ç1 + ÷ ç1 + ÷ ç1 + ÷ ..... ç1 +, 1ø è, 4øè, 9ø, è, n2 ø, è, , is equal to, (a), , (n + 1)2, , (n + 1)3, , (b), , 3, (d) None of these, , 2, (c) (n + 1)2, , 48. For all n Î N, the sum of, (a) a negative integer, (c) a real number, , n5, +, 5, (b), (d), , n3 7n, +, is, 3 15, a whole number, a natural number, , 49. For given series:, 12 + 2 × 22 + 32 + 2 × 42 + 52 + 2 × 62 + .....,, if Sn is the sum of n terms, then, n (n + 1), , 2, , (a) Sn =, (b) Sn =, , 2, , n 2 (n + 1), , , if n is even, , , if n is odd, 2, (c) Both (a) and (b) are true, (d) Both (a) and (b) are false, 50. If P(n) : “46n + 19n + k is divisible by 64 for n Î N” is true,, then the least negative integral value of k is., (a) – 1, (b) 1, (c) 2, (d) – 2

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PRINCIPLE OF MATHEMATICAL INDUCTION, , 47, , Exercise 2 : Exemplar & Past Year MCQs, 1., , 2., 3., , NCERT Exemplar MCQs, If 10n + 3.4n + 2 + k is divisible by 9, for all n Î N, then, the least positive integral value of k is, (a) 5, (b) 3, (c) 7, (d) 1, For all n Î N, 3.52n + 1 + 23n + 1 is divisible by, (a) 19, (b) 17, (c) 23, (d) 25, If xn – 1 is divisible by x – k, then the least positive integral, value of k is, (a) 1, (b) 2, (c) 3, (d) 4, , 5., , the following is true, [JEE MAIN 2004, A], (a) Principle of mathematical induction can be used to, prove the formula, , 6., , If an = 7 + 7 + 7 + ... ... having n radical signs then by, methods of mathematical induction which is true, [JEE MAIN 2002, A], , (b), , S ( K ) Þ S ( K + 1), , (c), , S (K ) Þ, / S ( K + 1), , (d), , S (1) is correct, , The greatest positive integer, which divides, n (n + 1)(n + 2)(n + 3) for all n Î N , is, [BITSAT 2014, C], (a) 2, (b) 6, (c) 24, (d) 120, , Past Year MCQs, 4., , Let S ( K ) = 1 + 3 + 5... + (2 K - 1) = 3 + K 2 . Then which of, , 7., , (2n)!, 4n, <, , then P(n) is true for [BITSAT 2017, A], n + 1 (n!)2, , If, , (a), , an > 7 " n ³ 1, , (b), , an < 7 " n ³ 1, , (a) n ³ 1, , (b) n > 0, , (c), , an < 4 " n ³ 1, , (d) an < 3 " n ³ 1, , (c) n < 0, , (d) n ³ 2, , Exercise 3 : Try If You Can, 1., 2., 3., , 4., , 5., , (n + 2)!, divisible by n, n Î N and 1 £ n £ 9, then n is, 6(n - 1)!, (a) 4, (b) 2, (c) 6, (d) 1, 2n, +, 1, 3n, +, 1, If n is a positive integer, then 2 . 4, +3, is divisible by :, (a) 2, (b) 7, (c) 11, (d) 27, Which of the following result is valid?, (a) (1 + x)n > (1 + nx), for all natural numbers n, (b) (1 + x)n ³ (1 + nx), for all natural numbers n, where, x > –1, (c) (1 + x)n £ (1 + nx), for all natural numbers n, (d) (1 + x)n < (1 + nx), for all natural numbers n, , If, , n5 n3 7n, +, +, is, 5, 3 15, (a) a negative integer, (b) a whole number, (c) a real number, (d) a natural number, Using mathematical induction, the numebrs an’s are, defined by a0 = 1, an+1 = 3n2 + n + an, (n ³ 0)., Then, an is equal to, (a) n3 + n2 + 1, (b) n3 – n2 + 1, 3, 2, (c) n – n, (d) n3 + n2, , 6., , For any n Î N , the value of the expression, , 2 + 2 + ..... + 2 is, n - roots, , (a), , 7., , æ p ö, 2 cos çç, ÷÷, è 2 n +1 ø, , (b), , (c), (d) None of these, 2 cos( 2 n +1 p), If n is a natural number, then, , n3, 3, , (b) 12 + 22 + ... + n 2 =, , n3, 3, , (c) 12 + 2 2 + ... + n 2 > n 3, , (d) 12 + 22 + ... + n 2 >, , n3, 3, , (a) 12 + 22 + ... + n 2 <, , For all n Î N, the sum of, , 8., , æ p ö, 2 sin çç, ÷÷, è 2 n +1 ø, , If, , 1, 1, 1, kn, +, +, + ...n terms =, , then k is equal to, 2´ 4 4´ 6 6´8, n +1, , (a), , 1, 4, , (b), , 1, 2, , (c) 1, , (d), , 1, 8

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EBD_7762, 48, , 9., , MATHEMATICS, , 10. If n Î N and n > 1, then, , Which one of the following is true ?, , æ n +1 ö, n! > ç, ÷, è 2 ø, , n, , (a), , æ n +1ö, n! ³ ç, ÷, è 2 ø, , n, , (b), , n, , æ 1ö, (a) ç 1 + ÷ < n 2 , n is a positive integer, è nø, n, , æ 1ö, (b) ç 1 + ÷ < 2, n is a positive integer, è nø, n, , (c), , æ 1ö, 3, ç 1 + ÷ < n , n is a positive integer, è nø, n, , æ 1ö, (d) ç 1 + ÷ > 2, n is a positive integer, è nø, , 1, 2, 3, 4, 5, , (c), (d), (b), (c), (d), , 6, 7, 8, 9, 10, , (c), (c), (c), (c), (a), , 11, 12, 13, 14, 15, , (b), (d), (c), (d), (d), , 1, , (a), , 2, , (b, c), , 3, , (a), , 1, , (d), , 2, , (c), , 3, , (b), , n, , æ n +1 ö, (c) n! < ç, ÷, è 2 ø, (d) None of these, , ANSWER KEYS, EXERCISE-1 : TOPIC-WISE MCQs, (c) 21, (c), (c) 31, (c), 16, 26, (a) 22, (a), (a) 32, (b), 17, 27, (b), (c) 33, (a), 18 (b) 23, 28, (a) 24, (c), (b) 34, (c), 19, 29, (d), (a) 35, (d), 20 (d) 25, 30, EXERCISE-2 : Exemplar & Past Year MCQs, (b), (b), (c), (d), 4, 5, 6, 7, EXERCISE-3 : Try If You Can, (d), (b), (a), (d), 4, 5, 6, 7, , 36, 37, 38, 39, 40, , (d), (a), (c), (c), (b), , 41, 42, 43, 44, 45, , (c), (b), (d), (b), (a), , 46, 47, 48, 49, 50, , (d), (c), (d), (c), (a), , 8, , (a), , 9, , (d), , 10, , (c)

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5, , COMPLEX NUMBERS, AND QUADRATIC EQUATIONS, , Chapter, , Trend, Analysis, , off JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 5, 4, 3, , JEE MAIN, BITSAT, , 2, 1, 0, , 2010 2011 2012 2013 2014 2015 2016 2017 2018, Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 6, 7, , Critical Concepts, , Algebra of complex numbers, Square, root of a negative real number,, Modulus & Conjugate of a complex, number, Argand plane and Polar, Representation of a complex number,, Geometry of Complex numbers,, Quadratic Equations-roots &, formation, condition for common roots., , Rating of, Difficulty Level, , CUS, (chapter utility score), Out of 10, , 4.5/5, , 8.6

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COMPLEX NUMBERS AND QUADRATIC EQUATIONS, , 51

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EBD_7762, 52, , MATHEMATICS, , Topic 1: Integral Powers of Iota, Algebraic Operations with, Complex Numbers, Conjugate and Modulus of a Complex, Number, Argument or Amplitude of a Complex, Number, Powers of a Complex Number., 2, , 1., , æ 2i ö, Value of ç, is, è 1 + i ÷ø, , (a) i, 2., , 3., , 4., , 5., , æ1- iö, If ç, è 1 + i ÷ø, , x=, , (c) 1 – i, , , when simplified has the value, i, (a) 0, (b) 2i, (c) – 2i, (d) 2, If z = 2 – 3i, then the value of z2 – 4z + 13 is, (a) 1, (b) –1, (c) 0, (d) None of these, If, , 25, , c∗i, = a + ib, where a, b, c are real, then a 2 + b2 is equal, c,i, , (b) 1, , (b) x =, , 3, 22, and y =, 4, 3, , 1+i, is 1., 1–i, , I., , Modulus of, , II., , p, 1+i, is ., Argument of, 2, 1–i, , IV. Argument of, , (d) – c2, , 11., , 12. Value of, , i592 + i590 + i588 + i 586 + i 584, , i582 + i580 + i578 + i 576 + i 574, (a) –2, (b) 0, (c) –1, , 13. Modulus of z =, (a), , 2., , p, 1, is ., 4, 1+i, , (a) I and II are correct (b) III and IV are correct, (c) I, II and III are correct (d) All are correct, , z1, is, z2, , purely imaginary., Reason : If z is purely imaginary, then z + z = 0., (a) Assertion is correct, Reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., If | z – 4 | < | z – 2 |, its solution is given by, (a) Re(z) > 0, (b) Re(z) < 0, (c) Re(z) > 3, (d) Re(z) > 2, , x=, , 1, III. Modulus of, is, 1+i, , (c) c2, , 10. Assertion : If |z1 + z2 | 2 = |z1 | 2 + |z2 | 2, then, , = a + ib then, , 3, 33, 3, 33, and y =, (d) x = and y =, 4, 4, 4, 5, Which of the following are correct?, , (c), , 1, , i 57 +, , to:, (a) 7, , (d) 1 – 2i, , 100, , 3, 33, and y =, 5, 4, , 8., , 9., , (a) a = 2, b = – 1, (b) a = 1, b = 0, (c) a = 0, b = 1, (d) a = – 1, b = 2, 2, 4, 6, 2n, 1 + i + i + i + ... + i is, (a) positive, (b) negative, (c) 0, (d) cannot be determined, If (x + iy) (2 – 3i) = 4 + i, then, (a) x = – 14/13, y = 5/13 (b) x = 5/13, y = 14/13, (c) x = 14/13, y = 5/13, (d) x = 5/13, y = – 14/13, If 4x + i(3x – y) = 3 + i(–6), where x and y are real numbers, then the values of x and y are, (a), , 6., , (b) 2i, , 7., , 1, 3, , - 1 is, , (d) 1, , (1 + i 3)(cos q + i sin q), is, 2(1 - i)(cos q - i sin q), , (b) -, , 14. If z = x - i y and, , 1, z3, , 1, 2, , (c), , 1, 2, , (d) 1, , = p + iq, then, , æ x yö, 2, 2, ç + ÷ ( p + q ) is equal to, è p qø, , (a) –2, (b) –1, (c) 2, (d) 1, 15. The polar form of the complex number (i25)3 is, (a), , cos, , p, p, + isin, 2, 2, , (b) cos, , p, p, - isin, 2, 2, , (c), , cos, , p, p, + isin, 3, 3, , (d) cos, , p, p, - isin, 3, 3

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COMPLEX NUMBERS AND QUADRATIC EQUATIONS, , 16. If z1 =, , 3 + i 3 and z2 = 3 + i , then in which quadrant, , æ z1 ö, ç ÷ lies?, è z2 ø, (a) I, (b) II, (c) III, (d) IV, 17. If z1 = 3 + i and z2 = i – 1, then, (a) | z1 + z2| > |z1| + |z2| (b) | z1+ z2 | < |z1| – |z2|, (c) |z1 + z2| £ |z1| + |z2|, (d) |z1 + z2| < |z1| + |z2|, 18. Let z be any complex number such that | z | = 4 and, , arg (z) =, (a), , 5p, , then value of z is, 6, , 5+ 2i, 1- 2 i, , (b) 2, , (c) 0, , x y, –, 21. If z = x + iy,, = a – ib and, = k(a2 – b2),, a b, then value of k equals, (a) 2, (b) 4, (c) 6, (d) 1, , (a), , 2 and, , (c) 1 and, , p, 6, , 1 + 2i, 1 - (1 - i)2, , are, , (b) 1 and 0, , p, 3, , (d) 1 and, , 24. If z =, (a) 0, , (b) 7, , p, 4, , (c) –1, , (d) –4, , 3-i 3+ i, +, , then value of arg (zi) is, 2+i 2-i, , (b), , p, 6, , (c), , 7 - 26i, 25, , (b), , p, 8, , 2 + 5i, is equal to :, 4 - 3i, , -7 - 26i, 25, , -7 + 26i, 7 + 26i, (d), 25, 25, If z = 1 + i, then the multiplicative inverse of z 2 is, , (where, i =, (a) 2 i, (c), , 28., , -, , p, 3, , -1 ), , (b) 1– i, , i, 2, , (d) i, 2, , 3 öæ 3 + 4i ö, æ 1, +, ç, ÷ç, ÷ is equal to :, è 1 - 2i 1 + i øè 2 - 4i ø, (a), , 1 9, + i, 2 2, , (b), , 1 9, - i, 2 2, , (c), , 1 9, - i, 4 4, , (d), , 1 9, + i, 4 4, , 30. Amplitude of, , (d), , p, 2, , 1 + 3i, 3 +1, , is :, , p, p, p, p, (b), (c), (d), 6, 3, 4, 2, Consider the following statements., I. Representation of z = x + iy in terms of r and q is, called polar form of the complex number., , (a), , 31., , æ z1 ö, arg ç z ÷ = arg (z1) – arg (z2), è 2ø, Choose the correct option., (a) Only I is incorrect., (b) Only II is correct., (c) Both I and II are incorrect., (d) Both I and II are correct., 32. If z1 = 2 + 3i and z2 = 3 + 2i, then z1 + z2 equals to a + ai., Value of ‘a’ is equal to, (a) 3, (b) 4, (c) 5, (d) 2, 33. If z1 = 2 + 3i and z2 = 3 – 2i, then z1 – z2 equals to –1 + bi., The value of ‘b’ is, (a) 1, (b) 2, (c) 3, (d) 5, , II., , 23. If z = 2 + i, then ( z – 1) ( z – 5) + ( z – 1) (z – 5) is equal to, (a) 2, , (a), , (d) 4, , 1, z3, , 22. The modulus and amplitude of, , (d) 28 cos, , æ1 ∗ 2i ö÷, 29. The complex number ççç, ÷ lies in:, è 1, i ø÷, (a) I quadrant, (b) II quadrant, (c) III quadrant, (d) IV quadrant, , a + ib, , then x2 + y2 =, a – ib, , (a) 1, , p, 4, , (b) 25, , 26. The conjugate of the complex number, , is 1 + 2 2 i., , Choose the correct option., (a) Only I and II are correct., (b) Only II and III are correct., (c) I, II and III are correct., (d) I, II and III are incorrect., 20. If x + iy =, , (c) 24 cos, , 27., , (c), (d) -2 3 + 2i, 2 + 3i, 19. Consider the following statements, I. Additive inverse of (1 – i) is equal to –1 + i., II. If z1 and z2 are two complex numbers, then z1 – z2, represents a complex number which is sum of z1 and, additive inverse of z2., III. Simplest form of, , 25. (1 + i)8 + (1 – i)8 equal to, (a) 28, , (c), , (b) 2 3 - i, , -2 3 - 2i, , 53

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EBD_7762, 54, , 34., 35., , MATHEMATICS, , z z, +, is, z z, (a) cos 2q (b) 2 cos 2q (c) 2 cos q (d) 2 sin q, The square root of i is, , If z = r(cos q + i sin q), then the value of, , (a), , ±, , (c), , ±, , 1, 2, 1, 2, , 1, , (1 + i), , ( –1 + i), , (b) ±, , (1 – i), , (d) None of these, , 2, , 4, , 36., 37., , 38., , 39., 40., , 41., , 42., , æ 1ö, The value of (1 + i)4 ç 1 + ÷, iø, è, (a) 12, (b) 2, (c), Evaluate: (1 + i)6 + (1 – i)3., (a) –2 – 10i, (b), (c) –2 + 10i, (d), , 8, , (d) 16, , 2 – 10i, 2 + 10i, , x y, If ( x + iy) = a + ib, where x, y, a, b Î R, then –, =, a b, 2, 2, 2, 2, (a) a – b, (b) –2(a + b ), (c) 2(a2 – b2), (d) a2 + b2, , i 4n + 1 – i 4n – 1, . is, 2, (a) i, (b) 2i, (c) –i, (d) –2i, If z(2 – i) = (3 + i), then z20 is equal to, (a) 210, (b) –210, 20, (c) 2, (d) –220, , The value of, , The real part of, (a), , 1, 3, , (c), , –, , (1 + i)2, (3 – i ), , (d) None of these, , The multiplicative inverse of, 8, 31, –, i, 25 25, , 3 + 4i, is, 4 – 5i, , (b) –, , 8 31, –, i, 25 25, , 8 31, +, i, (d) None of these, 25 25, Consider the following statements., I. The value of x3 + 7x2 – x + 16, when x = 1 + 2i is, –17 + 24i., II. If iz3 + z2 – z + i = 0 then |z| = 1, Choose the correct option., (a) Only I is correct., (b) Only II is correct., (c) Both are correct., (d) Both are incorrect., , (c), , –, , (b) 2, , (c) 0, , 45. If z1 = 6 + 3i and z2 = 2 – i, then, , (d) 3, , z1, 1, is equal to (9 + 12i)., z2, a, , The value of ‘a’ is, (a) 1, (b) 2, (c) 4, (d) 5, 46. Value of i4k + i4k+1 + i4k + 2 + i4k + 3 is, (a) 0, (b) 1, (c) 2, (d) 3, (a) |z + 3| 2, (c) z2 + 3, , (b) |z – 3|, (d) None of these, 5 + 12i + 5 – 12i, , 48. What is the conjugate of, , (a) –3i, 49. If z =, , (b) 3i, , 5 + 12i – 5 – 12i, , ?, , 3, i, 2, , 3, (d) – i, 2, , (c) –27, , (d) –27 i, , (c), , 7–i, , then |z | 14 =, 3 – 4i, , (a) 27, , (b) 27 i, , 50. Represent z = 1 + i 3 in the polar form., p, p, + i sin, 3, 3, , (a), , cos, , (c), , p, pö, æ, 2 ç cos + i sin ÷, 3, 3ø, è, , is, 1, (b), 5, , 1, 3, , (a) 1, , 47. The value of ( z + 3) ( z + 3) is equivalent to, , 1, 3, , (a), , 43., , is, , æ -3 ö, 44. If z = 5i ç i ÷ , then z is equal to 3 + bi. The value of ‘b’ is, è 5 ø, , 51. The modulus of, , (a) 2, , p, pö, æ, (d) 4 ç cos + i sin ÷, 3, 3ø, è, , (1 + i 3 ) (2 + 2i), ( 3 – i), , (b) 4, , p, p, – i sin, 3, 3, , (b) cos, , is, , (c) 3 2, , (d) 2 2, , æ i 2ö, 52. The argument of the complex number ç – ÷ is equal to, è2 i ø, (a), , p, 4, , (b), , 3p, 4, , (c), , p, 12, , (d), , p, 2, , 53. The square root of (7 – 24i) is, (a) ± (3 – 5i), , (b) ± (3 + 4i), , (c) ± (3 – 4i), , (d) ± (4 – 3i), , 54. If 1 – i, is a root of the equation x2 + ax + b = 0,, where a, b Î R, then the values of a and b are,, (a) 2, 2, , (b) –2, 2, , (c) –2, –2 (d) 1, 2

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COMPLEX NUMBERS AND QUADRATIC EQUATIONS, , 55. If z1 = 6 + 3i and z2 = 2 – i, then, (a), , 55, , 66. If z be the conjugate of the complex number z, then, which of the following relations is false?, , z1, is equal to, z2, , (a), , 1, (9 + 12i), 5, , (b) 9 + 12i, , I., II., , (d) arg z = arg z, z1 + z 2 = z1 + z2, 5, 5, 67. The value of (1 + i) (1 – i) is 2n. ‘n’ is equal to, (a) 2, (b) 3, (c) 4, (d) 5, 68. The value of (1 + i)8 + (1 – i)8 is 2n. Value of n is, (a) 2, (b) 3, (c) 4, (d) 5, , (d), , |z|2, , If z, z1, z2 be three complex numbers then z z =, The modulus of a complex number z = a + ib is, defined as |z| =, , a +b ., 2, , 2, , III. Multiplicative inverse of z = 3 – 2i is, , 3 2, + i, 13 13, , Choose the correct option., (a) Only I and II are correct., (b) Only II and III are correct., (c) Only I and III are correct., (d) All I, II and III are correct., 58. If z = i9 + i19, then z is equal to a + ai. The value of ‘a’ is, (a) 0, (b) 1, (c) 2, (d) 3, 59. If z = i–39, then simplest form of z is equal to a + i. The, value of ‘a’ is, (a) 0, (b) 1, (c) 2, (d) 3, 60. If (1 – i)n = 2n, then the value of n is, (a) 1, (b) 2, (c) 0, (d) None of these, 61. If z1 = 2 – i and z2 = 1 + i, then value of, (a) 2, 62. If, , (b) 2i, , (c), , (1 + i)3 (1 – i)3, –, (1 – i)3 (1 + i)3, , = x + iy, , 2, , z1 + z 2 + 1, is, z1 – z 2 + 1, , (d), , p, pù, é, 2 êcos + i sin ú and z =, 2, 4, 4û, ë, , 69. If z1 =, , then |z1 z2| is equal to, (a) 6, (b) 3, 70. If, , 64. If z is a complex number such that z2 = ( z )2 , then, z is purely real, z is purely imaginary, either z is purely real or purely imaginary, None of these, , æ z – 1ö, 65. If | z | = 1, (z ¹ –1) and z = x + iy, then ç, ÷ is, è z + 1ø, (a) purely real, (b) purely imaginary, (c) zero, (d) undefined, , p, pù, é, 3 êcos + i sin ú ,, 3, 3û, ë, , m . Value of m is, (c) 2, (d) 5, , a + ib = x + iy, then possible value of, , (b), , (c) x + iy, , (d) x – iy, , 71. The modulus of, , a – ib is, , x 2 + y2, , (a) x2 + y2, , 2i , ,2i is:, , (a) 2, (b), (c) 0, 2, 72. arg z + arg z; z ¹ 0 is equal to :, , (d) 2 2, , p, p, (b) p, (c) 0, (d), 4, 2, 73. If z = 2 –3i, then value of z2 – 4z + 13 is, (a) 0, (b) 1, (c) 2, (d) 3, 74. The modulus of the complex number z such that, | z + 3 – i | = 1 and arg(z) = p is equal to, (a) 3, (b) 2, (c) 9, (d) 4, , (a), , 1- i, , then polar form of Z is, p, p, cos + i sin, 3, 3, , 75. If Z =, , 2i, , (a) x = 0, y = –2, (b) x = –2, y = 0, (c) x = 1, y = 1, (d) x = –1, y = 1, 63. Additive inverse of 1 – i is, (a) 0 + 0i, (b) –1 – i, (c) –1 + i, (d) None of these, , (a), (b), (c), (d), , 2, , (c), , 1, (12 + 9i), 5, 56. The value of (1 + i)5 × (1 – i)5 is, (a) –8, (b) 8i, (c) 8, (d) 32, 57. Consider the following statements., , (c) 3 + 2i, , (b) z × z = z, , z = z, , (a), , 5p, 5p ö, æ, 2 ç cos, – i sin ÷ (b), 12, 12, è, ø, , 5p, 5p ö, æ, 2 ç cos, + i sin, ÷, 12, 12 ø, è, , p, pö, p, pö, æ, æ, (d) 2 ç cos – i sin ÷, 2 ç cos + i sin ÷, 4, 4ø, 4, 4, è, è, ø, 76. (x – iy) (3 + 5i) is the conjugate of (–6 – 24i), then x and y, are, (a) x = 3, y = –3, (b) x = –3, y = 3, (c) x = –3, y = –3, (d) x = 3, y = 3, , (c), , 77. If z is a complex number such that, imaginary, then, (a) | z | = 0 (b) | z | = 1, 78. The amplitude of sin, (a), , p, 5, , (b), , 2p, 5, , z –1, is purely, z +1, , (c) | z | > 1 (d) | z | < 1, , p, pö, æ, + i ç1 – cos ÷ is, è, 5, 5ø, (c), , p, 10, , (d), , p, 15

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EBD_7762, 56, , 79., , MATHEMATICS, , If x + iy =, , (a), , (c), , I., , a + ib, , then (x2 + y2 ) 2 =, c + id, , a 2 + b2, , (b), , c2 + d 2, , a+b, c+d, , æ a2 + b2 ö, ÷, (d) çç 2, 2÷, èc +d ø, , c2 + d 2, a 2 + b2, , II., , x2 + 3x + 5 = 0 is x =, 2, , Topic 2: Solution of Quadratic Equations, Sum and, Product of Roots, Nature of Roots, Relation between Roots and, Co-efficients, Formation of an Equation with Given Roots, 80., , The solutions of the quadratic equation ax2 + bx + c = 0,, where a, b, c Î R, a ¹ 0, b2 – 4ac < 0, are given by x = ?, (a), , b ± 4ac - b2i, 2a, , 81., , 82., 83., , 85., , (a), , 1 + 3i, 2, , (b), , (c), , -1 ± 3i, 3, , (d), , -1 ± 2i, 2, , 2x 2 - 11x + 30 = 0, , (b) - x 2 + 11x = 0, , 2, (c) x 2 - 11x + 30 = 0, (d) 2x - 5x + 30 = 0, If a < b < c < d, then the nature of roots of, ( x – a) (x – c) + 2 (x – b) (x – d) = 0 is, (a) real and equal, (b) complex, (c) real and unequal, (d) None of these, , 1, 1, 1, , if the product of, =, x+a x+b x+c, roots is zero, then sum of roots is, , For the equation, , -, , 2bc, b+c, , 2ca, (b), c+a, , bc, -bc, (d), c+a, b+c, Product of real roots of the equation t 2 x 2 +|x|+9 = 0, (a) is always positive, (b) is always negative, (c) does not exist, (d) None of these, Consider the following statements., , (c), , 87., , -1 ± 3i, 2, , The solution of 3x 2 - 2 = 2x - 1 are :, (a) (2, 4) (b) (1, 4), (c) (3, 4) (d) (1, 3), If a, b are roots of the equation x2 – 5x + 6 = 0, then the, equation whose roots are a + 3 and b + 3 is, , (a), , 86., , 2, , -b ± 4ac - b i, -b ± 4ab - c i, (d), 2a, 2a, 2, If x + x + 1 = 0, then what is the value of x ?, , (a), 84., , -b ± 4ac + b 2i, (b), 2a, , 2, , (c), , Let z1 and z2 be two complex numbers such that, |z1 + z2| = |z1| + |z2| then arg (z1) – arg (z2) = 0, Roots of quadratic equation, , -3 ± i 11, 2, , Choose the correct option., (a) Only I is correct., (b) Only II is correct., (c) Both are correct., (d) Neither I nor II is correct., 88. 2x2 – (p + 1) x + (p – 1) = 0. If a – b = ab, then what, is the value of p?, (a) 1, (b) 2, (c) 3, (d) –2, 89. If p and q are the roots of the equation x2+px+q = 0, then, (a) p = 1, q = –2, (b) p = 0, q = 1, (c) p = –2, q = 0, (d) p = – 2, q =1, 90. The roots of the given equation, (p – q) x 2 + (q – r) x + (r – p) = 0 are :, p-q, ,1, r-p, , (a), , (b), , q-r, ,1, p-q, , r-p, ,1, (d) None of these, p-q, 91. If a, b are the roots of ax2 + bx + c = 0, then ab2 + a2b +, ab equals, , (c), , (a), , (c), , c(a , b), , (b) 0, , a2, , ,bc, , (d) abc, , a2, , 92. The roots of equation x (a) one, (c) infinite, 5x 2 + x +, , 93. Solve, (a), (c), , ±, , 19, i, 5, , 2, 2, = 1is, x -1, x -1, (b) two, (d) None of these, , 5 = 0., (b) ±, , – 1 ± 19i, , (d), , 19i, 2, , – 1 ± 19i, , 2 5, 5, 94. If a and b are the roots of the equation x2 + 2x + 4 =, , 0, then, , 1, a, , 3, , +, , 1, b3, , is equal to, , 1, 1, 1, (b), (c) 32, (d), 2, 2, 4, 95. If a, b are the roots of the equation ax2 + bx + c = 0,, , (a), , –, , then the value of, (a), , ac, b, , 1, 1, +, equals, aa + b ab + b, , (b) 1, , (c), , ab, c, , (d), , b, ac

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COMPLEX NUMBERS AND QUADRATIC EQUATIONS, , 96. Roots of x2 + 2 = 0 are ± n i . The value of n is, (a) 1, (b) 2, (c) 3, (d) 4, 97. Assertion : Let f(x) be a quadratic expression such that, f(0) + f(1) = 0. If –2 is one of the root of f(x) = 0,, 3, ., 5, Reason : If a and b are the zeroes of f(x) = ax2 + bx + c,, , then other root is, , 98., , 99., , 100., 101., , 102., , b, c, then sum of zeroes = – , product of zeroes = ., a, a, (a) Assertion is correct, Reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., If ax2 + bx + c = 0 is a quadratic equation, then equation, has no real roots, if, (a) D > 0, (b) D = 0, (c) D < 0, (d) None of these, A value of k for which the quadratic equation, x2 – 2x(1 + 3k) + 7(2k + 3) = 0 has equal roots is, (a) 1, (b) 2, (c) 3, (d) 4, The roots of the equation 32x – 10.3x + 9 = 0 are, (a) 1, 2, (b) 0, 2, (c) 0, 1, (d) 1, 3, The equation whose roots are twice the roots of the equation,, x2 – 3x + 3 = 0 is:, (a) 4x2 + 6x + 3 = 0, (b) 2x2 – 3x + 3 = 0, 2, (c) x – 3x + 6 = 0, (d) x2 – 6x + 12 = 0, x, The roots of the equation 4 – 3 . 2 x + 3 + 128 = 0 are, (a) 4 and 5 (b) 3 and 4 (c) 2 and 3 (d) 1 and 2, , 103. If one root of the equation x 2 + px + 12 = 0 is 4, while the, equation x 2 + px + q = 0 has equal roots , then the value, 49, 4, 104. For the equation 3x2 + px + 3 = 0, p > 0, if one of the root, is square of the other, then p is equal to, (a) 1/3, (b) 1, (c) 3, (d) 2/3, , (b) 12, , (c) 3, , (d), , 3, , 1ö, 1ö, æ, æ, 105. The number of real roots of ç x + ÷ + ç x + ÷ = 0 is, xø, xø, è, è, , (a) 0, , 107. Find the value of a such that the sum of the squares of, the roots of the equation x2 – (a – 2)x – (a + 1) = 0 is, least., (a) 4, (b) 2, (c) 1, (d) 3, 108. If a, b are the roots of the equation (x – a) (x – b) = 5,, then the roots of the equation (x – a) (x – b) + 5 = 0 are, (a) a, 5, (b) b, 5, (c) a, a, (d) a, b, 109. If x2 + y2 = 25, xy = 12, then x =, (a) {3, 4}, (b) {3, –3}, (c) {3, 4, –3, –4}, (d) {–3, –3}, 110. If the roots of the equations px2 + 2qx + r = 0 and, qx2 – 2, , ( pr ) x + q = 0 be real, then, , (a) p = q, (b) q2 = pr, (c) p2 = qr, (d) r2 = pq, 111. If a > 0, b > 0, c > 0, then both the roots of the equation, ax2 + bx + c = 0., (a) Are real and negative (b) Have negative real parts, (c) Are rational numbers (d) None of these, 112. If a and b are the odd integers, then the roots of the, equation 2ax2 + (2a + b)x + b = 0, a ¹ 0, will be, (a) rational, (b) irrational, (c) non-real, (d) equal, 113. If 2 + i 3 is a root of the equation x2 + px + q = 0, where, p and q are real, then (p, q) =, (a) (–4, 7) (b) (4, –7) (c) (4, 7) (d) (–4, –7), 114. If the sum of the roots of the equation x2 + px + q = 0, is equal to the sum of their squares, then, (a) p2 – q2 = 0, (b) p2 + q2 = 2q, 2, (c) p + p = 2q, (d) None of these, 115. If the roots of the equation x2 – 2ax + a2 + a – 3 = 0 are, real and less than 3, then, (a) a < 2, (b) 2 £ a £ 3, (c) 3 < a £ 4, , of ‘ q’ is, (a) 4, , 57, , (b) 2, , 106. If the roots of the equation, , (c) 4, , (d) 6, , a, b, +, = 1 are equal, x–a x–b, , in magnitude and opposite in sign, then, (a) a = b, (b) a + b = 1, (c) a – b = 1, (d) a + b = 0, , (d) a > 4, , 116. If the equation (m – n)x2 + (n – l)x + l – m = 0 has equal, roots, then l, m and n satisfy, (a) 2l = m + n, (b) 2m = n + l, (c) m = n + l, (d) l = m + n, 117. If the product of the roots of the equation, (a + 1)x2 + (2a + 3)x + (3a + 4) = 0 be 2, then the sum, of roots is, (a) 1, (b) –1, (c) 2, (d) –2, 118. If a, b are the roots of the equation ax2 + bx + c = 0,, then, , a, b, +, =, ab + b aa + b, , (a), , 2, a, , (b), , 2, b, , (c), , 2, c, , (d) –, , 2, a

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EBD_7762, 58, , MATHEMATICS, , 119. If one root of ax2 + bx + c = 0 be square of the other,, then the value of b3 + ac2 + a2 c is, (a) 3abc, (b) –3abc, (c) 0, (d) None of these, 120. If a, b are the roots of (x – a) (x – b) = c, c ¹ 0, then, the roots of (x – a) (x – b) + c = 0 shall be, (a) a, c, (b) b, c, (c) a, b, (d) a + c, b + c, 121. If the roots of the equation ax2 + bx + c = 0 are a, b,, then the value of ab2 + a2 b + ab will be, (a), , c (a – b ), , (b) 0, , a2, , –, , bc, , ab, , then, 2, (a) p = 1, q = –56, (b) p = –1, q = –56, (c) p = 1, q = 56, (d) p = –1, q = 56, 125. If the roots of 4x2 + 5k = (5k + 1)x differ by unity, then, the negative value of k is, , a2 + b2,, , 1, 5, 126. Sum of all real roots of the equation, | x – 2 | 2 + | x – 2 | – 2 = 0 is, (a) 2, (b) 4, (c) 5, , (a) –3, , (c) –, , (b) –5, , 127. The value of 2 +, , 1, 1, 2+, 2 + ................ ¥, , (d) –, , (d) 6, is, , (a) 1 – 2, , (b) 1 + 2, , (c) 1 ± 2, , (d) None of these, , 128. If x =, (a), (b), (c), (d), , BEYOND NCERT, Topic 3: Rotational Theorem, De-moiver’s Theorem,, Geometry of Complex Numbers, Cube Roots of Unity., 130. If the complex numbers z1, z2,z3 represents the vertices of, an equilateral triangle such that | z1 | = | z2 | = | z3 |, then, value of z1 + z2 + z3 is, (a) 0, , (d) None of these, a2, 122. If a, b be the roots of the equation 2x2 – 35x + 2 = 0,, then the value of (2a – 35)3 . (2b – 35)3 is equal to, (a) 1, (b) 64, (c) 8, (d) None of these, 123. If the sum of the roots of the equation x2 + px + q = 0, is three times their difference, then which one of the, following is true?, (a) 9p2 = 2q, (b) 2q2 = 9p, 2, (c) 2p = 9q, (d) 9q2 = 2p, 124. If the roots of the equation x2 – 5x + 16 = 0 are a, b and, the roots of equation x2 + px + q = 0 are, (c), , 129. If the ratio of the roots of x2 + bx + c = 0 and, x2 + qx + r = 0 be the same, then, (a) r2 c = b2 q, (b) r2 b = c2 q, 2, 2, (c) rb = cq, (d) rc2 = bq2, , 6 + 6 + 6 + ..... to ¥ , then, , x is an irrational number, 2 <x < 3, x=3, None of these, , 3, 5, , (b) 1, , (c) 2, , (d), , 3, 2, , 131. If | z2 – 1 | = | z | 2 + 1, then z lies on, (a) imaginary axis, (b) real axis, (c) origin, (d) None of these, 132. If | z + 4 | £ 3, then the maximum value of | z + 1 | is, (a) 6, (b) 0, (c) 4, (d) 10, 133. Value of, , (cos q + i sin q ) 4, (cos q - i sin q) 3, , (a), , cos 5q + i sin 5q, , (c), , cos 4q + i sin 4q, , is, (b) cos 7q + i sin 7q, (d) cos q + i sin q, , 134. If centre of a regular hexagon is at origin and one of the, vertex on argand diagram is 1 + 2i, then its perimeter is, (a), , (c) 4 5, (d) 6 5, (b) 6 2, 2 5, 135. If iz + 1 = 0 , then z can take the value :, 1+ i, p, p, (a), (b) cos + i sin, 2, 8, 8, 1, (c), (d) i, 4i, 136. If n is a positive integer grater than unity and z is a complex, 4, , satisfying the equation z n = (z + 1) n , then, (a) Re(z) < 2, , (b) Re(z) > 0, , 1, 2, 137. Assertion : If P and Q are the points in the plane XOY, representing the complex numbers z 1 and z2 respectively,, then distance |PQ | = | z2 – z1 |., Reason : Locus of the point P(z) satisfying, | z – (2 + 3i)| = 4 is a straight line., (a) Assertion is correct, Reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., , (c) Re(z) = 0, , (d) z lies on x = –

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COMPLEX NUMBERS AND QUADRATIC EQUATIONS, , 59, , BEYOND NCERT, , 138. If z1 and z2 are two non-zero complex numbers such that, | z1 + z2 | = | z1 | + | z2 | , then argg z1 – argg z2 is equal to, p, 2, , (a), , (b) – p, , (c) 0, , (d), , -p, 2, , 139. If the cube roots of unity are 1, w , w 2 then the roots of the, equation ( x –1)3 + 8 = 0, are, (a) –1, 1 + 2 w , 1 – 2 w 2 (b) –1, – 1, – 1, (c) – 1, 1 – 2 w , 1 – 2 w, , 2, , æ 1+ i ö, 140. One of the values of ç, è 2 ÷ø, , (a), , 1, ( 3 + i), 2, , (d) None of these, is, , pü, ì, sin í(w13 + w2 ) p + ý is equal to, 4þ, î, , 142. If w =, , (b) –, z, , 1, z- i, 3, , 1, 2, , (c), , 1, 2, , (d), , 3, 2, , and | w | = 1, then z lies on, , (a) an ellipse, (c) a straight line, , (b) a circle, (d) a parabola, , -1 + 3i, then (3 + w + 3w2)4 is, 2, (a) 16, (b) – 16, (c) 16w, (d) 16w2, 1/3, 144. The value of i is :, 3+i, 3-i, (a), (b), 2, 2, 1+ i 3, 1-i 3, (c), (d), 2, 2, 145. Let z = 1 – t + i ( t 2 + t + 2) , where t is a real parameter.., The locus of z in the Argand plane is, (a) a hyperbola, (b) an ellipse, (c) a straight line, (d) None of these, 143. If w =, , 146. If 2x = –1 +, , 27, 4, , (c) 3,, , (d) – 3 + i, , 3, 2, , (b) –3, -, , 4, (d) –2, – 3, 27, 149. If a root of the equations x2 + px + q = 0 and, x2 + ax + b = 0 is common, then its value will be, (where p ¹ a and q ¹ b), , 2/3, , 141. If w is imaginary cube root of unity, then, , (a) –, , 148. Value of k such that equations 2x2 + kx – 5 = 0 and x2 – 3x – 4 = 0, have one common root, is, (a) –1, –2, , (b) –i, , (c) i, , Topic 4: Condition for Common Roots, Maximum and Minimum Values of Quadratic Equations, Sign of, Quadratic Expression, Quadratic Expression in Two, Variables, Solution of Quadratic Inequalities., , 3i, then the value of, , (1 – x2 + x)6 – (1 – x + x2)6 =, (a) 32, (b) –64, (c) 64, (d) 0, 147. The points 0, 2 + 3i, i, –2 –2i in the argand plane are the, vertices of a, (a) rectangle, (b) rhombus, (c) trapezium, (d) parallelogram, , pb – a q, q–b, , (a), , q–b, a–p, , (b), , (c), , q–b, pb – aq, or, a–p, q–b, , (d) None of these, , 150. The solution set of, (a), , x 2 - 3x + 4, > 1 , x Î R , is, x +1, , (3, + ¥), , (b) (-1, 1) È (3, + ¥), , (c) [ -1, 1] È [3, + ¥), (d) None of these, 151. If p and q be the roots of the quadratic equation, x2 – (a – 2) x – a – 1 = 0 then minimum value of p2 + q2 is, equal to, (a) 2, , (b) 3, , (c) 6, , (d) 5, , 152. For all x Î R, if m x 2 – 9mx + 5m + 1 > 0, then m lies, in the interval, ö, æ 4, (a) ç - , 0 ÷, 61, ø, è, 4ö, é, (c) ê0, ÷, ë 61 ø, , 4ù, (b) éê0,, ú, 61, ë, û, , (d) None of these, , 153. If both the roots of k (6 x 2 + 3) + rx + 2 x 2 - 1 = 0 and, 6k (2 x 2 + 1) + px + 4 x 2 - 2 = 0 are common, then 2r - p, is equal to, (a) – 1, (b) 0, (c) 1, (d) 2, , 154. The least integral value a of x such that, satisfies :, (a) a2 + 3a – 4 = 0, (c) a2 – 7a + 6 = 0, , x-5, 2, , x + 5x - 14, , (b) a2 – 5a + 4 = 0, (d) a2 + 5a – 6 = 0, , >0,

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EBD_7762, 60, , MATHEMATICS, , 155. If x2 + ax + 10 = 0 and x2 + bx – 10 = 0 have a common, root, then a2 – b2 is equal to, (a) 10, (b) 20, (c) 30, (d) 40, 156. If the equations k (6x2 + 3) + rx + 2x2 – 1 = 0 and, 6k (2x2 – 1) + px + 4x2 + 2 = 0 have both roots common,, then the value of (2r – p) is :, (a) 0, (b) 1/2, (c) 1, (d) None of these, , 158. A value of b for which the equations, x2 + bx – 1 = 0, x2 + x + b = 0, have one root in common is, (a), , (c) i 5, (d), (b) -i 3, - 2, 159. The number of real solutions of the equation, , 2, , 2, ( x – 1) 2 + ( x – 2) + ( x – 3) 2 = 0 is, , 157. If x 2 - 5 x + 6 > 0 , then x Î :, (a), , (-¥,2) È (3, ¥), , (c) (2, 3), , (a) 2, , (b) [2, 3], (d) None of these, , (b) 1, , (c) 0, , (d) 3, , 160. If x be real, then the minimum value of x 2 - 8 x + 17 is, (a) –1, , (b) 0, , (c) 1, , (d) 2, , Exercise 2 : Exemplar & Past Year MCQs, where, n Î N, , NCERT Exemplar MCQs, 1., , 2., , sin x + i cos 2x and cos x – i sin 2x are conjugate to each, other for, (a) x = np, , 1ö p, æ, (b) x = ç n + ÷, 2ø 2, è, , (c) x = 0, , (d) No value of x, , The real value of a for which the expression, , 6., , 7., , 1– i sin a, is, 1 + 2i sin a, , purely real is, (a), , (n + 1), , p, 2, , (c) np, 3., , If z = x + iy lies in the third quadrant, then, third quadrant, if, (a) x > y > 0, (c) y < x < 0, , 4., , p, 2, (d) None of these, , (b) (2 n + 1), , z, also lies in the, z, , (b) x < y < 0, (d) y > x > 0, , The value of (z + 3)( z + 3) is equivalent to, (a) |z + 3|2, (b) |z – 3|, (c) z2 + 3, (d) None of these, x, , 5., , æ 1+ i ö, If ç, = 1, then, è 1– i ÷ø, (a) x = 2n + 1, (c) x = 2n, , (b) x = 4n, (d) x = 4n + 1, , 8., , æ 3 – 4ix ö, A real value of x satisfies the equation ç, = a – ib, è 3 + 4ix ÷ø, (a, b Î R), if a2 + b2 is equal to, (a) 1, (b) – 1, (c) 2, (d) – 2, Which of the following is correct for any two complex, numbers z1 and z2?, (a) |z1 z2 | = |z1 | |z2 |, (b) arg(z1 z2) = arg(z1) . arg(z2), (c) |z1 + z2 | = |z1 | + |z2 |, (d) |z1 + z2 | ³ |z1 | – |z2 |, The point represented by the complex number (2 – i) is, rotated about origin through an angle, , p, in the clockwise, 2, , direction, the new position of point is, (a) 1 + 2 i, (b) – 1 – 2 i, (c) 2 + i, (d) – 1 + 2 i, 9., If x, y Î R, then x + iy is a non-real complex number, if, (a) x = 0 (b) y = 0, (c) x ¹ 0 (d) y ¹ 0, 10. If a + ib = c + id, then, (a) a2 + c2 = 0, (b) b2 + c2 = 0, (c) b2 + d2 = 0, (d) a2 + b2 = c2 + d2, 11. The complex number z which satisfies the condition, , i+z, i–z, , = 1 lies on, , (a) circle x2 + y2 = 1, (c) the y-axis, , (b) the x-axis, (d) the line x + y = 1

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COMPLEX NUMBERS AND QUADRATIC EQUATIONS, , 12. If z is a complex number, then, (a) |z2| > |z|, (b) |z2| = |z|2, 2, 2, (c) |z | < |z|, (d) |z2| ³ |z|2, 13. |z1 + z2 | = |z1 | + |z2 | is possible, if, (a), , z 2 = z1, , (b) z2 =, , (c) arg(z1) = arg(z2), , (c), , 20., , (d) |z1 | = |z2 |, , (c) 2np ±, , (b) np + (– 1)n, , p, 2, , 1+ i cos q, is, 1– 2i cos q, , p, 4, , (d) None of these, , 15. The value of arg (x), when x < 0 is, p, 2, (d) None of these, , (a) 0, , (b), , (c) p, 16. If f (z) =, , 7– z, 1– z 2, , , where z = 1 + 2i, then | f (z)| is equal to, , |z|, (a), 2, (c) 2|z|, , (d) None of these, Past Year MCQs, , 17. If z is a complex number such that z ³ 2, then the minimum, , 1, :, 2, , [JEE MAIN 2014, C], , (a) is strictly greater than, , 5, 2, , (b) is strictly greater than, , 3, 5, but less than, 2, 2, , 5, 2, (d) lie in the interval (1, 2), If a Î R and the equation, , (c) is equal to, 18., , i, (a) 0, , 1, 25, , , when simplified has the value [BITSAT 2014, C], (b) 2i, , (c) – 2i, (d) 2, x y, 21. If z = x + iy, z1 / 3 = a – ib, then – = k (a2 – b2) where k is, a b, equal to, [BITSAT 2014, A], (a) 1, (b) 2, (c) 3, (d) 4, 22. A complex number z is said to be unimodular if |z| = 1. Suppose z1, z1 - 2z 2, and z2 are complex numbers such that 2 - z z is unimodular, 1 2, and z2 is not unimodular. Then the point z1 lies on a:, [JEE MAIN 2015, A], (a) circle of radius 2., (b) circle of radius, , 2., (c) straight line parallel to x-axis, (d) straight line parallel to y-axis., 23. Let a and b be the roots of equation x2 – 6x – 2 = 0. If, , greatest integer £ x ) has no integral solution, then all, possible values of a lie in the interval:, [JEE MAIN 2014, S], (b), , ( -¥, -2 ) È ( 2, ¥ ), , a10 - 2a 8, is, 2a 9, , equal to :, [JEE MAIN 2015, A], (a) 3, (b) – 3, (c) 6, (d) – 6, 24. Universal set,, U = {x | x5 – 6x4 + 11x3 – 6x2 = 0}, A = {x | x2 – 5x + 6 = 0}, B = {x | x2 – 3x + 2 = 0}, What is (A Ç B)' equal to ?, [BITSAT 2015, A], (a) {1, 3}, (b) {1, 2, 3}, (c) {0, 1, 3}, (d) {0, 1, 2, 3}, 25. If complex number z1, z2 and 0 are vertices of equilateral, triangle, then z12 + z 22 - z1z 2 is equal to [BITSAT 2015, C], (a) 0, (b) z1 – z2, 26. The root of the equation, , (c) z1 + z2 (d) 1, , modulus is, , 2, , ( -2, -1), , i57 +, , 2 (1 + i ) x 2 - 4 ( 2 - i ) x - 5 - 3i = 0, , -3 ( x - [ x ]) + 2 ( x - [ x ]) + a 2 = 0 (where [x] denotes the, , (a), , (1, 2 ), , an = an – bn, for n ³ 1, then the value of, , (b) |z|, , value of z +, , (d), , of x2, , + x + a = 0 exceed a then, [BITSAT 2014, A], (a) 2 < a < 3, (b), a>3, (c) – 3 < a < 3, (d) a < – 2, , a real number is, p, 4, , ( -1, 0 ) È ( 0,1), , 19. If the roots, , 1, z1, , 14. The real value of q for which the expression, , (a) np +, , 61, , which has greater, [BITSAT 2015, C], , 3 - 5i, 5 - 3i, 3-i, (b), (c), (d) None, 2, 2, 2, 27. The sum of all real values of x satisfying the equation, , (a), , (x 2 - 5 x+ 5)x, , (a) 6, , 2, , + 4x - 60, , (b) 5, , = 1 is :, , [JEE MAIN 2016, C], (c) 3, , (d) – 4

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EBD_7762, 62, , 28., , 29., , 30., , MATHEMATICS, , 2 + 3i sinq, is purely imaginary, is:, 1 - 2i sinq, [JEE MAIN 2016, C], æ 3ö, æ 1 ö, (a) sin -1 ç ÷, (b) sin -1 çç ÷÷, ç 4 ÷, è 3ø, è ø, p, p, (c), (d), 3, 6, If a, b and c are real numbers then the roots of the equation, (x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 are, always, [BITSAT 2016, C], (a) real, (b) imaginary, (c) positive, (d) negative, If a, b are the roots of the equations x2 – 2x– 1 = 0, then what is the, value of a2 b–2+ a –2 b2, [BITSAT 2016, C], (a) –2, (b) 0, (c) 30, (d) 34, , A value ofqfor which, , 31., , If z1 = 3 + i 3 and z 2 = 3 + i , then the complex, , 32, , æ z1 ö, [BITSAT 2016, A], number çç ÷÷ lies in the :, è z2 ø, (a) first quadrant, (b) second quadrant, (c) third quadrant, (d) fourth quadrant, If, for a positive integer n, the quadratic equation,, , 34. If a and b are roots of the equation x 2 + px +, , that | a - b |= 10, then p belongs to the set :, , 35. If a, b Î C are the distinct roots, of the equation, x 2 - x + 1 = 0 , then a101 + b107 is equal to :, , [JEE MAIN 2018, A], (a) 0, (b) 1, (c) 2, (d) – 1, 36. If the amplitude of z – 2 – 3i is p/4, then the locus of, z = x + iy is, [BITSAT 2018, A], , 7-z, 1 - z2, , , where z = 1 + 2i, then |f(z)| is equal to :, [BITSAT 2017, A], , |z|, (a), 2, (c) 2 | z |, , (a), , x + y -1 = 0, , (b), , x - y -1 = 0, , (c), , x + y +1 = 0, , (d), , x - y +1 = 0, , 37. The roots of the equation x 4 - 2 x 3 + x = 380 are :, [BITSAT 2018, A], , x(x + 1) + (x + 1) (x + 2) + ..... + (x + n - 1 ) (x + n) = 10n has, two consecutive integral solutions, then n is equal to :, [JEE MAIN 2017, S], (a) 11, (b) 12, (c) 9, (d) 107, If f(z) =, , [BITSAT 2017, A], (b) {– 3, 2}, (d) {3, – 5}, , (a) {2, – 5}, (c) {– 2, 5}, , 50, , 33., , 3p, = 0 , such, 4, , -1± 5 - 3, 2, , 1± 5 - 3, 2, , (b), , - 5, 4,, , -1± 5 - 3, 2, , (d), , - 5, - 4,, , (a), , 5, - 4,, , (c), , 5, 4,, , 1± 5 - 3, 2, , 38. Roots of the equation x 2 + bx - c = 0(b, c > 0) are, (a) Both positive, (c) Of opposite sign, , (b) | z |, (d) None of these, , [BITSAT 2018, C], (b) Both negative, (d) None of these, , Exercise 3 : Try If You Can, 1., , Number of positive integral values of 'x', for which, ||x2 – 2x| – |3x – 20|| = |x2 + x – 20|, is, (a) 5, , 2., , Which of the following represent the subset of set of complex, number z satisfying log1/3 (log1/2 (|z|2 + 4|z| + 3)) > 0,, (a) [–1,3], (b) {z : Re(z) >1}, (c) {z : i (z) < 2} (d), None of these, , 4., , The equation 2 cos 2, , (b) 6, , (c) 7, (d) infinite, If the complex number is (1 + ri)3 = l (1 + i), when, i=, , 3., , -1, for some real l, the value of r can be, , (a) cos, , p, 5, , (b) cosec, , 3p, 2, , (c) cos, , p, 12, , (d) cosec, , p, 12, , x 2, 1, p, sin x = x 2 + 2 , 0 £ x £ has, 2, 2, x, , (a) one real solution, (b) no real solution, (c) more than one real solution, (d) None of these

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COMPLEX NUMBERS AND QUADRATIC EQUATIONS, , 5., , If, , 1, 1, 1, 1, 1, +, +, +, = , where a, b, c, d Î, a + w b + w c +w, d +w w, , R and w is a cube root of unity then S, (a) 1, 6., , 7., , (b) 2, , (c) 3, , 3, is equal to, aô - a + 1, , (d), , 1, 3, , If a, b, c are positive rational numbers such that a > b > c, and the quadratic equation, , (a) c + a < 2b, , (iii) Angle ABC must be right angle., , (b) Both roots of the given equation are rational, , (a) 0, , (b) 1, , (c) The equation ax2 + 2bx + c = 0 has both negative real, roots, , (c) 2, , (d) None of these, , (d) None of these, Which of the following is/are value of, 2, , Let z and w be two complex numbers such that, , z £ 1, w £ 1, z - iw = z - iw = 2 , then h ow many, , ABCD is necessarily rectangle, , (ii) ABCD is necessarily a rhombus, , p, p -i ö, 4 æ 1 - pi, +, ,, 13. If z = 4 (1 + i ) çç, ÷÷ where i = -1, then, è p + i 1 + pi ø, æ |z| ö, ç, ÷ equals to, è amp( z ) ø, , (a) 1, (c) 3, , (b) 2, (d) 4, , 14. Assume that A i (i = 1, 2, ........, n) are the vertices of a, , complex numbers z can satisfy, , regular polygon inscribed in a circle of radius unity then, , (a) 2, , the value of å A1A i +1, , (c) 4, , n –1, , (b) 3, , i =1, , (d) None of these, , If |z| = max {| z - 1 |, | z + 1 |} then, (a), , 1, | z + z |=, 2, , (b) z + z = 1, , (c), , |z+z| =, 1, , (d) None of these, , 10. Given that the two curves arg (z) = p and | z - 2 3i |= r, 6, , 11., , 12. z1, z2, z3, z4 correspond to the points A, B, C, D lie on a circle., If z1 + z2 + z3 + z4 = 0, then how many of the following, statements are correct ?, (i), , 2, , 9., , (a) a, b and c are rational, c, (b), is rational, ( a – b), b, (c), is rational, (c – a), (d) None of these, , (a + b – 2c)x2 + (b + c – 2a)x + (c + a – 2b) = 0 has a root, in the interval (– 1, 0), then which of the following is not, correct?, , sin ln (ii )i + cos ln (ii )i ?, (a) – 1, (b) 1, (c) 0, (d) None of these, 8., , 63, , intersect in two distinct points then ([r] represents integral, part of r) which of the following is not correct?, (a) r > 3, (b) r = 6, (c) 0 < r < 3 (d) [r] ¹ 2, Consider the quadratic equation, (a + c – b)x2 + 2cx + (b + c – a) = 0, where a, b, c are distinct, real numbers and a + c – b ¹ 0. Suppose that both the roots, of the equation are rational, then, , (a) 2n, , (b) n, , 2, , , is, (c) – n, , (d) – 2n, , 15. Let rk > 0 and zk = rk (cos ak + i sin ak) for k = 1, 2, 3 be such, that, , 1, 1, 1, +, +, = 0 Let Ak be the point in the complex, z3, z2, z1, , plane, , given, , by, , wk, , =, , k = 1, 2, 3. The origin, O is the, (a) incentre of DA1A2A3, (b) orthocentre of DA1A2A3, (c) circumcentre of DA1A2A3, (d) centroid of DA1A2A3, , cos 2a k + i sin 2a k, zk, , for

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EBD_7762, 64, , MATHEMATICS, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, , (b), (b), (d), (b), (c), (a), (a), (c), (b), (b), (c), (a), (c), (a), (b), (a), , 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, , (d), (d), (c), (a), (b), (b), (d), (d), (b), (c), (c), (d), (b), (c), (d), (c), , 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, , (d), (b), (b), (d), (a), (b), (a), (b), (d), (b), (c), (c), (d), (a), (a), (c), , 1, 2, 3, 4, , (d), (c), (b), (a), , 5, 6, 7, 8, , (b), (a), (a), (b), , 9, 10, 11, 12, , (d), (d), (b), (b), , 1, 2, , (a), (b), , 3, 4, , (d), (b), , 5, 6, , (c), (d), , ANSWER KEYS, Exercise-1 : Topic-wise MCQs, (b) 97 (a), 49 (a) 65 (b), 81, (d) 98 (c), 50 (c) 66 (d), 82, (c) 99 (b), 51 (d) 67 (d), 83, (c) 100 (b), 52 (d) 68 (d), 84, (a) 101 (d), 53 (d) 69 (a), 85, (a) 102 (b), 54 (b) 70 (d), 86, (c) 103 (d), 55 (a) 71 (a), 87, (b) 104 (c), 56 (d) 72 (c), 88, (a) 105 (a), 57 (d) 73 (a), 89, (c) 106 (d), 58 (a) 74 (a), 90, (a) 107 (c), 59 (a) 75 (b), 91, (b) 108 (d), 60 (c) 76 (a), 92, (c) 109 (c), 61 (c) 77 (b), 93, (d) 110 (b), 62 (a) 78 (c), 94, (d) 111 (b), 63 (c) 79 (a), 95, (b) 112 (a), 64 (c) 80 (c), 96, Exercise-2 : Exemplar & Past Year MCQs, (b) 21, (d) 25 (a), 13 (c) 17, (a) 26 (a), 14 (c) 18 (c), 22, (a) 27 (c), 15 (c) 19 (d), 23, (c) 28 (b), 16 (a) 20 (a), 24, Exercise-3 : Try If You Can, (b), (d), (b) 13 (d), 7, 9, 11, (a) 10 (c), (c) 14 (a), 8, 12, , 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, , (a), (c), (a), (b), (b), (d), (a), (c), (a), (b), (c), (b), (c), (b), (b), (c), , 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, , (c), (a), (a), (a), (b), (d), (b), (d), (c), (c), (c), (a), (b), (c), (c), (a), , 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, , (a), (d), (a), (b), (c), (b), (d), (c), (b), (a), (d), (a), (a), (b), (c), (c), , 29, 30, 31, 32, , (a), (d), (a), (a), , 33, 34, 35, 36, , (a), (c), (b), (d), , 37, 38, , (a), (c), , 15, , (d)

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6, , LINEAR INEQUALITIES, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2002-2018), , Number of Questions, , 2, , JEE MAIN, BITSAT, , 1, , 0, 2002, , 2006, , 2007, , 2008, , 2009, , 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 0, 0, , Critical Concepts, , Graphical solution of linear, inequalities in two variables, Solution, of system of linear inequalities in two, Variables, Equations & Inequations, involving Modulus of a Function,, Logarithm of a Function, Greatest, Integer Function & Fractional part, Function., , Rating of, Difficulty Level, , 2.5/5, , CUS, (chapter utility score), Out of 10, 4.5

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LINEAR INEQUALITIES, , 67

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EBD_7762, 68, , MATHEMATICS, , Exercise 1 : Topic-wise MCQs, Topic 1: Algebraic Solutions of linear Inequalities in One, Variable and their Graphical Representation, Graphical, Solution of Linear Inequalities in Two Variables, Solution, of System of Linear Inequalities in Two Variables., 1., , 2., , 3., , 4., , 5., , C F - 32, Let, . If C lies between 10 and 20, then :, =, 5, 9, , (a) 50 < F < 78 (b), 50 < F < 68, (c) 49 < F < 68 (d), 49 < F < 78, The solution set of the inequality 4x + 3 < 6x + 7 is, (a) [–2, ¥), (b) (–¥, –2), (c) (–2, ¥), (d) None of these, Which of the following is the solution set of, 3x – 7 > 5x – 1 " x Î R?, (a) (–¥, –3), (b) (–¥, –3], (c) (–3, ¥), (d) (–3, 3), The solution set of the inequality, 37 – (3x + 5) ³ 9x – 8 (x – 3) is, (a) (– ¥, 2), (b) (– ¥, –2), (c) (– ¥, 2], (d) (– ¥, – 2], The graph of the solution on number line of the inequality, 3x – 2 < 2x + 1 is, (a), (b), (c), (d), , 6., , 7., , 1 2 3, 1 2 3 4 5, 1 2 3, , 1 2 3 4 5, Consider the following statements about Linear Inequalities :, I. Two real numbers or two algebraic expressions related, by the symbols <, >, £ or ³ form an inequality., II. When equal numbers added to (or subtracted from), both sides of an inequality then the inequality does, not changed., III. When both sides of an inequality multiplied (or, divided) by the same positive number then the, inequality does not changed., Which of the above statements are true ?, (a) Only I, (b) Only II, (c) Only III, (d) All of the above, The solution set of the inequality 4x + 3 < 6x + 7 is, (–a, ¥). The value of ‘a’ is, , (a) 1, (c) 2, , (b) 4, (d) None of these, 5 - 2x x, £ - 5 is, 3, 6, , 8., , The set of real x satisfying the inequality, , 9., , [a, ¥). The value of ‘a’ is, (a) 2, (b) 4, (c) 6, (d) 8, The length of a rectangle is three times the breadth. If the, minimum perimeter of the rectangle is 160 cm, then what, can you say about breadth?, (a) breadth = 20, , (b) breadth £ 20, , (c) breadth ³ 20, (d) breadth ¹ 20, 10. The marks obtained by a student of class XI in first and, second terminal examinations are 62 and 48, respectively., The minimum marks he should get in the annual, examination to have an average of at least 60 marks, are, (a) 70, (b) 50, (c) 74, (d) 48, 11. Ravi obtained 70 and 75 marks in first two unit tests., Then, the minimum marks he should get in the third test, to have an average of at least 60 marks, are, (a) 45, (b) 35, (c) 25, (d) None of these, 12. Assertion : The inequality ax + by < 0 is strict inequality., Reason : The inequality ax + b ³ 0 is slack inequality., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 13. The solution set of the inequalities 6 £ –3(2x – 4) < 12 is, (a) (–¥, 1], (b) (0, 1], (c) (0, 1] È [1, ¥), (d) [1, ¥), 14. Which of the following is the solution set of linear, inequalities 2(x – 1) < x + 5 and 3(x + 2) > 2 – x?, a b, =, c c, a b, <, (c), c c, 15. If a < b and c < 0, then, , (a), , (b), , a b, >, c c, , (d) None of these, , (a), , a b, =, c c, , (b), , a b, >, c c, , (c), , a b, <, c c, , (d) None of these

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LINEAR INEQUALITIES, , 69, , 16. The graph of the inequality 40x + 20y £ 120, x ³ 0,, y ³ 0 is, , II, , (0, 6), , (a), , X¢, , (3, 0), , O, , I, , (0, 6), , X, , (b), , X¢, , Y¢, , Y, , Y, , X¢, , O, , (3, 0), , (d), , (3, 0), , X¢, , X, , O, , X, , I, , I, , O, , (b), , X, (2, 0), , X¢, , x=2, , (c), , X¢, , O, , Y¢, , Y¢, , Y, , Y, , O, , Y¢, , (d), , X, (2, 0), , X¢, , O, , x=2, Y¢, , X, (2, 0), x=2, , X, (2, 0), x=2, , 18. The solutions of the system of inequalities 3x – 7 < 5 + x, and 11 – 5x £ 1 on the number line is, (a), (b), , 3, , is (–¥, 2)., , Statement-II : The solution set of the inequality, , Choose the correct option., (a) Statement I is true, (b) Statement II is true, (c) Both are true, (d) Both are false, 21. The solution set of the inequality 3(2 – x) ³ 2(1 – x) is, (–¥, a]. The value of ‘a’ is, (a) 2, (b) 3, (c) 4, (d) 5, 22. The solution set of, , Y, , Y, , X¢, , 5 (2 – x ), , Y¢, , Y¢, , 17. The graphical solution of 3x – 6 ³ 0 is, , (a), , £, , 1 æ 3x, ö 1, ç + 4 ÷ø ³ (x – 6) is (–¥, 120]., 2è 5, 3, , II, , (0, 6), X, , 5, , II, , O I (3, 0), , Y¢, , (0, 6) II, , (c), , 3 (x – 2), , Y, , Y, , 20. Consider the following statements:, Statement-I : The solution set of the inequality, , 2, , 6, , 2, , 6, , (c), , 2, 6, (d) None of the above, 19. Consider the following statements:, Statement-I : The solution set of 7x + 3 < 5x + 9 is, (–¥, 3)., Statement-II : The graph of the solution of above, inequality is represented by, –2 –1 0 1 2 3 4 5, Choose the correct option., (a) Statement I is true, (b) Statement II is true, (c) Both are true, (d) Both are false, , 2x - 1, , æ 3x - 2 ö æ 2 - x ö, ÷-ç, ÷ is, è 4 ø è 5 ø, , ³ç, , 3, (–¥, a]. The value of ‘a’ is, (a) 2, (b) 3, (c) 4, (d) 5, 23. The pairs of consecutive even positive integers, both of, which are larger than 5 such that their sum is less than 23,, are, (a) (4, 6), (6, 8), (8, 10), (10, 12), (b) (6, 8), (8, 10), (10, 12), (c) (6, 8), (8, 10), (10, 12), (12, 14), (d) (8, 10), (10, 12), 24. A man wants to cut three lengths from a single piece of, board of length 91 cm. The second length is to be 3 cm, longer than the shortest and the third length is to be twice, as long as the shortest. The possible length of the shortest, board, if the third piece is to be at least 5 cm longer than, the second, is, (a) less than 8 cm, (b) greater than or equal to 8 cm but less than or equal, to 22 cm, (c) less than 22 cm, (d) greater than 22 cm, 25. Assertion : If 3x + 8 > 2, then x Î {–1, 0, 1, 2, …}, when, x is an integer., Reason : The solution set of the inequality 4x + 3 < 5x + 7, " x Î R is [4, ¥)., , (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct.

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EBD_7762, 70, , 26., , 27., , MATHEMATICS, , The solution set of the inequalities 3x – 7 > 2(x – 6) and, 6 – x > 11 – 2x, is, (a) (–5, ¥) (b) [5, ¥), (c) (5, ¥), (d) [–5, ¥), If, , If, , (c) [4, ¥), , 30., , (d) [8, ¥), , 3x – 4 x + 1, ³, – 1, then x Î, 2, 4, , (a) [1, ¥) (b) (1, ¥), 29., , 8–, 7–, 6–, 5–, 4–, 3–, 2–, 1–, , 5 – 2x x, £ – 5, then x Î, 3, 6, , (a) [2, ¥) (b) [–8, 8], 28., , Y, , If –5 £, , (c) (–5, 5), , X¢, , (d) [–5, 5], , (a), , é 11 ù, êë – 3 , 5úû, , (b) [–5, 5], , (c), , é 11 ö, êë – 3 , ¥ ÷ø, , (d) (–¥, ¥), , Which of the following is/are true?, I. The graphical solution of the system of inequalities, 3x + 2y £ 12, x ³ 1, y ³ 2 is, Y, , y=2, X, , O, , II., , The region represented by the solution set of the, inequalities 2x + y ³ 6, 3x + 4y £ 12 is bounded., III. The solution set of the inequalities x + y ³ 4,, 2x – y > 0 is, Y, , O, 2x – y = 0, , 31., , X, , 32., , 33., , 34., , 35., , (a) I, III and V, (b) I, IV and V, (c) I, III and IV (d), I, II, and IV, If 5x + 1 > –24 and 5x –1 < 24, then x Î (–a, a). The value of, ‘a’ is, (a) 2, (b) 3, (c) 4, (d) 5, If x satisfies the inequations 2x – 7 < 11 and 3x + 4 < –5, then, x lies in the interval (–¥, –m). The value of ‘m’ is, (a) 2, (b) 3, (c) 4, (d) 5, The length of a rectangle is three times the breadth. If the, minimum perimeter of the rectangle is 160 cm, then, (a) breadth > 20 cm, (b) length < 20 cm, (c) breadth ³ 20 cm, (d) length £ 20 cm, IQ of a person is given by the formula, IQ =, , 3x + 2y = 12, Y¢ x = 1, , X¢, , 1 2 3 4 5 6 7 8 9, , Y¢, , 5 – 3x, £ 8, then x Î, 2, , X¢, , O, , X, x+y=4, , Y¢, , (a) Only I is true, (b) I and II are true, (c) I and III are true, (d) Only III is true, Which of the following linear inequalities satisfy the shaded, region of the given figure., I. x + 2y £ 8, II. x ³ 0, y ³ 0, III. x £ 0, y £ 0, IV. 2x + y £ 8, V. 4x + 5y £ 40, , MA, ´ 100, CA, , where, MA is mental age and CA is chronological age., If 80 £ IQ £ 140 for a group of 12 years children, then, the range of their mental age is, (a) 9.8 £ MA £ 16.8, (b) 10 £ MA £ 16, (c) 9.6 £ MA £ 16.8, (d) 9.6 £ MA £ 16.6, 36. Solutions of the inequalities comprising a system in variable, x are represented on number lines as given below, then, , (a), (b), (c), (d), , x, x, x, x, , Î, Î, Î, Î, , –4, 3, (–¥, –4] È [3, ¥), [–3, 1], (–¥, –4) È [3, ¥), [–4, 3], , 37. The inequality, (a), , –3, , 1, , 2, < 3 is true, when x belongs to, x, , é2 ö, êë 3 , ¥ ÷ø, , æ2 ö, (c) (–¥, 0) È çè , ¥ ÷ø, 3, , (b), , 2ù, æ, çè – ¥, ú, 3û, , (d) None of these

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LINEAR INEQUALITIES, , 71, , 38. Consider the following statements., , II., , Solution set of the inequality 7 £, , III. Solution set of the inequality – 5 £, , 10, , x–, , y, , Y¢, , Y, , (c), , X¢, y, , X, 2y, , =, , 15, 0, , Y, , O, , x, , +, , X (d) X¢, 2y, =, , y, , O, , x, , X, +, , =, , 2y, =, , 10, , 1, , Y¢, , y, , Y¢, , x–, , 10, , =0, , +, , x–, , =0, , +, , 1, , Y¢, , =, , 3x, , O, , x, , Y¢, , X, , x–y=0, , =0, , =, , O, , 2y, , y, , X¢, , X (b) X¢, , +, , +, , (b), , x, , x, , Y¢, , X, , O, , 1, , O, , X¢, , =, , X¢, , (a), , y, , (a), , Y, , Y, , Y, , +, , Y, , 44. The graphical solution of the inequalities x + 2y £ 10,, x + y ³ 1, x – y £ 0, x ³ 0, y ³ 0 is, , x, , 43., , [–1, 1] È [3, 5], Choose the correct option, (a) Only I and II are true. (b) Only II and III are true., (c) Only I and III are true. (d) All are true., I. When x is an integer, the solution set of 3x + 8 > 2 is, {–1, 0, 1, 2, 3, …}., II. When x is a real number, the solution set of 3x + 8 >, 2 is {–1, 0, 1}., Choose the correct option., (a) Only I is incorrect., (b) Only II is incorrect., (c) Both I and II are incorrect., (d) Both I and II are correct., The longest side of a triangle is 3 times the shortest side, and the third side is 2 cm shorter than the longest side. If, the perimeter of the triangle is at least 61 cm, find the, minimum length of the shortest side., (a) 2, (b) 9, (c) 8, (d) 7, The solution of the inequality –8 £ 5x – 3 < 7 is [–a, b)., Sum of ‘a’ and ‘b’ is, (a) 1, (b) 2, (c) 3, (d) 4, The number of pairs of consecutive odd natural numbers, both of which are larger than 10, such that their sum is, less than 40, is, (a) 4, (b) 6, (c) 3, (d) 8, A furniture dealer deals in only two items — tables and, chairs. He has ` 15,000 to invest and a space to store atmost, 60 pieces. A table costs him ` 750 and chair ` 150. Suppose, he makes x tables and y chairs, The graphical solution of the inequations representing the, given data is, , Y¢, , 1, , 42., , 2 - 3x, £ 9 is, 4, , (d) None of these, , X, , O, , y=, , 41., , X¢, , x+, , 40., , (c), , 3x + 11, £ 11 is, 2, , é 11 ù, ê1, 3 ú, ë, û, , 39., , Y, , 45. Linear inequalities for which the shaded region for the, given figure is the solution set, are, Y, (0, 8), y=5, , x=5, , I., , 3(x - 2), £ 0 is, Solution set of the inequality –15 <, 5, (–23, 2], , (0, 4), X¢, , O (4, 0), Y¢, , x+, , y=, , 8, , (8, 0), x+, y, , X, =4, , (a) x + y £ 8, x + y £ 4, x £ 5, y £ 5, x ³ 0, y ³ 0, (b) x + y £ 8, x + y ³ 4, x £ 5, y £ 5, x ³ 0, y ³ 0, (c) x + y ³ 8, x + y ³ 4, x ³ 5, y ³ 5, x ³ 0, y ³ 0, (d) None of the above, 46. A solution of 8% boric is to be diluted by adding a 2%, boric acid solution to it. The resulting mixture is to be, more than 4% but less than 6% boric acid. If we have 640, L of the 8% solution, of the 2% solution will have to be, added is, (a) more than 320 and less than 1000, (b) more than 160 and less than 320, (c) more than 320 and less than 1280, (d) more than 320 and less than 640

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EBD_7762, 72, , 47., , 48., , 49., , MATHEMATICS, , A company manufactures cassettes. Its cost and revenue, functions are C(x) = 26000 + 30x and R(x) = 43x,, respectively, where x is the number of cassettes produced, and sold in a week., The number of cassettes must be sold by the company to, realise some profit, is, (a) more than 2000, (b) less than 2000, (c) more than 1000, (d) less than 1000, A manufacturer has 600 litres of a 12% solution of acid., How many litres of a 30% acid solution must be added to it, so that acid content in the resulting mixture will be more, than 15% but less than 18%?, (a) more than 120 litres but less than 300 litres, (b) more than 140 litres but less than 600 litres, (c) more than 100 litres but less than 280 litres, (d) more than 160 litres but less than 500 litres, Which of the following linear inequalities satisfy the shaded, region of the given figure?, æ 9ö, (a) 2x + 3y ³ 3, ç 0, ÷, è, , 2ø, , (b) 3x + 4y £ 18, (c) x – 6y £ 3, (d) All of these, , O, , 51., , If |x + 3| ³ 10, then, (a), , x Î( -13, 7], , (c), , x Î (- ¥,13] È [ -7, ¥) (d), , (b), , x Î ( -13, 7), x Î ( - ¥, -13] È [7, ¥), , The inequality representing the following graph is, Y, , X¢, , –3, , |x – 2 | + |x + 2| < 4 is, (a) 1, , (b) 2, , (c) 4, , (d) 0, , 55. Solution of | x – 1 | ³ | x – 3 | is, (a) x £ 2, , (b) x ³ 2, , (c) [1, 3], , (d) None of these, , 56. The least positive integer x satisfying the inequality, | x + 1 | + | x - 4 | > 7 is, (a) 6, , (b) 5, , (c) –2, , (d) –1, , 57. The set of real values of x satisfying |x – 1| £ 3 and |x – 1| ³ 1, is, (a) [2, 4], (c), , [-2, 0] È [2, 4], , (b), , (-¥, 2] È [4, + ¥), , (d) None of these, , 58. The set of real values of x satisfying | x - 1 |£ 3 and, | x - 1 |³ 1 is, (-¥, 2] È [4, + ¥), , (a) [2, 4], , (b), , (c) [-2, 0] È [2, 4], , (d) None of these, , 59. The set of values of x satisfying 2 £ | x – 3 | < 4 is, , Topic 2: Linear Inequalities Involving Modulus Functions., 50., , 54. The number of solutions of the inequation, , (a) (–1, 1] È [5, 7), , (b) –4 £ x £ 2, , (c) –1 < x < 7 or x ³ 5 (d) x < 7 or x ³ 5, 60. Solution set of the inequality | 5 – 2x| < 1 is, (a), , (-¥, 2), , (b) (2, 3), , (c), , (3, ¥), , (d) none of these, , –4 ö æ 14 ö, æ, 61. Assertion : | 3x – 5 | > 9 Þ x Î ç – ¥, ÷ È ç , ¥ ÷ ., è, ø, 3 ø è3, , O, , 3, , Reason : The region containing all the solutions of an, inequality is called the solution region., , X, , (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., Y¢, (a) | x | < 3 (b) | x | £ 3, 52., , 53., , (c) | x | > 3, , (d) | x | ³ 3, , The solution set of the inequality | x - 2 | £ | x + 4 | is, (a), , [1, ¥), , (b), , (-¥, - 1), , (c), , [ -1, ¥), , (d) (–1, 1), , Solution of | 3x + 2 | < 1 is, (a), (c), , ì 1, ü, í – , –1ý, î 3, þ, , 1ù, é, êë –1, – 3 úû, , (b), , 1ö, æ, çè –1, – ÷ø, 3, , (d) None of these, , (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 62. The integral value(s) of x satisfying the inequality, 2, 8, > is /are, x - 13 9, , (a) 0, , (b) 1, , (c) 0 or 1, , (d) None of these

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LINEAR INEQUALITIES, , 63. If, , x+3 +x, x+2, , 73, , 72. The equation | x + 1 || x - 1 |= a 2 - 2a - 3 can have all real, , > 1, then x Î, , (a) (–5, –2), (c) (–5, –2) È (–1, ¥), , solutions for x if a belongs to, , (b) (–1, ¥), (d) None of these, , 64. The root(s) of the equation | x + 2 |= 2(3 - x) is /are, , [1 -, , ], , (a), , (-¥,-1] È [3, ¥), , (c), , [1 - 5,-1] È [3, 1 + 5 ] (d) None of these, , (b), , 5 ,1 + 5, , 1/ x, , (a), , 4, 3, , (c), , - 1,, , 73. The solution set of the inequality 5 x + 2 > æç 1 ö÷, è 25 ø, , (b) –1, 4, 3, , (d), , 2,, , 4, 3, , 65. The set of real values of x satisfying | x - 1 | -1 | £ 1 is, (a), , [-1, 3], , (b) [0, 2], , (a) (–3, 5) (b) (5, 9), , 3, 67. Solution of 1 +, x, , 2ö, æ, (d) ç –8, ÷, è, 3ø, , > 2 is, , (a) (0, 3], (c) (–1, 0) È (0, 3), , (b) [–1, 0), (d) None of these, , 68. The solution set of equation | x | - | x - 2 |= 2 is, (a) {2}, , (b) {2, 4, 6}, , (c), , (d) [2, ¥), , (-¥, 2], , 69. Solution of | 2x – 3 | < | x + 2 | is, (a), , 1ö, æ, çè – ¥, ÷ø, 3, , (c) (5, ¥), , 1, 70. Solution of x +, x, (a) R – {0}, (c) R – {1}, , (b), , æ1 ö, çè , 5 ÷ø, 3, , (d), , 1ö, æ, çè – ¥, ÷ø È (5, ¥), 3, , > 2 is, , (b), , (-2, 2), , (c), , (-5, 5), , (d), , (0, ¥), , 74. The number of real roots of the equation, , (a) 1, , (b) R – {–1, 0, 1}, (d) R – {–1, 1}, , Topic 3: Miscellaneous Equalities and Inequalities., x 2 + 6x - 7, Set of values of x satisfying the inequality, <0, | x+4|, , (a), , (-¥, - 7), , (b), , ( -7, 4), , (c), , (-4, 1), , (d), , (1, ¥), , (c) 5, , (d) 6, , 2, , 2x + 2 × 56 – x = 10 x ?, (a) log10 4 – 3, , (b) 2, , (c) – log10 (250) (d), , None of these, , 76. If [ x ]2 = [ x + 2] , where [x ] = the greatest integer less than, or equal to x, then x must be such that, (a), , x = 2, - 1, , (b), , (c), , x Î [-1, 0) È [2, 3), , (d) None of these, , x Î {2, 3}, , 77. The solution set of the equation 4{x} = x + [x ], where {x}, and [x] denote the fractional and integral parts of a real, number ‘x’ respectively, is, ì 5ü, (a) { 0}, (b) í0, ý, î 3þ, (c) [0, ¥), (d) None of these, 78. The minimum value of the expression 2 log10x – logx 0.01;, where x > 1 is, (b) 0.1, , (c) 4, , (d) 1, , 79. The solution of x - 1 = (x - [x ])(x - {x}) (where [ x ] and, {x} are the integral and fractional part of x is, (a), , xÎR, , (b), , x Î R ~ [1, 2), , (c), , x Î [1, 2), , (d), , x Î R ~ [1, 2], , 80. The values of x satisfying the inequality | x 3 - 1 | ³ 1 - x, belong to, (a), , is/are, , (b) 3, , 75. Which of the following is not the solution of equation, , (a) 2, , BEYOND NCERT, , 71., , (-2, 0), , | 2- | 1- | x |||= 1 is, , (c) [–1, 1], (d) None of these, 66. If | 2x – 3 | < | x + 5|, then x belongs to, æ 2 ö, (c) ç – , 8 ÷, è 3 ø, , (a), , is, , (-¥, - 1], , (c) ( –¥, 0), , (b) [–1, 0], (d) None of these

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EBD_7762, 74, , MATHEMATICS, , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, 1., , 2., , If x < 5, then, (a) – x, < –5, (b) – x, < –5, (c) – x < –5, (d) – x < –5, Given that x, y and b are real numbers and x < y , b < 0,, then, x y, x y, <, £, (a), (b), b b, b b, x y, >, b b, If –3x + 17 < –13, then, (a) x Î (10, ¥), (c) x Î (–¥, 10], If x is real number and |x| <, (a) x ³ 3, , (c), , 3., , 4., , 5., , 6., , 7., , 8., , (d), , 9., , Solution of a linear inequality in variable x is represented, on number line is, 5, , (a), , x Î ( -¥,5), , (b), , x Î (-¥,5], , (c), , x Î[5, ¥), , (d), , x Î (5, ¥), , 10. Solution of linear inequality in variable x is represented, on number line is, , x y, ³, b b, , 9, 2, , (b) x Î [10, ¥), (d) x Î [–10, 10), 3, then, (b) –3 < x < 3, , (c) x £ -3, (d) -3 £ x £ 3, x and b are real numbers. If b > 0 and | x | > b, then, (a) x Î (–b, ¥), (b) x Î (–¥, b), (c) x Î (–b, b), (d) x Î (–¥, –b) È (b, ¥), If |x – 1 > 5, then, (a) x Î (–4, 6), (b) x Î [–4, 6], (c) x Î (–¥, –4) È (6, ¥), (d) x Î (–¥, –4) È (6, ¥), If | x + 2 | £ 9, then, (a) x Î (–7, 11), (b) x Î [–11, 7], (c) x Î (–¥, –7) È (11, ¥), (d) x Î (–¥, –7) È [11, ¥), The inequality representing the following graphs is, Y, , (a), , æ9 ö, x Î ç , ¥÷, è2 ø, , (b), , é9 ö, x Îê ,¥÷, ë2 ø, , (c), , 9, x Î (-¥, ), 2, , (d), , 9ù, æ, x Î ç -¥, ú, 2û, è, , 11., , 7, 2, , (a), , 7ö, æ, x Î ç -¥, ÷, 2ø, è, , (b), , æ, 7ù, x Î ç -¥, ú, 2û, è, , (c), , é7, x Î ê , -¥ ), ë2, , (d), , æ7, ö, x Î ç , -¥ ÷, ø, è2, , 12., –2, , (a), , x Î (-¥, -2), , (b), , x Î (-¥, -2], , (c), , x Î (-2, ¥], , (d), , x Î[-2, ¥), , Past Year MCQs, –5, , O, , 5, , X, , (a) | x |< 5, , (b) | x | £ 5, , | x |> 5, , (d) | x | ³ 5, , (c), , 13. If a, b, c are distinct +ve real numbers and a2 + b2 + c2 = 1, then ab + bc + ca is, [JEE MAIN 2002, C], (a) less than 1, (b) equal to 1, (c) greater than 1, (d) any real no., 14. If x is real, the maximum value of, , (a), , 1, 4, , (b) 41, , (c) 1, , 3x 2 + 9 x + 17, , is, 3x 2 + 9 x + 7, [JEE MAIN 2006, A], (d), , 17, 7

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LINEAR INEQUALITIES, , 75, , 15. Statement-1 : For every natural number n³ 2,, 1, 1, 1, +, + ......... +, > n., 1, 2, n, Statement-2 :For every natural number n ³ 2,, , n(n + 1) < n + 1., , [JEE MAIN 2008, C], , (a) Statement -1 is false, Statement-2 is true, (b) Statement -1 is true, Statement-2 is true; Statement 2 is a correct explanation for Statement-1, , (c) Statement -1 is true, Statement-2 is true; Statement -2, is not a correct explanation for Statement-1, (d) Statement -1 is true, Statement-2 is false, 16. The necessary condition for third quadrant region in, xy-plane, is, [BITSAT 2014, C], (a) x > 0, y < 0, (b) x < 0, y < 0, (c) x < 0, y > 0, (d) x < 0, y = 0, , Exercise 3 : Try If You Can, 1., , 2., , 3., , The set of all real numbers x for which x2 – [x + 2] + x > 0, is, (a), , (- ¥,-2) È (2, ¥ ), , (b), , (c), , (- ¥,-1) È (1, ¥ ), , (d), , (- ¥,- 2 )È (, ( 2 , ¥), , 2,¥, , Then the solution set of the equation x, , ), , (a) {–1, 1} (b) [2, ¥), 8., , 5., , The solution set of the inequality, , (-¥, 1), , (b), , (c) [10-9/2 ,100], , (c), , (-¥, 1], , (d) None of these, , | 9 x - 3 x +1 - 15 |< 2.9 x - 3x is, , (d) (–1, ¥), , Number of real roots of the equation, is, (a) 0, (b) 1, (c) 2, , x + x - 1- x = 1, , 9., , (d) 3, , æ 1 ö, The solution set of the inequality 5 x + 2 > ç ÷, è 25 ø, , (a), , (-2, 0), , (b), , (c), , (-5, 5), , (d) (0, ¥), , is, , æ xö (x, log 2 ç ÷, è 2ø, , 2, , Solution of 2 x + 2| x| ³ 2 2 is, (a), , (-¥, log 2 ( 2 + 1)) (b) (0, ¥), , (c), , æ1, ö, é1, ö, ç , log 2 ( 2 - 1) ÷ (d) (-¥, log 2 ( 2 - 1)] È ê 2 , ¥ ÷, 2, ë, ø, è, ø, , (c) 8, , If R ³ r > 0 and d > 0, then 0 <, , (b) (p – a) (p – b) (p – c) ³ 8abc, (c), , (d) 9, , d2 + R 2 - r 2, £1, 2dR, , (a) is satisfied if | d – R | £ r, , 8 3, p, 27, , (a) (p – a) (p – b) (p – c) £, , - 10 x + 22) > 0 is equal to, , (b) 6, , (1, ¥), , 10. For positive real numbers a, b, c such that a + b + c = p,, which one does not hold?, , (-2, 2), , The least positive integer x, which satisfies the inequality, , (a) 5, , 7., , (d) {0, 2}, , (a), , log, , 6., , (c) [0, 2), , + x = x + x 2 is, , If x Î R satisfies (log10 (100)x))2 +(log10 10 x)2, + log10 x £ 14 then x contains the interval., (a) (10, 100), (b) [10–9/2, 10], , 1/ x, , 4., , 2, , 11., , bc ca ab, +, +, £p, a, b, c, , (d) None of these, Solution set of the inequality, , (b) is satisfied if | d - R | £ 2r, , log3 (x + 2) (x + 4) + log1/3 (x + 2) <, , (c) is satisfied if | d – R | ³ r, (d) is not satisfied at all, , (a) (– 2, – 1), , 1, log 3 7 (1) is, 2, (b) (– 2, 3), , (c) (– 1, 3), , (d) (3, ¥), , For x Î R ,, , x is defined as follows :, , ì x + 1, 0 £ x < 2, x =í, x³2, î | x - 4 |,, , 12. The solution set of ( x – 2) x, (a) (2, ¥), (c) (4, 5) È (5, ¥), , 3, , – 6x + 8, , (b), (d), , > 1 is, (2, 3) È (4, ¥), (2, 3) È (4, 5)

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EBD_7762, 76, , MATHEMATICS, , The system of equation | x - 1 | +3y = 4, x - | y - 1 |= 2 has, , 13., , (a) No solution, (b) A unique solution, (c) Two solutions, (d) More than two solutions, If a, b, c are distinct positive real numbers then the expression (b + c – a) (c + a – b) (a + b – c) – abc is, (a) positive, (b) negative, (c) non-positive, (d) non-negative, Consider the following statements., , 14., , 15., , I., , Solution set of the inequality –15 <, (–23, 2], , 1, 2, 3, 4, 5, 6, 7, 8, , (b), (c), (a), (c), (a), (d), (c), (d), , 9, 10, 11, 12, 13, 14, 15, 16, , (c), (a), (b), (b), (b), (c), (b), (d), , 17, 18, 19, 20, 21, 22, 23, 24, , (a), (b), (a), (b), (c), (a), (b), (b), , 1, 2, , (c), (c), , 3, 4, , (a), (b), , 5, 6, , (d), (c), , 1, 2, , (b), (b), , 3, 4, , (b), (d), , 5, 6, , (c), (a), , 3(x - 2), £ 0 is, 5, , II., , Solution set of the inequality 7 £, , 3 x + 11, £ 11 is, 2, , é 11 ù, ê1, 3 ú, ë, û, III. Solution set of the inequality – 5 £, , 2 - 3x, £ 9 is, 4, , [–1, 1] È [3, 5], Choose the correct option, (a) Only I and II are true. (b) Only II and III are true., (c) Only I and III are true. (d) All are true., , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (c) 33, (b), (c) 49, 25, 41, (c) 34, (c), (a) 50, 26, 42, (c), (c) 51, 27 (d) 35, 43, (a) 36, (a), (d) 52, 28, 44, (a) 37, (c), (b) 53, 29, 45, (a) 38, (a), (c) 54, 30, 46, (b), (a) 55, 31 (d) 39, 47, (b), (a) 56, 32 (d) 40, 48, Exercise 2 : Exemplar & Past Year MCQs, (b), (d), (a) 13, 7, 9, 11, (a) 10, (b), (b) 14, 8, 12, Exercise 3 : Try If You Can, (d), (d), (b) 13, 7, 9, 11, (b) 10 (c), (b) 14, 8, 12, , (d), (d), (a), (c), (c), (d), (b), (a), , 57, 58, 59, 60, 61, 62, 63, 64, , (c), (c), (a), (b), (b), (d), (c), (a), , (a), (b), , 15, 16, , (b), (b), , (b), (b), , 15, , (a), , 65, 66, 67, 68, 69, 70, 71, 72, , (a), (c), (c), (d), (b), (b), (c), (c), , 73, 74, 75, 76, 77, 78, 79, 80, , (d), (c), (d), (c), (b), (c), (c), (a)

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7, , PERMUTATIONS AND, COMBINATIONS, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 4, , 3, JEE MAIN, BITSAT, , 2, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 3, 4, , Critical Concepts, , Fundamental Principle of Counting,, Factorials , Counting Formula for, Permutation , Circular Permutation,, Counting Formula for Combination,, Application of Combination,, Dearrangement Theorem., , Rating of, Difficulty Level, , CUS, (chapter utility score), Out of 10, , 3.5/5, , 6

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PERMUTATIONS AND COMBINATIONS, , 79

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EBD_7762, 80, , MATHEMATICS, , Topic 1 : Fundamental Principle of Counting, Factorials., 1., , 2., , 3., , 4., , In an examination, there are three multiple choice questions, and each question has 4 choices. Number of ways in which, a student can fail to get all answers correct is :, (a) 11, (b) 12, (c) 27, (d) 63, There are four chairs with two chairs in each row. In how, many ways can four persons be seated on the chairs, so, that no chair remains unoccupied ?, (a) 6, (b) 12, (c) 24, (d) 48, Total number of four digit odd numbers that can be formed, using 0, 1, 2, 3, 5, 7 (using repetition allowed) are, (a) 216, (b) 375, (c) 400, (d) 720, The value of 2n [1.3.5...(2n – 3) (2n – 1)] is, (2n)!, (2n)!, (b), n!, 2n, n!, (c), (d) None of these, (2n )!, In how many ways can this diagram be coloured subject, to the following two conditions?, (i) Each of the smaller triangle is to be painted with one, of three colours: red, blue or green., (ii) No two adjacent regions have the same colour., , (a), , 5., , 6., , 7., , 8., , 9., , (a) 20, (b) 24, (c) 28, (d) 30, There are four bus routes between A and B; and three bus, routes between B and C. A man can travel round-trip in, number of ways by bus from A to C via B. If he does not, want to use a bus route more than once, in how many, ways can he make round trip?, (a) 72, (b) 144, (c) 14, (d) 19, 2, 2, 2, 2, 2, 2, If P = n(n – 1 ) (n – 2 ) (n – 3 ) ...... (n 2 – r 2),, n > r, n Î N then P is necessarily divisible by, (a) (2r + 2) !, (b) (2r + 4) !, (c) (2r + 1) !, (d) None of these, The number of ways in which 3 prizes can be distributed, to 4 children, so that no child gets all the three prizes, are, (a) 64, (b) 62, (c) 60, (d) None of these, 4 buses runs between Bhopal and Gwalior. If a man goes, from Gwalior to Bhopal by a bus and comes back to, Gwalior by another bus, then the total possible ways are, (a) 12, (b) 16, (c) 4, (d) 8, , 10. Consider the following statements., I. The continued product of first n natural numbers is, called the permutation., II. L.C.M of 4!, 5! and 6! is 720., Choose the correct option., (a) Only I is true., (b) Only II is true., (c) Both are true., (d) Both are false., 11. An n - digit number is a positive number with exactly n, digits. Nine hundred distinct n-digit numbers are to be, formed using only the three digits 2, 5 and 7. The smallest, value of n for which this is possible is, (a) 6, (b) 7, (c) 8, (d) 9, 1 1, x, + = , then the value of x = 2m. The value of m is, 6! 7! 8!, (a) 2, (b) 4, (c) 6, (d) 5, 13. There are 5 roads leading to a town from a village., The number of different ways in which a villager can go, to the town and return back, is, (a) 25, (b) 20, (c) 10, (d) 5, 14. There are 5 historical momuments, 6 gardens and 7 shopping, malls in the city. In how many ways a tourist can visit the, city, if he visits at least one shopping mall?, (a) 25.26. (27 – 1), (b) 24.26 (27 – 1), (c) 25.26(26 –1), (d) None of these, 15. There are 10 true-false questions in a examination. Then, these questions can be answered in:, (a) 100 ways, (b) 20 ways, (c) 512 ways, (d) 1024 ways, , 12. If, , 16. Value of, , n!, when n = 6, r = 2 is 5 m. The value of m is, (n - r)!, , (a) 2, (b) 4, (c) 6, (d) 5, 17. Let A = {x | x is a prime number and x < 30}. The number of, different rational numbers whose numerator and, denominator belong to A is, (a) 90, (b) 180, (c) 91, (d) 100, (n + 2)!+ (n + 1)!(n - 1)!, equal to ?, (n + 1)!(n - 1)!, (a) 1, (b) Always an odd integer, (c) A perfect square, (d) None of these, 19. The number of combinations of 4 different objects A, B, C,, D taken 2 at a time is, (a) 4, (b) 6, (c) 8, (d) 7, 20. Number of 5 digit numbers that can be made using the digits, 1 and 2 and in which at least one digit is different., (a) 30, (b) 25, (c) 28, (d) 31, , 18. What is

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PERMUTATIONS AND COMBINATIONS, , 21. In an examination, there are three multiple choice questions, and each question has 4 choices. Number of ways in, which a student can fail to get all answers correct is, (a) 11, (b) 12, (c) 27, (d) 63, 22. Given 4 flags of different colours, how many different, signals can be generated, if a signal requires the use of, 2 flags one below the other?, (a) 12, (b) 13, (c) 14, (d) 15, 23. The number of 3 digit numbers having at least one of their, digit as 5 are, (a) 250, (b) 251, (c) 252, (d) 253, , 81, , different storeys. The number of ways they can do so if the, lift does not stop to the second storey is, (a) 78, (b) 112, (c) 720, (d) 132, 33. In how many ways 3 mathematics books, 4 history books,, 3 chemistry books and 2 biology books can be arranged on, a shelf so that all books of the same subjects are together?, (a) 41472 (b) 41470, (c) 41400, (d) 41274, 34. The number of ways of distributing 50 identical things, among 8 persons in such a way that three of them get, 8 things each, two of them get 7 things each, and, remaining 3 get 4 things each, is equal to, , Topic 2 : Permutation, Counting Formula for Permutation, Number of Permutations Under Certain Conditions., , (a), , 24. If 56Pr + 6 : 54Pr + 3 = 30800 : 1, then the value of r is, (a) 41, (b) 14, (c) 10, (d) 51, 25. How many arrangements can be made out of the letters of, the word “ MOTHER” taken four at a time so that each, arrangement contains the letter ´M´?, (a) 240, (b) 120, (c) 60, (d) 360, 26. How many numbers lying between 500 and 600 can be, formed with the help of the digits 1, 2, 3, 4, 5, 6 when the, digits are not be repeated?, (a) 20, (b) 40, (c) 60, (d) 80, 27. Total number of eight digit numbers in which all digits are, different is, , (b), , 8.7!, 5.8!, 8.9!, 9.9!, (b), (c), (d), 3, 3, 2, 2, Number of words from the letters of the words BHARAT in, which B and H will never come together is, (a) 210, (b) 240, (c) 422, (d) 400, Six identical coins are arranged in a row. The number of, ways in which the number of tails is equal to the number of, heads is, (a) 20, (b) 9, (c) 120, (d) 40, How many different nine digit numbers can be formed from, the number 223355888 by rearranging its digits so that the, odd digits occupy even positions ?, (a) 16, (b) 36, (c) 60, (d) 180, The number of 3 letters words, with or without meaning, which can be formed out of the letters of the word, ‘NUMBER’., Statement I : When repetition of letters is not allowed, is 120., Statement II : When repetition of letters is allowed is 216., Choose the correct option., (a) Only Statement I is correct, (b) Only Statement II is correct, (c) Both I and II are correct, (d) Both I and II are false, In a 12 - storey house ten people enter a lift cabin. It is, known that they will leave in groups of 2, 3 and 5 people at, , (a), 28., , 29., , 30., , 31., , 32., , (c), , (d), , (50!)(8!), (8!) (3!)2 (7!)2 (4!)3 ( 2!), 3, , (50!)(8!), (8!)3 (7!)3 (4!)3, (50!), 3, (8!) (7!)2 (4!)3, (8!), (3!)2 (2!), , 35. Let 1 £ m < n £ p. The number of subsets of the set, A = {1, 2, 3, ........p} having m, n as the least and the greatest, elements respectively, is, (a) 2n – m – 1 – 1, (b) 2n – m – 1, (c) 2n – m, , 36., , 37., , 38., , 39., , 40., , 2p -n + m -1, If the letters of the word RACHIT are arranged in all, possible ways as listed in dictionary. Then, what is the, rank of the word RACHIT?, (a) 479, (b) 480, (c) 481, (d) 482, The number of 4 letter words that can be formed from, letters of the word ‘PART’, when:, Statement I : Repetition is not allowed is 24., Statement II : Repetition is allowed is 256., Which of the above statement(s) is/are true?, (a) Only I, (b) Only II, (c) Both I and II, (d) Neither I nor II, If a man and his wife enter in a bus, in which five seats are, vacant, then the number of different ways in which they can, be seated is :, (a) 2, (b) 5, (c) 20, (d) 40, Six identical coins are arranged in a row. The number of, ways in which the number of tails is equal to the number of, heads is, (a) 20, (b) 9, (c) 120, (d) 40, The figures 4, 5, 6, 7, 8 are written in every possible order., The number of numbers greater than 56000 is, (a) 72, (b) 96, (c) 90, (d) 98, (d), , 41. If m1 and m2 satisfy the relation m+5 Pm+1 =, then m1 + m2 is equal to, (a) 10, (b) 9, (c) 13, , (, , ), , 11, m+3, Pm ,, (m -1), 2, , (d) 17

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EBD_7762, 82, , MATHEMATICS, , 42. The number of numbers that can be formed with the help, of the digits 1, 2, 3, 4, 3, 2, 1 so that odd digits always, occupy odd places, is, (a) 24, (b) 18, (c) 12, (d) 30, 43. In a circus, there are ten cages for accommodating ten, animals. Out of these, four cages are so small that five out, of 10 animals cannot enter into them. In how many ways will, it be possible to accommodate ten animals in these ten cages?, (a) 66400, (b) 86400, (c) 96400, (d) None of these, 44. How many different nine digit numbers can be formed from, the number 223355888 by rearranging its digits so that the, odd digits occupy even positions ?, (a) 16, (b) 36, (c) 60, (d) 180, 45. Consider the following statements., I. Three letters can be posted in five letter boxes in 35, ways., II. In the permutations of n things, r taken together, the, number of permutations in which m particular things, occur together is n–mPr–m ´ rPm ., Choose the correct option., (a) Only I is false., (b) Only II is false., (c) Both are false., (d) Both are true., 46. Assertion : A five digit number divisible by 3 is to be formed, using the digits 0, 1, 2, 3, 4 and 5 with repetition. The total, number formed are 216., , 47., , 48., , 49., 50., , 51., , Reason : If sum of digits of any number is divisible by 3, then the number must be divisible by 3., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., The numbers of ways in which the letters of the word, 'VOWEL' can be arranged so that the letters O, E occupy, only even places is, (a) 12, (b) 24, (c) 18, (d) 36, The total number of 3-digit numbers, the sum of whose digits, is even, is equal to, (a) 450, (b) 350, (c) 250, (d) 325, Find n if n – 1P3 : nP4= 1 : 9, (a) 2, (b) 6, (c) 8, (d) 9, If a denotes the number of permutations of x + 2 things, taken all at a time, b the number of permutations of x things, taken 11 at a time and c the number of permutations of x –, 11 things taken all at a time such that a = 182 bc, then the, value of x will be, (a) 12, (b) 10, (c) 8, (d) 6, The number of permutations of the letters of the word, HINDUSTAN such that neither the pattern 'HIN' nor 'DUS', nor 'TAN' appears, are, (a) 166674, (b) 169194, (c) 166680, (d) 181434, , 52. In a chess tournament where the participants were to play, one game with one another, two players fell ill having played, 6 games each, without playing among themselves. If the, total number of games is 117, then the number of participants, at the beginning was :, (a) 15, (b) 16, (c) 17, (d) 18, 53. In how many ways can 10 lion and 6 tigers be arranged in a, row so that no two tigers are together?, (a) 10! × 11P6, (b) 10! × 10P6, 10, (c) 6! × P7, (d) 6! × 10P6, 54. The number of numbers of 9 different non-zero digits such, that all the digits in the first four places are less than the, digit in the middle and all the digits in the last four places, are greater than the digit in the middle is, (a) 2 (4 !), (b) (4 !) 2, (c) 8 !, (d) None of these, 55. Four writers must write a book containing 17 chapters., The first and third writer must write 5 chapters each,, the second writer must write 4 chapters and fourth writer, must write three chapters. The number of ways that can, be found to divide the book between four writers, is, (a), (c), , 17!, , (5!), , 2, , 4! 3! 2!, , 17!, , (b), (d), , 17!, 5! 4!3! 2!, 17!, , (5!) ´ 4 ´ 3, (5!) 4!3!, 56. The number of 4-digit numbers that can be formed with, the digits 1, 2, 3, 4 and 5 in which at least 2 digits are, identical, is, (a) 505, (b) 45 – 5!, (c) 600, (d) None of these, 57. How many numbers lying between 999 and 10000 can be, formed with the help of the digits 0, 2, 3, 6, 7, 8, when the, digits are not repeated?, (a) 100, (b) 200, (c) 300, (d) 400, 2, , 2, , Topic 3 : Combination, Counting Formula for Combination,, Number of combinations Under Certain Conditions,, Distribution & Division of Objects, Rank of a Word., 58. If n + 2C8 : n – 2P4 = 57 : 16, then the value of n is:, (a) 20, (b) 19, (c) 18, (d) 17, 59. If 30Cr + 2 = 30Cr – 2, then r equals:, (a) 8, (b) 15, (c) 30, (d) 32, 60. In how many ways can a bowler take four wickets in a single, 6-ball over ?, (a) 6, (b) 15, (c) 20, (d) 30, n, n, n, 61. If C9 = C8, what is the value of C17 ?, (a) 1, (b) 0, (c) 3, (d) 17, 62. If 10Cx= 10Cx + 4, then the value of x is, (a) 5, (b) 4, (c) 3, (d) 2, n, n, n, 63. Cr + 2 Cr–1 + Cr–2 is equal to:, (a) n+2Cr, (b) nCr+1, n–1, (c), Cr+1, (d) None of these

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PERMUTATIONS AND COMBINATIONS, , 83, , 64. If nCr denotes the number of combination of n things taken, r at a time, then the expression, equals, (a), , n +1, , (c), , n+2, , (b), , Cr +1, Cr +1, , (d), , n +1, , n, , Cr +1 + nC r -1 + 2´n Cr, , n+ 2, , Cr, , Cr, , 65. If a secretary and a joint secretary are to be selected from a, committee of 11 members, then in how many ways can they, be selected ?, (a) 110, (b) 55, (c) 22, (d) 11, 66. A bag contains 3 black, 4 white and 2 red balls, all the balls, being different. Number of selections of atmost 6 balls, containing balls of all the colours is, (a) 1008 (b) 1080, (c) 1204, (d) 1130, 67. Number of different seven digit numbers that can be written, using only the three digits 1, 2 and 3 with the condition that, the digit 2 occurs twice in each number is, (a) 652, (b) 650, (c) 651, (d) 640, 68. Consider the following statements., I. Value of 15C8 + 15C9 – 15C6 – 15C7 is zero., II. The total number of 9 digit numbers which have all, different digits is 9!, Choose the correct option., (a) Only I is true, (b) Only II is true., (c) Both are true., (d) Both are false., 69. Total number of ways of selecting five letters from letters of, the word INDEPENDENT is, (a) 70, (b) 72, (c) 75, (d) 80, 70. Assertion : If the letters W, I, F, E are arranged in a row, in all possible ways and the words (with or without, meaning) so formed are written as in a dictionary, then the, word WIFE occurs in the 24th position., Reason : The number of ways of arranging four distinct, objects taken all at a time is C(4, 4)., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 71. Given 12 points in a plane, no three of which are collinear., Then number of line segments can be determined, are:, (a) 76, (b) 66, (c) 60, (d) 80, 72. The total number of ways of selecting six coins out of 20, one rupee coins, 10 fifty paise coins and 7 twenty five paise, coins is:, (a) 37C6, (b) 56, (c) 28, (d) 29, 73. Four couples (husband and wife) decide to form a committee, of four members, then the number of different committees, that can be formed in which no couple finds a place is, (a) 15, (b) 16, (c) 20, (d) 21, , 74. Number of different ways in which 8 persons can stand in a, row so that between two particular person A and B there are, always two person is, (a) 11, (b) 13, (c) 15, (d) 16, 75. If the letters of the word SACHIN are arranged in all possible, ways and these words are written out as in dictionary, then, the word SACHIN appears at serial number, (a) 601, (b) 600, (c) 603, (d) 602, 76. How many different words can be formed by jumbling the, letters in the word MISSISSIPPI in which no two S are, adjacent?, (a) 8. 6C4. 7C4, (b) 6.7. 8C4, 7, (c) 6. 8. C4., (d) 7. 6C4. 8C4, 77. Consider the following statements., I. If there are n different objects, then nCr = nCn–r,, 0 £ r £ n., II. If there are n different objects, then nCr + nCr–1, = n+1Cr, 0 £ r £ n, Choose the correct option., (a) Both are false., (b) Both are true., (c) Only I is true., (d) Only II is true., 78. Determine n if 2nC3 : nC2 = 12 : 1, (a) 5, (b) 3, (c) 4, (d) 1, 79. The number of values of r satisfying the equation, 39, , C3r -1 - 39 C, , r2, , =, , 39, , C, , r 2 -1, , - 39 C3r is, , (a) 1, (b) 2, (c) 3, (d) 4, 80. In how many ways can the letters of the word, CORPORATION be arranged so that vowels always occupy, even places ?, (a) 120, (b) 2700, (c) 720, (d) 7200, 81. The number of ways in which four letters of the word, MATHEMATICS can be arranged is given by, (a) 136, (b) 192, (c) 1680, (d) 2454, 82. The number of ways a student can choose a programme, out of 5 courses, if 9 courses are available and 2 specific, courses are compulsory for every student is, (a) 35, (b) 40, (c) 24, (d) 120, 83. Consider the following statements., I. If nPr = nPr+1 and nCr = nCr–1, then the values of, n and r are 3 and 2 respectively., II. From a class of 32 students, 4 are to be chosen for, a competition. This can be done in 32C2 ways., Choose the correct option., (a) Only I is true., (b) Only II is true., (c) Both are false., (d) Both are true., 84. If nC9 = nC8, what is the value of nC17 ?, (a) 1, (b) 0, (c) 3, (d) 17, 85. If 10Cx= 10Cx + 4, then the value of x is, (a) 5, (b) 4, (c) 3, (d) 2, 86. Number of 6 digit numbers that can be made with the digits, 1, 2, 3 and 4 and having exactly two pairs of digits is, (a) 978, (b) 1801, (c) 1080, (d) 789

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EBD_7762, 84, , MATHEMATICS, , 87. Five balls of different colours are to be placed in three boxes, of different sizes. Each box can hold all five balls, Number or, ways in which we can place the balls in the boxes (order is, not considered in the box) so that no box remains empty is, (a) 150, (b), 160 (c), 12 (d) 19, 88. A father with 8 children takes them 3 at a time to the, zoological garden, as often as he can without taking the, same 3 children together more than once. The number of, times he will go to the garden is, (a) 56, (b) 100, (c) 112, (d) None of these, 89. On the occasion of Deepawali festival, each student of a, class sends greeting cards to the others. If there are, 20 students in the class, then the total number of greeting, cards exchanged by the students is, (a) 20C2, (b) 2 . 20C2, ., 20, (c) 2 P2, (d) None of these, 90. To fill 12 vacancies, there are 25 candidates of which five are, from scheduled caste. If 3 of the vacancies are reserved for, scheduled caste candidates while the rest are open to all,, then the number of ways in which the selection can be made, (a) 5C3 × 22C9, (b) 22C9 – 5C3, 22, 5, (c), C3 + C3, (d) None of these, 91. If the ratio 2nC3 : nC3 is equal to 11 : 1, n equals, (a) 2, (b) 6, (c) 8, (d) 9, 92. If 12Pr = 11P6 + 6. 11P5, then r is equal to:, (a) 6, (b) 5, (c) 7, (d) None of these, 93. (8C1 – 8C2 + 8C3 – 8C4 + 8C5 – 8C6 + 8C7 – 8C8) equals:, (a) 0, (b) 1, (c) 70, (d) 256, 94. A student has to answer 10 questions, choosing at least, 4 from each of parts A and B. If there are 6 questions in, Part A and 7 in Part B, in how many ways can the student, choose 10 questions?, (a) 266, (b) 260, (c) 256, (d) 270, 95. Consider the following statements., I. If n is an even natural number, then the greatest, among nC0, nC1 , nC2, …, nCn is nCn/2., II. If n is an odd natural number, then the greatest, n, n, C, among nC , nC , nC …, nC is n -1 or C n +1 ., 0, , 1, , 2, , n, , 2, , 2, , Choose the correct option., (a) Only I is false., (b) Only II is true., (c) Both are true., (d) Both are false., 96. A boy has 3 library tickets and 8 books of his interest in, the library. Of these 8, he does not want to borrow, Mathematics Part II, unless Mathematics Part I is also, borrowed. In how many ways can he choose the three, books to be borrowed?, (a) 40, (b) 45, (c) 42, (d) 41, 97. There were two women participants in a chess tournament., The number of games the men played between themselves, exceeded by 52 the number of games they played with, women. If each player played one game with each other,, the number of men in the tournament, was, (a) 10, (b) 11, (c) 12, (d) 13, , 98. For a game in which two partners play against two other, partners, six persons are available. If every possible pair, must play with every other possible pair, then the total, number of games played is, (a) 90, (b) 45, (c) 30, (d) 60, 99. A house master in a vegetarian boarding school takes, 3 children from his house to the nearby dhaba for, non-vegetarian food at a time as often as he can, but he, does not take the same three children more than once., He finds that he goes to the dhaba (road side hotel), 84 times more than a particular child goes with him., Then the number of children taking non-vegetarian food, in his hostel, is, (a) 15, (b) 5, (c) 20, (d) 10, 100. Five balls of different colours are to be placed in three, boxes of different sizes. Each box can hold all five balls., In how many ways can we place the balls so that no box, remains empty?, (a) 50, (b) 100, (c) 150, (d) 200, 101. The number of ways of dividing 52 cards amongst four, players so that three players have 17 cards each and the, fourth player have just one card, is, (a), , 103., , 104., , 105., , 106., , (17!)3, , (b) 52!, , 52!, (d) None of these, 17!, If the letters of the word KRISNA are arranged in all, possible ways and these words are written out as in a, dictionary, then the rank of the word KRISNA is, (a) 324, (b) 341, (c) 359, (d) None of these, Eighteen guests are to be seated, half on each side of a, long table. Four particular guests desire to sit on one, particular side and three others on the other side of the, table. The number of ways in which the seating arrangement, can be done equals, (a) 11C4 (9!)2, (b) 11C6 (9!)2, 6, 5, (c) P0 × P0, (d) None of these, At an election, a voter may vote for any number of, candidates not greater than the number to be elected., There are 10 candidates and 4 are to be elected. If a voter, votes for at least one candidate, then the number of ways, in which he can vote, is, (a) 6210 (b) 385, (c) 1110, (d) 5040, A student is to answer 10 out of 13 questions in an, examination such that he must choose at least 4 from the, first five questions. The number of choices available to, him is, (a) 140, (b) 196, (c) 280, (d) 346, Ten persons, amongst whom are A, B and C to speak at, a function. The number of ways in which it can be done, if A wants to speak before B and B wants to speak before, C is, , (c), , 102., , 52!

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PERMUTATIONS AND COMBINATIONS, , 85, , 10!, (b) 3! 7!, 6, (d) None of these, (c) 10P3 . 7!, 107. A car will hold 2 in the front seat and 1 in the rear seat., If among 6 persons 2 can drive, then the number of ways, in which the car can be filled is, (a) 10, (b) 20, (c) 30, (d) None of these, BEYOND NCERT, , (a), , Topic 4 : Circular Permutations, De-arrangement Theorem,, Sum of Numbers, Number of Integral Solutions, Number, of Divisors, Geometrical Application of Combinations., 108. Number of ways in which 20 different pearls of two colours, can be set alternately on a necklace, there being 10 pearls of, each colour., (a) 6 × (9!)2, (c) 4 × (8!)2, , (b), (d), , 12!, 5 × (9!)2, , 109. If eleven members of a committee sit at a round table so, that the President and Secretary always sit together, then, the number of arrangements is, (a) 10! × 2 (b) 10!, (c) 9! × 2, (d) 11! × 2!, 110. In how many ways vertices of a square can be coloured, with 4 distinct colour if rotations are considered to be, equivalent, but reflections are distinct?, (a) 65, (c) 71, , (b) 70, (d) None of these, , 111. The number of ways in which 5 beads of different colours, form a necklace is, (a) 12, (b) 24, (c) 120, (d) 60, 112. ABC is a triangle. 4, 5, 6 points are marked on the sides, AB, BC, CA, respectively, the number of triangles on, different side is, (a) (4 + 5 + 6)!, (c) 5! 4! 6!, , (b) (4 – 1) (5 – 1) (6 – 1), (d) 4 × 5 × 6, , 113. The number of chords that can be drawn through 21, points on a circle, is, (a) 200, (b) 190, (c) 210, (d) None of these, 114. In how many ways can 5 keys be put in a ring?, (a), , 1, 4!, 2, , (b), , 1, 5!, 2, , (c) 4!, , (d) 5!, , 115. 12 persons are to be arranged to a round table. If two, particular persons among them are not to be side by side,, the total number of arrangements is, (a) 9(10!) (b) 2(10!), (c) 45(8!), 116. Consider the following statements:, , (d) 10!, , Statement I : The number of diagonals of n-sided polygon, is nC2 – n., , Statement II : A polygon has 44 diagonals. The number, of its sides are 10., Choose the correct option from the choices given below., (a) Only I is true, (b) Only II is true, (c) Both I and II are true (d) Both I and II are false, 117. There are 10 points in a plane, no three are collinear, except, 4 which are collinear. All points are joined . Let L be the, number of different straight lines and T be the number of, different triangles, then, (a) T = 120, (b), L = 40, (c) T = 3 L – 5, (d) None of these, 118. Three boys and three girls are to be seated around a table,, in a circle. Among them, the boy X does not want any girl, neighbour and the girls Y does not want any boy neighbour., The number of such arrangements possible is, (a) 4, (b) 6, (c) 8, (d) None of these, 119. If x, y, z are integers and x ³ 0, y ³ 1, z ³ 2, x + y + z = 15, then the number of values of the ordered triplet (x, y, z) is, (a) 91, (b) 455, (c) 17C15, (d) None of these, 120. Let Tn denote the number of triangles which can be formed, using the vertices of a regular polygon of n sides. If, Tn + 1 - Tn = 21, then n equals`, (a) 5, (b) 7, (c) 6, (d) 4, 121. Given 12 points in a plane, no three of which are collinear., Then number of line segments can be determined, are:, (a) 76, (b) 66, (c) 60, (d) 80, 122. How many triangles can be drawn by means of 9, non-collinear points?, (a) 84, (b) 72, (c) 144, (d) 126, 123. ABCD is a convex quadrilateral. 3, 4, 5 and 6 points are, marked on the sides AB, BC, CD and DA respectively. Find, the number of triangles with vertices on different sides., (a) 215, (b) 342, (c) 225, (d) 424, 124. A person writes letters to six friends and addresses the, corresponding envelopes. Let x be the number of ways so, that at least two of the letters are in wrong envelopes and y, be the number of ways so that all the letters are in wrong, envelopes. Then x – y =, (a) 719, (b) 265, (c) 454, (d) None, 125. The number of circles that can be drawn out of 10 points, of which 7 are collinear, is, (a) 120, (b) 113, (c) 85, (d) 86, 126. The number of divisors of 9600 including 1 and 9600 are, (a) 60, (b) 58, (c) 48, (d) 46, 127. The straight lines l1 , l 2 , l 3 are parallel and lie in the same, plane. A total number of m points are taken on l1 , n points, on l 2 , k points on l 3 . The maximum number of triangles, formed with vertices at these points are :, (a) m+n+kC 3, (b) m+n+kC3 – mC3 – nC3 – kC3, (c) mC3 + mC3 + kC3, (d) None of these

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EBD_7762, 86, , MATHEMATICS, , 128. If n = 2p – 1 (2p – 1), where 2p – 1 is a prime, then the sum of, the divisors of n is equal to, (a) n, (b) 2n, (c) pn, (d) p n, 129. A parallelogram is cut by two sets of m lines parallel to its, sides. The number of parallelograms thus formed is, (a) (mC2)2, (b) (m + 1C2)2, m, +, 2, 2, (c) (, C2), (d) None of these, , 130. Two straight line intersect at a point O. Points A1, A2, .... An, are taken on one line and points B1, B2, ...., Bn on the other., If the point O is not to be used, the number of triangles that, can be drawn using these points as vertices, is, (a) n (n – 1), (b) n (n – 1)2, 2, (c) n (n – 1), (d) n2 (n – 1)2, , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, If nC12 = nC8, then n is equal to, (a) 20, (b) 12, (c) 6, (d) 30, 2., The number of possible outcomes when a coin is tossed 6, times is, (a) 36, (b) 64, (c) 12, (d) 32, 3., The number of different four-digit numbers that can be, formed with the digits 2, 3, 4, 7 and using each digit only, once is, (a) 120, (b) 96, (c) 24, (d) 100, 4 . The sum of the digits in the unit place of all the numbers, formed with the help of 3, 4, 5, 6 taken all at a time is, (a) 432, (b) 108, (c) 36, (d) 18, 5., Total number of words formed by 2 vowels and, 3 consonants taken from 4 vowels and 5 consonants is, equal to, (a) 60, (b) 120, (c) 7200, (d) 720, 6., Five digit number divisible by 3 is to be formed by using 0, 1, 2,, 3, 4, 5 without repetition. Total number of such numbers are, (a) 216, (b) 600, (c) 240, (d) 3125, 7., Every body in a room shakes hands with every body else. If, total number of hand-shaken is 66, then the number of, persons in the room is, (a) 11, (b) 12, (c) 13, (d) 14, 8., The number of triangles that are formed by choosing the, vertices from a set of 12 points, seven of which lie on the, same line is, (a) 105, (b) 15, (c) 175, (d) 185, 9., The number of parallelograms that can be formed from a, set of four parallel lines intersecting another set of three, parallel lines is, (a) 6, (b) 18, (c) 12, (d) 9, 10. The number of ways in which a team of eleven players can, be selected from 22 players always including 2 of them and, excluding 4 of them is, (a) 16C11 (b) 16C5, (c) 16C9, (d) 20C9, 11. The number of 5-digit telephone numbers having atleast, one of their digits repeated is, (a) 90000 (b) 10000, (c) 30240, (d) 69760, 12. The number of ways in which we can choose a committee, from four men and six women so that the committee, include at least two men and exactly twice as many, women as men is, 1., , (a) 94, (b) 126, (c) 128, (d) None of these, 13. The total number of 9-digit numbers which have all, different digits is, (a) 10!, (b) 9!, (c) 9 × 9!, (d) 10 × 10!, 14. The number of words which can be formed out of the, letters of the word ‘ARTICLE’, so that vowel occupy the, even place is, (a) 1440 (b) 144, (c) 7!, (d) 4C4 × 3C3, 15. Given five different green dyes, four different blue dyes, and three different red dyes. The number of combinations, of dyes which can be chosen, taking at least one green and, one blue dye, is, (a) 3600, (b) 3720, (c) 3800, (d) 3600, 16., , 17., , 18., , 19., , Past Year MCQs, In how many ways can a committee of 5 made out 6 men, and 4 women containing atleast one woman?, [BITSAT 2014, C], (a) 246, (b) 222, (c) 186, (d) None of these, The number of all three elements subsets of the set {a 1, a2,, a3 . . . an} which contain a3 is, [BITSAT 2014, C], (a) nC3, (b) n – 1C3, (c) n – 1C2, (d) None of these, The number of integers greater than 6,000 that can be, formed, using the digits 3, 5, 6, 7 and 8, without repetition,, is :, [JEE MAIN 2015, C], (a) 120 (b) 72, (c) 216, (d) 192, The total number of 4-digit numbers in which the digits, are in descending order, is, [BITSAT 2015, C], (a) 10C4 × 4!, (b) 10C4, , 10!, (d) None of these, 4!, 20. If all the words (with or without meaning) having five, letters, formed using the letters of the word SMALL and, arranged as in a dictionary; then the position of the word, SMALL is :, [JEE MAIN 2016, A], (a) 52 nd (b) 58 th, (c) 46 th, (d) 59 th, 21. All the words that can be formed using alphabets A, H, L,, U and R are written as in a dictionary (no alphabet is, repeated). Rank of the word RAHUL is [BITSAT 2016, A], (a) 71, (b) 72, (c) 73, (d) 74, , (c)

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PERMUTATIONS AND COMBINATIONS, , 87, , 22. The number of values of r satisfying the equation, 39, , 23., , C 3r -1 -, , 39, , 25., , r2, , =, , C, , r 2 -1, , -, , 39, , C 3r is [BITSAT 2016, C], , (a) 1, (b) 2, (c) 3, (d) 4, A man X has 7 friends, 4 of them are ladies and 3 are, men. His wife Y also has 7 friends, 3 of them are ladies, and 4 are men. Assume X and Y have no common, friends. Then the total number of ways in which X and, Y together can throw a party inviting 3 ladies and 3 men, so, that 3 friends of each of X and Y are in this party, is :, [JEE MAIN 2017, A], (a) 484, (b) 485, (c) 468, (d) 469, n, , 24., , C, , 39, , n, , C, , 3, , then a – n is equal to, a+3, r=0, r, [BITSAT 2017, S], (a) 0, (b) 1, (c) 2, (d) None of these, Statement 1 : A five digit number divisible by 3 is to be, formed using the digits 0, 1, 2, 3, 4 and 5 with repetition., The total number formed are 216., Statement 2 : If sum of digits of any number is divisible, by 3 then the number must be divisible by 3., [BITSAT 2017, C], (a) Statement-1 is true, Statement-2 is true and is a correct, explanation for Statement -1, If, , å (-1)r r +3 Cr, , 26., , 27., , =, , 28., , 29., , (b) Statement -1 is true, Statement -2 is true and is NOT, a correct explanation for Statement - 1, (c) Statement - 1 is true, Statement- 2 is false, (d) Statement -1 is false, Statement -2 is true, How many different nine digit numbers can be formed from, the number 223355888 by rearranging its digits so that the, odd digits occupy even positions ?, [BITSAT 2017, A], (a) 16, (b) 36, (c) 60, (d) 180, From 6 different novels and 3 different dictionaries, 4, novels and 1 dictionary are to be selected and arranged in a, row on a shelf so that the dictionary is always in the middle., The number of such arrangements is :, [JEE MAIN 2018, C], (a) less than 500, (b) at least 500 but less than 750, (c) at least 750 but less than 1000, (d) at least 1000, In how many ways can 12 gentlemen sit around a round, table so that three specified gentlemen are always together?, [BITSAT 2018, A], (a) 9!, (b) 10!, (c) 3! 10! (d) 3! 9!, The number of ways in which first, second and third prizes, can be given to 5 competitors is, [BITSAT 2018, A], (a) 10, , (b) 60, , (c) 15, , (d) 125, , Exercise 3 : Try If You Can, 1., , 2., , 3., , Messages are conveyed by arranging four white, one blue,, and three red flags on a pole. Flags of the same colour are, alike. If a message is transmitted by the order in which the, colours are arranged, the total number of messages that can, be transmitted if exactly six flags are used is, (a) 45, (b) 65, (c) 125, (d) 185, In a particular batch of Pioneer career Kolkata there are 4, boys and certain number of girls. In every mock test only 5, students including at least 3 boys can appear. If different, group of students write the Mock exam every time, if number, of times test conducted is 66 then find the total number of, students in the class., (a) 5, (b) 6, (c) 9, (d) None of these, , åååå n, , 5., , The number of numbers, that can be formed by using all, digits 1, 2, 3, 3, 3, 2,1 so that odd digits always occupy odd, places, is, (a) 3! 4! (b) 34, (c) 30, (d) 12, , 6., , Let n and k be positive integers such that n ³, , 7., , is equal to, , 0£ i < j < k < l £ n, , (a), 4., , n+1C, 4, n+1C, 3, , 8., , (b) n. n+1C4, , (c), (d) n.(n + 1), The number of ordered 4-tuples, (x, y, z, w); x, y, z, w Î [0, 10]) which satisfies the inequality,, 2, , 2, , 2, , 9., , 2, , 2sin x × 3cos y × 4sin z × 5cos w × ³120 is, (a) 0, (b) 144, (c) 81, , (d) infinite, , k ( k +1), , . The, 2, number of solutions (x1, x2, …, xk), x1 e ³ 2,… , xk e ³ k all, integers satisfying x1 + x2 + … + xk = n is given by which, one of the following assume that 2n = 2p + k2 + k?, (a) k + p +1Cp, (b) k + p – 1 Cp, k, +, p, –, 1, (c), Ck, (d) None of these, In how many ways Ram can distribute 40 apples in his six, children named A, B, C, D, E and F such that A gets two, more than B, C gets 3 more than F and D gets five less than, E and every one must have atleast one fruit, (a) 101, (b) 91, (c) 96, (d) 136, Anil have tiled his square bathroom wall with congruent, square tiles. All the tiles are red, except those along the two, diagonals, which are all blue. If he used 121 blue tiles, then, the number of red tiles used are, (a) 900, (b) 1800, (c) 3600, (d) 7200, In a small village, there are 87 families, of which 52 families, have at most 2 children. In a rural development programme, 20 families are to be chosen for assistance, of which at

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EBD_7762, 88, , MATHEMATICS, , least 18 families must have at most 2 children. In how, many ways can the choice be made?, (a) 52C18× 35C2, (b) 52C18 × 35C2 + 52C19 × 35C1 + 52C20, (c) 52C18 + 35C2 + 52C19, (d) 52C18 × 35C2 + 35C1 × 52C19, If the number of functions f : {1, 2, 3, ..... 1999} ® {2000, 2001,, 2002, 2003} satisfy the condition that f (1) + f (2) + .... f (1999), is odd, are 2P. Then sum of digits of P is given by, (a) 19, (b) 18, (c) 28, (d) 30, Three digit numbers in which the middle one is a perfect, square are formed using the digits 1 to 9. Their sum is, (a) 134055, (b) 270540, (c) 170055, (d) None of these, Find the number of non negative solutions of the system, of equations, a + b = 10, a + b + c + d = 21, a + b + c + d +e, + f = 33, a + b + c + d + e + f + g + h = 46, and so on till a + b +, c + d + … + x + y + z = 208., (a) 23P10, (b) 22P11, (c) 23P13, (d) None of these, , 10., , 11., , 12., , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, , (d), (c), (d), (a), (b), (a), (c), (c), (a), (b), (b), (c), (a), , 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, , (a), (d), (c), (c), (c), (b), (a), (d), (a), (c), (a), (a), (a), , 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, , (d), (b), (a), (c), (c), (c), (a), (d), (b), (c), (c), (c), (a), , 1, 2, 3, , (a), (b), (c), , 4, 5, 6, , (b), (c), (a), , 7, 8, 9, , (b), (d), (b), , 1, 2, , (d), (d), , 3, 4, , (b), (b), , 5, 6, , (c), (b), , 13. Number of seven digit numbers, which can be formed with, non-zero distinct digits such that it starts and end with a, prime digit and middle digit is an even digit, is(a) 17280 (b) 15120, (c) 12960, (d) 21600, n, –, 1, 2, n, 14. If, Cr = (K – 3) × Cr + 1 and K is positive, then K belongs, to the interval, (a) (– 3 ,, (c) [0, 3 ], , 3 ) (b), , ( 3 , 2], , (d), , ( 3 , 2), , 15. Seven people leave their bags outside temple and while, returning after worshiping the deity, picked one bag each at, random. In how many ways at least one and at most three of, them get their correct bags?, (a), , 7, , C3 × 9 + 7 C5 × 44 + 7 C1 × 265, , (b), , 7, , (c), , 7, , C6 × 265 + 7 C5 × 9 + 7 C7 × 44, C5 × 9 + 7 C2 × 44 + 7 C1 × 265, , (d) None of these, , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (a) 79 (b), 40 (a) 53 (a), 66, (a) 80 (d), 41 (b) 54 (b), 67, (a) 81 (d), 42 (c) 55 (c), 68, (b) 82 (a), 43 (c) 56 (a), 69, (c) 83 (a), 44 (a) 57 (c), 70, (b) 84 (a), 45 (c) 58 (b), 71, (c) 85 (c), 46 (c) 59 (b), 72, (b) 86 (c), 47 (b) 60 (b), 73, (d) 87 (a), 48 (b) 61 (a), 74, (a) 88 (a), 49 (b) 62 (c), 75, (d) 89 (b), 50 (a) 63 (a), 76, (b) 90 (a), 51 (c) 64 (c), 77, (a) 91 (b ), 52 (a) 65 (b), 78, Exercise 2 : Exemplar & Past Year MCQs, (a), 10 (c) 13 (c), 16, 19 (b), (c) 20 (b), 11 (d) 14 (b), 17, (d) 21 (d), 12 (a) 15 (b), 18, Exercise-3 : Try If You Can, (b), (b), (a) 13 (b), 7, 9, 11, (c) 10 (c), (c) 14 (b), 8, 12, , 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, , (a), (b), (a), (c), (d), (d), (b), (d), (c), (a), (a), (b), (b), , 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, , (b), (a), (d), (d), (c), (b), (a), (d), (c), (a), (a), (a), (b), , 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, , (a), (a), (b), (b), (a), (b), (c), (c), (c), (b), (b), (c), (c), , 22, 23, 24, , (b), (b), (a), , 25, 26, 27, , (d), (c), (d), , 28, 29, , (d), (b), , 15, , (a)

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8, , BINOMIAL THEOREM, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 5, 4, 3, , JEE MAIN, BITSAT, , 2, 1, 0, , 2010 2011 2012 2013 2014 2015 2016 2017 2018, Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 3, 4, , Critical Concepts, , Binomial Theorem for positive integral, indices, Binomial Expansion, Binomial, Coefficients and Their properties, General, and Middle terms, Multinomial Theorem, Binomial Theorem for any Rational Index,, Exponential and Logarithmic Series., , Rating of, Difficulty Level, , 3.5/5, , CUS, (chapter utility score), Out of 10, 6.5

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BINOMIAL THEOREM, , 91

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EBD_7762, 92, , MATHEMATICS, , Topic 1 : Expansion of Binomial, 1., 2., , 3., , The total number of terms in the expansion of, (x + a)51 – (x – a)51 after simplification is, (a) 102, (b) 25, (c) 26, (d) None of these, The coefficient of xp and xq (p and q are positive integers), in the expansion of (1 + x)p + q are, (a) equal, (b) equal with opposite signs, (c) reciprocal of each other, (d) None of these, The formula, , (a + b)m = a m + ma m-1b +, (a) b < a, (c) | a | < | b |, 4., , 5., 6., , m(m - 1) m - 2 2, a, b + ... holds when, 1.2, , (b) a < b, (d) | b | < | a |, , 1, , can be expanded by binomial theorem, if, 5 + 4x, (a) x < 1, (b) | x | < 1, 5, 4, (c) | x | <, (d) | x | <, 4, 5, If x = 9950 + 10050 and y = (101)50 then, (a) x = y, (b) x < y, (c) x > y, (d) None of these, How many terms are present in the expansion of, 11, , æ 2 2 ö, çx + 2 ÷ ?, x ø, è, (a) 11, (b) 12, , 7., , 9., 10., , 11., , (d) 11!, , The expansion of, by binomial theorem will, (4 - 3x)1/ 2, be valid, if, (a) x < 1, (b) | x | < 1, 2, 2, <x<, (d) None of these, 3, 3, The minimum positive integral value of m such that, (1073)71 – m may be divisible by 10, is, (a) 1, (b) 3, (c) 7, (d) 9, , (c), , 8., , (c) 10, 1, , -, , What is the coefficient of x3 in (3 - 2x) ?, (1 + 3x)3, (a) – 272 (b) – 540, (c) – 870 (d) – 918, If 'n' is positive integer and three consecutive coefficient, in the expansion of (1 + x)n are in the ratio 6 : 33 : 110,, then n is equal to :, (a) 9, (b) 6, (c) 12, (d) 16, 4n, , The last digit of 33, (a) 0, (b) 4, , + 1 , is, (c) 8, , (d) 2, , 50, 50, 5 [( 5 + 1) – ( 5 – 1) ] is, (a) an irrational number (b) 0, (c) a natural number, (d) None of these, 13. The number of term in the expansion of, , 12., , [( x + 4 y)3 (x – 4 y) 3 ]2 is, (a) 6, (b) 7, (c) 8, (d) 32, 14. The last two digits of the number 3400 are, (a) 00, (b) 01, (c) 21, (d) 81, 15. If (1 + x)n = C0 + C1x + C2x2 +.....+ Cnxn, then, , value of, , (C0 + C1 ) (C1 + C 2 ).....(C n -1 + C n ), is, C0 C1C2 .....Cn -1, , (2n)!, (n + 3)3, (n + 1)n, (n - 1)n, (b), (d), (c), (n + 1)!, (2n)!, n!, n!, n, 16. Notation form of (a + b) is, (a), , n, , (a), , å, , k=0, , n, , n, , Ck a n + k bk, , (b), , å n Ck a n - k bk, , k=0, , n, , (c), , å n Ck b n + k a k, , (d) None of these, , k= 0, , 17. The number of dissimilar terms in the expansion of (a +, b)n is n + 1, therefore number of dissimilar terms in the, expansion of (a + b + c)12 is, (a) 13, (b) 39, (c) 78, (d) 91, 18. In every term, the sum of indices of a and b in the, expansion of (a + b)n is, (a) n, (b) n + 1, (c) n + 2 (d) n – 1, 19. The approximation of (0.99)5 using the first three terms, of its expansion is, (a) 0.851 (b) 0.751, (c) 0.951 (d) None of these, 20. The approximate value of (1.0002)3000 is, (a) 1.6, (b) 1.4, (c) 1.8, (d) 1.2, 21. The number of zero terms in the expansion of, (1 + 3 2 x)9 + (1 - 3 2 x)9 is, (a) 2, (b) 3, (c) 4, 22. Number of terms in the expansion of, , (1 + 5 2x) + (1 - 5 2x ), 9, , (a) 2, , 9, , (b) 3, , 23. The fractional part of, (a), , 1, 15, , (b), , 2, 15, , (d) 5, , is, (c) 4, , (d) 5, , 2 4n, is, 15, , (c), , 4, 15, , (d) None of these

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BINOMIAL THEOREM, , 93, , 24. Assertion : If (1 + ax)n = 1 + 8x + 24x2 + ..., then the, values of a and n are 2 and 4 respectively., n ( n - 1) 2, x + ... for all, Reason : (1 + x)n = 1 + nx +, 2!, , 25., 26., 27., 28., , n Î Z+ ., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., After simplification, what is the number of terms in the, expansion of [(3x + y)5]4 – [(3x–y)4]5?, (a) 4, (b) 5, (c) 10, (d) 11, The last digit in 7300 is :, (a) 7, (b) 9, (c) 1, (d) 3, If 79 + 97 is divided by 64 then the remainder is, (a) 0, (b) 1, (c) 2, (d) 63, Expand by using binomial and find the degree of, polynomial, , (, , x + x3 - 1, , ) (, , (a) 7, , 5, , + x - x3 - 1, , (b) 6, , ), , 15, 28, , (b), , 5, 28, , (c), , 19, 28, , n, 2, , terms., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 32. If number of terms in the expansion of ( x – 2y + 3z ), 45 then n =, (a) 7, (b) 8, (c) 9, (d) 610, , 33. The coefficient of xn in expansion of (1 + x )(1 - x )n is, , (c), , ( -1)n -1 n, ( -1), , n -1, , (n - 1), , (b) ( -1)n (1 - n), 2, , (d) (n - 1), , (d) None of these, , 38. In the binomial expansion (a + bx)-3 =, the value of a and b are :, (a) a = 2, b = 3, (c) a = 3, b = 2, , 39., , 30. If (1 + ax)n = 1 + 8x + 24 x2 + … then the values of a and, n are, (a) n = 4, a = 2, (b) n = 5, a = 1, (c) n = 8, a = 3, (d) n = 8, a = 2, 31. Assertion: Number of terms in the expansion of, [(3x + y)8 – (3x – y)8] is 4., , (a), , (c) 101, , 1 9, + x + .... , then, 8 8, , (b) a = 2, b = – 6, (d) a = – 3, b = 2, , æ, 1 ö, The term independent of x in the expansion of ç 2 x + 2 ÷, è, 3x ø, , is, (a) 2nd, , n, , a r –1, , Topic 2 : General Term, Middle Term, and Independent Term, , 9, 28, , Reason: If n is even, then {(x + a)n – (x – a)n} has, , r =0, , n, 100, Õ b r = (101), , then n is, r =1, 100!, (a) 99, (b) 100, , (d) 4, , (d), , ar, , and, , is, , (c) 5, , n, , r, 37. If (1 + x)n = å a r . x and b r = 1 +, , 5, , ìï 32003 üï, ý , where {.} denotes the fractional, 29. The value of í, îï 28 þï, part, is equal to, , (a), , 34. The total number of terms in the expansion of, (x + a)51 – (x – a)51 after simplification is, (a) 102, (b) 25, (c) 26, (d) None of these, 35. If 2515 is divided by 13, then the remainder is, (a) 1, (b) 4, (c) 9, (d) 12, 36. The remainder when 2740 is divided by 12 is, (a) 3, (b) 7, (c) 9, (d) 11, , is, , (b) 3rd, , (c) 4th, , 9, , (d) 5th, , 10, , æ x, 3ö, 40. In the expansion of ç 3 ÷ , x > 0, the constant, ç 3, x ÷ø, è, , term is, (a) –70, (b) 70, (c) 210, (d) –210, 41. If the 7th term in the binomial expansion of, 9, , æ 3, ö, + 3 ln x ÷ , x > 0, is equal to 729, then x can be :, ç3, è 84, ø, , (a) e2, , (b) e, , 42. The coefficient of, (a), , 20C, 8, , (b), , (c), , e, 2, , (d) 2e, , x–12, , æ, yö, in the expansion of ç x + 3 ÷, è, x ø, , 20C, 8, , y8, , (c), , 20C, 12, , (d), , 20C, 12, , 20, , is, , y12, , 43. In the binomial expansion of (a – b)n, n ³ 5 the sum of the, 5th and 6th terms is zero. Then a/b equals :, (a), , n -5, 6, , (b), , n-4, 5, , (c), , 5, n-4, , (d), , 6, n -5

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EBD_7762, 94, , MATHEMATICS, 10, , 44., , æ1, ö, If the middle term in the expansion of ç + x sin x ÷, èx, ø, equals to 7, , 7, then x is equal to (n Î I), 8, , (a), , p, 2np ±, 6, , (c), , n p + (-1)n, , 46., , 12, , n, (d) n p + ( -1), , 5p, 6, , xö, æ, If the coefficients of x7 and x8 in ç 2 + ÷ are equal, then, è, 3ø, n is, (a) 56, (b) 55, (c) 45, (d) 15, The coefficient of the term independent of x in the, , 47., , 10, , 48., , 15, , (d) None of these, , The middle term in the expansion of æç 10 + x ö÷, è x 10 ø, (a), , 10C, 5, , (b), , 10C, 6, , (c), , 10C, 5, , 1, , 10, , 58., is, , 20x, 19, , (b) 83 x, , (c) 19 x, , 59., , 83x, (d), 19, , 15, , C6 .26, , (b), , 15, , (c), , 15, , C4 .24, , (d), , None of these, 2n, , 50., , 51., , 52., , 53., , (c) ± 5, , (d) ± 4, equals the, , 11, , é, -7, æ 1 öù, coefficient of x in ê ax - ç, ú , then a and b satisfy, è bx 2 ÷ø û, ë, the relation, (a) a – b = 1, (b) a + b = 1, , C5 .25, , æ, 1 ö, If the expansion of ç x - 2 ÷ contains a term, x ø, è, independent of x them n must be of the type ( here k Î N), (a) 2 k, (b) 3 k, (c) 4 k, (d) 5 k, If in the binomial expansion of (1 + x)n where n is a natural, number, the coefficients of the 5th, 6th and 7th terms are, in A.P., then n is equal to:, (a) 7 or 13 (b) 7 or 14 (c) 7 or 15 (d) 7 or 17, The coefficient of the middle term in the expansion of, (2 + 3x)4 is :, (a) 6, (b) 5!, (c) 8!, (d) 216, t6, Let tn denote the nth term in a binomial expansion. If t, 5, t5, n+4, in the expansion of (a + b) and t in the expansion of, 4, (a + b)n are equal, then n is, (a) 9, (b) 11, (c) 13, (d) 15, , is 405?, , 11, , 2 ö, æ, In the expansion of ç x + 2 ÷ , the term independent of, è, x ø, x is :, , (a), , 10, , 7, 60. If the coefficient of x in é ax 2 + 1 ù, êë, bx úû, , 15, , 49., , æ 4 1 ö, ç x + 3 ÷ , then what is the value of r ?, x ø, è, (a) 3, (b) 5, (c) 7, (d) 9, What is the coefficient of x3 y4 in (2x + 3y2)5 ?, (a) 240, (b) 360, (c) 720, (d) 1080, What are the values of k if the term independent of x in the, æ, k ö, expansion of ç x + 2 ÷, è, x ø, (a) ± 3, (b) ± 6, , (d) 10C5 x10, , x10, If tr is the rth term in the expansion of (1+ x)101, then the, t 20, ratio t equal to, 19, (a), , (b) C(12, 6) x–3 y3, (d) C(12, 6) x3 y–3, , æ x, 3ö, 56. In the expansion of ç 3 ÷ , x > 0, the constant, ç 3, x ÷ø, è, term is, (a) –70, (b) 70, (c) 210, (d) –210, 57. If x18 occurs in the rth term in the expansion of, , 10, , æ x, 3 ö, + 2 ÷÷ is, expansion of çç, è 3 2x ø, (a) 5/4, (b) 7/4, (c) 9/4, , æ 4 1 ö, ç x + 3 ÷ , then the value of r is equal to :, x ø, è, (a) 4, (b) 8, (c) 12, (d) 16, What is the middle term in the expansion of, æx y, 3 ö, –, ç, ÷ ?, ç 3, y x ÷ø, è, (a) C(12, 7) x3 y–3, (c) C(12, 7) x–3 y3, , n, , 45., , 15, , 55., , p, (b) n p +, 6, p, 6, , 54. If x 11 occurs in the r th term in the expansion of, , a, =1, (d) ab = 1, b, The term independent of x in the expansion of, , (c), 61., , 9, , æ6, 1 ö, çè x - 3 ÷ø is, x, (a) – 9 C3 (b) – 9 C4, (c) – 9 C5 (d) – 8 C3, 62. If 'n' is positive integer and three consecutive coefficient, in the expansion of (1 + x)n are in the ratio 6 : 33 : 110,, then n is equal to :, (a) 9, (b) 6, (c) 12, (d) 16, 63. If the coefficients of r th and (r + 1)th terms in the expansion, of (3 + 7x)29 are equal, then the value of r is, (a) 31, (b) 11, (c) 18, (d) 21, 11, , 64. The coefficient, , of x–7 in the expansion, , be :, (a), , 462, b5, , a 6 (b), , 462a 5, b6, , (c), , é, 1 ù, of ê ax - 2 ú will, bx û, ë, , -462a 5, b6, , (d), , -462a 6, b5

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BINOMIAL THEOREM, , 95, , -3, is, . Other value of ‘a’ is, 10, (a) 0, (b) 1, (c) 2, (d) 3, 79. Number of terms involving x 6 in the expansion of, , 65. The number of real negative terms in the binomial, 4n - 2 ,, , n Î N, x > 0 is, expansion of (1 + ix), (a) n, (b) n + 1, (c) n – 1 (d) 2n, 7, , 11, , 1ö, æ, 66. The coefficient of x3 in the expansion of ç x - ÷ is :, è, xø, (a) 14, (b) 21, (c) 28, (d) 35, , æ 2 3ö, ç 2x - ÷ , r ¹ 0, is, xø, è, (a) 1, (b) 2, , 67. If the third term in the expansion of [ x + x log10 x ]5 is 106,, then x may be, (a) 1, (b) 10, (c) 10, (d) 10 -2 / 5, 68. Consider the following statements., I. General term in the expansion of (x2 – y)6 is, ( – 1)r x12 – 2r · yr, II. 4th term in the expansion of (x – 2y)12 is –1760x9y3., Choose the correct option., (a) Only I is false, (b) Only II is false, (c) Both are false, (d) Both are true, 69. If the second, third and fourth terms in the expansion of, (a + b)n are 135, 30 and 10/3 respectively, then the value of, n is, (a) 6, (b) 5, (c) 4, (d) None of these, æ 4 1ö, 70. If x4 occurs in the tth term in the expansion of ç x + 3 ÷, è, x ø, then the value of t is equal to :, (a) 7, (b) 8, (c) 9, (d) 10, 71. Coefficient of x13 in the expansion of, (1 – x)5 (1 + x + x2 + x3 )4 is, (a) 4, (b) 6, (c) 32, (d) 5, , 15, , ,, , 10, , 72., , 73., 74., 75., 76., , æx 2 ö, If the term in the expansion of ç - 2 ÷ contains x4,, è3 x ø, then r is equal to, (a) 2, (b) 3, (c) 4, (d) 5, In the expansion of (1 + x)18 , if the coefficients of, (2r + 4)th and (r – 2)th terms are equal, then the value of r is :, (a) 12, (b) 10, (c) 8, (d) 6, A positive value of m for which the coefficient of x2 in the, expansion (1 + x)m is 6, is, (a) 3, (b) 4, (c) 0, (d) None of these, If the coefficient of x in (x2 + k/x)5 is 270, then the value of, k is, (a) 2, (b) 3, (c) 4, (d) 5, If (1 + x)2n = a0 + a1x + a2x2 + ..... + a2nx2n, then, , rth, , (a) a0 + a2 + a4 + .... =, , 1, (a + a1 + a2 + a3 + ....), 2 0, , (b) an+1 < an, (c) an–3 = an+3, (d) All of these, 77. Value of ‘a’, if 17th and 18th terms in the expansion of, (2 + a)50 are equal, is, (a) 1, (b) 2, (c) 3, (d) 4, 78. One value of a for which the coefficients of the middle, terms in the expansion of (1 + ax)4 and (1 – ax)6 are, equal,, , (c) 6, , (d) 0, n, , 1ö, 5, æ, 80. If the fourth term in the expansion of ç ax + ÷ is , then, 2, x, è, ø, the value of a × n is, (a) 2, (b) 6, (c) 3, (d) 4, 81. Assertion : The term independent of x in the expansion, , ( 4m )!, 1, æ, ö, of ç x + + 2 ÷ is, ., x, è, ø, ( 2m!)2, Reason : The coefficient of x 6 in the expansion of, (1 + x)n is nC6., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 82. The term independent of x in the expansion of, m, , 18, , æ, 1 ö, çè 9x ÷ , x > 0 , is ‘a’ times the corresponding binomial, 3 xø, , coefficient. Then ‘a’ is, (a) 3, (b) 1/3, , (c) –1/3, , (d) None of these, 2, , æ1- xö, 83. The term independent of x in the expansion of ç, is, è 1 + x ÷ø, (a) 4, (b) 3, (c) 1, (d) None of these, 84. The middle term in the expansion of, , 1 ö, æ, 2 n, is, çè1 + 2 ÷ø 1 + x, x, (a) 2nCn x2n (b) 2nCn x-2n (c) 2nCn, (d) 2nCn – 1, 85. What are the values of k if the term independent of x in the, , (, , ), , 10, , æ, k ö, expansion of ç x + 2 ÷ is 405?, è, x ø, (a) ± 3, (b) ± 6, (c) ± 5, (d) ± 4, 86. If x is positive, the first negative term in the expansion of, , (1 + x)27 5 is, (a) 6th term, (c) 5th term, , (b) 7th term, (d) 8th term, , æ, 1ö, 87. The middle term in the expansion of ç1 + 2 ÷, è x ø, , is, (a), (c), , 2nC, n, 2nC, n, , x2n, , (b), (d), , 2nC x–2n, n, 2nC, n–1, , n, , (1 + x 2 ), , n

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EBD_7762, 96, , 88., , MATHEMATICS, , If the rth term in the expansion of, , (a) a – b = 1, , 10, , æx 2 ö, çè 3 - 2 ÷ø contains x7, then the value of r is, x, (a) 1, (b) 2, (c) 3, (d) 4, , 89., , 96., , (a), , –15, , C3 (b), , 15, , C4, , (c), , –15, , C5 (d), , 15, , C2, , Topic 3 : Coefficient of any power of x, Exponential, Series and Logarithmic Series, 3, , If x is so small that x and higher powers of x may be, , neglected, then, , (1 +, , 3, x) 2, , æ 1 ö, - ç 1 + x÷, è 2 ø, , 3, , may be approximated, , 1, (1 - x ) 2, , x 3 2, 3, (d), - x, - x2, 2, 8, 8, The coefficient of x5 in (1 + 2x + 3x2 + ....)–7/2 is, (a) 15, (b) 21, (c) 12, (d) 30, , 92. The sum of the series, , 12, 28, 50, 78, +, +, +, + ........, 2!, 3!, 4!, 5!, , .....is, (a) e, (b) 3 e, (c) 4 e, (d) 5 e, If x is so small that its second and higher power can be, neglected and (1 – 2x)–1/2 (1 – 4x) – 5/2 = 1 + kx then k is, equal to, 7, 2, If x is very small in magnitude compared with a, then, , (a) 4, , (b) 7, , 1, ö2, , 1, ö2, , æ a, æ a, ç, ÷ +ç, ÷, èa+xø, èa-xø, 1 x, (a) 1 +, 2 a, , (c) 1 +, , 3 x2, 4 a2, , (b) 4e, , (c) 11, , (d), , can be approximately equal to, , (c) If a > 0, then, , (d) e2, , n, , ¥, , æ a ö, ç, ÷ = a +1, a +1ø, n = 0è, , å, , (d) All are correct, , p+q, nq, , (b) 1 +, , 1/ 4, , æ 3 ö, ç1 – x ÷, è 5 ø, , 11, , 1/ 2, , p, q, , (1+ 2x ), , (a) 1 +, , –6, , 1/ 5, , æ 1 ö, + ç1 – x ÷, è 2 ø, , 343, x, 120, , =, , (b) 1 –, , 343, x, 120, , 275, x, (d) None of these, 120, 100. If m = (2013)! then the value of, , (c) 1 –, , 1, 1, 1, +, + ...... +, is equal to, log 2 m log3 m, log 2013 m, (a) 1, (b) (2013)!, , (d) None of these, , +, , equals the, , é, ù, -7, coefficient of x in ê ax - æç 1 ö÷ ú , then a and b satisfy, 2, è bx ø û, ë, the relation, , (d) (n + 1), , æ 5 ö, + ç1+ x ÷, è 6 ø, , ( x - y)( x + y) +, , 7, If the coefficient of x in éê ax 2 + æç 1 ö÷ ùú, è bx ø û, ë, , p-q, nq, , 99. Neglecting x 2 and higher power of x,, , 1, (2013)!, 101. The value of, 3 x2, 4 a2, , (n + 1) p + (n -1) q, (n - 1) p + (n +1) q, , is approximately equal to, , 11, , 95., , 4, e, 3, , 97. Find out the correct statement, (a) The value of (7.995)1/3 correct to four decimal places, is 1.9996, (b) If x is very small compared to 1 then, (1 – 2x)–1/2 (1 – 4x)–5/2 = 1 + 11x, , (c), , x, (b), a, , (d) 2 +, , (c), , p q, (c) 1 + nq + np, , (c), , 94., , 2.4 3.5, 4.6, +, +, + .......¥ =, 1.2 1.2.3 1.2.3.4, , (a), , 3, (b) 3 x + x 2, 8, , 3, (a) 1 - x 2, 8, , 93., , 1.3 +, , (d) ab = 1, , 98. If p is nearly equal to q and n > 1 then, , as, , 91., , a, =1, b, , (a) 3e, , æ 4 1 ö, ç x – 3 ÷ is:, x ø, è, , 90., , (c), , The coefficient of x32 in the expansion of :, 15, , (b) a + b = 1, , 1, (x - y)(x + y)( x 2 + y 2 ), 2!, 1, (x - y)(x + y)( x 4 + y 4 + x 2 y 2 ) + ... to, 3!, , infinity is, 2, , (b) e x - e y, , (a) ex – ey, (c), , 2, , ex + ey, , 2, , (d) ex + ey, , 2

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BINOMIAL THEOREM, , 97, , 102. The sum of the series, log 4 2 – log 8 2 + log16 2 + ..................to ¥ is, , 103., , (a) 1 – log e 2, , (b) 1 + log e 2, , (c), , (d) log 2 e, , log e 2 – 1, , 108. The co-efficient of x n in the expansion of, , (a), , 4 5, 3 4, 2 3, 1 2, x + .................is, x + x + x +, 5, 3, 4, 2, , (a), , x, + log (1 + x), 1+ x, , (c) –, , (b), , (c), , x, + log (1 + x), 1– x, , (c), , å, , n, , (a), , æ 4ö, (3) 25 ´ 25C10 ç ÷, è 3ø, , (c), , (2)8 ´, , 25, , æ 5ö, C11 ç ÷, è 2ø, , [(t -1 - 1) x + (t -1 + 1)-1 x -1 ]8 is :, 3, , æ 1- t ö, æ 1+ t ö, (d) 70 ç, 70 ç, ÷, ÷, è 1+ t ø, è 1- t ø, 106. Which of the following is correct, , 4, , 4, , (c) (n + 1) (, 112. The value of, , 2n, , then y = 2 x 2 - x, , (c), , (a), , 3, , 1æa-bö, æa-bö 1æa-bö, ç, ÷+ ç, ÷ + ç, ÷ + .....¥ = log ab, 3è a ø, è a ø 2è a ø, , 2, é, ù, 107. The coefficient of x in the expansion of ê 1 + x - x ú, ë, û, ascending powers of x, when | x | < 1, is:, , (a) 0, , (b), , -1, 2, , (c), , 1, 2, , n, , 11, , C r then, , 50, , n, (d) (n + 1)( C n / 2 ), , C4 +, , 6, , å 56 - r C3, , is, , r =1, , (n + 1) n, n!, , nn, (b), (n - 1)!, , (n + 1) n, (d) None of these, (n - 1) !, 114. In the expansion of (1 + x) 50, the sum of the coefficients, of odd powers of x is :, (a) 0, (b) 249, (c) 250, (d) 251, 115. The value of the sum, (c), , x, 1 2 2 3 3 4, + log(1 - x ), x + x + x + ..¥ =, 4, 3, 2, 1- x, , (d) log 4 2 - log 8 2 + log16 2 - .....¥ = - log 2, , æ 3ö, (d) (4)25 ´ 25 C11 ´ ç ÷, è 4ø, , 55, C3, (a) 55 C4 (b), (c) 56 C3 (d) 56 C4, n, 113. If (1 + x) = C0 + C1 x + C2 x2 + .... Cn xn, then, (C0 + C1) (C1 + C2) ..... (Cn – 1 + Cn) = k . C1C2C3....Cn, where k =, , æ, ö, x3 x5, y3 y5, +, + ....¥ ÷, +, + ....¥ = 2çç x +, ÷, 3, 5, 3, 5, è, ø, , 2, , 14, , (b) n ( 2n C n ), , Cn ), , (c), , (b), , 11, , (a) (n – 1) ( 2n C n ), , æ 1+ t ö, (b) 56 ç, ÷, è 1- t ø, , æ 4ö, 25, (b) 20 ´ C11 ç ÷, è 3ø, , 2, 2, 2, a 0 C20 + a1C1 + a 2 C 2 + ............. a n C n =, , 105. The term independent of x in the expansion of, , (a) If y +, , 11, , 111. If a n = 2n + 1 and C r =, , n=2, , 3, , 4n –1 + 2n, n!, , 4 n –1 + ( – 2) n -1, 4 n + (– 2) n, (d), n!, n!, BEYOND NCERT, , C, , å (n + 12)! = e - 2, , æ 1- t ö, (a) 56 ç, ÷, è 1+ t ø, , is, , 109. If the sum of the coefficients in the expansion of (a + b)n is, 4096, then the greatest coefficient in the expansion is, (a) 1594 (b) 792, (c) 924, (d) 2924, 110. Find the largest coefficient in the expansion of (4 + 3x)25., , 1 1 1, + + + ......¥, 2! 4! 6!, e -1, =, 1 1 1, e +1, + + + ....¥, 1! 3! 5 !, ¥, 3n - 2, n, 2, C2, = e3, n, !, n=2, ¥, , (d), , (b), , e3 x, , Topic 4 : Greatest Coefficient of Binomial Expansion and, Properties of Binomial Coefficients, , x, x, + log (1 + x) (d), + log (1 – x), 1+ x, 1- x, , 104. Which of the following is not correct?, 2 4 6, + + + ....¥ = e, (a), 1! 3! 5!, (b), , 4 n – 1 + ( – 2) n, n!, , e 7x + e x, , -1, , in, , 1 æ 99 ö 1 æ 99 ö, 1 æ 99 ö, ç ÷ is equal to, 1 - çç ÷÷ + çç ÷÷ - ........2è 1 ø 3è 2 ø, 100 çè 99 ÷ø, , (d) 1, (a), , 1, 99, , (b), , 1, 100, , (c), , 2, 90, , (d), , 2, 100

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EBD_7762, 98, , MATHEMATICS, 10, , 116. Value of, , n, , Cr, , å r. n C, r =1, , (a) 210, (c) 10.210, 119. The sum of the series, , is, , r -1, , (a) 10 n – 45, (b) 10n + 45, (c) 10n – 35, (d) 10n2 – 35, 117. The largest term in the expansion of (3 + 2x)50, where, 1, x = , is, 5, I. 5th, II. 3rd, th, III. 7, IV. 6th, Choose the correct option, (a) Only I, (b) Only II, (c) Both I and IV, (d) Both III and IV, 118. The value of (10C0) + (10C0+ 10C1) + (10C0+10C1 +10C2) +, ... + (10C0+ 10C1 10C2+ ...+10C9 ) is, , 20, , C0 -, , 20, , C1 +, , 20, , C2 -, , (b) 10.29, (d) None of these, 20, , C3 + ..... -..... +, , 20, , C10 is, , 1 20, C10, 2, 120. The greatest value of the term independent of x in the, expansion of (x sin p + x –1 cos p )10 , p Î R is, , (a) 0, , 20, , (b), , (c) - 20 C10 (d), , C10, , (a) 25, , (c), , 10!, , (b), , 10!, , 5, , 2 (5!) 2, , (d) None of these, , (5!) 2, , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, 1., , 2., , 3., , 4., , 5., 6., , The total number of temrs in the expansion of, (x + a)100 + (x – a)100 after simplification is, (a) 50, (b) 202, (c) 51, (d) None of these, If the integers r > 1, n > 2 and coefficients of (3r)th and, (r+2)nd terms in binomial expansion of (1 + x)2n are equal,, then, (a) n = 2r, (b) n = 3r, (c) n = 2r + 1, (d) none of these, The two successive temrs in the expansion of (1 + x)24, whose coefficients are in the ratio 1 : 4 are, (a) 3rd and 4th, (b) 4th and 5th, th, th, (c) 5 and 6, (d) 6th and 7th, n, The coefficients of x in the expansions of (1 + x)2n and, (1 + x)2n – 1 are in the ratio, (a) 1 : 2, (b) 1 : 3, (c) 3 : 1, (d) 2 : 1, If the coefficients of 2 nd, 3rd and the 4th terms in the, expansion of (1 + x)n are in A.P., then value of n is, (a) 3, (b) 7, (c) 11, (d) 14, If A and B are coefficients of x n in the expansion of, (1+ x)2n and (1 + x)2n – 1 respectively, then, (a) 1, , 7., , (b) 2, , (c), , 1, 2, , (d), , 2np +, , p, 6, , (b), , np +, , p, 6, , 8., , p, 6, , n p+ (-1)n, , (d), , p, 3, , {, , }, , If X = 4n - 3n - 1 : n Î N and, Y = {9 ( n - 1) : n Î N } , where N is the set of natural, , numbers, then X È Y is equal to: [JEE MAIN 2014, A], (a) X, (b) Y, (c) N, (d) Y – X, 9., , If T0 , T1 , T2 ....Tn represent the terms in the expansion of, (x + a)n, then (T0 –T2 + T4 – .......)2 + (T1 – T3 + T5 – .....)2, =, [BITSAT 2014, C], (a), , ( x2 + a2 ), , (b) ( x 2 + a 2 )n, , (c) ( x 2 + a 2 )1/ n, (d) ( x 2 + a 2 )-1/ n, 10. The coefficient of x4 in the expansion of, (1 + x + x2 + x3)11, is, [BITSAT 2014, A], (a) 440 (b) 770, (c) 990 (d) 1001, 11. The sum of coefficients of integral power of x in the, , (, , binomial expansion 1 - 2 x, , 1, n, 10, , n p + (-1)n, , Past Year MCQs, , A, equals, B, , If the middle term in the expansion of æç 1 + x sin x ö÷, èx, ø, 7, to 7 then the value of x is, 8, (a), , (c), , equal, 12., , ), , 50, , is : [JEE MAIN 2015, A], , (a), , 1 50, 3 -1, 2, , ), , (b), , 1 50, 2 +1, 2, , (c), , 1 50, 3 +1, 2, , (d), , 1 50, 3, 2, , (, (, , ), , (, , ), , ( ), , æ 2 4 ön, If the number of terms in the expansion of çç1 - + 2 ÷÷ ,, è x x ø, x ¹ 0, is 28, then the sum of the coefficients of all the terms, in this expansion, is, [JEE MAIN 2016, C], (a) 243, (b) 729, (c) 64, (d) 2187

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BINOMIAL THEOREM, , 99, , 13. If the third term in the expansion of [ x + x log10 x ]5 is 106,, then x may be, [BITSAT 2016, A], -2 / 5, (a) 1, (b), 10 (c) 10 (d) 10, 14. If the sum of odd numbered terms and the sum of even, numbered terms in the expansion of (x + a)n are A and B, respectively, then the value of (x2 – a2)n is, [BITSAT 2016, C], (a) A2 – B2, (b) A2 + B2, (c) 4AB, (d) None of these, , 8, r+2 n, C r = 2 – 1 , then n = [BITSAT 2016, A], r = 0 r +1, 6, (a) 8, (b) 4, (c) 6, (d) 5, 16. The value of, (21C1 – 10C1) + (21C2 – 10C2) + (21C3 – 10 C3) + (21C4, – 10C4) + .... + (21C10 – 10C10) is :[JEE MAIN 2017, A], (a) 220 – 210, (b) 221 – 211, (c) 221 – 210, (d) 220 – 29, n, , 15. If å, , (a) 2n–1, (b) 2n+1, n+2, (c) 2, (d) Not divisible by 2, 18. The sum of the co-efficients of all odd degree terms in the, expansion of (x + x3 - 1)5 + (x - x3 - 1)5 ,(x > 1) is :, [JEE MAIN 2018, C], (a) 0, (b) 1, (c) 2, , (d) – 1, 7, , 1ö, æ, 19. The coefficient of x3 in the expansion of ç x - ÷ is :, xø, è, [BITSAT 2018, A], (a) 14, (b) 21, , (c) 28, , (d) 35, , 20. If x > 0, the 1 +, , 17. The integer just greater than (3 + 5)2 n is divisible by (n, Î N), [BITSAT 2017, A], , log, , e2, , 1!, , x, , +, , (log, , e2, , x) 2, , + ........=, [BITSAT 2018, A], , 2!, , (a) x, , (b), , x2, , (c) 2x, , (d), , x, , Exercise 3 : Try If You Can, 1., , Let N = 21224 – 1, a = 2153 + 277 + 1 and b = 2408 – 2204 + 1., Then which of the following statement is correct ?, (a) a divides N but b does not, , 5., , (b) b divides N but a does not, (c) a and b both divides N, (d) neither a nor b divides N, 2., , 2, , å, , 3 æç, , (b), , (c), , n (n + 1) 2 (n + 2), 6, , (d) None of these, , 4., , If I is integral part of (2 + 3) and f is its fractional part., Then (I + f ) (1 – f ) is, , If x +, , 1 ö, æ x, +, In the expansion of ç, ÷, è cos q x sin q ø, , (a) 8, n, , (c) n, , (d) 2n, , 1, 1, = 1 and p = x1000 + 1000 and q be the digit at, x, x, 2n, , (d) 4, , if l1 is the least, , p, p, < q < and l, 2, 8, 4, is the least value of the term independent of x, l2, p, p, when < q < , then the value of, is, l1, 16, 8, , n (n + 1) 2 (n + 2), 12, , n (n + 1)(n + 2), 12, , (b) 1, , (c) 3, , value of the term independent of x when, , (a), , (a) I + 1, , (b) 2, , 16, , 6., , C ö, n, k ç k ÷÷ , where Ck = Ck, is, C, k, 1, ø, k =1 è, , 3., , (a) 1, , If n is a positive integer and k is a positive integer not, exceeding n, then, n, , é nCr + 4. nCr +1 + 6. nCr +2 ù, ê, ú, êë, n+l, + 4. nCr +3 + nCr +4 úû, =, the value of l is., n, n, n, r+l, é Cr + 3. Cr +1 + 3. Cr +2 ù, ê, ú, êë, + nCr +3 úû, , unit place in the number 2 + 1, n Î N and n > 1, then, p+q=, (a) 8, (b) 6, (c) 7, (d) 0, , 7., , (b) 32, , (c) 16, 2, , (d) 64, n, , Let f (x) = a0 + a1x + a2x + … + anx + …, and, , f ( x), = b0 + b1 x + b2 x 2 + ¼ + bn x n + ¼., 1- x, , If a0 = 1 and b1 = 3, then find the unit digit of b10 = ., Given that, (a) 5, , a0 a1, =, = ...... = Constant, a1 a2, , (b) 6, , (c) 7, , (d) 8

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EBD_7762, 100, , MATHEMATICS, , 1, 1, 3 4 5, , then sum, + + + ..... + 50 term =, 3! (k + 3)!, 4! 5! 6!, of coefficients in the expansion, , 8., , If, , (1 + 2x1 + 3x2 + ..... + 100x100)k is :, (where x1, x2, x3, ......, x100 are independent variable), (a) (5050)49, , (b) (5050)51, , (c) (5050)52, , (d) (5050)50, , If (1 + x)n = C0 + C1x + C2x2 + ...... + CnXn, where C0, C1,, C2 , ..... are binomial coefficients and Cr = nCr. Then, 2(C0 + C3 + C6 + ..... + Cn) + (C1+ C4 + C7 + ...... + Cn –, 2, 2)(1 + w) + (C2 + C5 + C8 + ..... Cn – 1)(1 + w ), (where w is the non real complex cube root of unity and n, is an odd multiple of 3), is equal to, (a) 2n + 1, (b) 2n – 1 + 1, (c) 2n + 1 – 1, (d) 2n – 1, , 9., , Let (1+ X)15 = a0 X P0 + a1 X P1 + a2 X P1 + a2 X P2 + ..... +, , 10., , a15 X P15 where a0 ³ a1 ³ a2 ³ a3 .... ³ a15 > 0, , ìïPi +1 > Pi , if a i = a i+1 = 2n + 1, n ÎI, i Î{ 0,1, 2,...,14}, and í, ïîPi +1 < Pi , if a i = a i+1 = 2n, n ÎI, i Î{ 0,1, 2,...,14}, a + P + a + P + 869, then the value of 5 5 11 11, is, a 9 + P9 + P14 + a14 + 32, (a) 5, , (b) 6, , If an = ( loge 3), , 11., , is equal to, , n, , n, , (c) 7, k2, , å k !(n - k )! then, , k =1, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, , (c), (a), (d), (c), (b), (b), (d), (c), (d), (c), (b), (c), , 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, , (b), (b), (b), (b), (d), (a), (c), (a), (d), (d), (a), (a), , 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, , (c), (c), (a), (a), (c), (a), (a), (b), (b), (c), (d), (c), , 1, 2, , (c), (a), , 3, 4, , (c), (d), , 5, 6, , (b), (b), , 1, 2, , (c), (b), , 3, 4, , (b), (b), , 5, 6, , (d), (c), , (d) 8, a1 + a2 + a3 + ..... ¥, , (a) 3 loge 9, (b) 9 loge3, (c) 9 loge3 (loge3 + 1), (d) (loge 9 )2, 12. Which of the following is NOT CORRECT?, (a) The greatest integer less than or equal to ( 2 + 1) 6 is 197., (b) The integer next above ( 3 + 1) 2 m contains 2 m +1 as, factor, (c) The greatest integer less than or equal to the number, (7 + 4 3 ) m is a multiple of 2., , (d) If R = (6 6 + 14) 2n+1 and f = R – [R] where [R] is, integer and 0 £ f < 1 then Rf = 202n + 1., 13. If k be positive integer and Sk = 1k + 2k +...+ nk, then find, m, , the value of, , å m+1Cr sr, , if n = 10, m = 11., , r =1, , (a) 1112 – 11, (b) 1011 – 10, 12, (c) 10 – 11, (d) None of these, 14. Which of the following is the greatest?, (a) 31C02 – 31C12 + 31C22 – ... – 31C312, (b) 32C02 – 32C12 + 32C12 – ...+ 32C322, (c) 32C02 + 32C12 + 32C22 –...+ 32C322, (d) 34C02 – 34C12 + 34C22 –...+ 34C322, 15. The interval in which x must lies so that the numerically, greatest term in the expansion of (1 – x) 21 has the greatest, coefficient is, (x > 0)., (a), , æ5 6ö, é5 6ù, ê 6 , 5 ú (b) ç 6 , 5 ÷, è, ø, û, ë, , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (b), (a) 73, 37 (b) 49, 61, (b), (c) 74, 38 (b) 50, 62, (c) 51, (b), (d) 75, 39, 63, (c) 52, (d), (b) 76, 40, 64, (d), (a) 77, 41 (b) 53, 65, (b), (b) 78, 42 (b) 54, 66, (d), (c) 79, 43 (b) 55, 67, (c) 56, (c), (a) 80, 44, 68, (c), (b) 81, 45 (b) 57, 69, (a) 58, (c), (c) 82, 46, 70, (a) 59 (a), (a) 83, 47, 71, (d), (b) 84, 48 (d) 60, 72, Exercise 2 : Exemplar & Past Year MCQs, (c), (b), (c), 7, 9, 11, 13, (b), (c), (b), 8, 10, 12, 14, Exercise 3 : Try If You Can, (c), (d), (c) 13, 7, 9, 11, (d) 10, (d), (c) 14, 8, 12, , æ4 5ö, , é4 5ù, , (c) ç 5 , 4 ÷ (d) ê 5 , 4 ú, è, ø, û, ë, , (d), (a), (b), (d), (a), (a), (d), (c), (d), (d), (c), (c), , 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, , (a), (d), (c), (b), (b), (c), (b), (d), (c), (d), (d), (b), , 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, , (d), (b), (d), (a), (b), (a), (d), (d), (c), (c), (d), (c), , 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, , (c), (d), (c), (d), (a), (b), (b), (a), (d), (b), (d), (b), , (c), (a), , 15, 16, , (d), (a), , 17, 18, , (b), (c), , 19, 20, , (b), (d), , (a), (c), , 15, , (b)

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9, , SEQUENCES AND SERIES, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , 5, , Number of Questions, , 4, , 3, JEE MAIN, BITSAT, , 2, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 5, 8, , Critical Concepts, , Arithmetic Progression, Arithmetic, Mean, Geometric Progression,, Geometric Mean, Harmonic, Progression, Harmonic mean,, Sum to n terms of G.P, Geometric, Mean , Relation Between A.M & G.M, & H.M, Sum of Special Series, , Rating of, Difficulty Level, , 4/5, , CUS, (chapter utility score), Out of 10, , 7.8

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SEQUENCES AND SERIES, , 103

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EBD_7762, 104, , MATHEMATICS, , Topic 1 : Sequences, Series, Arithmetic Progression (A.P.),, General term of A.P., Sum to n terms of A.P.,, Arithmetic Mean., 1., , 2., , 3., , 4., , 5., , 6., , In an AP. the pth term is q and the (p + q)th term is 0., Then the qth term is, (a) – p, (b) p, (c) p + q, (d) p – q, If a, b, c, d, e, f are in A.P., then e – c is equal to:, (a) 2(c – a), (b) 2(d – c), (c) 2(f – d), (d) (d – c), 2, , (a) form an A.P., (b) form a G.P., (c) form an H.P., (d) do not form any progression, If pth term of an AP is q, and its qth term is p, then what is, the common difference ?, (a) – 1, (b) 0, (c) 2, (d) 1, Which term of the sequence (– 8 +18i), (–6 + 15i),, (–4 + 12i) is purely imaginary, (a) 5th, (b) 7th, (c) 8th, (d) 6th, The following consecutive terms, , 1, 1, ,, of a series are in:, 1+ x 1- x 1- x, , 8., , 9., , 10., , If a1, a2, a3, ...... are in A.P. and, a 12 - a 22 + a 32 - a 24 + .........+ a 22k -1 - a 22 k, , = M (a12 - a 22 k ) . Then M =, , ,, , (a) A.P., (b) G.P., (c) A.P., G.P., (d) None of these, If the sum of the first 2n terms of 2, 5, 8, ....... is equal to, the sum of the first n terms of 57, 59, 61......., then n is, equal to, (a) 10, (b) 12, (c) 11, (d) 13, Let (1 + x)n = 1 + a1x + a2x2 + ... + anxn. If a1, a2 and a3, are in A.P., then the value of n is, (a) 4, (b) 5, (c) 6, (d) 7, There are four arithmetic means between 2 and, –18. The means are, (a) –4, –7, –10, –13, (b) 1, –4, –7, –10, (c) –2, –5, –9, –13, (d) –2, –6, –10, –14, The arithmetic mean of three observations is x. If the values, of two observations are y, z; then what is the value of the, third observation ?, (a) x, (b) 2x – y – z, (c) 3x – y – z, (d) y + z – x, , k -1, (b), k +1, , (a), 12., , k, 2k - 1, , k +1, 2k + 1, , (c), , (d) none, , 1, 1, 1, ,, ,, are in A.P. then,, q+r r+ p p+q, , (b) p2, q2, r2 are in A.P, , (a) p, q, r are in A.P, , The roots of the equation ( x - 1) - 4 | x - 1 | +3 = 0, , 1, , 7., , 11., , (c), , 1 1 1, , , are in A.P, p q r, , (d) p + q + r are in A.P, , 13. Let a1, a2 , a3 ............ be terms of an A.P. If, a1 + a2 + ¼¼ a p, a1 + a2 + ¼¼+ aq, , (a), , 41, 11, , (b), , =, , P2, , a, , p ¹ q , then 6 equals, a21, q, 2, , 7, 2, , 2, 7, , (c), , (d), , 11, 41, , 14. The value of, 1, 10 - 9, , -, , 1, 11 - 10, , +, , is equal to, (a) –10, (b) 11, , 1, 12 - 11, , (c) 14, , 15. If arithmetic mean of a and b is, , 1, , - ... -, , 121 - 120, , (d) – 8, , (a n +1 + b n +1 ), a n + bn, , , then the, , value of n is equal to, (a) –1, (b) 0, (c) 1, (d) 2, 16. If m arithmetic means are inserted between 1 and 31 so, that the ratio of the 7th and (m – 1)th means 5 : 9, then the, value of m is, (a) 10, (b) 11, (c) 12, (d) 14, 17. If the angles A < B < C of a triangle are in A. P., then, (a), , c 2 = a 2 + b 2 – ab, , (b), , (c), , c2 = a 2 + b 2, , (d) None of these, , 18. I., II., , b 2 = a 2 + c 2 – ac, , 18th term of the sequence 72, 70, 68, 66, ... is 40., 4th term of the sequence 8 – 6i, 7 – 4i, 6 – 2i, ... is, purely real.

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SEQUENCES AND SERIES, , 105, , Choose the correct option., (a) Only I is true, (b) Only II is true, (c) Both are true, (d) Both are false, 19. If the jth term and kth term of an A.P. are k and j respectively,, the ( k + j) th term is, (a) 0, (b) 1, (c) k + j + 1, (d) k + j –1, 20. The perimeter of a triangle whose sides are in A.P. is 21, cm and the product of lengths of the shortest side and the, longest side exceeds the length of the other side by 6 cm., The longest side of the triangle is, (a) 1 cm, , (b) 7 cm, , (c) 13 cm, , (d) none, , 21. The Fibonacci sequence is defined by 1 = a1 = a2 and, , a n +1, an = an – 1 + an – 2, n > 2. Then value of a for n = 2, is, n, (a) 1, (b) 2, (c) 3, (d) 4, 22. If the sum of a certain number of terms of the A.P. 25, 22,, 19, ........ is 116. then the last term is, (a) 0, (b) 2, (c) 4, (d) 6, 23. If cos x = b. For what b do the roots of the equation form, an A.P. ?, 1, 3, (b), 2, 2, (c) –1, (d) None of these, 24. If the sum of first p terms of an A.P. is equal to the sum of, the first q terms then the sum of the first (p + q) terms, is, (a) 0, (b) 1, (c) 2, (d) 3, , 28. The number of common terms to the two sequences, 17, 21, 25, ....., 417 and 16, 21, 26, ........, 466 is –, (a) 19, (b) 20, (c) 21, (d) 91, 29. Let Tr be the rth term of an A.P. whose first term is a and, common difference is d. If for some positive integers, m, n, m ¹ n, Tm =, , (a), , (c) 83660, 32. 51+x + 51–x,, (a) a < 12, 33., , 34., , is the A.M. between a and b, then the value, a n -1 + b n -1, of n is, (a) 1, (b) 2, (c) 3, (d) 4, 26. The difference between any two consecutive interior angles, of a polygon is 5°. If the smallest angle is 120°. The number, of the sides of the polygon is, (a) 6, (b) 9, (c) 8, (d) 5, , 25. If, , 27. Assertion: If, , 2, , k, 5 are in A.P, then the value of k is, 3, 8, , 31, ., 48, Reason: Three numbers a, b, c are in A.P. iff 2b = a + c, (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., , 1 1, + (b) 1, m n, , (c), , 1, mn, , (d) 0, , 30. 4th term from the end of the G.P. 3, 6, 12, 24., .........., 3072, is, (a) 348, (b) 843, (c) 438, (d) 384, 31. The sum of all odd numbers between 1 and 1000 which, are divisible by 3 is, (a) 83667, (b) 90000, , (a), , an + bn, , 1, 1, and Tn = , then a – d equals, n, m, , 35., , 36., , 37., , (d) None of these, a 2x –2x, , 5 + 5 are in A.P., then the value of a is:, 2, , (b) a £ 12, (d) None of these, , (c) a ³ 12, If in a series Sn = an2 + bn + c, where Sn denotes the sum, of n terms, then, (a) The series is always arithmetic, (b) The series is arithmetic from the second term onwards, (c) The series may or may not be arithmetic, (d) The series cannot be arithmetic, I. 37 terms are there in the sequence 3, 6, 9, 12, ..., 111., II. General term of the sequence 9, 12, 15, 18, ... is 3n + 8., Choose the correct option., (a) Only I is true, (b) Only II is true., (c) Both are true, (d) Both are false, If the sum of the first ten terms of an arithmetic progression, is four times the sum of the first five terms, then the ratio, of the first term to the common difference is :, (a) 1 : 2 (b) 2 : 1, (c) 1 : 4, (d) 4 : 1, There are 25 trees at equal distances of 5 meters in a line, with a well, the distance of the well from the nearest tree, being 10 metres. A gardener waters all the trees separately, starting from the well and he returns to the well after, watering each tree to get water for the next. The total, distance the gardener will cover in order to water all the, trees is, (a) 3550 m(b) 3434 m (c) 3370 m (d) 3200 m, If the nth term of an arithmetic progression is 3n + 7, then, what is the sum of its first 50 terms?, (a) 3925 (b) 4100, (c) 4175, (d) 8200

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EBD_7762, 106, , 38., , 39., , MATHEMATICS, , The A.M. of the series 1, 2, 4, 8, 16, ..., 2n is :, (a), , 2n - 1, n, , (b), , 2n +1 - 1, n +1, , (c), , 2n + 1, n, , (d), , 2n - 1, n +1, , 45. If b2, a2, c2 are in A.P., then, , (a) A.P., (b) G.P., (c) H.P., (d) None of these, th, 46. If the 9 term of an A.P. is zero, then the ratio of 29th term, to 19th term is, (a) 1 : 2 (b) 1 : 3, (c) 2 : 1, (d) 3 : 1, 47. The sum of 11 terms of an A.P. whose middle term is 30,, (a) 320, (b) 330, (c) 340, (d) 350, , The 10 th common term between the series, 3 + 7 + 11 + .... and 1 + 6 + 11 + .... is, , 40., , 41., , (a) 191, , (b) 193, , (c) 211, , (d) None of these, , A man saves ` 135/- in the first year, ` 150/- in the second, year and in this way he increases his savings by ` 15/every year. In what time will his total savings be ` 5550/-?, (a) 20 years, , (b) 25 years, , (c) 30 years, , (d) 35 years, , If, , 1 1 1, 1 1 1, are A. P., then æç + - ö÷ æç 1 + 1 - 1 ö÷ is, , ,, a b c, èa b cø èb c aø, , equal to, , 42., , b 2 – ac, , (a), , 4, 3, –, ac, b2, , (b), , (c), , 1, 4, –, ac, b2, , (d) None of these, , a 2b 2c2, , 43., , (b) 2, , 2, (a) 1 + 2a + g = 0, , 2, (b) 1 + 2a – g = 0, , 2, (c) 1 – 2a – g = 0, , 2, (d) 1 – 2a + g = 0, , (c), , -, , 1, 2, , (d), , (c) 0, , 3, 2, , (d) –2, , ABC is a right angled triangle in which ÐB = 90° and, BC = a. If n points L1, L2, ¼, Ln on AB are such that AB is, divided in n + 1 equal parts and L1M1, L2M2, ¼, LnMn are, line segments parallel to BC and M1, M2, ¼, Mn are on AC,, then the sum of the lengths of L1M1, L2M2, ¼, LnMn is, (a), , a (n + 1), 2, , (b), , (c), , an, 2, , (d) None of these, , a (n –1), 2, , 1, 1, ,, , ........ is, 2 -1 2 - 2 2, , (a), (c), , 1 1 1, If , , are the pth, qth, rth terms respectively of an A.P.., a b c, then the value of ab(p – q) + bc (q – r) + ca (r – p) is, , (a) –1, 44., , (b) 0, , 48. The fourth, seventh and tenth terms of a G.P. are p, q, r, respectively, then :, (a) p2 = q2 + r2, (b) q2 = pr, (c) p2 = qr, (d) pqr + pq + 1 = 0, 49. If 1, a and P are in A. P. and 1, g and P are in G. P., then, , 2 +1, , S, S1, S, (n 2 - n 3 ) + 2 (n 3 - n1 ) + 3 (n1 - n 2 ) is, n1, n2, n3, 1, 2, , Topic 2 : Geometric Progression (G.P), General Term of, G.P., Sum to n terms of G.P., Geometric Mean, Relation, Between G.M. & A.M., Infinite Geometric Series., , 50. The sum of infinite terms of the geometric progression, , If S1,S2 and S3 denote the sum of first n1, n2 and n3 terms, respectively of an A.P., then value of, , (a), , 1, 1, 1, ,, ,, will be in, a+b b+c c+a, , 2( 2 + 1) 2, , (b), , ( 2 + 1)2, , (d) 3 2 + 5, , 5 2, , 51. For a, b, c to be in G.P. What should be the value of, (a) ab, (c), , a-b, ?, b-c, , (b) bc, , a b, or, b c, , (d) None of these, , 52. The first and eight terms of a G.P. are x–4 and x52, respectively. If the second term is xt, then t is equal to:, (a) –13, , (b) 4, , (c), , 5, 2, , (d) 3, , 53. A series is such that its every even term is 'a' times the, term before it and every odd term is c times the term before, it. The sum of 2n term of the series is (the first term is, unity), (a), , (1 - c n )(1 - a n ), 1 - ac, , (b), , (1 + a ) (1 - c n a n ), 1 - ac, , (c), , (1 + c n ) (1 + a n ), 1 - ac, , (d), , (1 + a ) (1 + c n a n ), 1 + ac

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SEQUENCES AND SERIES, , 107, , 54. If the pth, qth and r th terms of a G.P. are again in G.P., then, which one of the following is correct?, (a) p, q, r are in A.P., (b) p, q, r are in G.P., (c) p, q, r are in H.P., (d) p, q, r are neither in A.P. nor in G.P. nor in H.P., 55. If a, b, c are in geometric progression and a, 2b, 3c are in, arithmetic progression, then what is the common ratio r, such that 0 < r < 1 ?, (a), , 1, 3, , (b), , 1, 2, , (c), , 1, 4, , (d), , p, q, , (b), , q, p, , (c) pq, , (d), , (a), , pq, , 59. An infinite G.P. has first term ‘x’ and sum ‘5’, then x belongs, to, (a) 0 £ x £ 10, (b) –10 < x < 0, (c) 0 < x < 10, , (d) –10 £ x £ 0, , 60. The series ( 2 + 1),1, ( 2 - 1) ... is in :, (a) A.P., (b) G.P., (c) H.P., (d) None of these, 61. Three numbers form an increasing G.P. If the middle, number is doubled, then the new numbers are in A.P. The, common ratio of the G.P. is:, (a) 2 –, , 3, , (b) 2 +, , 3, , (d) 3 + 2, 62. Let An be the sum of the first n terms of the geometric, 704, 704, 704, series 704 +, +, +, + ¼ and Bn be the sum, 2, 4, 8, 1984, of the first n terms of the geometric series 1984 –, +, 2, 1984, 1984, +, + ¼ If An = Bn, then the value of n is, 4, 8, (where n Î N)., (a) 4, (b) 5, (c) 6, (d) 7, (c), , 64. What is the sum of the series 1 –, , 1, 8, , 56. If 1, x, y, z, 16 are in geometric progression, then what is, the value of x + y + z ?, (a) 8, (b) 12, (c) 14, (d) 16, 57. The product of first nine terms of a GP is, in general, equal, to which one of the following?, (a) The 9th power of the 4th term, (b) The 4th power of the 9th term, (c) The 5th power of the 9th term, (d) The 9th power of the 5th term, 58. In a G.P. if (m + n)th term is p and (m – n)th term is q, then, mth term is:, (a), , 11th terms of the G.P. 5, 10, 20, 40, ... is 5120, If A.M. and G.M. of roots of a quadratic equation, are 8 and 5, respectively, then obtained quadratic, equation is x2 – 16x + 25 = 0, Choose the correct option., (a) Only I is true, (b) Only II is true., (c) Both are true, (d) Both are false., , 63. I., II., , 1, 2, , (b), , 3, 4, , (c), , 1 1 1, + – + .....?, 2 4 8, 3, 2, , (d), , 2, 3, , 65. If G be the geometric mean of x and y, then, 1, 1, +, =, 2, 2, 2, G -x, G - y2, (a) G2, , (b), , 1, G, , 2, , (c), , 2, , (d) 3G2, , G2, , 66. If y = 3x - 1 + 3- x - 1 (x real), then the least value of y is :, (a) 2, , (b) 6, , 2, (d) None of these, 3, In a Geometric Progression with first term a and common, ratio r, what is the Arithmetic Mean of the first five terms?, , (c), 67., , (a) a + 2r, (b) a r2, (c) a (r 5 – 1)/[5(r – 1)], (d) a (r 4 – 1)/[5(r – 1)], 68. If p, q, r are in A.P., a is G.M. between p & q and b is G.M., between q and r, then a2, q2, b2 are in, (a) G.P., (b) A.P., (c) H.P, (d) None of these, 69. In a geometric progression consisting of positive terms,, each term equals the sum of the next two terms. Then the, common ratio of its progression equals, (a), (c), , 5, , (, , 1, 1- 5, 2, , ), , 3–2, , 70. The minimum value of, , (, , (b), , 1, 2, , (d), , 1, 5., 2, , ), , 5 -1, , x4 + y 4 + z 2, for positive real, xyz, , number x, y, z is, (b) 2 2, (c) 4 2, (d) 8 2, (a), 2, 71. The first two terms of a geometric progression add up to, 12. the sum of the third and the fourth terms is 48. If the, terms of the geometric progression are alternately positive, and negative, then the first term is, (a) –4, (b) –12, (c) 12, (d) 4

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EBD_7762, 108, , 72., , 73., , MATHEMATICS, , Assertion: If a, b, c are in A.P., then b + c, c + a, a + b are in, A.P., Reason: If a, b, c are in A.P., then 10a, 10b, 10c are in G.P., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., Which term of the following sequence, 1 1 1, 1, , ,, .......is, ?, 3 9 27, 19683, , (a) 3, (c) 6, 74., , 75., , 76., , 77., , The value of 0.037 where 0.037 stands for the number, .0373737.........., is :, (a) 37/ 1000, (b) 37/990, (c) 1/37, (d) 1/27, How many terms of G.P. 3, 32, 33, .......... are needed to, give the sum 120?, (a) 3, (b) 4, (c) 5, (d) 6, A G.P. consists of an even number of terms. If the sum of, all the terms is 5 times the sum of terms occupying odd, places, then the common ratio is, (a) 5, (b) 1, (c) 4, (d) 3, The sum of an infinite geometric series is 2 and the sum, of the geometric series made from the cubes of this infinite, sereis is 24. Then the series is, (a) 3 +, (c), , 78., , 79., , 80., , (b) 9, (d) None of these, , 3 3 3, - + - ...., 2 4 8, , 3 3 3, 3 - + - + ..., 2 4 8, , (b) 3 +, , 3 3 3, + + + ...., 2 4 8, , (d) None of these, , Consider an infinite geometric series with first term a and, 3, common ratio r. If its sum is 4 and the second term is ,, 4, then :, (a), , 4, 3, a = ,r =, 7, 7, , (b), , 3, a = 2, r =, 8, , (c), , a=, , 3, 1, ,r =, 2, 2, , (d), , a = 3, r =, , 1, 4, , 4th term from the end of the G.P. 3, 6, 12, 24., .........., 3072, is, (a) 348, (b) 843, (c) 438, (d) 384, If log a, log b, and log c are in A.P. and also log a – log 2b,, log 2b – log 3c, log 3c – log a are in A.P., then :, (a) a, b, c, are in H.P., , (b) a, 2b, 3c are in A.P., (c) a, b, c are the sides of a triangle, (d) None of these, 81. If ax = by = cz, where a, b, c are in G.P. and a,b, c, x, y, z ¹ 0;, then, , 1 1 1, , , are in:, x y z, , (a) A.P., (b) G..P., (c) H.P, (d) None of these, 82. The product of n positive numbers is unity, then their sum, is :, (a) a positive integer, (b) divisible by n, (c) equal to n +, , 1, n, , (d) never less than n, , 83. An infinite G.P has first term x and sum 5, then, (a) x < – 10, (b) – 10 < x < 0, (c) 0 < x < 10, (d) x > 10, 84. In a G.P. of even number of terms, the sum of all terms is 5, times the sum of the odd terms. The common ratio of the, G.P. is, (a), , -4, 5, , (b), , 1, 5, , (c) 4, (d) None of these, 85. The first term of an infinite G.P. is 1 and each term is twice, the sum of the succeeding terms. then the sum of the series, is, (a) 2, , (b) 3, , (c), , 3, 2, , (d), , 5, 2, , 86. x and y are positive number. Let g and a be G. M. and AM, of these numbers. Also let G be G. M. of x + 1 and y + 1. If, G and g are roots of equation x 2 – 5x + 6 = 0, then, (a) x = 2, y =, (c) x =, , 3, 4, , 5, 8, ,y=, 2, 5, , (b) x =, , 3, , y = 12, 4, , (d) x = y = 2, , 87. There are four numbers of which the first three are in G.P., and the last three are in A.P., whose common difference is, 6. If the first and the last numbers are equal then two other, numbers are, (a) –2, 4, (b) –4, 2, (c) 2, 6, (d) None of these, 88. Let x be one A.M and g1 and g2 be two G.Ms between y, and z. What is g13 + g32 equal to ?, (a) xyz, , (b) xy2z, , (c) xyz2, , (d) 2 xyz

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SEQUENCES AND SERIES, , 109, , 89. Let a, b, c, be in A.P. with a common difference d. Then, 1/ c, , e, , , e, , b / ac, , 1/ a, , ,e, , (a), , are in :, , (a) G.P. with common ratio ed, (b) G.P with common ratio e1/d, 2, , -d 2 ), , y = b-, , (d) A.P., 1ö, æ, çx+ ÷, xø, è, , 2, , æ, 1 ö, + çç x 2 +, ÷÷, x2 ø, è, , 2, , 2, , æ, 1 ö, + çç x 3 +, ÷÷ ....upto n terms, x3 ø, è, , (a), , (b), (c), , x, , –1, , x 2 –1, x 2n + 1, x 2 +1, x 2n – 1, x 2 –1, , x 2n, , ×, , +1, , x 2n, , ×, ×, , +2, , x 2n + 2 – 1, x 2n, x 2n – 1, , (a), + 2n, , 1+, , – 2n, , 1, b+ c, , ,, , 1, c+ a, , (b), ,, , 1, , are, , a+ b, , in A.P. then, , 9ax+1 , 9bx +1 , 9cx+1 , x ¹ 0 are in :, , (a) ( 2 + 3) : ( 2 – 3), , 3 ) : (2 –, , (c), (d), , (a) G.P., (b) G.P. only if x < 0, (c) G.P. only if x > 0, (d) None of these, 92. The A. M. between two positive numbers a and b is twice, the G. M. between them. The ratio of the numbers is, , (b) (2 +, , 3), , (c) ( 3 + 1) : ( 3 – 1), , respectively a, b and c then a b -c × b c -a × c a - b =, (a) 1, (b) abc, (c) pqr, (d) ap bq cr, 4, 3, and its first term is then, 3, 4, , its common ratio is :, (a), , 7, 16, , 9, (b), 16, , (c), , 97, 12, , a a, + + ....to ¥ ,, r r2, , b b, + - ....to ¥, r r2, , ab, c, , ac, b, , (b), , (c), , bc, a, , (d) 1, , 2 6 10 14 ... is, +, + +, +, 3 32 33 34, , 1, 9, , (d), , 7, 9, , 95. If the sum of an infinitely decreasing GP is 3, and the sum, of the squares of its terms is 9/2, then sum of the cubes of, the terms is, , A ± A2 – G2, A±, , (A + G ) (A – G ), , A±, , (A + G ) (A – G ), 2, , 99. The value of 91 / 3 ´ 91 / 9 ´ 91 / 27 ´ ..........¥ is :, (a) 9, (b) 1, (c) 3, (d) None of these, 100. If a, b, c are in A.P. and a, b, d in G.P., then a, a – b,, d – c will be in, (a) A.P., (b) G.P., (c) H.P., (d) None of these, , Topic 3 : Sum to n terms of Special Series, , (d) None of these, 93. If the pth, qth and rth terms of both an A.P. and a G.P. be, , 94. If sum of the infinite G.P. is, , (d), , (a) 3, (b) 4, (c) 6, (d) 2, 98. If the arithmetic mean of two numbers be A and geometric, mean be G, then the numbers will be, (a) A ± (A2 – G2), , (d) None of these, 91. If, , 108, 13, , 97. The sum to infinite term of the series, , – 2n, , x 2n, , (c), , xy, c, c, =, and z = c + 2 + 4 + ...to ¥ , then, z, r, r, , is, 2n, , 105, 17, , (b), , 96. If | r | > 1and x = a +, , (c) G.P. with common ratio e d /( b, , 90., , 107, 12, , 101. Sum of n terms of series 1.3+3.5+5.7+.......... is, (a), , 1, n(n + 1)(2n + 1) - n, 3, , (b), , 3, n(n + 1)(2n + 1) - n, 2, , (c), , 4, n(n + 1)(2n + 1) - n, 5, , (d), , 2, n(n + 1)(2n + 1) - n, 3, , 102. The sum of n terms of the series 22 + 42 + 62 +........ is, (a), (c), , n( n + 1)(2n + 1), 3, n( n + 1)(2n + 1), 6, , (b), (d), , 2n(n +1)(2n +1), 3, n(n + 1)(2n + 1), 9

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EBD_7762, 110, , MATHEMATICS, , 103. If [x] stands for the greatest integer functions, then the, , 112. The sum of the first n terms of the series, , 1 ù é1, 2 ù, é1, +ê +, + ...... + é 1 + 999 ù is, value of ê +, ú, êë 2 1000 úû, ë 2 1000 û ë 2 1000 úû, equal to:, (a) 497, (b) 498, (c) 500, (d) 502, 104. Sum of the first n terms of the series, , 12 + 2.2 2 + 3 2 + 2.4 2 + 5 2 + 2.6 2 + ..., , 1 3 7 15, + + + + ......... is equal to :, 2 4 8 16, , (a) 2n – n – 1, (c) n + 2–n – 1, 105. The sum, , 3, 12, , +, , (b) 1 – 2–n, (d) 2n + 1, , 5, 12 + 22, , +, , 7, 12 + 22 + 32, , + .... upto 11-terms is:, , is, , n(n + 1)2, when n is even. When n is odd the sum is, 2, , (a), , é n(n + 1) ù, êë 2 úû, , (c), , n(n + 1)2, 4, , (a), , 18, 11, , (b), , 2, , æ 1ö, æ 1ö, 1 + 2 ç1 + ÷ + 3 ç1 + ÷ + .... is, è nø, è nø, , (a), , n2, , (c), , æ 1ö, n ç1 + ÷, è nø, , (b), 2, , æ 1ö, (d) ç1 + ÷, è nø, , 2, , 109. Find the sum up to 16 terms of the series, 13 13 + 23 13 + 23 + 33, +, +, + ...., 1, 1+ 3, 1+ 3 + 5, (a) 448, (b) 445, (c) 446, (d) None of these, 110. The sum of the series :, (2)2 + 2(4)2 + 3(6)2 + ... upto 10 terms is :, (a) 11300 (b) 11200, (c) 12100 (d) 12300, , 111. If the sum of the first ten terms of the series, , æ 3 ö2 æ 2 ö2 æ 1 ö2 2 æ 4 ö2, çç1 ÷÷ + çç2 ÷÷ + çç3 ÷÷ + 4 + çç4 ÷÷ + .......,, è 5ø è 5ø è 5ø, è 5ø, 16, m, then m is equal to :, is, 5, (a) 100, (b) 99, (c) 102, (d) 101, , 3n( n + 1), s, 2, , (d), , 16, 9, , Topic 4 : Harmonic Progression, Harmonic mean,, Relationship Between A.M., G.M. & H.M., Arithmetic - Geometric Sequences., 114. The harmonic mean of, , (a) a, , (c), n(n + 1), , (d), , 20, 22, (c), 11, 13, BEYOND NCERT, , 1, (2n3 + 12n2 + 10n – 84), 6, , (c) n3 + n2 + n, (d) None of these, 108. The sum of n terms of the series, , n2 (n + 1), 2, , 1, 1, +, + ....... upto 10 terms, is :, 1+ 2 1+ 2 + 3, , (a), , (b), , (b), , 113. The sum of the series :, 1+, , 7, 11, 11, 60, (b), (c), (d), 2, 4, 2, 11, 106. What is the sum of the first 50 terms of the series, (1 × 3) + (3 × 5) + (5 × 7) + .... ?, (a) 1,71,650, (b) 26,600, (c) 26,650, (d) 26,900, 107. The sum of the series 3.6 + 4.7 + 5.8 + ......upto (n – 2), terms, (a) n3 + n2 + n + 2, , 2, , a, a, and, is :, 1 - ab, 1 + ab, , (b), , 1, , (d), , 2 2, , 1- a b, , a, 1 - a 2 b2, a, 1 + a2b2, , 115. The H. M between roots of the equation, x2 – 10x + 11 = 0 is equal to :, (a), , 1, 5, , (b), , 5, 21, , (c), , 21, 20, , (d), , 11, 5, , 116. If x1 and x2 are the arithmetic and harmonic mean of the, roots of the equations ax2 + bx + c = 0 then quadratic, equation whose roots are x1 and x2, is –, (a) abx2 + (b2 + ac) x + bc = 0, (b) 2abx2 + (b2 + 4ac) x + 2bc = 0, (c) 2abx2 + (b2 + ac) x + bc = 0, (d) None of these, 117. The sum of, 0.2 + 0.004 + 0.00006 + 0.0000008 + ¼ ¥ is, (a), , 200, 891, , (b), , 2000, 9801, , (c), , 1000, 9801, , (d) None of these

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SEQUENCES AND SERIES, , 111, , 118. The value of x + y + z is 15 if a, x, y, z, b are in A.P. while, the value of, , 1 1 1 5, + + is if a, x, y, z, b are in H.P. Then, x y z 3, , the value of a and b are, (a) 2 and 8, (c) 3 and 7, , (b) 1 and 9, (d) None of these, , 1, 28, , (b), , 1, 39, , (c), , 1, 6, , (d), , 1, 17, , 144, ,, 15, 15 and 12 , but not necessarily in this order then, HM, GM, and AM respectively are, , 120. The AM , HM and GM between two numbers are, , (a), , 144, ,12, 15, 15, , (c) 15, 12,, , (d) 12, 15, , 121. Sum of n terms of series 12 + 16 + 24 + 40 + ... will be, (a) 2 (2n – 1) + 8n, (b) 2(2n – 1) + 6n, (c) 3 (2 n – 1) + 8n, (d) 4(2n – 1) + 8n, 1 1, 122. The fifth term of the H.P., 2, 2 , 3 , ................... will be, 2 3, , (a) 5, , 1, 5, , (b) 3, , 1, 5, , 123. If the 7th term of a H.P. is, , (c), , 1, 10, , (d) 10, , 1, 1, ,, and the 12th term is, 10, 25, , then the 20th term is, (a), , 1, 37, , (b), , 1, 41, , (c), , 1, 45, , (d), , 1, 49, , 124. If a, b, c are the sides of a triangle, then the minimum, value of, (a) 3, , a, b, c, +, +, is equal to, b+c – a c+a –b a+b–c, , (b) 6, , (c) 9, , (d) 12, , æ n, ö, 125. If a1, a2, a3 ¼ an are in H.P. and f (k) = ç å ar ÷ – ak,, çè r = 1 ÷ø, , then, , a, a, a, a1, , 2 , 3 , ¼, n are in, f (n), f (1) f (2) f (3), , (a) A.P., (c) H.P., , (b) G.P., (d) None of these, , a, , a, , (b), , 2 2, , 1– a b, , (c) a, , 1 – a 2 b2, , 1, , (d), , 1 – a 2 b2, , 128. If the arithmetic, geometric and harmonic means between, two distinct positive real numbers be A, G and H, respectively, then the relation between them is, (a) A > G > H, (b) A > G < H, (c) H > G > A, (d) G > A > H, , (a) 1, (b) 2, 130. The nth term of the series, , 144, 15, , (d) 5/2, , a, a, and, is, 1, +, ab, 1 – ab, , 129. If a, b, c, d are in H.P., then value of, , 144, (b), , 15, 12, 15, , 144, 15, , 127. The harmonic mean of, , (a), , 1, ,, 119. If sixth term of a H. P. is 1 and its tenth term is, 105, 61, then the first term of that H.P. is, , (a), , 126. 21/4. 22/8. 23/16. 24/32......¥ is equal to(a) 1, (b) 2, (c) 3/2, , 1+, , a– 2 – d – 2, b– 2 – c – 2, , (c) 3, , is, , (d) 4, , (1 + 2) (1 + 2 + 3), +, + ... is equal to, 2, 3, , (a) n2(n – 1), n +1, 2, , (c), , (b), , (n + 1)(2n + 1), 2, , (d), , n(n + 1), 2, , 131. If the arithmetic, geometric and harmonic means between, two positive real numbers be A, G and H, then, (a) A2 = GH, (b) H2 = AG, (c) G = AH, (d) G2 = AH, 132. If, , 1, 1, 1 1, +, = + , then a, b, c are in, b–a b–c a c, , (a) A.P., (c) H.P., , (b) G.P., (d) In G.P. and H.P. both, , 133. If a, b, c are in A.P. and a 2 , b2 , c 2 are in H.P., then, (a), (c), 134. If x =, , (b), , a=b=c, 2, , b = (ac / 8), , 2b = 3a + c, , (d) None of these, , ¥, , ¥, , ¥, , n=0, , n=0, , n=0, , å a n , y = å bn , z = å c n, , where a, b, c are in, , A.P and |a | < 1, | b | < 1, | c | < 1 then x, y, z are in, (a) G. P., (b) A.P., (c) H.P., (d) None of these

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EBD_7762, 112, , MATHEMATICS, , 135. If the ratio of H.M. and G.M. of two quantities is 12 : 13,, then the ratio of the numbers is, (a) 1 : 2, (b) 2 : 3, (c) 3 : 4, (d) None of these, 136. If the ratio of H.M. and G.M. between two numbers a and b, is 4 : 5, then the ratio of the two numbers will be, (a) 1 : 2, (b) 2 : 1, (c) 4 : 1, (d) 1 : 4, 137. If 0.272727...., x and 0.727272 ... are in H. P., then x must, be, (a) rational, (b) integer, (c) irrational, (d) None of these, 138. The sum of the infinite series, to, (a), , e4 + 1, (e2 - 1)2, e2 + 1, (b), (c), 2e, 2e 2, 2e2, , (d), , 2., , 3., , 4., , 5., , 7., , (c) 2, , 1, , + ..... + ¥ =, , p4, , then, 90, , the value of, , p4, p4, (b), 96, 45, 89 4, p, (c), (d) None of these, 90, 140. If the harmonic mean between a and b be H, then the, 1, 1, +, value of, is, H -a H -b, , (a), , 8., , (d), , +, , a+b, , (b) ab, , (c), , 1 1, +, a b, , (d), , 1 1, a b, , 2e2, , 1, 2, , If in an AP, Sn = qn2 and Sm = qm2, where Sr denotes the, sum of r terms of the AP, then Sq equals to, q3, (b) mnq, (c) q3, (d) (m + n)q2, 2, Let Sn denote the sum of first n terms of an A.P. If S2n = 3, Sn, then the ratio S3n/ Sn is equal to :, (a) 4, (b) 6, (c) 8, (d) 10, x, 1, –, x, The minimum value of 4 + 4 , x Î R is, (a) 2, (b) 4, (c) 1, (d) 0, , (a), , 6., , 1, 3, , 1, , (e2 + 1)2, , If the sum of n terms of an AP is given by Sn = 3n +, then the common difference of the AP is, (a) 3, (b) 2, (c) 6, (d) 4, The third term of a geometric progression is 4. The product, of the first five terms is :, (a) 43, (b) 45, (c) 44, (d) 47, If 9 times the 9th term of an AP is equal to 13 times the 13th, term, then the 22nd term of the AP is, (a) 0, (b) 22, (c) 198, (d) 220, If x, 2y and 3z are in AP where the distinct numbers, x, y, and z are in GP, then the common ratio of the GP is, (b), , +, , (a), , 2n2,, , (a) 3, , 1, , 14 2 4 34, 1 1, 1, + 4 + 4 + ......¥ is, 4, 1, 3, 5, , 2 2 24 26, +, +, + ... is equal, 2! 4! 6!, , NCERT Exemplar MCQs, 1., , 139. If, , Let Sn denote the sum of the cubes of the first n natural, numbers and Sn denote the sum of the first n natural, n, , numbers, then, , S, , å sr, , equals to, , r =1 r, , (a), , n(n + 1)(n + 2), 6, , (b), , n(n + 1), 2, , (c), , n2 + 3n + 2, 2, , (d) None of these, , If tn denotes the nth term of the series 2 + 3 + 6 + 11 + 18 + ...,, then t50 is, (a) 492 – 1 (b) 492, (c) 502 + 1 (d) 492 + 2, 10. The lengths of three unequal edges of a rectangular solid, block are in GP. If the volume of the block is 216 cm3 and, the total surface area is 252 cm2, then the length of the, longest edge is, (a) 12 cm (b) 6 cm, (c) 18 cm (d) 3 cm, 9., , Past Year MCQs, 11., , Let a and b be the roots of equation px2 + qx + r = 0,, 1 1, p ¹ 0. If p, q, r are in A.P and + = 4, then the value of, a b, | a – b| is:, [JEE MAIN 2014, A]

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SEQUENCES AND SERIES, , (a), , 34, 9, , 2 13, 9, , (b), , 9, , 1, , ( ), , 113, , (c), 2, , 61, 9, , (d), , 7, , 8, 12. If (10 ) + 2 (11) 10 + 3 (11) (10 ) + ....., 9, , (a) 1, (c) 0, , 2 17, 9, , (b) 2, (d) None of these, , 20. The value of, , 3 15 63, + +, +... upto n terms is, 4 16 64, , 9, , +10 (11) = k (10 ) , then k is equal to:, , [BITSAT 2015, A], , [JEE MAIN 2014, A], (a) 100, , (b) 110, , (c), , 121, 10, , (d), , 441, 100, , 13. Three positive numbers form an increasing G. P. If the, middle term in this G.P. is doubled, the new numbers are, in A.P. then the common ratio of the G.P. is:, [JEE MAIN 2014, C], (a), , 2- 3, , (b), , 2+ 3, , (c), (d) 3 + 2, 2+ 3, 14. The product of n positive numbers is unity, then their sum, is :, [BITSAT 2014, A], (a) a positive integer, (b) divisible by n, (c) equal to n +, , 1, n, , (d) never less than n, , (a), , (c), , (b), , [JEE MAIN 2015, C], (a) 4, (b) 4, (c) 4 l2 mn (d) 4 lm2n, 19. If binomial coefficients of three consecutive terms of, (1 + x)n are in HP, then the maximum value of n is, [BITSAT 2015, C], lmn2, , l2 m2 n 2, , is, , [BITSAT 2015, A], , (b), , 7, 4, , (c), , 8, 5, , (d), , 4, 3, , 23. If the sum of the first ten terms of the series, , æ 3 ö2 æ 2 ö2 æ 1 ö2 2 æ 4 ö2, çç1 ÷÷ + çç2 ÷÷ + çç3 ÷÷ + 4 + çç4 ÷÷ + .......,, è 5ø è 5ø è 5ø, è 5ø, is, , 16, m, then m is equal to :, 5, , (a) 100, , a b c, , between l and n, then G14 + 2G 24 + G 34 equals., , 4- n 1, 3, 3, , n+, , (a) 1, , 2 2 2, , 16. If the (2p)th term of a H.P. is q and the (2q)th term is p,, then the 2(p + q)th term is[BITSAT 2014, C], pq, 2pq, (a), (b), 2(p + q), p+q, pq, p+q, (c), (d), p+q, pq, 17. The sum of first 9 terms of the series., 13 13 + 23 13 + 23 + 33, +, +, + ...., [JEE MAIN 2015, A], 1 1+ 3, 1+ 3 + 5, (a) 142, (b) 192, (c) 71, (d), 96, 18. If m is the A.M. of two distinct real numbers l and, n(l, n > 1) and G1, G2 and G3 are three geometric means, , n+, , (a) – 1, (b) 1, (c) – i, (d) i, nd, th, th, 22. If the 2 , 5 and 9 terms of a non-constant A.P. are in, G.P., then the common ratio of this G.P. is :, [JEE MAIN 2016, C], , b 2 – ac, , (d) None of these, , (b), , æ 1 3 9 27, ö, +.... ÷, ç + + +, 2 8 32 128, è, ø, w+ w, , 1 1 1, 1 1 1, are A. P., then æç + - ö÷ æç 1 + 1 - 1 ö÷ is, , ,, a b c, èa b cø èb c aø, equal to, [BITSAT 2014, A], , 4, 3, –, ac, b2, 1, 4, –, ac, b2, , 4, 1, 3 3, , 4n 1, 4- n 1, +, (d) n 3 3, 3, 3, 21. If w is the complex cube root of unity, then the value of, , (c), , 15. If, , (a), , n-, , n, , (b) 99, , (c) 102, 2, , 2, , 24., , [JEE MAIN 2016, A], , 1ö æ 2 1, æ, ç x + ÷ + çç x + 2, xø è, è, x, , (d), , 101, , 2, , ö, æ, 1 ö, ÷÷ + çç x 3 +, ÷÷ ....upto n terms is, ø, x3 ø, è, , [BITSAT 2016, C], (a), , (b), , (c), , x 2n – 1, 2, , x –1, x 2n + 1, x 2 +1, x 2n – 1, x 2 –1, , ×, , ×, , ×, , x 2n, x, , +2, , +1, , 2n, , x 2n + 2 – 1, x 2n, x 2n – 1, x 2n, , + 2n, , – 2n, , – 2n, , (d) None of these, 25. If log a, log b, and log c are in A.P. and also log a – log 2b,, log 2b – log 3c, log 3c – log a are in A.P., then, [BITSAT 2016, A], (a) a, b, c, are in H.P., (b) a, 2b, 3c are in A.P., (c) a, b, c are the sides of a triangle, (d) None of these

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EBD_7762, 114, , 26., , MATHEMATICS, , The sum to n terms of the series, 32. If, , 1, 3, 7, 15, +, +, +, + .............. is, 2, 4, 8, 16, , (a) n – 1 – 2 – n, , [BITSAT, T 2016, A], , –n, , (c) n – 1 + 2, (d) 1 + 2, The fourth term of an A.P. is three times of the first term, and the seventh term exceeds the twice of the third term by, one, then the common difference of the progression is, [BITSAT 2016, C], (a) 2, , (b) 3, , (c), , 3, 2, , (d) –1, , 28. For any three positive real numbers a, b and c,, 9(25a2 + b2) + 25(c2 – 3ac) = 15b(3a + c). Then :, [JEE MAIN 2017, A], (a) a, b and c are in G.P., (b) b, c and a are in G.P., (c) b, c and a are in A.P., (d) a, b and c are in A.P., 29. Let a, b, c Î R. If f(x) = ax2 + bx + c is such that a + b + c, = 3 and f(x + y) = f(x) + f(y) + xy, " x, y Î R, then, , 10, , å f (n), , n =1, , 30., , is equal to :, [JEE MAIN 2017, S], (a) 255, (b) 330, (c) 165, (d) 190, 1/4, 2/8, 3/16, 4/32, 2 . 2 . 2 . 2 ......¥ is equal to-[BITSAT 2017, A], (a) 1, , 31., , (b) 2, , å, , (c), , 3, 2, , k =1, , (a), , 4, th of its, 5, height from which it has fallen. The total distance that the, ball travels before coming to rest if it is gently released, from a height of 120 m is, [BITSAT 2017, A], (a) 960 m (b) 1000 m (c) 1080 m (d) Infinite, , After striking the floor a certain ball rebounds, , -, , 1, 4, , (b), , -, , 1, 2, , (c), , 1, 2, , (d), , 1, 4, , 33. If A, B, C are the angles of a triangle and e iA , e iB , e iC are, in A.P. Then the triangle must be, BITSAT 2017, A], (a) right angled, (b) isosceles, (c) equilateral, (d) None of these, 12, , 34. Let a1,a 2 ,a 3 ,..., a 49 be in A.P. such that, , å a 4k +1 = 416, , k =0, , 2, and a 9 + a 43 = 66. If a12 + a 22 + ... + a17, = 140m , then m is, equal to :, [JEE MAIN 2018, S], (a) 68, (b) 34, (c) 33, (d) 66, 35. Let A be the sum of the first 20 terms and B be the sum of, the first 40 terms of the series, , 12 + 2 × 22 + 32 + 2.4 2 + 52 + 2.62 + ........., , If B - 2A = 100 l , then l is equal to :, [JEE MAIN 2018, A], (a) 248, (b) 464, (c) 496, (d) 232, 36. If a, b, c are in G.P., then, [BITSAT 2018, C], (a) a2, b2, c2 are in G.P., (b), , (d) 5/2, , k ( k + 1) (k - 1) = pn4 + qn3 + tn2 + sn, where p, q, t, , and s are constants, then the value of s is equal to, [BITSAT 2017, A], , (b) 1, , –n, , 27., , n, , a 2 (b + c), c 2 (a + b), b2 (a + c) are in G.P.., , a, b, c, ,, ,, are in G.P.., b+c c + a a +b, (d) None of these, , (c)

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SEQUENCES AND SERIES, , 115, , Exercise 3 : Try If You Can, 1., , If the roots of cubic equation x3 – 6x2 + b1x – b2 = 0 are in, A.P. with positive integral common difference, then the, maximum value of (b1 + b2), is, (a) 17, , 2., , (b) 18, , (c) 19, , 8., , Let (an), n = 1, 2, 3... be the sequence of real numbers such, , æ n + 1ö, that a1 = 2 and an = ç, (a + a2 + .... + an – 1), (n ³ 2),, è n –1÷ø 1, , (d) 20, , then which of the following is not correct ?, , The numbers 1, 4, 16 can be three terms (not necessarily, consecutive) of, (a) no A.P., (b) only one G.P., , (a), , é a2016 ù, ê 2016 ú = 1008, ë2, û, , (b), , é a2016 ù, ê 2016 ú = 2016, ë2, û, , (c) Infinite number of A.P.'s (d) only one A.P., 3., , It is given that, , 1, 2 n sin a, , , 1,, , 2n, , sin a are in A.P. for some, , (c) a2015 is divisible by 22019, , value of a. Let say for n = 1, the a satisfying the above A.P., , (d) If a1 + a2 + ..... + an > 1008, then least value of n is 8., , ¥, , is a1, for n = 2, the value is a2 and so on. If S =, , å sin ai ,, , i =1, , [Where [.] denotes greatest integer function], 9., , then the value of S is, , 4., , 5., , 6., , 7., , Consider the sequence of numbers 121, 12321, 1234321,...., Each term in the sequence is, (a) a prime number, (b) square of an odd number, , 1, 2, , (a), , 1, , (b), , (c), , 2, , (d) None of these, , (c) divisible by 11, (d) form a G.P., , If three successive terms of a G..P. with common ratio, r (r > 1) form the sides of a D ABC and [r] denotes greatest, integer function, then [r] + [– r] =, (a) 0, , (b) 1, , (c) – 1, , (d) None of these, , {(sin, , 10. If e, , 2, , ) }, , x + sin 4 x + sin6 x + ¼ up to ¥ ln 2, , x2 – 17x + 16 = 0 and k =, , satisfies the equation, , 2 cos x, (0 < x < p ) , then, sin x + 2 cos x, 2, , ‘k’ belongs to, , If the numbers 32a – 1, 14, 34 – 2a (0 < a < 1) are the first, three terms of an AP, then its fifth term is equal to, , é 71 61 ù, (a) ê150 , 99 ú, ë, û, , é 71 61 ù, (b) ê140 , 79 ú, ë, û, , (a) 33, , é 71 61 ù, (c) ê190 , 130 ú, ë, û, , (d) None of these, , (b) 43, , (c) 53, , (d) 63, , An infinite G.P. has first term ‘x’ and sum ‘5’, then x belongs, to, (a) x < – 10, , (b) – 10 < x < 0, , (c) 0 < x < 10, , (d) x > 10, , 1, 2, The maximum sum of the series 20 + 19 + 18 + 18 + ...., 3, 3, , is, (a) 300, , 11., , Let ABCDEF be a convex hexagon in which the diagonals, AD, BD, CF are concurrent at O. Suppose the area of the, triangle OAF is the geometric mean of those of OAB and, OCD, then area of the triangle OCD, OED and OEF are in, (a) AP, , (b) HP, , (c) GP, , (d) None of these, n, , 12. If Sn = (1 + 3-1 )(1 + 3-2 )(1 + 3-4 )(1 + 3-8 )...(1 + 3-2 ), then, (b) 310, , (c) 311, , 2, 3, , (d) 333, , 1, 3, , S¥ is equal to, (a) 1, , (b), , 1, 2, , (c), , 3, 2, , (d) None

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EBD_7762, 116, , MATHEMATICS, , 2, 13. The sequence {xk} is defined by x k + 1 = xk + xk and, , 1, é 1, 1, 1 ù, +, + ... +, . Then ê, ú (where [.], 2, x100 + 1 û, ë x1 + 1 x2 + 1, , x1 =, , denotes the greatest integer function) is equal to, (a) 0, , (b) 2, , (c) 4, , (d) 1, , (a) 21, , (b) 20, , (c) 19, , (d) None, , 15. If f is a function satisfying f (x + y) = f (x) f (y) for all, , x, y Î N . such that f (1) = 3 and, , find the, , x =1, , value of n., (a) 2, , n, , å f ( x) = 120 ,, , (b) 4, , (c) 6, , (d) 8, , 14. The least value of n (a natural number), for which the sum, S of the series 1 +, , 1, 1, 1, + 2 + 3 + ....... differs from Sn, 2, 2, 2, , by a quantity < 10 – 6 , is, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, , (b), (b), (a), (a), (a), (c), (c), (d), (d), (c), (b), (b), (d), (d), , 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, , (b), (d), (b), (b), (a), (c), (b), (c), (c), (a), (a), (b), (a), (b), , 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, , (d), (d), (a), (d), (b), (a), (a), (c), (c), (b), (a), (a), (a), (b), , 1, 2, 3, 4, , (d), (c), (a), (b), , 5, 6, 7, 8, , (c), (b), (b), (a), , 9, 10, 11, 12, , (d), (a), (b), (a), , 1, 2, , (a), (c), , 3, 4, , (a), (c), , 5, 6, , (c), (c), , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (c) 57, (d), (b) 85, 43, 71, (c) 58, (d), (b) 86, 44, 72, (a) 59, (c), (b) 87, 45, 73, (c) 60, (b), (b) 88, 46, 74, (b), (b) 89, 47 (b) 61, 75, (b), (c) 90, 48 (b) 62, 76, (c), (c) 91, 49 (d) 63, 77, (a) 64, (d), (d) 92, 50, 78, (c) 65, (b), (d) 93, 51, 79, (c), (c) 94, 52 (b) 66, 80, (c), (a) 95, 53 (b) 67, 81, (a) 68, (b), (d) 96, 54, 82, (a) 69, (b), (c) 97, 55, 83, (c) 70, (b), (c) 98, 56, 84, Exercise 2 : Exemplar & Past Year MCQs, (d), (a) 25, 13 (b) 17, 21, (d), (d) 26, 14 (d) 18, 22, (a) 19, (d), (d) 27, 15, 23, (b), (a) 28, 16 (d) 20, 24, Exercise 3 : Try If You Can, (b), (b), (c) 13, 7, 9, 11, (b) 10 (a), (c) 14, 8, 12, , (c), (d), (b), (d), (c), (a), (a), (b), (a), (a), (c), (a), (a), (c), , 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, , (c), (b), (d), (b), (c), (c), (c), (a), (b), (a), (c), (c), (d), (b), , 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, , (c), (a), (d), (b), (b), (b), (c), (a), (d), (d), (d), (a), (c), (b), , (c), (c), (a), (c), , 29, 30, 31, 32, , (b), (b), (c), (b), , 33, 34, 35, 36, , (c), (b), (a), (a), , (d), (a), , 15, , (b), , 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, , (c), (a), (c), (c), (d), (c), (a), (c), (d), (c), (a), (c), (a), (c)

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10, , STRAIGHT LINES, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 4, , 3, JEE MAIN, BITSAT, , 2, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 2, 2, , Critical Concepts, , Slope of a line, Angle between two, lines, Collinearity of Three points,, Point- slope form, Two point form,, Slope- Intercept form, Distance Of a, point from a line, Distance between, two parallel lines, Pair of Straight Lines, , Rating of, Difficulty Level, , 3.7/5, , CUS, (Chapter Utility Score), Out of 10, 5.8

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STRAIGHT LINES, , 119

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EBD_7762, 120, , MATHEMATICS, , Topic 1 : Cartesnia Co-ordinate System, Distance Formula,, Section Formula, Area of Triangle, 1., , 2., , 3., , Which one of the following is the nearest point on the line, 3x– 4y = 25 from the origin?, (a) ( –1, –7), (b) (3, – 4), (c) ( –5, –8), (d) (3, 4), If the point P(x, y) is equidistant from the points A(a + b, b, – a) and B(a – b, a + b), then, (a) ax = by, (b) bx = ay and P can be (a, b), (c) x2 – y2 = 2(ax + by), (d) None of the above, If the point [x1 + t (x2 – x1), y1 + t (y2 – y1)] divides the join, , of (x1, y1) and (x2 , y2) internally then, 4., , 5., , (a) t < 0 (b) 0 < t < 1 (c) t > 1, (d) t = 1, Let A (1, k), B(1, 1) and C (2, 1) be the vertices of a right, angled triangle with AC as its hypotenuse. If the area of the, triangle is 1square unit, then the set of values which 'k' can, take is given by, (a) {–1, 3} (b) {–3, –2} (c) {1, 3}, (d) {0, 2}, The point A divides the join of P (-5, 1) and Q (3, 5) in the, ratio k : 1. The two values of k for which the area of DABC,, where B (1, 5), C (7, - 2) is equal to 2 sq. units, are, 30, (b) 7, 31, (c) 4, 31, (d) 7, 31, 9, 9, 3, 9, Assertion: If A (– 2, –1), B (4, 0), C (3, 3) and D (–3, 2) are the, vertices of a parallelogram, then mid-point of, AC = Mid-point of BD, Reason: The points A, B and C are collinear Û Area of, DABC = 0., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., ABC is an isosceles triangle. If the coordinates of the base, are B(1, 3) and C(–2, 7), the coordinates of vertex A can be, (a) (1, 6), (b) (–1/2, 5), (c) (5/6, 6), (d) None of these, , (a), , 6., , 7., , 8., , 7,, , The three points [(a + b)(a + 2b), (a + b)] ,, [(a + 2b)(a + 3b), (a + 2b)] and [(a + 3b)(a + 4b), (a + 3b)], (a) are collinear, , 9., , (b) form a traingle whose area is independent of a, (c) form a triangle whose area is independent of b, (d) form a triangle whose area is independent of a, b, If x1, y1 are roots of x2 + 8x – 20 = 0, x2, y2 are the roots of 4x2, + 32x – 57 = 0 and x3, y3 are the roots of 9x2 + 72x – 112 = 0,, then the points (x1, y1), (x2, y2) and (x3, y3), (a) are collinear, (b) form an equilateral triangle, (c) form a right angled isosceles triangle, (d) are concyclic, , Topic 2 : Slope of a Line, Parallel and Perpendicular Lines,, Various Forms of Equations of a Line, 10. Slope of non-vertical line passing through the points (x1, y1), and (x2, y2) is given by :, (a), , m=, , y2 - y1, x2 - x1, , (b), , m=, , x2 - x1, y2 - y1, , (c), , m=, , x2 + x1, y2 + y1, , (d), , m=, , y2 + y1, x2 + x1, , If a line makes an angle a in anti-clockwise direction with, the positive direction of x-axis, then the slope of the line is, given by :, (a) m = sin a, (b) m = cos a, (c) m = tan a, (d) m = sec a, 12. If p1, p2 are the lengths of the normals drawn from the origin, on the lines, x cos q + y sin q = 2a cos 4q and, x sec q + y cosec q = 4a cos 2q, 11., , respectively, and mp12 + np22 = 4a 2 . Then, (a) m = 1, n = 1, (b) m = 1, n = 4, (c) m = 4, n = 1, (d) m = 1, n = – 1, 13. The point (x, y) lies on the line with slope m and through the, fixed point (x0, y0) if and only if its coordinates satisfy the, equation y – y0 is equal to ......... ., (a) m(x – x0), (b) m(y – x0), (c) m(y – x), (d) m(x – y0), 14. If a line with slope m makes x-intercept d. Then equation of, the line is :, (a) y = m(d – x), (b) y = m(x – d), (c) y = m(x + d), (d) y = mx + d, 15. If the medians from A and B of the triangle with vertices, A (0, b), B (0, 0) and C (a, 0) are mutually perpendicular then, (a) a2 = b2, (b) a2 = 2b2, 2, 2, (c) a = 4b, (d) 2a2 = b2

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STRAIGHT LINES, , 121, , 16. The inclination of the line x – y + 3 = 0 with the positive, direction of x-axis is, (a) 45°, (b) 135°, (c) –45°, (d) –135°, 17. Equation of the straight line making equal intercepts on the, axes and passing through the point (2, 4) is :, (a) 4x – y – 4 = 0, (b) 2x + y – 8 = 0, (c) x + y – 6 = 0 (d), x + 2y – 10 = 0, 18. ABCD is a rhombus. Its diagonals AC and BD intersect at, the point M and satisfy BD = 2AC. If the coordinates of D, and M are (1, 1) and (2, –1) respectively, then the coordinates, of A are, 1ö, æ 3, (a) (1, 3) or ç - , - ÷, 2ø, è 2, , (b), , (c) (1, 2) or (1, –1), , (d), , 3ö, 1ö, æ, æ, ç1, - ÷ or ç 3, - ÷, 2ø, 2ø, è, è, 1ö, 3ö, æ, æ, ç1, - ÷ or ç 3, - ÷, 2, 2ø, ø, è, è, , 19. A line passes through P (1, 2) such that its intercept between, the axes is bisected at P. The equation of the line is, (a) x + 2y = 5, (b) x – y + 1 = 0, (c) x + y – 3 = 0 (d), 2x + y – 4 = 0, 20. The intercept cut off by a line from y-axis twice than that, from x-axis, and the line passes through the point (1, 2). The, equation of the line is, (a) 2x + y = 4, (b) 2x + y + 4 = 0, (c) 2x – y = 4, (d) 2x – y + 4 = 0, 21. Consider the equation, , 3x + y - 8 = 0, , I., , Normal form of the given equation is, cos 30°x + sin 30° y = 4, II. Values of p and w are 4 and 30° respectively., Choose the correct option., (a) Only I is true, (b) Only II is true, (c) Both are true, (d) Both are false, 22. Line through the points (–2, 6) and (4, 8) is perpendicular to, the line through the points (8, 12) and (x, 24). Find the value, of x., (a) 2, (b) 3, (c) 4, (d) 5, 23. The lines x + 2y – 5 = 0, 2x – 3y + 4 = 0, 6x + 4y – 13 = 0, (a) are concurrent., (b) form a right angled triangle., (c) form an isosceles triangle., (d) form an equilateral triangle., 24. Locus of mid point of the portion between the axes of, x cos a + y sina = p whre p is constant is, (a) x2 + y2 =, (c), , 1, x, , 2, , +, , 1, y, , 2, , 4, , (a) (– 4, 2), , 2, p2, , (d), , 1, x2, , +, , 1, y2, , =, , 4, , (b) (4, – 2), , (c) (0, 4), , (d) (0, – 4), , 26. The relation between a, b, a' and b' such that the two lines, ax + by = c and a'x + b'y = c' are perpendicular is, (a) aa¢ – bb¢ = 0, , (b) aa¢ + bb¢ = 0, , (c) ab + a¢b¢ = 0, , (d) ab – a¢b¢ = 0, , 27. The equation of a straight line which cuts off an intercept of, 5 units on negative direction of y-axis and makes an angle, of 120º with the positive direction of x-axis is, (a), , 3x + y + 5 = 0, , (b), , 3x + y - 5 = 0, , (c), , 3x - y - 5 = 0, , (d), , 3x - y + 5 = 0, , 28. The equation of the straight line that passes through the, point (3, 4) and perpendicular to the line 3x + 2y + 5 = 0 is, (a) 2x + 3y + 6 = 0, , (b) 2x – 3y – 6 = 0, , (c) 2x – 3y + 6 = 0, , (d) 2x + 3y – 6 = 0, , 29. Assertion: Slope of the line passing through the points, (3, –2) and (3, 4) is 0., Reason: If two lines having the same slope pass through a, common point, then these lines will coincide., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 30. If the mid-point of the section of a straight line intercepted, between the axes is (1, 1), then what is the equation of this, line?, (a) 2x + y = 3, , (b) 2x – y = 1, , (c) x – y = 0, , (d) x + y = 2, , 31. If the points ( x, y), (1, 2) and ( –3, 4) are collinear, then, (a) x + 2y – 5 = 0, , (b) x + y – 1 = 0, , (c) 2x + y – 4 = 0, (d) 2x – y + 10 = 0, 32. The equation of the straight line passing through the point, (4, 3) and making intercepts on the co-ordinate axes whose, sum is –1 is, (a), , x y, x y, - = 1 and, + =1, 2 3, -2 1, , (b), , x y, x y, - = -1 and, + = -1, 2 3, -2 1, , (c), , x y, x y, + = 1 and + = 1, 2 3, 2 1, , (d), , x y, x y, + = -1 and, + = -1, 2 3, -2 1, , (b) x2 + y2 = 4p2, , p2, =, , 25. A triangle ABC is right angled at A has points A and B as, (2, 3) and (0, –1) respectively. If BC = 5, then point C may be, , p2

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EBD_7762, 122, , 33., , MATHEMATICS, , The length of the perpendicular from the origin to a line is 7, and line makes an angle of 150° with the positive direction, of y-axis then the equation of the line is, (a) 4x + 5y = 7, (b) –x + 3y = 2, , 3x - y = 10 2, , (c), 34., , 35., , 36., , 37., , (d), , 3x + y = 14, , If the lines 2(sin a + sin b) x – 2sin(a – b) y = 3 and 2(cos a, + cos b)x + 2cos(a – b)y = 5 are perpendicular, then sin2a +, sin2b is equal to, (a) sin(a – b) – 2sin (a + b), (b) sin 2(a – b) – 2sin (a + b), (c) 2sin(a – b) – sin(a + b), (d) sin2(a – b) – sin(a + b), A straight line makes an angle of 135° with x-axis and cuts, y-axis at a distance of – 5 from the origin. The equation of the, line is, (a) 2x + y + 5 = 0, (b) x + 2y + 3 = 0, (c) x + y + 5 = 0 (d), x + y+ 3 = 0, The lines a1x + b1y + c1= 0 and a2x + b2y + c2 = 0 are, perpendicular to each other, (a) a1b1 – b1a2 = 0, (b) a12b2 + b12a2 = 0, (c) a1b1 + a2b2 = 0, (d) a1a2 + b1b2 = 0, If the coordinates of the points A and B be (3, 3) and, (7, 6), then the length of the portion of the line AB, intercepted between the axes is, (a), , 5, 4, , (b), , 10, 4, , 13, (d) None of these, 3, The line (3x – y + 5) + l (2x – 3y – 4) = 0 will be parallel, to y-axis, if l =, (c), , 37., , 1, –1, 3, –3, (b), (c), (d), 3, 3, 2, 2, The equation of a straight line passing through (–3, 2), and cutting an intercept equal in magnitude but opposite, in sign from the axes is given by, (a) x – y + 5 = 0, (b) x + y – 5 = 0, (c) x – y – 5 = 0, (d) x + y + 5 = 0, The points A(1, 3) and C(5, 1) are the opposite vertices, of rectangle. The equation of line passing through other, two vertices and of gradient 2, is, (a) 2x + y – 8 = 0, (b) 2x – y – 4 = 0, (c) 2x – y + 4 = 0, (d) 2x + y + 7 = 0, If p1, p2 are the lengths of the normals drawn from the origin, on the lines, x cos q + y sin q = 2a cos 4q and, x sec q + y cosec q = 4a cos 2q, , (a), , 38., , 39., , 40., , respectively, and mp12 + np22 = 4a 2 . Then, (a) m = 1, n = 1, (c) m = 4, n = 1, , (b) m = 1, n = 4, (d) m = 1, n = – 1, , 41. Consider the following statements., The three given points A, B, C are collinear i.e., lie on the, same straight line, if, I. area of DABC is zero., II. slope of AB = Slope of BC., III. any one of the three points lie on the straight line joining, the other two points., Choose the correct option, (a) Only I is true, (b) Only II is true, (c) Only III is true, (d) All are true, 42. The value of x for which the points (x, –1), (2, 1) and (4, 5) are, collinear, is, (a) 1, (b) 2, (c) 3, (d) 4, 43. The values of k for which the line, (k – 3) x – (4 – k2)y + k2 – 7k + 6 = 0 is parallel to the x-axis,, is, (a) 3, (b) 2, (c) 1, (d) 4, 45. The line joining (–1, 1) and (5, 7) is divided by the line, x + y = 4 in the ratio 1 : k. The value of ‘k’ is, (a) 2, (b) 4, (c) 3, (d) 1, 46. Equation of the hour hand at 4 O' clock is, (a), , x - 3y = 0, , (b), , 3 x- y =0, , (c), , x + 3y = 0, , (d), , 3x+ y =0, , 47. If three points (h, 0), (a, b) and (0, k) lies on a line, then the, value of, , a b, + is, h k, , (a) 0, (b) 1, (c) 2, (d) 3, 48. Value of x so that 2 is the slope of the line through (2, 5) and, (x, 3) is, (a) 0, (b) 1, (c) 2, (d) 3, 49. The point (t2 + 2t + 5, 2t2 + t – 2) lies on the line x + y = 2 for, (a) All real values of t, (b) Some real values of t, (c), , t=, , -3 ± 3, 6, , (d) None of these, , 50. Equation of a line is 3x – 4y + 10 = 0, 3, I. Slope of the given line is ., 4, II. x-intercept of the given line is - 10 ., 3, 5, III. y-intercept of the given line is ., 2, Choose the correct option., (a) Only I and II are true, (b) Only II and III are true, (c) Only I and III are true, (d) All I, II and III are true

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STRAIGHT LINES, , 123, , 51. What is the value of y so that the line through (3, y) and, (2, 7) is parallel to the line through (– 1, 4) and (0, 6)?, (a) 6, (b) 7, (c) 5, (d) 9, 52. Reduce the equation 3x + y - 8 = 0 into normal form. The, value of p is, (a) 2, (b) 3, (c) 4, (d) 5, 53. In what ratio does the line y – x + 2 = 0 cut the line joining, (3, –1) and ( 8, 9)?, (a) 2 : 3 (b) 3 : 2, (c) 3 : –2, (d) 1 : 2, 54. A line passes through (2, 2) and is perpendicular to the line, 3x + y = 3. Its y – intercept is:, 4, 2, 1, (b), (c) 1, (d), 3, 3, 3, 55. The straight lines x + 2y – 9 = 0, 3x + 5y – 5 = 0 and, ax + by = 1 are concurrent if the straight line 35x – 22y + 1 = 0, passes through :, (a) (a, b) (b) (b, a), (c) (a, – b) (d) (– a, b), 56. The equation of straight line passing through (–a, 0) and, making a triangle with the axes of area T is, (a) 2Tx + a2y + 2aT = 0, (b) 2Tx – a2y + 2aT = 0, (c) 2Tx – a2y – 2aT = 0, (d) None of these, 57. The reflection of the point (4, – 13) in the line, 5x + y + 6 = 0 is, (a) (–1, –14), (b) (3, 4), (c) (0, 0), (d) (1, 2), 58. If a, b, c are in A.P., then the straight lines ax + by + c = 0 will, always pass through, (a) (1, – 2), (b) (1, 2), (c) (– 1, 2), (d) (– 1, – 2), 59. Let (h, k) be a fixed point where h > 0, k > 0. A straight line, passing through this point cuts the positive direction of the, coordinate axes at the points P and Q. Then the minimum, area of the DOPQ. O being the origin, is, (a) 4hk sq. units, (b) 2hk sq. units, (c) 3hk sq. units, (d) None of these, 60. If (– 4, 5) is one vertex and 7x – y + 8 = 0 is one diagonal of, a square, then the equation of second diagonal is, (a) x + 3y = 21, (b) 2x – 3y = 7, (c) x + 7y = 31, (d) 2x + 3y = 21, 61. A ray of light coming from the point (1, 2) is reflected at a, point A on the x-axis and then passes through the point, (5, 3). The co-ordinates of the point A is, , 63., , 64., , (a), , (a), , æ 13 ö, ç , 0÷, ø, è 5, , (b), , ö, æ 5, ç , 0÷, 13, ø, è, , (c) (–7, 0), (d) None of these, 62. A line L intersects the three sides BC, CA and AB of a DABC, at P, Q and R respectively. Then,, , BP CQ AR, ×, ×, is equal to, PC QA RB, , 65., , 66., , (a) 1, (b) 0, (c) –1, (d) None of these, The equation of two equal sides of an isosceles triangle are, 7x – y + 3 = 0 and x + y – 3 = 0 and its third side passes through, the point (1, – 10), then the equation of the third side is (are), (a) 3x + y + 7 = 0, x – 3y – 31 = 0, (b) 2x + y + 5 = 0, x – 2y + 3 = 0, (c) 3x + y + 7 = 0, x + y = 0, (d) 3x – y = 7, x + 3y = 15, The lines p(p2 +1) x – y + q = 0 and, (p2 + 1)2x + (p2 + 1) y + 2q = 0 are perpendicular to a common, line for, (a) exactly one value of p, (b) exactly two values of p, (c) more than two values of p, (d) no value of p, The equations of the lines which cuts off an intercept, –1 from y-axis and equally inclined to the axes are, (a) x – y + 1 = 0, x + y + 1 = 0, (b) x – y – 1 = 0, x + y – 1 = 0, (c) x – y – 1 = 0, x + y + 1 = 0, (d) None of these, If the coordinates of the points A, B, C be (–1, 5), (0, 0), and (2, 2) respectively and D be the middle point of BC,, then the equation of the perpendicular drawn from B to, the line AD is, (a) x + 2y = 0, (b) 2x + y = 0, (c) x – 2y = 0, (d) 2x – y = 0, x y, + = 1 meets the axis of y and axis of x at A and, 3 4, B respectively. A square ABCD is constructed on the line, segment AB away from the origin, the coordinates of the, vertex of the square farthest from the origin are, (a) (7, 3) (b) (4, 7), (c) (6, 4), (d) (3, 8), The line parallel to the x-axis and passing through, the intersection of the lines ax + 2by + 3b = 0 and, bx – 2ay – 3a = 0, where (a, b) ¹ (0, 0) is, , 67. The line, , 68., , 3, from it, 2, 2, (b) Above the x-axis at a distance of, from it, 3, , (a) Above the x-axis at a distance of, , (c) Below the x-axis at a distance of, , 3, from it, 2, , 2, from it, 3, 69. If A and B are two points on the line 3x + 4y + 15 = 0 such, that OA = OB = 9 units, then the area of the triangle OAB, is, , (d) Below the x-axis at a distance of, , (a) 18 sq. units, (c), , 18, 2, , sq. units, , (b) 18 2 sq. units, (d) None of these

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EBD_7762, 124, , MATHEMATICS, , Topic 3 : Distance Between Two Lines, Angle Between, Two Lines, Perpendicular, Distance of a Point From a Line, 70. The perpendicular distance (d) of a line Ax + By + C = 0, from a point (x1, y1) is given by :, (a), , d=, , Ax1 + By1 + C, A 2 + B2, 2, , (b) d =, , Ax1 - By1 + C, A 2 + B2, , 2, , 2, , 2, , A +B, A +B, (d) d =, Ax1 + By1 + C, Ax1 - By1 + C, 71. Distance between the parallel lines, , (c) d =, , Ax + By + C1 = 0 and Ax + By + C2 = 0, is given by:, (a), , d=, , (c) d =, , A 2 + B2, C1 - C2, C1 - C 2, A 2 + B2, , (b) d =, (d) d =, , A 2 - B2, C1 - C2, C1 + C2, A 2 + B2, , 72. Given a family of lines a(2x + y + 4) + b(x – 2y – 3) = 0,, the number of lines belonging to the family at a distance, , 10 from P(2, –3) is, (a) 0, , (b) 1, , (c) 2, , (d) 4, , (a) x – y + 1 = 0, , (b) 2x – y + 3 = 0, , (c) x + 2y – 2 = 0, , (d) x + y – 2 = 0, , 74. Let the perpendiculars from any point on the line, 7x + 56y = 0 upon 3x + 4y = 0 and 5x – 12y = 0 be p and p¢,, then, (a) 2p = p¢, , (b) p = 2p¢, , (c) p = p', , (d) None of these, , Assertion: Equation of the horizontal line having distance, ‘a’ from the x-axis is either y = a or y = –a., Reason: Equation of the vertical line having distance b from, the y-axis is either x = b or x = –b., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., , 76., , What is the angle between the two straight lines, y = (2 - 3 )x + 5 and y = (2 + 3) x - 7 ?, (a) 60°, , 77., , (b) 45°, , (c) 30°, , (d) 15°, , If p be the length of the perpendicular from the origin on the, straight line x + 2by = 2p ,then what is the value of b?, (a), , 1, p, , (b) p, , (a) 2x + y – 5 =0 (b), x – 3y + 6 = 0, (c) x + 2y – 7 = 0, (d) x + 3y + 8 = 0, 80. The distances of the point (1, 2, 3) from the coordinate axes, are A, B and C respectively. Now consider the following, equations:, I. A2 = B2 + C2 II., B2 = 2C2, 2, 2, 2, III. 2A C = 13 B, Which of these hold(s) true?, (a) Only I (b) I and III (c) I and II (d) II and III, 81. Locus of mid point of the portion between the axes of, x cos a + y sina = p whre p is constant is, (a) x2 + y2 =, (c), , 73. Which of the following lines is farthest from the origin?, , 75., , 78. The number of lines drawn from the point (4, –5) so that, its distance from (–2, 3) will be equal to 12 is, (a) 2, (b) 1, (c) 4, (d) None of these, 79. The equation of a line through the point of intersection of, the lines x – 3y + 1 = 0 and 2x + 5y – 9 = 0 and whose distance, from the origin is 5 is, , (c), , 1, 2, , (d), , 3, 2, , 1, x2, , +, , 1, y2, , 4, , (b) x2 + y2 = 4p2, , p2, =, , 2, p2, , (d), , 1, x, , 2, , +, , 1, y, , 2, , =, , 4, p2, , 82. The distance of the point (–1, 1) from the line, 12(x + 6) = 5 (y – 2) is, (a) 2, (b) 3, (c) 4, (d) 5, 83. The distance between the parallel lines 3x – 4y + 7 = 0 and, a, 3x – 4y + 5 = 0 is . Value of a + b is, b, (a) 2, (b) 5, (c) 7, (d) 3, 84. If the line y = 3 x cut the curve x3 + y3 + 3xy + 5x2 + 3y2 +, 4x + 5y – 1 = 0 at the points A, B, C then OA.OB.OC is, (where O is origin), (a), , 4, (3 3 – 1), 13, , 2, , (b) 3 3 + 1, , (d) None of these, +7, 3, 85. Assertion: The distance between the parallel lines, 33, 3x – 4y + 9 = 0 and 6x – 8y – 15 = 0 is, ., 10, Reason: Distance between the parallel lines Ax + By + C1 = 0, and Ax + By + C2 = 0, is given by, | C1 - C 2 |, d=, A 2 + B2, (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., (c)

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STRAIGHT LINES, , 125, , 86. The distance between the lines 3x + 4y = 9 and 6x + 8y =, 15 is:, 3, 3, 9, (b), (c) 6, (d), 2, 10, 4, The angle between the lines whose intercepts on the axes, are a, –b and b, –a respectively , is, , (a), , 87., , (a), , tan -1, , a 2 - b2, ab, , (b), , tan -1, , b2 - a 2, 2, , b2 - a 2, (d) None of these, 2ab, 88. The distance of the line 2x + y = 3 from the point (–1, 3) in the, direction whose slope is 1, is, , (c), , tan -1, , (a), , 2, 3, , (b), , 2, 3, , (c), , 2 2, 3, , (d), , 2 5, 3, , 89. What is the image of the point (2, 3) in the line y = – x ?, (a) (–3, –2), (b), (– 3, 2), (c) (–2, –3), (d), (3, 2), 90. If 2p is the length of perpendicular from the origin to the, x y, lines + = 1 , then a2, 8p2, b2 are in, a b, (a) A.P., (b) G.P., (c) H.P., (d) None of these, 91. If p be the length of the perpendicular from the origin on the, , 3, , then what is the angle, 2, between the perpendicular and the positive direction of xaxis?, (a) 30°, (b) 45°, (c) 60°, (d) 90°, , straight line ax + by = p and b =, , 92. For what value of k are the two straight lines 3x + 4y = 1 and, 4x + 3y + 2k = 0 equidistant from the point (1, 1) ?, , 1, 1, (b) 2, (c) –2, (d) 2, 2, 2, 93. The lengths of the perpendicular from the points (m , 2m),, (mm', m + m') and (m'2, 2m') to the line x + y + 1 = 0 form, (a), , (a) an A.P., (b) a G.P., (c) an H.P., (d) None of these, 94. The locus of a point that is equidistant from the lines, , x + y - 2 2 = 0 and x + y - 2 = 0 is, (a), , x+ y -5 2 =0, , (b), , x+ y -3 2 = 0, , (c), , 2x + 2 y - 3 2 = 0, , (d), , 2x + 2 y - 5 2 = 0, , 95. The straight line ax + by + c = 0, where abc ¹ 0, will pass, through the first quadrant if, (a) ac > 0, bc > 0, (c) bc > 0 and/or ac > 0, , (b) c > 0 and bc < 0, (d) ac < 0 and/or bc < 0, , BEYOND NCERT, Topic 4 : Centroid Orthocentre, Incentre, Locus and its, Equation, Transformation of axes, Bisector of Angle, Between Two Lines, Position of a Point w.r.t a Line,, Foot of Perpendicular, Pedal Points,, Pair of Straight Lines., 96. The coordinates of the foot of the perpendicular from the, point (2, 3) on the line x + y – 11 = 0 are, (a) (–6, 5), (b) (5, 6), (c) (–5, 6), (d) (6, 5), 97. The perpendicular from the origin to the line y = mx + c, meets it at the point (–1, 2). Find the value of m + c., (a) 2, (b) 3, (c) 4, (d) 5, 98. The incentre of a triangle with vertices (7, 1), (–1, 5) and, , (3 + 2 3,3 + 4 3) is, (a), , æ, 2, 4 ö, ,3+, ç3+, ÷, 3, 3ø, è, , (b), , æ, 2, 4 ö, ,1 +, ç1 +, ÷, 3 3ø, è 3 3, , (c) (7, 1), (d) None of these, 99. The vertices of a triangle ABC are (1, 1), (4, – 2) and (5, 5), respectively. Then equation of perpendicular dropped from, C to the internal bisector of angle A is, (a) y – 5 = 0, (b) x – 5 = 0, (c) 2x + 3y –7 = 0, (d) None of these, 100. The line L has intercepts a and b on the coordinate axes., When keeping the origin fixed, the coordinate axes are, rotated through a fixed angle, then same line has intercepts, p and q on the rotated axes, then, (a) a2 + b2 = p2 + q2, (c), , (b), , a 2 + b2 = b2 + q 2, , (d), , 1, a, , 2, , +, , 1, b, , 2, , =, , b2 + q 2 =, , 1, p, , 2, , +, , 1, q2, , 1 1, +, b2 q 2, , 101. The orthocentre of triangle formed by lines 4x – 7y +10 = 0,, x + y = 5 and 7x + 4y = 15 is, (a) (1, 2) (b) (1 , – 2), (c) (–1, –2) (d) (–1, 2), 102. The bisector of the acute angle formed between the lines, 4x – 3y + 7 = 0 and 3x – 4y + 14 = 0 has the equation, (a) x + y +3 = 0, (b) x – y – 3 = 0, (c) x – y + 3 = 0, (d) 3x + y – 7 = 0, 103. Equation of angle bisector between the lines 3x + 4y – 7 = 0, and 12x + 5y + 17 = 0 are, (a), , (b), , (c), , 3x + 4y – 7, 25, 3x + 4y + 7, 25, 3x + 4y + 7, 25, , =±, =, , 169, , 12x + 5y + 17, , =±, , (d) None of these, , 12x + 5y + 17, , 169, 12x + 5y + 17, 169

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EBD_7762, 126, , MATHEMATICS, , 104. In DABC, if the orthocenter is (1, 2) and the circumcenter is, (0, 0), then centroid of DABC is, (a) (1/2, 2/3), (b) (1/3, 2/3), (c) (2/3, 1), (d) None of these, 105. The equation of the line which bisects the obtuse angle, between the lines x – 2y + 4 = 0 and 4x – 3y + 2 = 0, is, (a), (b), (c), , (4 – 5 ) x – (3 – 2 5 ) y + (2 – 4 5 ) = 0, (4 + 5 ) x – (3 + 2 5 ) y + (2 + 4 5 ) = 0, (4 + 5 ) x + (3 + 2 5 ) y + (2 + 4 5 ) = 0, , (d) None of these, 106. Choose the correct statement which describe the position, of the point (–6, 2) relative to straight lines 2x + 3y –4 = 0, and 6x + 9y + 8 = 0., (a) Below both the lines (b) Above both the lines, (c) In between the lines (d) None of these, 107. If a vertex of a triangle is (1, 1) and the mid points of two, sides through this vertex are (–1, 2) and (3, 2) then the, centroid of the triangle is, (a), , 7ö, æ, ç - 1, ÷, 3ø, è, , (b), , æ -1 7 ö, ç , ÷, è 3 3ø, , æ 7ö, 1 7, (d) æç , ö÷, ç1, ÷, è 3ø, è3 3ø, 108. The coordinates of the points A and B are (a, 0) and (–a, 0), respectively. If a point P moves so that PA2 – PB2 = 2k2,, when k is constant , then the equation to the locus of the, point P, is, (a) 2ax – k2 = 0, (b) 2ax + k2 = 0, 2, (c) 2ay – k = 0, (d) 2ay + k2 = 0, 109. The centroid of the triangle formed by the pair of lines, 2x2 – 27y2 – 3xy + 4x – 3y + 2 = 0 and the line 4x – 3y – 26 = 0 is, (a) (3, –2), (b) (4, 2), (c) (4, 0), (d) None of these, 110. If the equation of the locus of a point equidistant from the, , (c), , points (a1 , b1 ) and (a2 , b2 ) is, (a1 - a2 ) x + (b1 - b2 ) y + c = 0 , then the value of `c` is, , (a), , a12 + b12 - a22 - b22, , (b), , 1, (a2 2 + b22 - a12 - b12 ), 2, , (c), , a12 - a22 + b12 - b2 2, , (d), , 1 2, (a1 + a22 + b12 + b22 ) ., 2, , 111. ABC is a variable triangle such that A is (1, 2), and B and C, lie on the line y = x + l (l is a variable). Then the locus of the, orthocenter of DABC is, (a) x + y = 0, (b) x – y = 0, 2, 2, (c) x + y = 4, (d) x + y = 3, , 112. Area of the triangle formed by the line x + y = 3 and the angle, bisectors of the pairs of straight lines x2 – y2 + 2y = 1 is, (a) 2 sq. units, (b) 4 sq. units, (c) 6 sq. units, (d) 8 sq. units, 113. Let A(2, - 3) and B (–2, 1) be vertices of a triangle ABC. If, the centroid of this triangle moves on the line, 2 x + 3 y = 1, then the locus of the vertex C is the line, (a) 3x – 2y = 3, (b) 2x – 3y = 7, (c) 3x + 2y = 5, (d) 2x + 3y = 9, 114. All points lying inside the triangle formed by the points, (1, 3), (5, 0) and (–1, 2) satisfy, , 115., , (a), , 3x + 2 y ³ 0, , (b), , 2 x + y - 13 £ 0, , (c), , 2 x - 3 y - 12 £ 0, , (d) All the above, , If one of the lines of my2 + (1– m2) xy – mx2= 0 is a bisector, , of the angle between the lines xy = 0, then m is, (a) 1, (b) 2, (c) –1/2, (d) –2., 116. P(m, n) (where m, n are natural numbers) is any point in the, interor of the quadrilateral formed by the pair of lines xy = 0, and the two lines 2x + y – 2 = 0 and 4x + 5y = 20. The possible, number of positions of the point P is, (a) six, (b) five, (c) four, (d) eleven, 117. A variable line 'L' is drawn through O(0, 0) to meet the lines, L1 : y – x – 10 = 0 and L2 : y – x – 20 = 0 at the points A and, B respectively. A point P is taken on 'L' such that, 2, 1, 1, =, +, . Locus of 'P' is, OP OA OB, , (a) 3x + 3y = 40 (b), 3x + 3y + 40 = 0, (c) 3x – 3y = 40, (d) 3y – 3x = 40, 118. The image of the pair of lines represented by:, ax2 + 2hxy + by2 = 0 by the line mirror y = 0 is, (a) ax2 – 2hxy – by2 = 0, (b) bx2 – 2hxy + ay2 = 0, (c) bx2 + 2hxy + ay2 = 0, (d) ax2 – 2hxy + by2 = 0, 119. The equation, 8 x 2 + 8 xy + 2 y 2 + 26 x + 13 y + 15 = 0 represents a pair of, straight lines. The distance between them is, , (a), , (b), , 7/ 5, , 7/2 5, , (d) None of these, (c), 7 /5, 120. The coordinate axes are rotated about the origin O in the, counter-clockwise direction through an angle 60°. If p and q, are the intercepts made on the new axes by a stright line, whose equation refered to the original axes is x + y = 1, then, 1, 1, +, =, p2 q 2, , (a) 2, , (b) 4, , (c) 6, , (d) 8, , 121. The pedal points of a perpendicular drawn from origin on, the line 3x + 4y – 5 = 0, is

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STRAIGHT LINES, , 127, , (a), , æ3 ö, ç , 2÷, è5 ø, , (b), , æ3 4ö, ç , ÷, è5 5ø, , (c), , æ 3 4ö, ç - 5 ,- 5 ÷, è, ø, , æ 30 19 ö, (d) ç , ÷, è 17 17 ø, , æ 1 ö æ 1ö æ 1ö, 122. Let A ç a, ÷ , B ç b, ÷ , C ç g, ÷ be the vertices of a, è aø è bø è gø, DABC, where a, b are the roots of the equation, , x 2 - 6 p1 x + 2 = 0 , b, g are the roots of the equation, x 2 - 6 p2 x + 3 = 0 and g, a are the roots of the equation, x 2 - 6 p3 x + 6 = 0 , p1 , p2, p3 being positive. Then, the, coordinates of the centroid of DABC is, æ 11 ö, æ 11 ö, (a) ç 1, ÷, (b) ç 0, ÷, è 18 ø, è 8ø, , (c), , 1., , æ 11 ö, ç 2, ÷, è 8ø, , (d) None of these, , NCERT Exemplar MCQs, A line cutting off intercept – 3 from the Y-axis and the tangent, 3, , its equation is, 5, (a) 5y – 3x + 15 = 0, (b) 3y – 5x + 15 = 0, (c) 5y – 3x – 15 = 0, (d) None of the above, Slope of a line which cuts off intercepts of equal lengths on, the axes is, , at angle to the X-axis is, , 2., , 3., , 4., , 5., , (a) –1, (b) 0, (c) 2, (d), 3, The equation of the straight line passing through the point, (3, 2) and perpendicular to the line y = x is, (a) x – y = 5, (b) x + y = 5, (c) x + y = 1, (d) x – y = 1, The equation of the line passing through the point (1, 2), and perpendicular to the line x + y + 1 = 0 is, (a) y – x + 1 = 0, (b) y – x – 1 = 0, (c) y – x + 2 = 0 (d), y–x–2=0, The tangent of angle between the lines whose intercepts on, the axes are a, –b and b, –a respectively, is, (a), , a 2 - b2, ab, , (c), , b 2 - a2, 2ab, , (b), , b2 - a2, 2, , (d) None of these, , 123. The number of equilateral triangles with y = 3( x - 1) + 2, and y = - 3x as two of its sides is, (a) 0, (b) 1, (c) 2, (d) None of these, 124. Let PQR be a right-angled isosceles triangle, right angled at, P(2, 1). If the equation of the line QR is 2x + y = 3, then the, equation representing the pair of lines PQ and PR is, (a) 3x2 – 3y2 + 8xy + 20x + 10y + 25 = 0, (b) 3x2 – 3y2 + 8xy – 20x – 10y + 25 = 0, (c) 3x2 – 3y2 + 8xy + 10x + 15y + 20 = 0, (d) 3x2 – 3y2 – 8xy – 15y – 20 = 0, 125. Let O be the origin. If A(1, 0) and B(0, 1) and P(x, y) are, points such that xy > 0 and x + y < 1, then, (a) P lies either inside the triangle OAB or in the third, quadrant, (b) P cannot lie inside the triangle OAB, (c) P lies inside the triangle OAB, (d) P lies in the first quadrant only, , 6., , 7., , x y, + = 1 passes through the points (2, – 3) and, a b, (4, – 5), then (a, b) is, (a) (1, 1) (b) (– 1, 1), (c) (1, – 1) (d) (– 1, – 1), The distance of the point of intersection of the lines, 2x – 3y + 5 + 0 and 3x + 4y = 0 from the line 5x – 2y = 0 is, , If the line, , (a), , 130, , (b), , 17 29, , 13, 7 29, , 130, (d) None of these, 7, The equation of the lines which pass through the point, , (c), , 8., , (3, – 2) and are inclined at 60° to the line, , 3x + y = 1 is, , (a) y + 2 = 0, 3x – y – 2 – 3 3 = 0, (b) x – 2 = 0, 3x – y + 2 + 3 3 = 0, , 9., , (c), 3x – y – 2 – 3 3 = 0, (d) None of the above, The equations of the lines passing through the point(1, 0), and at a distance, , 3, from the origin, are, 2, , (a), , 3 = 0,, , 3x + y –, , 3x – y –, , 3 =0, , (b), 3x + y + 3 = 0, 3x – y + 3 = 0, (c) x + 3 y – 3 = 0, x – 3 y – 3 = 0, (d) None of the above

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EBD_7762, 128, , 10., , MATHEMATICS, , The distacne between the lines y = mx + c1 and y = mx + c2 is, c1 – c2, , (a), , m2 + 1, c2 – c1, , (c), 11., , 1 + m2, , (b), , | c1 – c2 |, 1 + m2, , (d) 0, , The coordinates of the foot of perpendiculars from the point, (2, 3) on the line y = 3x + 4 is given by, (a), , æ 37 –1ö, çè , ÷ø, 10 10, , (b), , æ 1 37 ö, çè – , ÷ø, 10 10, , (c), , æ 10, ö, çè , –10÷ø, 37, , (d), , æ 2 1ö, çè , – ÷ø, 3 3, , 12., , If the coordinates of the middle point of the portion of a line, intercepted between the coordinate axes is (3, 2) then the, equation of the line will be, (a) 2x + 3y = 12, (b) 3x + 2y = 12, (c) 4x – 3y = 6, (d) 5x – 2y = 10, , 13., , Equation of the line passing through (1, 2) and parallel to, the line y = 3x – 1 is, (a) y + 2 = x + 1 (b), y + 2 = 3(x + 1), (c) y – 2 = 3(x – 1), (d) y – 2 = x – 1, Equations of diagonals of the square formed by the lines, x = 0, y = 0, x = 1 and y = 1 are, (a) y = x, y + x = 1, (b) y = x, x + y = 2, , 14., , 1, (d) y = 2xy + 2x = 1, 3, For specifying a straight line, how many geometrical, parameters should be known?, (a) 1, (b) 2, (c) 4, (d) 3, The point (4, 1) undergoes the following two success ive, transformations, (i) Reflection about the line y = x, (ii) Translation through a distance 2 units along the positive, X-axis., Then, the final coordinates of the point are, , (c) 2y = x, y + x =, 15., , 16., , (a) (4, 3), 17., , 18., , (b) (3, 4), , (c) (1, 4), , (d), , æ 7 7ö, çè , ÷ø, 2 2, , A point equidistant from the lines 4x + 3y + 10 = 0,, 5x – 12y + 26 = 0 and 7x + 24y – 50 = 0, (a) (1, – 1) (b) (1, 1), (c) (0, 0), (d) (0, 1), A line passes through (2, 2) and is perpendicular to the line, 3x + y = 3. Its y-intercept is, , 19. The ratio in which the line 3x + 4y + 2 = 0 divides the distance, between the lines 3x + 4y + 5 = 0 and 3x + 4y – 5 = 0 is, (a) 1 : 2, (b) 3 : 7, (c) 2 : 3, (d) 2 : 5, 20. One vertex of the equilateral triangle with centroid at the, origin and one side as x + y – 2 = 0 is, (a) (– 1, –1), (b) (2, 2), (c) (– 2, – 2), , Past Year MCQs, 21. Let PS be the median of the triangle vertices, P(2, 2), Q(6, –1) and R(7, 3). The equation of the line passing, through (1, –1) and parallel to PS is:, [JEE MAIN 2014, A], (a) 4x + 7y + 3 = 0, (b) 2x – 9y – 11 = 0, (c) 4x – 7y – 11 = 0, (d) 2x + 9y + 7 = 0, 22. Let a, b, c and d be non-zero numbers. If the point of, intersection of the lines 4ax + 2ay + c = 0 and 5bx + 2by + d =0, lies in the fourth quadrant and is equidistant from the two, axes then, [JEE MAIN 2014, C], (a) 3bc – 2ad = 0, (b) 3bc + 2ad = 0, (c) 2bc – 3ad = 0, (d) 2bc + 3ad = 0, 23. If P1 and P2 be the length of perpendiculars from the origin, upon the straight lines x secq + y cosecq = a and, x cosq – y sinq = a cos2q respectively, then the value of, [BITSAT 2014, C], 4P12 + P22., 2, 2, 2, (a) a, (b) 2a, (c) a /2, (d) 3a2, 24. Two sides of a rhombus are along the lines,, x – y + 1 = 0 and 7x – y – 5 = 0. If its diagonals intersect at, (–1, –2), then which one of the following is a vertex of this, rhombus?, [JEE MAIN 2016, S], , æ 1 -8 ö, ç , ÷, è3 3 ø, (c) (–3, –9), , (a), , æ -10 -7 ö, , ÷, ç, è 3 3 ø, (d) (– 3, – 8), (b), , 25. The eq. x 2 - 2 3xy + 3 y 2 - 3x + 3 3 y - 4 = 0 represents, [BITSAT 2016, C,BN], (a) a pair of intersecting lines, (b) a pair of parallel lines with distance between, 5, them, 2, (c) a pair of parallel lines with distance between, them 5 2, (d) a conic section, which is not a pair of straight lines, 26. A ray of light coming from the point (1, 2) is reflected at a, point A on the x-axis and then passes through the point (5,, 3). The co-ordinates of the point A is [BITSAT 2016, A], , 1, 3, , (b), , 2, 3, , (a), , (c) 1, , (d), , 4, 3, , (c) (–7, 0), , (a), , (d) (2, – 2), , æ 13 ö, ç , 0÷, ø, è 5, , ö, æ 5, (b) ç , 0 ÷, è 13 ø, , (d) None of these

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STRAIGHT LINES, , 129, , 27. The angle between the lines whose intercepts on the axes, are a, –b and b, –a respectively , is, [BITSAT 2017, C], (a), (c), , a 2 - b2, ab, 2, b - a2, tan -1, 2ab, tan -1, , (b) tan -1, , b2 - a 2, 2, , 29. A straight the through a fixed point (2, 3) intersects the, coordinate axes at distinct points P and Q. If O is the origin, and the rectangle OPRQ is completed, then the locus of R is :, [JEE MAIN 2018, C], , (d) None of these, , 28. Given the system of straight lines a(2x + y – 3) +, b(3x + 2y – 5) = 0, the line of the system situated farthest, from the point (4, –3) has the equation, [BITSAT 2017, A], (a) 4x + 11y – 15 = 0, (b) 7x + y – 8 = 0, (c) 4x + 3y – 7 = 0, (d) 3x – 4y + 1 = 0, , (a), , 2x + 3y = xy, , (b), , 3x + 2y = xy, , (c), , 3x + 2y = 6xy, , (d), , 3x + 2y = 6, , 30. Let the orthocentre and centroid of a triangle be A(–3, 5), and B(3, 3) respectively. If C is the circumcentre of this triangle, then the radius of the circle having line segment AC, as diameter, is :, [JEE MAIN 2018, C, BN], (a), , 5, 2 10 (b) 3 2, , (c), , 3 5, 2, , (d), , 10, , Exercise 3 : Try If You Can, 1., , If l1 : ax + by + p = 0 makes angle, , p, with, 4, , 5., , D(0, 2), undergoes the following transformations, successively, i., f1 ( x, y ) ® ( y , x ), , l2, l1 , l 2 and the line, xsina – ycosa = 0 are concurrent, then value of a2 + b2 is, : xcosa + ysina = p, p Î R+ such that, , p, æ, ö, equal to çè a ¹ , n ÎI ÷ø, 2, 2., , 3., , 4., , ii., , (a) 1, (b) 2, (c) 3, (d) 4, The equation of straight line passing through the point of, intersection of family of lines x(3 + 4l) + y(4 + 3l) + 1 + 6l = 0, which is at a minimum distance from the point (1, 1) can be, expressed as x + by – c = 0, where b and c are natural numbers,, then the value of (b + c) is, (a) 5, (b) 8, (c) 9, (d) 10, L1 and L2 are two mutually perpendicular straight lines, intersecting at the point O. A fixed point P lies on the line L1, and Q is any point on the line L2. If PQR is an equilateral, triangle with vertex R being on the opposite side of PQ with, respect to O, then the locus of R is, (a) a straight line through O, (b) a straight line inclined at 60° to L1, (c) a straight line inclined at 60° to L2, (d) not a straight line, The line x + y = a meets the x-axis of A and y-axis at B., A triangle AMN is inscribed in the triangle OAB, O, being the origin, with right angle at N; M and N lie, respectively on OB and AB. If area of D AMN is, the area of D OAB, then, (a) 3, , (b), , 1, 3, , A rectangle ABCD, where A(0, 0), B (4, 0), C (4, 2),, , AN, is equal to, BN, 1, (d), (c) 3 or, 3, , 6., , iii. f 3 ( x, y ) ® (( x - y) / 2, ( x + y) / 2), The final figure will be, (a) a square, (b) a rhombus, (c) a rectangle, (d) a parallelogram, Form the point of intersection (P) of lines given by, x2 – y2 – 2x + 2y = 0, points A, B, C, D are taken on the lines, at a distance of 2 2 units to form a quadrilateral whose, area is A1 and the area of the quadrilateral formed by joining, the circumcentres of DPAB, DPBC, DPCD, DPDA is A2, then, , 7., , A1, equals, A2, (a) 2, (b) 4, (c), (d) 1, 3, A system of lines is given as y = mix + ci, where mi can take, any value out of 0, 1, – 1 and when mi is positive then ci can, be 1 or –1 when mi equal 0, ci can be 0 or 1 and when mi, equal – 1, ci can take 0 or 2. Then the area enclosed by all, these straight lines is, , (a), , 3, of, 8, , 3, 2, , ( 2 - 1), , (b), , 3, 2, , 3, (d) 2, 2, The circumcentre of a triangle lies at the origin and its, centroid is the mid point of the line segment joining the, points (a2 + 1, a2 + 1) and (2a, – 2a), a ¹ 0. Then for any a,, the orthocentre of this triangle lies on the line:, (a) y – 2ax = 0, (b) y – (a2 + 1)x = 0, (c) y + x = 0, (d) (a – 1)2x – (a + 1)2y = 0, , (c), , 8., 2, 3, , f 2 ( x, y ) ® ( x + 3 y, y )

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EBD_7762, 130, , 9., , MATHEMATICS, , The area of the region bounded by the locus of a point P, satisfying d (P, A) = 4, where A is (1, 2) is, (a) 64 sq. unit, (b) 54 sq. unit, (c) 16p sq. unit, (d) None of these, Let coordinates of the points A and B are (5, 0 ) and (0, 7), respectively. P and Q are the variable points lying on the, x- and y-axis respectively so that PQ is always perpendicular, to the line AB. Then locus of the point of intersection of, BP and AQ is, (a) x2 + y2 – 5 x + 7y = 0 (b) x2 + y2 + 5x – 7y = 0, (c) x2 + y2 + 5x + 7y = 0 (d) x2 + y2 – 5x – 7y = 0, The number of intergral points (integral point means both, the coordinates should be integer) exactly in the interior of, the triangle with vertices (0,0), (0,21) and (21,0), is, (a) 133, (b) 190, (c) 233, (d) 105, The value of l for which the lines joining the points of, intersection of curves C1 and C2 to the origin are equally, inclined to the axis of x, where C2 : l x2 + 3y2 – 2l xy + 9x =0, and C1 : 3x2 – 4y2 + 8xy – 3x = 0, is, , 10., , 11., , 12., , (a), , l=, , 4, (b) l = 12, 3, , (c) l = 1, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, , (b), (b), (b), (a), (b), (b), (c), (b), (a), (a), (c), (c), (a), , 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, , (b), (b), (a), (c), (b), (d), (a), (c), (c), (b), (d), (c), (b), , 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, , (a), (c), (c), (d), (a), (a), (d), (b), (c), (d), (a), (b), (a), , 1, 2, 3, , (a), (a), (b), , 4, 5, 6, , (b), (c), (d), , 7, 8, 9, , (a), (a), (a), , 1, 2, , (b), (c), , 3, 4, , (c), (a), , 5, 6, , (d), (a), , (d) l =, , 13. For a > b > c > 0, the distance between (1, 1) and the point of, intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is, less than 2 2 . Then, (a) a + b – c > 0 (b), a–b+c<0, (c) a – b + c > 0 (d), a+b–c<0, 14. The combined equation of three sides of a triangle is, (x2 –y2) (2x + 3y – 6) = 0. If ( – 2, a) is an interior and (b, 1) is, an exterior point of the triangle, then, (a) 2 < a <, , 10, 3, , (b) – 2 < a <, , 10, 3, , 9, (d) – 1 < b < 1, 2, 15. A light ray coming along the line 3x + 4y = 5 gets reflected, from the line ax + by = 1 and goes along the line, 5x – 12y = 10. Then,, (a) a = 64/115, b = 112/15, (b) a = 14/15, b = –8/115, (c) a = 64/115, b = –8/115, (d) a = 64/15, b = 14/15, , (c) – 1 < b <, , 5, 6, , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (a), (c) 79, 40 (b) 53, 66, (d), (b) 80, 41 (b) 54, 67, (a), (c) 81, 42 (d) 55, 68, (a), (b), (b), 43, 56, 69, 82, (a) 57, (a), (a) 83, 44, 70, (a) 58, (a), (c) 84, 45, 71, (c) 59, (b), (b) 85, 46, 72, (c), (d) 86, 47 (b) 60, 73, (a), (c) 87, 48 (b) 61, 74, (c), (b) 88, 49 (d) 62, 75, (a), (a) 89, 50 (d) 63, 76, (d), (a), (d), 51, 64, 77, 90, (c) 65, (c), (d) 91, 52, 78, Exercise 2 : Exemplar & Past Year MCQs, (c), (b) 19, 10 (b) 13, 16, (a), (c) 20, 11 (b) 14, 17, (a) 15, (b), (d) 21, 12, 18, Exercise 3 : Try If You Can, (c), (a), (b) 13, 7, 9, 11, (d) 10, (d), (b) 14, 8, 12, , (a), (d), (d), (d), (c), (a), (a), (b), (c), (c), (a), (c), (c), , 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, , (d), (b), (c), (d), (b), (b), (a), (b), (b), (a), (c), (a), (b), , 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, , (a), (a), (c), (b), (c), (b), (d), (a), (d), (d), (a), (a), (d), , 118, 119, 120, 121, 122, 123, 124, 125, , (d), (b), (a), (b), (c), (d), (b), (a), , (b), (c), (d), , 22, 23, 24, , (a), (a), (a), , 25, 26, 27, , (b), (a), (c), , 28, 29, 30, , (d), (b), (b), , (a), (d), , 15, , (c)

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11, , CONIC SECTIONS, , Chapter, , Trend, Analysis, , off JEE MAIN and BITSAT (Year 2010-2018), , 7, , Number of Questions, , 6, 5, , JEE MAIN, , 4, , BITSAT, , 3, 2, 1, 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 10, 11, , Critical Concepts, , Circle: Equation of Circle, Parabola:, Equation of Parabola, Latus Rectum,, Length of Latus Rectum, Ellipse:, Equation of an Ellipse, Length of, Latus Rectum, Eccentricity of an, Ellipse, Hyperbola: Equation of an, Hyperbola, , Rating of, Difficulty Level, , CUS, (chapter utility score), Out of 10, , 4.5/5, , 8.9

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CONIC SECTIONS, , 133

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EBD_7762, 134, , MATHEMATICS, , Topic 1 : Circle: Different Equations of a Circle, Position of a, Point with Respect to a Circle., 1., , 2., , 3., , The equation of the circle which passes through the point, (4, 5) and has its centre at (2, 2) is, (a) (x – 2) + (y – 2) = 13, (b) (x – 2)2 + (y – 2)2 = 13, 2, 2, (c) (x) + (y) = 13, (d) (x – 4)2 + (y – 5)2 = 13, Point (1, 2) relative to the circle x2 + y2 + 4x – 2y – 4 = 0 is, a/an, (a) exterior point, (b) interior point, but not centre, (c) boundary point, (d) centre, If the equation of a circle is, (4a - 3) x 2 + ay 2 + 6 x - 2 y + 2 = 0 ,, then its centre is, (a), , 4., , 5., , 6., , 7., , (3, - 1) (b) (3, 1), , (c) (–3, 1) (d) None of these, , For what value of k, does the equation, 9x2 + y2 = k (x2 – y2 – 2x), represent equation of a circle ?, (a) 1, (b) 2, (c) –1, For the equation, ax2 + by2 + 2hxy + 2gx + 2fy + c = 0, , The centre of circle inscribed in square formed by the lines, x2 – 8x + 12 = 0 and y2 – 14y + 45 = 0, is, (a) (4, 7), (b) (7, 4), (c) (9, 4) (d) (4, 9), 2, 2, 10. The circle x + y – 8x + 4y + 4 = 0 touches :, (a) x-axis only, (b) y-axis only, (c) both (a) and (b), (d) None of these, 11., , If the equation of the circle is x2 + y2 – 8x + 10y – 12 = 0, then, I., , Centre of the circle is (4, –5)., , II., , Radius of the circle is, , (a) Only I is true., , (b) Only II is true., , (c) Both are true., , (d) Both are false., , 12. Radius of the circle (x + 5)2 + (y – 3)2 = 36 is, (a) 2, , (d) 4, , (b), , x 2 + y 2 - 8 x + 6 y + 16 = 0, 2, , 2, , (c), , x + y - 8 x - 6 y - 16 = 0, , (d), , x 2 + y 2 - 8 x - 6 y + 16 = 0, , (c) 6, , (d) 5, , P is, (a) a pair of straight lines (b) a circle, (c) a straight line, (d) None of these, 14. The equation of the circle with centre (0, 2) and radius 2 is, x2 + y2 – my = 0. The value of m is, (a) 1, , The point diametrically opposite to the point P(1, 0) on the, circle x2 + y2 + 2x + 4y – 3 = 0 is, (a) (3, – 4) (b) (–3, 4), (c) (–3, –4) (d) (3, 4), , (b) 2, , (c) 4, , (d) 3, , 15. What is the radius of the circle passing through the points, (0, 0), (a, 0) and (0, b) ?, (a), , a2 - b2, , (c), , 1 2, a + b2, 2, , A circle has radius 3 and its centre lies on the line y = x - 1 ., x 2 + y 2 + 8 x - 6 y + 16 = 0, , (b) 3, , internally in the ratio 2 : 3 at P. If q varies, then the locus of, , where a ¹ 0 , to represent a circle, the condition will be, (a) a = b and c = 0, (b) f = g and h = 0, (c) a = b and h = 0, (d) f = g and c = 0, The equation of a circle with centre at (1, 0) and, circumference 10p units is, (a) x2 + y2 – 2x + 24 = 0, (b) x2 + y2 – x – 25 = 0, 2, 2, (c) x + y – 2x – 24 = 0, (d) x2 + y2 + 2x + 24 = 0, , (a), , 53 ., , 13. The line joining (5, 0) to ( (10 cos q, 10 sin q) is divided, , The equation of the circle, if it passes through (7, 3), is, , 8., , 9., , (b), , a2 + b2, , 2, 2, (d) 2 a + b, , 16. The equation of the circle, which touches the line y = 5 and, passes through (–1, 2) and (1, 2) is, (a), , 9 x 2 + 9 y 2 - 60 y + 75 = 0, , (b), , 9 x 2 + 9 y 2 - 60 x - 75 = 0, , (c), , 9 x 2 + 9 y 2 + 60 y - 75 = 0, , (d), , 9 x 2 + 9 y 2 + 60 x + 75 = 0

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CONIC SECTIONS, , 135, , 17. The parametric equations of the circle x2 + y2 + x + 3y = 0, are, (a) x = 1 + cosq, y =, , (b) x = -, , (c) x =, , (d) x =, , 3, + sinq, 2, , (a) (2, 8), , 1, 3, + cosq, y = –, + sinq, 2, 2, , 1, 3, + cosq, y = –, + sinq, 2, 2, 1, 1, 1, 3, + cosq, y =, + sinq, 2, 2, 2, 2, , (a), , y ± 2x = 0, , (b) 2 y ± x = 0, , (c), , x ± 2y = 0, , (d) 2 x ± y = 0, , the vertex at, , 21. A conic section with eccentricity e is a parabola if:, (a) e = 0, (b) e < 1, (c) e > 1, (d) e = 1, 22. The focal distance of a point on the parabola y2 = 12x is 4., What is the abscissa of the point ?, (c) 2 3, , 9ö, æ 9ö, æ, (c) ç 4, ÷ (d) ç -4, - ÷, è 2ø, è, 2ø, , 28. The equation y 2 + 3 = 2(2 x + y ) represents a parabola with, , Topic 2 : Parabola: Different equations of a Parabola, Terms, Related to Parabola., , (b) – 1, , (b) (7, 2), , 27. The equations of the lines joining the vertex of the parabola, y2 = 6x to the points on it which have abscissa 24 are, , 18. The equation of the circle in the first quadrant touching, each coordinate axis at a distance of one unit from the, origin is, (a) x2 + y2 – 2x – 2y + 1 = 0, (b) x2 + y2 – 2x – 2y – 1 = 0, (c) x2 + y2 – 2x – 2y = 0, (d) x2 + y2 – 2x + 2y – 1 = 0, 19. Four distinct points (2k, 3k), (1, 0), (0, 1) and (0, 0) lie on a, circle for, (a) only one value of k, (b) 0 < k < 1, (c) k < 0, (d) all integral values of k, 20. Find the equation of a circle which passes through the, origin and makes intercepts 2 units and 4 units on x-axis, and y-axis respectively., (a) x2 + y2 – 2x – 4y = 0, (b) x2 + y2 – 4y = 0, 2, 2, (c) x + y + 2x = 0, (d) x2 + y2 – 4x – 2y = 0, , (a) 1, , 25. The focal distance of a point on the parabola y2 = 8x is 4. Its, ordinates are:, (a) ± 1, (b) ± 2, (c) ± 3, (d) ± 4, 2, 26. The vertex of the parabola (x – 4) + 2y = 9 is :, , (d) – 2, , 23. What is the length of the smallest focal chord of the parabola, y2 = 4ax ?, (a) a, (b) 2a, (c) 4a, (d) 8a, 24. The equation of the parabola with vertex at origin, which, passes through the point (–3, 7) and axis along the x-axis is, (a) y2 = 49x, (b) 3y2 = – 49x, (c) 3y2 = 49x, (d) x2 = – 49y, , (a), , æ1 ö, ç , 1÷ and axis parallel to y-axis, è2 ø, , (b), , æ 1ö, ç1, ÷ and axis parallel to x-axis, è 2ø, , (c), , æ1 ö, æ3 ö, ç , 1÷ and focus at ç , 1÷, è2 ø, è2 ø, , (d), , æ 1ö, æ3 ö, ç1, ÷ and focus at ç , 1÷, 2, ø, è, è2 ø, , 29. If a ¹ 0 and the line 2bx + 3cy + 4d = 0 passes through, the points of intersection of the parabolas y2 = 4ax and, x2 = 4ay, then, (a), , d 2 + (3b - 2c )2 = 0, , (b) d 2 + (3b + 2c )2 = 0, , (c), , d 2 + (2b - 3c )2 = 0, , (d) d 2 + (2b + 3c )2 = 0, , 30. The focus of the curve y2 + 4x – 6y + 13 = 0 is, (a) (2, 3), (b) (–2, 3), (c) (2, –3) (d) (–2, –3), 31. The vertex of the parabola x2 + 8x + 12y + 4 = 0 is:, (a) (– 4, 1), , (b) (4, –1), , (c) (– 4, –1), , (d) (4, 1), , 32. The focal distance of a point on the parabola y2 – 12x is 4., The abscissa of this point is, (a) 0, , (b) 1, , (c) 2, , (d) 4, , 33. The equation of parabola whose vertex (0, 0) and focus, (3, 0) is y2 = 4ax. The value of ‘a’ is, (a) 2, , (b) 3, , (c) 4, , (d) 1

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EBD_7762, 136, , 34., , MATHEMATICS, , For the parabola y2 = 8x, the length of the latus-rectum is, (a) 4, , 35., , (b) 2, , (c) 8, , (d) None of these, , For the parabola y2 = –12x, equation of directrix is x = a. The, value of ‘a’ is, (a) 3, , 36., , (c) 2, , (d) 6, , The equation of the parabola having axis parallel to y-axis, and which passes through the points (0, 4), (1, 9) and (4, 5) is, (a), , (c), 37., , (b) 4, , y=, , y=, , -19 2 79, -19 2 79, x +, x + 4 (b) y =, x +, x-4, 12, 12, 12, 12, 19 2 79, x +, x+4, 12, 12, , segment makes an angle q to the x-axis is, (d) None of these, , The equation of the directrix of the parabola, y2 + 4y + 4x + 2 = 0 is :, (a) x = –1, , 38., , (b) x = 1, , 45. The cable of a uniformly loaded suspension bridge hangs, in the form of a parabola. The roadway which is horizontal, and 100 m long is supported by vertical wires attached to, the cable, the longest wire being 30 m and the shortest being, 6 m. Then, the length of supporting wire attached to the, roadway 18 m from the middle is, (a) 10.02 m (b) 9.11 m, (c) 10.76 m (d) 12.06 m, 46. The length of the line segment joining the vertex of the, parabola y2 = 4ax and a point on the parabola where the line, , (c) x =, , -3, 3, (d) x =, 2, 2, , The value of p such that the vertex of y = x 2 + 2 px + 13 is, , and n respectively are, (a) sin q, cos q, (c) cos q, sin2q, , (b) cos q, sin q, (d) sin2 q, cos q, , Topic 3: Ellipse: Different Equations of an Ellipse, Terms, related to Ellipse, Eccentricity., 47. For the ellipse 3x2 + 4y2 = 12 length of the latus rectum is:, , 4 units above the y-axis is, (a) 2, 39., , 43., , (c) (0, 1), , (d) (2, 0), , (b) (– 2, 0), , (c) (4, 0), , (d) (– 4, 0), , (b) 5, , 5, (c), 4, , 5, (d), 2, , Which points on the curve x2 = 2y are closest to the point, (0, 5) ?, , (a) 3, , (b) 4, , (c), , (a), , 2, 3, , 2, 3, , (b), , (a), , 3, 5, , (b), , 50. Eccentricity of ellipse, , (c), , (± 3,9 / 2), , (d) (± 2 ,1), , (a), , 12, 5, , (c), , 6, 5, , (d) 1, 5, , If a parabolic reflector is 20 cm in diameter and 5 cm deep,, then the focus is, (a) (2, 0), , (b) (3, 0), , (c) (4, 0), , (d) (5, 0), , 1, 3, , (c) 4, 5, , point (9, 5) and (12, 4) is, , (b), , (c), , 1, 2, , (b) (± 2, 2), , 24, 5, , (d), , 2, 5, , (d), , 1, 2, , 49. In an ellipse, the distance between its foci is 6 and minor, axis is 8. Then its eccentricity is, , (± 2 2, 4), , The latus rectum of the parabola y2 = 4ax whose focal chord, is PSQ such that SP = 3 and SQ = 2 is given by :, , 3, 5, , 48. The length of the semi-latus rectum of an ellipse is one third, of its major axis, its eccentricity would be, , (a), , (a), 44., , (b) (1, 0), , The latus rectum of parabola y2 = 5x + 4y + 1 is:, (a) 10, , 42., , (d) ± 3, , If (2, 0) is the vertex and the y-axis is the directrix of a, parabola, then its focus is, (a) (0, 0), , 41., , (c) 5, , A parabola has the origin as its focus and the line x = 2 as, the directrix. Then the vertex of the parabola is at, (a) (0, 2), , 40., , (b) ± 4, , 4am, . Here, m, n, , 3/ 4, , (b), , x2, a2, , 4/5, , +, , y2, b2, , (c), , (d), , 1, 5, , = 1 , if it passes through, , 5 / 6 (d), , 6/7, , 51. The equation of the ellipse with focus at (± 5, 0) and x =, as one directrix is, (a), , x2 y 2, +, =1, 36 25, , (b), , x2 y 2, +, =1, 36 11, , (c), , x2 y 2, +, =1, 25 11, , (d) None of these, , 36, 5

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CONIC SECTIONS, , 137, , 52. The foci of the ellipse 25 (x + 1)2 + 9(y + 2)2 = 225 are at :, (a) (–1, 2) and (–1, –6), , (b) (–2, 1) and (–2, 6), , (c) (–1, –2) and (–2, –1), , (d) (–1, –2) and (–1, –6), , 53. The eccentricity of an ellipse, with its centre at the origin, is, 1, . If one of the directrices is x = 4, then the equation of the, 2, , 59. The eccentricity of the curve 2x2 + y2 – 8x – 2y + 1= 0 is:, (a), , 1, 2, , (b), , 1, 2, , 2, 3, , (c), , 2, , 2, , 2, , (a), , 4x + 3 y = 1, , (b) 3x + 4 y = 12, , (c), , 4 x 2 + 3 y 2 = 12, , (d) 3x 2 + 4 y 2 = 1, , 54. A focus of an ellipse is at the origin. The directrix is the line, 1, x = 4 and the eccentricity is . Then the length of the semi2, major axis is, (a), , 8, 3, , 2, (b), 3, , 4, (c), 3, , 5, (d), 3, , 55. Equation of the ellipse whose axes are the axes of, coordinates and which passes through the point (–3, 1), and has eccentricity, , 2, is, 5, , (a) 5x2 + 3y2 – 48 = 0, , (b) 3x2 + 5y2 – 15 = 0, , (c) 5x2 + 3y2 – 32 = 0, , (d) 3x2 + 5y2 – 32 = 0, , 56. The eccentricity of the ellipse whose major axis is three, times the minor axis is:, (a), , 2, 3, , (b), , 3, 2, , (c), , 2 2, 3, , (d), , 16x2 + 25y2 = 400, then PF1 +PF2 equals, , (a) 8, , (b) 6, , (c) 10, , 57. The equation of an ellipse with one vertex at the point (3, 1),, 2, the nearer focus at the point (1, 1) and e = is :, 3, , (a), , (c), , (x + 3)2 (y - 1)2, (x - 3) 2 (y + 1) 2, =1, +, = 1 (b), +, 36, 20, 20, 36, (x - 3)2 (y - 1) 2, (x - 3) 2 (y + 1) 2, = 1 (d), +, =1, +, 36, 20, 36, 20, , 58. An ellipse has OB as semi minor axis, F and F ' its focii and, the angle FBF ' is a right angle. Then the eccentricity of, the ellipse is, (a), , 1, 2, , (b), , 1, 2, , 1, (c), 4, , (d), , 61. The eccentricities of the ellipse, , 1, , (d) 12, , x2, , y2, +, = 1, a > b; and, a 2 b2, , x 2 y2, +, = 1 are equal. Which one of the following is, 9 16, , correct ?, (a), , 4a = 3b, , (b) ab = 12, , (c), , 4b = 3a, , (d) 9a = 16b, , 62. The conic represented by, x = 2 (cos t + sin t), y = 5 (cos t – sin t) is, (a) a circle, (b) a parabola, (c) an ellipse, (d) a hyperbola, 63. Equation of the ellipse whose axes are along the coordinate, axes, vertices are (± 5, 0) and foci at (± 4, 0) is, (a), , x 2 y2, +, =1, 16 9, , (b), , x 2 y2, +, =1, 25 9, , (c), , x 2 y2, +, =1, 4 25, , (d), , x 2 y2, +, =1, 25 16, , 2, 3, , 3, 4, , 60. If P º (x, y), F1 º (3, 0), F2 º (–3, 0) and, , ellipse is:, 2, , (d), , 64. If equation of the ellipse is, , x2, y2, +, = 1 , then, 100 400, , I., , Vertices of the ellipse are (0, ± 20), , II., , Foci of the ellipse are (0, ± 10 3), , III. Length of major axis is 40., IV., , 3, ., 2, , Eccentricity of the ellipse is, , (a) I and II are true., (c) II, III, IV are true., , (b) III and IV are true., (d) All are true., , 65. The foci of an ellipse are (±2, 0) and its eccentricity is, , then the equation of ellipse is, , 3, , (a) 3, , (b) 4, , x2, a2, , +, , 1, 2, , y2, = 1 . The value of ‘a’ is, 12, , (c) 6, , (d) 2

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EBD_7762, 138, , 66., , MATHEMATICS, , The equation of the ellipse whose axes are along the, co-ordinate axes, vertices are (±5, 0) and foci at (±4, 0), is, x2 y 2, +, = 1 . The value of b2 is, 25 b 2, , 67., , (a) 3, (b) 5, (c) 9, (d) 4, Let the centre of an ellipse is at (0, 0), Assertion: If major axis is on the y-axis and ellipse passes, through the points (3, 2) and (1, 6), then the equation of, ellipse is, , Reason:, , 68., , x 2 y2, +, = 1., 10 40, x2, b2, , +, , y2, a2, , = 1 is an equation of ellipse if major axis, , is along y-axis., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., A bar of given length moves with its extremities on two, fixed straight lines at right angles. Any point of the bar, describes, , 70. A rod AB of length 15 cm rests in between two coordinate, axes in such a way that the end point A lies on x-axis and, end point B lies on y-axis. A point P(x, y) is taken on the rod, in such a way that AP = 6 cm. Then, the locus of P is a/an., (a) circle, (b) ellipse, (c) parabola, (d) hyperbola, 71. An arch is in the form of semi-ellipse. It is 8 m wide and 2 m, high at the centre. Then, the height of the arch at a point, 1.5 m from one end is, (a) 1.56 m (b) 2.4375 m (c) 2.056 m (d) 1.086 m, 72. A man running a race course notes that sum of its distance, from two flag posts from him is always 10 m and the distance, between the flag posts is 8 m. Then, the equation of the, posts traced by the man is, x 2 y2, +, =1, 25 9, , (a), , (c) x2 + y2 = 9, , 73. The equation of a hyperbola with foci on the x-axis is, , x2, a, , q, , y, O, , 69., , L, , the ellipse, , a2, , +, , y2, b2, , x, , 2, , -, , -, , y2, b, , 2, , b2, y, , 2, , =1, , (b), , x y, - =1, a b, , =1, , (d), , a b, - =1, x y, , 74. Length of the latus rectum of the hyperbola :, , q, A, , X, , (a) parabola, (b) ellipse, (c) hyperbola, (d) circle, The eccentric angles of the extremities of the latus rectum of, x2, , 2, , a2, , (c), x, , x 2 y2, +, =1, 9 25, , Topic 4: Hyperbola: Different equations of a Hyperbola, Terms, Related to Hyperbola, Eccentricity., , Y, , P(x,y), , (d), , v, , (a), , B, , (b) x2 + y2 = 25, , = 1 are given by, , (a), , æ ae ö, tan -1 ç ± ÷, è bø, , -1 æ be ö, (b) tan ç ± ÷, è a ø, , (c), , æ bö, tan -1 ç ± ÷, è ae ø, , -1 æ a ö, (d) tan ç ± ÷, è be ø, , x2, a2, , (a), , -, , y2, b2, b2, a, , = 1 is, , (b), , 2b 2, a, , (c), , a2, b, , (d), , 2a 2, b, , 75. What is the difference of the focal distances of any point on, the hyperbola?, (a) Eccentricity, (b) Distance between foci, (c) Length of transverse axis, (d) Length of semi-transverse axis, 76. The equation of the hyperbola with vertices (3, 0), (–3, 0) and, semi-latus rectum 4 is given by :, (a) 4x2 – 3y2 + 36 = 0, (b) 4x2 – 3y2 + 12 = 0, 2, 2, (c) 4x – 3y – 36 = 0, (d) 4x2 + 3y2 – 25 = 0

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CONIC SECTIONS, , 139, , 77. The eccentricity of the hyperbola x 2 - 3 y 2 = 2 x + 8 is, (a), , 2, 3, , (b), , 1, 3, , 2, , (c), , 3, , (d), , 3, 2, , (c) conjugate axis is along y-axis of length 6, (d) None of the above, 84. The length of transverse axis of the hyperbola 3x2 – 4y2 = 32,, is, , 78. The equation of the hyperbola with vertices at (0, ± 6) and, (a), , 5, e = is, 3, (a), , x2 y 2, =1, 36 64, , y 2 x2, (b), =1, 36 64, , (c), , x2 y 2, =1, 64 36, , y 2 x2, (d), =1, 64 36, , 79. The vertices of the hyperbola, 9 x 2 - 16 y 2 - 36 x + 96 y - 252 = 0 are, , (a), , (6, 3), (-2,3), , (b) (6, 3), (-6,3), , (c), , (-6,3), (-6, -3), , (d) (2,3), (-2,3), , 80. If the equation of the hyperbola is, , y2 x2, = 1 , then, 9 27, , I., , the coordinates of the foci are (0, ± 6), , II., , the length of the latus rectum is 18 units., , III. the eccentricity is, , 4, ., 5, , (c), , – x2 + 3y2 = 3, , (d) – 3x2 + y2 = 3, , 10, 13, and, units respectively, then what is the length, 3, 3, , 7, unit (b) 12 unit, 2, , 16 2, 3, , (c), , 3, 32, , (d), , 64, 3, , 85. If the equation of the hyperbola is, 9y2 – 4x2 = 36, then, the coordinates of foci are (0, ± 13), 2, II. the eccentricity is, ., 13, III. the length of the latus rectum is 8., , I., , (a) Only I is true., , (b) Only II is true., , (c) Only III is true., , (d) None of them is true., , 86. The length of the transverse axis along x-axis with centre at, origin of a hyperbola is 7 and it passes through the point, (5, – 2). Then, the equation of the hyperbola is, (a), , 4 2 196 2, x y =1, 49, 51, , (b), , 49 2 51 2, x y =1, 4, 196, , (c), , 4 2 51 2, x y =1, 49, 196, , (d) None of these, , ( x - 1) 2 - 3( y - 1) 2 = 1 is, , (a) 3, , (b) 2, , (c), , 2, 3, , (c), , 15, 15, unit (d), unit, 2, 4, , x2 y2, = 1 , then, 83. If the equation of hyperbola is, 9 16, , (a) transverse axis is along x-axis of length 6, (b) transverse axis is along y-axis of length 8, , (d), , 3, 2, , 88. The equation of the hyperbola whose conjugate axis is 5, and the distance between the foci is 13, is, (a) 25x2 – 144y2 = 900, , (b) 144x2 – 25y2 = 900, , (c) 144x2 + 25y2 = 900, , (b) 25x2 + 144y2 = 900, , 89. If e1 is the eccentricity of the ellipse, , of the transverse axis?, (a), , (b), , (b) 3x2 – y2 = 3, , 82. If the eccentricity and length of latus rectum of a hyperbola, are, , 3, , 87. The eccentricity of the hyperbola conjugate to, , (a) Only I is true., (b) Only II is true., (c) Only I and II is true., (d) Only II and III is true., 81. The equation of the hyperbola whose foci are (– 2, 0) and (2, 0), and eccentricity is 2 is given by :, (a) x2 – 3y2 = 3, , 8 2, , x2 y 2, +, = 1 and e2 is, 16 25, , the eccentricity of the hyperbola passing through the foci, of the ellipse and e1e2 = 1, then equation of the hyperbola is :, (a), , x2 y 2, =1, 9 16, , (b), , x2 y 2, = -1, 16 9, , (c), , x2 y 2, =1, 9 25, , (d), , x2 y 2, =1, 9 36

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EBD_7762, 140, , 90., , MATHEMATICS, , The equation of the conic with focus at (1, – 1) directrix, along x – y + 1 = 0 and with eccentricity, , 2 is, , x2 – y2 = 1, , 91., , (a), (b) xy = 1, (c) 2xy – 4x + 4y + 1 = 0, (d) 2xy + 4x – 4y – 1 = 0, Equation of the hyperbola whose directrix is 2x + y = 1,, focus (1, 2) and eccentricity 3 is, (a), (b), (c), (d), , 7x2 – 2y2 + 12xy – 2x + 9y – 22 = 0, 5x2 – 2y2 + 10xy + 2x + 5y – 20 = 0, 4x2 + 8y2 + 8xy + 2x – 2y + 10 = 0, None of these, 2, , 92., , x2 y 2, 1, =, coincide. Then the value of b2 is, 144 81 25, , (a) 9, 93., , (b) 1, , (c) 5, , (d) 7, , The equation of the hyperbola whose vertices are (± 2, 0), and foci are (± 3, 0) is, (a) 5, , x, , 2, , a, , 2, , (b) 4, , -, , y, , 2, , b2, , = 1 . Sum of a2 and b2 is, , (c) 9, , (d) 1, , BEYOND NCERT, Topic 5: Circle: Line and a Circle, Tangents and Normals,, Director circle, chord of contact, Pole and Polar, Common, Chord of Two Circles, Angle of Intersection of Two Circles,, Radical Axis and Radical Centre, Family of Circles., 94., , 95., , A circle of radius 5 touches another circle, x2 + y2 –2x – 4y – 20 = 0 at (5, 5) then its equation is :, (a) x2 + y2 + 18x + 16y + 120 = 0, (b) x2 + y2 – 18x – 16y + 120 = 0, (c) x2 + y2 – 18x + 16y + 120 = 0, (d) None of these, If the two circles ( x - 1) 2 + ( y - 3) 2 = r 2 and, x 2 + y 2 - 8 x + 2 y + 8 = 0 intersect in two distinct point,, , 96., , then, (a) r > 2, (b) 2 < r < 8, (c) r < 2 (d) r = 2, The limiting points of the coaxial system determined by the, circles x2 + y2 – 2x – 6y + 9 = 0 and x2 + y2 + 6x – 2y + 1 = 0, æ 3 14 ö, æ 3 - 14 ö, (a) ( -1, 2), ç ,, (b) (-1, 2), ç , ÷, ÷, è5 5 ø, è5 5 ø, (c), , æ - 3 14 ö, ( -1, 2), ç, , ÷, è 5 5ø, , (a), , (c) 3, (d) 2, (b), 2, 3, 2, 2, 98. If y = 2x is a chord of the circle x + y – 10x = 0, then the, equation of a circle with this chord as diameter, is, x2 + y2 – ax – by = 0. Sum of a and b is, (a) 4, (b) 2, (c) 6, (d) 0, 99. The shortest distance between the circles, (x–1)2 +(y+2)2 =1 and (x + 2)2 + (y – 2)2 = 4 is, (a), , 2, , x, y, +, = 1 and the hyperbola, 16 b2, , The foci of the ellipse, , 97. If one of the diameters of the circle x2 + y2 – 2x – 6y + 6 = 0, is a chord to the circle with centre (2, 1), then the radius of, the circle is, , (d) None of these, , 1, , (b) 2, , (c) 3, , (d) 4, , 100. A pair of tangents are drawn from the origin to the circle, x2 + y2+ 20 (x + y) + 20 = 0, then the equation of the pair of, tangent are, (a), , x 2 + y2 - 5xy = 0, , (b) x 2 + y2 + 2x + y = 0, , (c), , x 2 + y 2 – xy + 7 = 0, , (d) 2x 2 + 2y2 + 5xy = 0, , 101. The area of an equilateral triangle inscribed in the circle, x 2 + y 2 + 2gx + 2fy + c = 0 is, , (, (, , (, , ), ), , ), , 3 3 2, 3 3 2, g +f 2 -c, g +f 2 -c, (b), 4, 2, 3 3 2, g +f 2 +c, (c), (d) None of these, 4, 102. Equation of the circle passing through the origin and through, the points of intersection of the circle, (a), , x2 + y2– 2x + 4y – 20 = 0 and the line x + y – 1 = 0 is, (a), , x 2 + y2 - 20x + 15y = 0, , (b), , x 2 + y2 + 33x + 33y = 0, , (c), , x 2 + y2 – 22x –16y = 0, , (d), , 2x 2 + 2y 2 – 4x – 5y = 0, , 103. Consider the following statements :, I., , Circle x2 + y2 – x – y – 1 = 0 is completely inside the, circle x2 + y2 – 2x + 2y – 7 = 0., , II., , Number of common tangents of the circles, x2 + y2 + 14x + 12y + 21 = 0 and x2 + y2 + 2x – 4y – 4 = 0 is 4, , Which of these is/are correct?, (a) Only (1), , (b) Only (2), , (c) Both of these, , (d) None of these

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CONIC SECTIONS, , 141, , æ3 3 ö, ,1÷ is a point on the ellipse, 104. Assertion : If P çç, ÷, è 2, ø, , 4x2 + 9y2 = 36. Circle drawn AP as diameter touches another, circle x2 + y2 = 9, where A º (- 5, 0), Reason : Circle drawn with focal radius as diameter touches, the auxiliary circle., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 105. If the circle x2 + y2 + 2gx + 2fy + c = 0 bisects the circumference, of the circle x2 + y2 + 2g¢x + 2f¢y + c¢ =0, then,, (a), , 2g¢(g - g¢) + 2f ¢(f - f ¢) = c - c¢, , (b), , 2g(g - g¢) + 2f (f - f ¢) = c - c¢, , (c), , g¢(g - g¢) + f ¢(f - f ¢) = c - c¢, , (a), , x 2 + y 2 - 3x - 4y = 0 (b) x 2 + y 2 - 3x + 4y = 0, , (c), , x 2 + y 2 + 3x + 4y = 0 (d) x 2 + y 2 – 7x + 7y = 0, , 107. If the line x + y = 1 is a tangent to a circle with centre, (2, 3), then its equation is, , x 2 + y 2 – 4x – 6y + 5 = 0, , (c), , x 2 + y2 – x – y + 3 = 0, , (d), , x 2 + y 2 + 5x + 2y = 0, , (c), , 3, 5, , (d), , 1, 7, , (b), , 4x 2 + 4y 2 + 30x - 13y - 25 = 0, , (c), , 2x 2 + 2y2 + 3x - 4y = 0, , (d), , 4x 2 + 4y2 - 8x + 7y + 10 = 0, , circle having area as 154 sq.units.Then the equation of the, circle is, 2, 2, (a) x 2 + y 2 - 2 x + 2 y = 62 (b) x + y + 2 x - 2 y = 62, , (c) x 2 + y 2 + 2 x - 2 y = 47 (d) x 2 + y 2 - 2 x + 2 y = 47 ., 114. If a > 2b > 0 then the positive value of m for which, 2, 2, 2, y = mx - b 1 + m 2 is a common tangent to x + y = b, , and ( x - a)2 + y 2 = b2 , is, , 2b, a 2 - 4b2, 2b, a - 2b, , (b), , (d), , a 2 - 4b2, 2b, b, a - 2b, , 115. If the lines 2 x + 3 y + 1 = 0 and 3x - y - 4 = 0 lie along, 2, , x 2 + y 2 - 12 x + 27 = 0, x 2 + y 2 - 12 y + 8 = 0 is, , (a) (13, 33/4), (b) (33/4, –13), (c) (33/4, 13), (d) None of these, 109. If the lines 3x – 4y + 4 = 0 and 6x – 8y – 7 = 0 are tangents to, a circle, then radius of the circle is, 2, 3, , 3, 4, , x 2 + y2 + 5x + y = 0, , (c), , 108. The radical centre of the circles x + y - 16 x + 60 = 0,, , (b), , (b), , (a), , (a), , 2, , 3, 4, , 2, 5, , 111. The slope of the tangent at the point (h, h) of the circles, x2 + y2 = a2 is, (a) 0, (b) 1, (c) –1, (d) depends on h, 112. Equation of the circle which passes through the intersection, of x2 + y2 + 13x – 3y = 0 and 2x2 + 2y2+ 4x – 7y – 25 = 0, whose centre lies on 13x + 30y = 0 is, , x 2 + y 2 + 2x + 2y + 5 = 0, , (b), , (a), , (a), , 113. The lines 2 x - 3 y = 5 and 3x - 4 y = 7 are diameters of a, , (d) None of these, 106. Equation of the circle concentric with the circle, x2 + y2 – 3x + 4y – c = 0 and passing through the point, (–1, – 2), is, , (a), , 110. A.M. of the slopes of two tangents which can be drawn, from the point (3, 1) to the circle x2 + y2 = 4 is, , (c), , 1, 4, , (d), , 5, 2, , diameter of a circle of circumference 10p, then the equation, of the circle is, (a), , x 2 + y 2 + 2 x - 2 y - 23 = 0, , (b), , x 2 + y 2 - 2 x - 2 y - 23 = 0, , (c), , x 2 + y 2 + 2 x + 2 y - 23 = 0, , (d), , x 2 + y 2 - 2 x + 2 y - 23 = 0

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EBD_7762, 142, , MATHEMATICS, , 116. Intercept on the line y = x by the circle x 2 + y 2 - 2 x = 0 is, AB. Equation of the circle on AB as a diameter is, (a), , x2 + y 2 + x - y = 0, , (b) x 2 + y 2 - x + y = 0, , (c), , x2 + y 2 + x + y = 0, , (d) x 2 + y 2 - x - y = 0, , 117. The equation of chord of the circle x2 + y2 = 8x bisected, at the point (4, 3) is, (a) x = 3, (b) y = 3, (c) x = –3 (d) y = –3, 118. The locus of the centre of a circle, which touches externally, the circle x2 + y2 – 6x – 6y + 14 = 0 and also touches the, y-axis, is given by the equation:, (a) x2 – 6x – 10y + 14 = 0, , (b) x2 – 10x – 6y + 14 = 0, , (c) y2 – 6x – 10y + 14 = 0, , (d) y2 – 10x – 6y + 14 = 0, , 119. Two circles S1 = x2 + y2 + 2g1x + 2f1y + c1 = 0 and, S2 = x2 + y2 + 2g2x + 2f2y + c2 = 0 cut each other orthogonally,, then :, (a) 2g1g2 + 2f1f2 = c1 + c2 (b) 2g1g2 – 2f1f2 = c1 + c2, (c) 2g1g2 + 2f1f2 = c1 – c2 (d) 2g1g2 – 2f1f2 = c1 – c2, 120. In the given figure, the equation of the larger circle is, x 2 + y 2 + 4 y - 5 = 0 and the distance between centres is, , 122. A variable circle passes through the fixed point A( p, q ) and, touches x-axis . The locus of the other end of the diameter, through A is, (a), , ( y - q)2 = 4 px, , (b) ( x - q)2 = 4 py, , (c), , ( y - p )2 = 4qx, , (d) ( x - p)2 = 4qy, , 123. A circle is given by x2 + (y–1)2 = 1, another circle C touches, it externally and also the x-axis, then the locus of its centre is, (a) {(x, y) : x2 = 4y} È {(x, y) : y £ 0}, (b) {(x, y) : x2 + (y – 1)2 = 4}È {(x, y) : y £ 0}, (c) {(x, y) : x2 = y} È {(0, y) : y £ 0}, (d) {(x, y) : x2 = 4y} È {(0, y) : y < 0}, 124. The sum of the minimum distance and the maximum distance, from the point (4, – 3) to the circle x2 + y2 + 4x – 10y – 7 = 0 is, (a) 20, (b) 12, (c) 10, (d) 16, 125. a, b and g are parametric angles of three points P, Q and R, respectively, on the circle x2 + y2 = 1 and A is the point, (–1, 0). If the lengths of the chords AP, AQ and AR are in, G.P., then cos, (a) A.P., (c) H.P., , 4. Then the equation of smaller circle is, , C2, , (a), , C1(0,–2), , Topic 6 : Parabola: Position of a Point and a Line with, Respect to a Parabola, Tangent, Subtangent, Normal,, Subnormal, Co-normal Points, Chord of contact, Pole and, Polar, Geometrical Properties of a Parabola., , X, , 126. Tangents are drawn from the point (–2, –1) to the parabola, y2 = 4x. If a is the angle between these tangent then the, value of tan a is, (a) 3, (b) 4, (c) –5, (d) 5, 127. From the point (15, 12) three normals are drawn to the, parabola y2 = 4x, then centroid of triangle formed by three, co-normal points is –, , ( x - 7 ) 2 + ( y - 1) 2 = 1, , (b) ( x + 7 ) 2 + ( y - 1) 2 = 1, (c), , x 2 + y 2 = 2 7 x + 2y, , (d) None of these, 121. If the straight line ax + by = 2 ; a, b ¹ 0 touches the circle, x2 + y2 – 2x = 3 and is normal to the circle x2 + y2 – 4y = 6,, then the values of a and b are, (a), , 3, , 2, 2, , 4, (b) - , 1, 3, , (c), , (b) G.P., (d) None of these, BEYOND NCERT, , Y, , O, , g, a, b, , cos and cos are in, 2, 2, 2, , 1, ,2, 4, , (d), , 2, , –1, 3, , (a), , æ 16 ö, çè , 0÷ø (b) (4, 0), 3, , æ 26 ö, (c) ç , 0÷ (d) (6, 0), è 3 ø, , 128. The middle point of the chord x + 3y = 2 of the conic, x 2 + xy - y 2 = 1 is, , (a) (5, –1), , (b) (1, 1), , (c) (2, 0), , (d) (–1, 1), , 129. The area of triangle made by the chord of contact and, tangents drawn from point (4, 6) to the parabola y2 = 8x, is, (a) 1, , (b) 2, , (c) 3, , (d) 4

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CONIC SECTIONS, , 143, , 130. If the normal drawn from the point on the axis of the parabola, y2 = 8ax whose distance from the focus is 8a and which is, not parallel to either axis, makes an angle. q with the axis of, x, then q is equal to, (a), , p, 6, , (b), , p, 4, , (c), , p, 3, , (d), , 2p, 3, , 131. Two common tangents to the circle x2 + y2 = 2a2 and parabola, y2 = 8ax are, (a) x = ±(y + 2a), , (b) y = ±(x + 2a), , (c) x = ±(y + a), (d) y = ±(x + a), 2, 132. If the line x + my + am = 0 touches the parabola y2 = 4ax,, then the point of contact is, (a) (–am2, 2am), (b) (–am2, – 2am), (c) (am2, –2am), (d) (am2, 2am), 133. The point of intersection of the tangents to the parabola, y2 = 4ax at the points 't1' and 't2' is, (a) (at1t2, a(t1 + t2)), (b) (at1t2, at1t2(t1 + t2)), (c) (at1t2 (t1 +t2), a (t1 + t2) (d) None of these, 134. If a chord which is normal to the parabola at one end and, subtends a right angle at the vertex, then slope of the chord is, (a) 1, , (b) –2, , (c), , 2, , (d), , 1, 2, , 2, , 135. The length of the chord of the parabola x = 4 y passing, through the vertex and having slope cot a is, (a), , 4 cos acosec 2 a, , (b) 4 tan a sec a, , (c), , 4 sin a sec 2 a, , (d) None of these, , 138. The equation of the parabola whose focus is (0, 0) and the, tangent at the vertex is x – y + 1 = 0 is, (a), , x 2 + y 2 + 2xy - 4x + 4y - 4 = 0, , (b), , x 2 - 4x + 4y - 4 = 0, , (c), , y 2 - 4x + 4y - 4 = 0, , (d), , 2x 2 + 2 y 2 - 4xy - x + y - 4 = 0, , 139. If x + y = k is normal to y2 = 12 x, then the value of k is, (a) 3, (b) 9, (c) –9, (d) –3, 2, 140. If the parabola y = 4ax passes through the point (1, –2),, then the tangent at this point is, , (c), , Reason : Length of focal chord of a parabola y2 = 4ax making, an angle a with x-axis is 4a cosec2 a., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., y2, , x2, , 137. If the two parabolas = 4a (x – k1) and = 4a (y – k2), always touch each other, k1 and k2 being variable parameters,, then their point of contact lies on the curve, (a) xy = a2, (b) xy = 2a2, 2, (c) xy = 4a, (d) None of these, , x + y +1 = 0, , (d) x – y – 1 =0, , BEYOND NCERT, Topic 7: Ellipse: Position of a Point with respect to an, Ellipse, Auxillary Circle, Director Circle, Tangents and, Normals, Chord of Contact, Pole and Polar,, Diameter of an Ellipse., 141. The value of l does the line y = x + l touches the ellipse, 9x2 + 16y2 =144 is/are, (a), , ± 2 2 (b) 2 ± 3, , (c) ± 5, , (d) 5 ± 2, , 142. If the chords of contact of tangent from two points (x1, y1), and (x2, y2) to the ellipse, then the value of, (a), , 136. Assertion : Length of focal chord of a parabola y2 = 8x, making an angle of 60° with x-axis is 32., , (b) x - y - 1 = 0, , (a) x + y – 1 = 0, , a4, , (b), , b4, , x1 x2, is, y1 y2, b4, a4, , x2, , y2, = 1 are at right angles,, +, a 2 b2, , (c), , –a 4, b4, , (d), , –b 4, a4, , æ, ö, 16, sin q ÷÷ to the ellipse, 143. If the tangent at the point çç 4 cos q,, 11, è, ø, 16x 2 + 11y2 = 256 is also a tangent to the circle, x2 + y2 – 2x = 15, then the value of q is, , (a), , ±, , p, 4, , (b) ±, , p, 3, , (c) ±, , p, 6, , (d) ±, , p, 2, , 144. If p is the length of the perpendicular from the focus S of the, ellipse, , x2, , +, , y2, , a 2 b2, 2a, -1 =, then, SP, , (a), , a2, p2, , (b), , = 1 to a tangent at a point P on the ellipse,, , b2, p2, , (c), , p2, , (d), , a 2 + b2, p2

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CONIC SECTIONS, , 145, , 155. The equation 9x2 – 16y2 – 18x + 32y – 151 = 0 represents a, hyperbola, (a) The length of the transverse axes is 4, (b) Length of latus rectum is 9, (c) Equation of directrix is x =, , 21, 11, and x = –, 5, 5, , (d) None of these, 156. The equation of the chord of the hyperbola, , 158. The combined equation of the asymptotes of the hyperbola, 2x2 + 5xy + 2y2 + 4x + 5y = 0 is –, (a) 2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0, (b) 2x2 + 5xy + 2y2 + 4x + 5y – 2 = 0, (c) 2x2 + 5xy + 2y2 = 0, (d) None of these, 159. The locus of a point P (a, b) moving under the condition, that the line y = ax + b is a tangent to the hyperbola, , 25 x 2 - 16 y 2 = 400 that is bisected at point (5, 3) is:, , x2, , -, , y2, , = 1 is, , (a) 135 x – 48 y = 481, , (b) 125 x – 48 y = 481, , a2, , (c) 125 x – 4 y = 48, , (d) None of these, , (a) an ellipse, (c) a parabola, , 157. Assertion : Ellipse, , x 2 y2, +, = 1 and 12x2 – 4y2 = 27 intersect, 25 16, , b2, , (b) a circle, (d) a hyperbola, , 160. The value of m, for which the line y = mx +, , each other at right angle., Reason : Whenever focal conics intersect, they intersect, each other orthogonally., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, , normal to the conic, , (d) Assertion is incorrect, reason is correct., , (a), , -, , (c), , -, , 2, 3, 3, 2, , 25 3, is a, 3, , x2 y2, = 1 , is, 16 9, , (b), , 3, , (d) None of these, , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, 1., , 2., , 3., , The area of the circle centred at (1, 2) and passing through, the point (4, 6) is, (a) 5p, (b) 10p, (c) 25p, (d) None of these, Equation of a circle which passes through (3, 6) and touches, the axes is, (a) x2 + y2 + 6x + 6y + 3 = 0, (b) x2 + y2 – 6x – 6y – 9 = 0, (c) x2 + y2 – 6x – 6y + 9 = 0, (d) None of these, Equation of the circle with centre on the Y-axis and passing, throught the origin and the point (2, 3) is, (a) x2 + y2 + 13y = 0, (b) 3x2 + 3y2 + 13x + 3 = 0, (c) 6x2 + 6y2 – 13y = 0, (d) x2 + y2 + 13x + 3 = 0, , 4., , 5., , 6., , The equation of a circle with origin as centre and passing, through the vertices of an equilateral triangle whose median, is of length 3a is, (a) x2 + y2 = 9a2, (b) x2 + y2 = 16a2, (c) x2 + y2 = 4a2, (d) x2 + y2 = a2, If the focus of a parabola is (0, – 3) and its directrix is, y = 3, then its equation is, (a) x2 = – 12y, (b) x2 = 12y, 2, (c) y = – 12x, (d) y2 = 12x, 2, If the parabola y = 49x passes through the point (3, 2) , then, the length of its latusrectum is, (a), , 2, 3, , (b), , 4, 3, , (c), , 1, 3, , (d) 4

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EBD_7762, 146, , 7., , 8., , MATHEMATICS, , If the vertex of the parabola is the point (–3, 0) and the, directrix is the line x + 5 = 0, then its equation is, (a) y2 = 8(x + 3), (b) x2 = 8(y + 3), 2, (c) y = – 8(x + 3), (d) y2 = 8(x + 5), If equation of the ellipse whose focus is (1, – 1), then directrix, 1, is, 2, (a) 7x2 + 2xy + 7y2 – 10x + 10y + 7 = 0, (b) 7x2 + 2xy + 7y2 + 7 = 0, (c) 7x2 + 2xy + 7y2 + 10x – 10y – 7 = 0, (d) None of the above, The length of the latusrectum of the ellipse 3x2 + y2 = 12 is, (a) 4, (b) 3, , the line x – y – 3 = 0 and eccentricity, , 9., , (c) 8, , 10., , 11., , x2, a2, , +, , y2, , = 1 (where,, , b2, , a < b), then, (a) b2 = a2 (1 – e2), (b) a2 = b2 (1 – e2), 2, 2, 2, (c) a = b (e – 1), (d) b2 = a2 (e2 – 1), The eccentricity of the hyperbola whose latus rectum is 8, and conjugate axis is equal to half the distance between the, foci is :, 4, 3, , 4, , (b), , 3, , 2, , (c), , (d) None of these, 3, The distance between the foci of a hyperbola is 16 and its, 2. Its equation is, , eccentricity is, , x2 y 2, =1, 4, 9, (d) None of these, , (a) x2 – y2 = 32, , (b), , (c) 2x – 3y2 = 7, 13., , 3, , If e is eccentricity of the ellipse, , (a), , 12., , 4, , (d), , 3, and foci at (±, 2, , Equation of the hyperbola with eccentricity, 2, 0) is, , (a), , 1, 2, , (b), , 1, 4, , 3, , (c), , 2, , (d), , 3, 2, , 16. The slope of the line touching both the parabolas y 2 = 4 x, and x 2 = -32 y is, (a), , 1, 8, , (b), , [JEE MAIN 2014, A, BN], 2, 3, , 1, 2, , (c), , (d), , 3, 2, , 17. An arch of a bridge is semi-elliptical with major axis, horizontal. If the length the base is 9 meter and the highest, part of the bridge is 3 meter from the horizontal; the best, approximation of the height of the arch. 2 meter from the, centre of the base is, [BITSAT 2014, C], (a) 11/4 m (b) 8/3 m, (c) 7/2 m (d) 2 m, 18. The angle of intersection of the two circles, x2 + y2 – 2x – 2y = 0 and x2 + y2 = 4, is [BITSAT 2014, C, BN], (a) 30º, (b) 60º, (c) 90º, (d) 45º, 19. Let O be the vertex and Q be any point on the parabola,, x2 = 8y. If the point P divides the line segment OQ internally in, the ratio 1 : 3, then locus of P is :, [JEE MAIN 2015, C], 2, 2, 2, (a) y = 2x (b) x = 2y (c) x = y (d) y2 = x, 20. The number of common tangents to the circles, x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is, [JEE MAIN 2015, C, BN], (a) 3, (b) 4, (c) 1, (d) 2, 21. The area (in sq. units) of the quadrilateral formed by the, tangents at the end points of the lat us rectum to the ellipse, , x 2 y2, +, = 1, is, 9, 5, , [JEE MAIN 2015, A, BN], , 27, 27, (b) 27, (c), (d) 18, 2, 4, 22. Locus of the image of the point (2, 3) in the line, (2x – 3y + 4) + k (x – 2y + 3) = 0, k Î R, is a :, [JEE MAIN 2015, A], , (a), , (a), , x2 y 2 4, –, =, 4, 5 9, , (b), , x2 y 2 4, –, =, 9, 9 9, , (c), , x2 y 2, –, =1, 4, 9, , (d) None of these, Past Year MCQs, , 14., , 15. Let C be the circle with centre at (1, 1) and radius = 1. If T is, the circle centred at (0, y), passing through origin and, touching the circle C externally, then the radius of T is equal, to, [JEE MAIN 2014, A], , (a), , ( x2 + y ), , (c), , ( x2 - y2 ), , 2, , (, , 2 2, , (, , ), , = 6 x 2 + 2 y 2 (b) x 2 + y, , 2., , (b) circle of radius, , The locus of the foot of perpendicular drawn from the centre, of the ellipse x2 + 3y2 = 6 on any tangent to it is, [JEE MAIN 2014, A, BN], 2 2, , (a) circle of radius, , = 6 x 2 + 2 y 2 (d) x 2 - y 2, , ), , 2, , = 6 x2 - 2 y 2, = 6 x2 - 2 y 2, , 3., (c) straight line parallel to x-axis, (d) straight line parallel to y-axis, 23. Area of the circle in which a chord of length 2 makes an, angle p/2 at the centre, is, [BITSAT 2015, A], (a) p/2 sq units, (b) 2p sq units, (c) p sq units, (d) p/4 sq units

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CONIC SECTIONS, , 147, , 24. The eccentricity of an ellipse, with its centre at the origin, is, 1/2. If one of the directrices is x = 4 , then the equation of, the ellipse is:, [BITSAT 2015, C], (a) 4x2 + 3y2 = 1, (b) 3x2 + 4y2 = 12, (c) 4x2 + 3y2 = 12, (d) 3x2 + 4y2 = 1, 25. Let S be the focus of the parabola y2 = 8x and PQ be the, common chord of the circle x2 + y2 – 2x – 4y = 0 and the, given parabola. The area of DPQS is [BITSAT 2015, A], (a) 4 sq units, (b) 3 sq units, (c) 2 sq units, (d) 8 sq units, 26. The centres of those circles which touch the circle,, x2 + y2 – 8x – 8y – 4 = 0, externally and also touch the x-axis, lie on:, [JEE MAIN 2016, A], (a) a hyperbola, (b) a parabola, (c) a circle, (d) an ellipse which is not a circle, 27. The eccentricity of the hyperbola whose length of the latus, rectum is equal to 8 and the length of its conjugate axis is, equal to half of the distance between its foci, is :, [JEE MAIN 2016, C], , 2, , (a), , (b), , 3, , (c), , 4, 3, , (d), , 4, , 3, 3, 28. If one of the diameters of the circle, given by the equation,, x2 + y2 – 4x + 6y – 12 = 0, is a chord of a circle S, whose, centre is at (–3, 2), then the radius of S is:, [JEE MAIN 2016, A], (a) 5, , (c) 5 2, (d) 5 3, 2, 29. Let P be the point on the parabola, y = 8x which is at a, minimum distance from the centre C of the circle,, x2 + (y + 6)2 = 1. Then the equation of the circle, passing, through C and having its centre at P is:, [JEE MAIN 2016, A], x, + 2y - 24 = 0, 4, (b) x2 + y2 – 4x + 9y + 18 = 0, (c) x2 + y2 – 4x + 8y + 12 = 0, (d) x2 + y2 – x + 4y – 12 = 0, The parabola having its focus at (3, 2) and directrix along, the y-axis has its vertex at, [BITSAT 2016, C], , (a), , 30., , (b) 10, , x 2 + y2 -, , æ3, è2, , ö, ø, , æ2, , ö, , (b) ç , 2 ÷, , (a) (2, 2), æ1 ö, ç , 2÷, è2 ø, , (d) ç 3 , 2 ÷, è, ø, 31. The locus of the point of intersection of two tangents to the, parabola y2 = 4ax, which are at right angle to one another is, [BITSAT 2016, A, BN], (a) x2 + y2 = a2, (b) ay2 =x, (c) x + a = 0, (d) x + y ± a = 0, 32. The length of the chord x + y = 3 intercepted by the circle, (c), , x 2 + y 2 - 2x - 2 y - 2 = 0 is, , (a), , 7, 2, , (b), , 3 3, 2, , [BITSAT 2016, C], (c), , 14, , (d), , 7, 2, , 33. The lengths of the tangent drawn from any point on the circle, 15x 2 + 15y 2 - 48x + 64 y = 0 to the two circles, 5x2 + 5y2 – 24x + 32y + 75 = 0 and 5x2 + 5y2 – 48x + 64y + 300 =, 0 are in the ratio of, [BITSAT 2016, A, BN], (a) 1 : 2, (b) 18, (c) 16 (d) None of these, 34. The number of integral values of l for which, , 35., , x 2 + y 2 + lx + (1 - l) y + 5 = 0 is the equation of a circle, whose radius cannot exceed 5, is, [BITSAT 2016, C], (a) 14, (b) 18, (c) 16, (d) None of these, The line joining (5, 0) to ( (10 cos q, 10 sin q) is divided, internally in the ratio 2 : 3 at P. If q varies, then the locus of P is, [BITSAT 2016, C], (a) a pair of straight, (b) a circle lines, (c) a straight line, (d) None of these, , 36. A hyperbola passes through the point P, , (, , ), , 2, 3 and, , has foci at (± 2, 0). Then the tangent to this hyperbola at, P also passes through the point :, [JEE MAIN 2017, A, B N], (a), (c), , ((2, , ), 3), , 2, - 3, , (b), , 2,3, , (d), , (3 2, 2 3 ), ( 3, 2 ), , 37. The radius of a circle, having minimum area, which touches, the curve y = 4 – x2 and the lines, y = |x| is :, [JEE MAIN 2017, S], (a), (c), , (, 2(, 4, , ), 2 - 1), 2 +1, , (, 4(, , (b) 2, (d), , ), 2 - 1), 2 +1, , 38. The equation of one of the common tangents to the parabola, y2 = 8x and x 2 + y 2 - 12x + 4 = 0 is, [BITSAT 2017, A, BN], (a) y = –x + 2, (b) y = x – 2, (c) y = x + 2, (d) None of these, 39. An equilateral triangle is inscribed in the circle x2 + y2 = a2, with one of the vertices at (a, 0). What is the equation of the, side opposite to this vertex ?, [BITSAT 2017, A], (a) 2x – a = 0, (b) x + a = 0, (c) 2x + a = 0, (d) 3x – 2a = 0, 40. The length of the semi-latus rectum of an ellipse is one thrid, of its major axis, its eccentricity would be, [BITSAT 2017, C], (a), , 2, 3, , (b), , 2, 3, , (c), , 1, 3, , (d), , 1, 2, , 41. If the tangent at (1, 7) to the curve x 2 = y - 6 touches the, circle x 2 + y2 + 16x + 12y + c = 0 then the value of c is :, [JEE MAIN 2018, A, BN], (a) 185, (b) 85, (c) 95, (d) 195

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EBD_7762, 148, , MATHEMATICS, , 42. Tangent and normal are drawn at P(16, 16) on the parabola, 2, , y = 16x , which intersect the axis of the parabola at A and, B, respectively. If C is the centre of the circle through the, points P, A and B and ÐCPB = q , then a value of tan q is :, [JEE MAIN 2018, A, BN], , (a) 2, 43., , 44., , (b) 3, , (c), , 4, 3, , (d), , 1, 2, , Tangents are drawn to the hyperbola 4x 2 - y2 = 36 at the, points P and Q. If these tangents intersect at the point, T(0, 3) then the area (in sq. units) of DPTQ is :, [JEE MAIN 2018, A, BN], (a) 54 3 (c) 36 5, (b) 60 3 (d) 45 5, Two sets A and B are as under :, A = {(a, b) Î R ´ R : | a - 5 | < 1 and | b - 5 | < 1};, 2, 2, B = {(a, b) Î R ´ R : 4(a - 6) + 9(b - 5) £ 36}. Then :, [JEE MAIN 2018, C], , 45. The locus of the point of intersection of the lines, æ1- t2 ö, 2at, ÷, x = aç, ç 1 + t 2 ÷ and y = 1 + t 2 r epresent (t being a, è, ø, parameter), [BITSAT 2018, A], (a) circle, (b) parabola, (c) ellipse, (d) hyperbola, 46. The equation of the circle which passes through the point, (4, 5) and has its centre at (2, 2) is, [BITSAT 2018, A], (a) (x – 2) + (y – 2) = 13, (b) (x – 2)2 + (y – 2)2 = 13, (c) (x)2 + (y)2 = 13, (d) (x – 4)2 + (y – 5)2 = 13, , x2, , 47. Eccentricity of ellipse 2, a, (9, 5) and (12, 4) is, (a), , +, , y2, b2, , = 1 if it passes through point, [BITSAT 2018, A], , (b), , 3/ 4, , 4/5, , (c), , 48., , (a) A Ì B, (b) A Ç B = f (an empty set), (c) neither A Ì B nor B Ì A, (d) B Ì A, , (d), 5/ 6, 6/7, Consider the equation of a parabola y2 + 4ax = 0, where, a > 0 which of the following is/are correct? [BITSAT 2018, C], (a) Tangent at the vertex is x = 0, (b) Directrix of the parabola is x = 0, (c) Vertex of the parabola is not at the origin, (d) Focus of the parabola is at (a, 0), , Exercise 3 : Try If You Can, 1., , The minimum area (sq. units) of triangle formed by the, tangent to the, , (a) ab, , (c), 2., , ( a + b) 2, 2, , x2, a2, , +, , y2, b2, , 3., , = 1 and coordinate axes is, , (b), , a 2 + b2, 2, , (d), , a 2 + ab + b2, 3, , The equation of the circle having th e lines, 2, , 4., , 5., , The equation of a circle is x2 + y2 – lax – ay – a2 (l2 + 1) = 0., If two chords, each bisected by the x-axis can be drawn to, the circle from the point (a (l– 1), a (l + 1)), which lies on the, circle then l satisfies, (a) l2 – 4l – 12 > 0, (b) l2 – 4l – 12 < 0, 2, (c) l – 12l – 4 > 0, (d) l2 – 12l – 4 < 0, A ray of light is coming along the line which is parallel to, y-axis and strikes a concave mirror whose intersection with, the x - y plane is parabola (x – 4)2 = 4 (y + 2). After reflection,, the ray must pass through the point, (a) (4, –1), (b) (0, 1), (c) (–4, 1), (d) None of these, The locus of the middle points of the chords of the ellipse, , x + 2xy + 3x + 6y = 0 as its normal and having size just, , x2, , sufficient to contain the circle x ( x - 4) + y( y - 3) = 0 is, , a2, , +, , y2, b2, , = 1 , which pass through the positive end of the, , (a), , x 2 + y 2 + 3x - 6 y - 40 = 0, , major axis, is, , (b), , x 2 + y 2 + 6 x - 3y - 45 = 0, , (a), , (c), , x 2 + y 2 + 8x + 4 y - 20 = 0, , (d), , x 2 + y 2 + 4x + 8y + 20 = 0, , (c), , x2, a, , 2, , x2, a2, , +, , +, , y2, b, , 2, , y2, b2, , =, , x, a, , (b), , =, , x y, +, a b, , (d), , x2, a, , 2, , x2, a2, , +, , +, , y2, b, , 2, , y2, b2, , =, , y, b, , =, , x y, a b

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CONIC SECTIONS, , 6., , 7., , 149, , If the two parabolas y2 = 4a (x – k1) and x2 = 4a (y – k2), always touch each other, k1 and k2 being variable parameters,, then their point of contact lies on the curve, (a) xy = a2, (b) xy = 2a2, 2, (c) xy = 4a, (d) None of these, One of the roots of the equation f(x) = 0 is an even prime, integer where f(x) = x2 – ax + b (‘a’ is an odd positive integer)., 5, , Also known that, , å f ( i ) = 20 and two curves are defined, i =1, , as C1 : y2 = f(x) and C2 : y2 = – f(x). Number of distinct normal, to the curve C1 which passes through the centre of the curve, C2 is/are, (a) 1, (b) 2, (c) 3, (d) 4, 8., , In the ellipse, , x2, , y2, +, = 1, the length of the perpendiculars, a2, b2, , from the centre upon all chords, which join the ends of, perpendicular diameter, is, (a), , ab, 2, , a +b, , 2, , (b), , a 2 + b2, , x 2 y2, = 1 and passing through the two foci of the, 49 9, hyperbola. The co-ordinates of the focus of the parabola, are, , (a), (c), , æ 11 ö, ç 0, ÷, è 12 ø, , Tangents drawn from P(1, 8) to the circle x2 + y2 – 6x – 4y, – 11 = 0 touches the circle at the points A and B, respectively., The radius of the circle which passes through the points of, intersection of circles x2 + y 2 – 2x + 6y – 6 = 0 and, x2 + y2 – 2x – 6y + 6 = 0 and intersects the circumcircle of the, DPAB orthogonally is equal to, , 73, 71, (b), (c) 3, (d) 2, 4, 2, 12. If tangents are drawn to the parabola y2 = 4ax at points, whose abscissae are in the ratio m2 : 1, then the locus of, their point of intersection is the curve (m > 0)., (a) y2 = (m½ – m–½)2 ax, (b) y2 = (m½ + m–½)2 ax, 2, ½, –½, 2, (c) y = (m + m ) x, (d) None of these, (a), , 13. Let S is a circle with centre (0, 2 ) . Then, (a) There cannot be any rational point on S, (b) There can be infinitely many rational points on S, (c) There can be at most two rational points on S, (d) There are exactly two rational points on S, [A rational point is a point both of whose coordinates are, rational numbers], 14. If the conics whose equations are, S1 : (sin 2 q) x 2 + (2h tan q) xy + (cos 2 q) y 2, , (c), ab, (d) None of these., 9., The point ([P + 1], [P]) (where [x] is the greatest integer less, than or equal to x), lying inside the region bounded by the, circle x2 + y2 – 2x – 15 = 0 and x2 + y2 – 2x – 7 = 0, then, (a) P Î [–1, 0) È [0, 1) È [1, 2), (b) P Î [–1, 2) – {0, 1}, (c) P Î (–1, 2), (d) none of these, 10. A parabola is drawn with its vertex at (0, – 3), the axis of, symmetry along the conjugate axis of the hyperbola, , æ 11 ö, ç 0, ÷, è 6 ø, , 11., , (b), , 11 ö, æ, ç 0, - ÷, 6ø, è, , (d), , 11 ö, æ, ç 0, - ÷, 12 ø, è, , + 32 x + 16 y + 19 = 0, S2 : (cos2 q) x 2 – (2h ¢ cot q) xy + (sin 2 q) y 2, , + 16 x + 32 y + 19 = 0, æ pö, intersect in four concyclic points, where q Î ç 0, ÷ , then, è 2ø, the incorrect statement(s) can be, (a) h + h ¢ = 0, (b) h - h¢ = 0, (c), , q=, , p, 4, , (d) None of these, , 15. If the circle x 2 + y 2 = a 2 intersects the hyperbola xy = c 2, in four points P(x1, y1),Q(x 2 , y 2 ), R(x 3 , y3 ),, S(x 4 , y 4 ) then, (a), , x1 + x 2 + x 3 + x 4 = 0 (b), , y1 + y 2 + y 3 + y 4 = 2, , (c), , x 1 x 2 x 3 x 4 = 2c 4, , y1 y 2 y 3 y 4 = 2c 4, , (d)

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EBD_7762, 150, , MATHEMATICS, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, , (b), (a), (c), (d), (c), (c), (d), (c), (a), (b), (c), (c), (b), (c), (c), (a), , 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, , (b), (a), (a), (a), (d), (a), (c), (b), (d), (c), (b), (c), (d), (b), (a), (b), , 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, , (b), (c), (a), (a), (d), (d), (b), (c), (b), (a), (a), (d), (b), (c), (a), (c), , 1, 2, 3, 4, 5, , (c), (c), (c), (c), (a), , 6, 7, 8, 9, 10, , (b), (a), (a), (d), (b), , 11, 12, 13, 14, 15, , (c), (a), (a), (a), (b), , 1, 2, , (a), (b), , 3, 4, , (c), (a), , 5, 6, , (a), (c), , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (c), 49 (a) 65 (b) 81 (b) 97, (c), 50 (d) 66 (c) 82 (c) 98, (b), 51 (b) 67 (a) 83 (a) 99, 52 (a) 68 (b) 84 (a) 100 (d), 53 (b) 69 (c) 85 (a) 101 (b), 54 (a) 70 (b) 86 (c) 102 (c), 55 (d) 71 (a) 87 (b) 103 (a), 56 (c) 72 (a) 88 (a) 104 (a), 57 (d) 73 (a) 89 (b) 105 (a), 58 (a) 74 (b) 90 (c) 106 (b), 59 (b) 75 (c) 91 (a) 107 (b), 60 (c) 76 (c) 92 (d) 108 (d), 61 (a) 77 (c) 93 (c) 109 (a), 62 (c) 78 (b) 94 (b) 110 (c), 63 (b) 79 (a) 95 (b) 111 (c), 64 (d) 80 (c) 96 (b) 112 (b), Exercise 2 : Exemplar & Past Year MCQs, (c), 16 (c) 21 (b) 26 (b) 31, (c), 17 (b) 22 (a) 27 (a) 32, (a), 18 (d) 23 (c) 28 (d) 33, (c), 19 (b) 24 (b) 29 (c) 34, (b), 20 (a) 25 (a) 30 (b) 35, Exercise 3 : Try If You Can, (a), (d) 11 (a) 13, (c), 7, 9, (a) 10 (a) 12 (b) 14, (a), 8, , 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, , (d), (a), (d), (d), (b), (d), (a), (a), (b), (d), (d), (a), (b), (a), (c), (d), , (c), 36, 37 (None), (c), 38, (c), 39, (c), 40, 15, , (a), , 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, , (b), (c), (b), (c), (a), (c), (a), (d), (c), (a), (b), (c), (c), (c), (b), (b), , 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, , (c), (d), (a), (a), (a), (d), (c), (c), (b), (c), (c), (b), (a), (a), (d), (a), , 41, 42, 43, 44, 45, , (c), (a), (d), (a), (a), , 46, 47, 48, , (b), (d), (a)

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12, Chapter, , INTRODUCTION TO THREE, DIMENSIONAL GEOMETRY, , This chapter appears in NCERT but no questions are asked in JEE Main/ BITSAT. The JEE Main/, BITSAT questions from the chapter Three Dimensional Geometry are covered in Chapter No. 27.

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EBD_7762, 152, , MATHEMATICS, , III. Q divides PR internally in the ratio 1 : 2, , Topic 1 : Co-ordinates of a Point, Distance Formula,, Co-ordinates of a Division Point, Centroid of a Triangle, 1., , 2., , 3., , Which of the statements given above is/are correct ?, , For every point P(x, y, z) on the xy-plane,, , (a) Only I, , (b) Only II, , (a) x = 0, , (b) y = 0, , (c) I and III, , (d) II and III, , (c) z = 0, , (d) None of these, , 8., , The perpendicular distance of the point P(6, 7, 8) from, , For every point P(x, y, z) on the x-axis (except the origin),, , xy-plane is, , (a) x = 0, y = 0, z ¹ 0, , (b) x = 0, z = 0, y ¹ 0, , (a) 8, , (b) 7, , (c) y = 0, z = 0, x ¹ 0, , (d) None of these, , (c) 6, , (d) None of these, , The distance of the point P(a, b, c) from the x-axis is, , 9., , The ratio in which the join of points (1, –2, 3) and (4, 2, –1) is, divided by XOY plane is, , 4., , 5., , 2, , 2, , (a), , b +c, , (c), , a 2 + b2, , (b), , 2, , a +c, , 2, , (d) None of these, , (a) 1 : 3, , (b) 3 : 1, , (c) –1 : 3, , (d) None of these, , 10. Which of the following statement is true ?, , Point (–3, 1, 2) lies in, , (a) The point A(0, –1), B(2,1), C(0,3) and D(–2, 1) are, , (a) Octant I, , (b) Octant II, , (c) Octant III, , (d) Octant IV, , vertices of a rhombus., (b) The points A(–4, –1), B(–2, –4), C(4, 0) and D(2, 3) are, vertices of a square., , The three vertices of a parallelogram taken in order are, (–1, 0), (3, 1) and (2, 2) respectively. The coordinate of the, , (c) The points A(–2, –1), B(1, 0), C(4, 3) and D(1, 2) are, , fourth vertex is, (a) (2,1), 6., , (b) (–2,1), , vertices of a parallelogram., (c) (1,2), , (d) (1,–2), , The point equidistant from the four points (0,0, 0), (3/2, 0, 0),, (0,5/2, 0) and (0, 0, 7/2) is:, (a), , æ 2 1 2 ö÷, çç , , ÷, çè 3 3 5 ø÷, , æ, 3ö, (b) çç3, 2, ÷÷÷, çè, 5ø, , (d) None of these., 11., , The ratio in which the line joining the points (2,4, 5) and, (3, 5, – 4) is internally divided by the xy-plane is, (a), , 5:4, , (b) 3 : 4, , (c) 1 : 2, , (d) 7 : 5, , 12. L is the foot of the perpendicular drawn from a point P(6, 7, 8), on the xy-plane. The coordinates of point L is, , (c), 7., , æ 3 5 7 ö÷, çç , , ÷, çè 4 4 4 ø÷, , æ1, ö, (d) çç , 0, ,1÷÷÷, çè 2, ø, , P(a, b, c); Q (a + 2, b + 2, c – 2) and R (a + 6, b + 6, c – 6) are, , (a) (6, 0, 0), , (b) (6, 7, 0), , (c) (6, 0, 8), , (d) None of these, , 13. If the sum of the squares of the distance of the point ( x, y, z), , collinear., , from the points ( a, 0, 0) and ( –a , 0, 0) is 2c2, then which one, , Consider the following statements :, , of the following is correct?, , I., , R divides PQ internally in the ratio 3 : 2, , (a) x2 + a2 = 2c2 – y2 – z2, , (b) x2 + a2 = c2 – y2 – z2, , II., , R divides PQ externally in the ratio 3 : 2, , (c) x2 – a2 = c2 – y2 – z2, , (d) x2 + a2 = c2 + y2 + z2

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INTRODUCTION TO THREE DIMENSIONAL GEOMETRY, , 153, , 14. The equation of set points P such that PA2 + PB2 = 2K2,, , (a) Only I is true., , (b) Only II is true., , where A and B are the points (3, 4, 5) and (–1, 3, –7),, , (c) Both are true., , (d) Both are false., , respectively is, (a) K2 – 109, (c), , 3K2 – 109, , 21. If the point A (3, 2, 2) and B(5, 5, 4) are equidistant from P,, (b) 2K 2 – 109, (d) 4K, , which is on x-axis, then the coordinates of P are, , 2 – 10, , 15. The octant in which the points (– 3, 1, 2) and (– 3, 1, – 2) lies, , (a), , æ 39, ö, ç , 2, 0 ÷, è 4, ø, , (b), , æ 49, ö, ç , 2, 0 ÷, è 4, ø, , (c), , æ 39, ö, ç , 0, 0 ÷, 4, è, ø, , (d), , æ 49, ö, ç , 0, 0 ÷, 4, è, ø, , respectively is, (a) second, fourth, , (b) sixth, second, , (c) fifth, sixth, , (d) second, sixth, , 16. The co-ordinates of the point which divides the join of the, points (2, –1, 3) and (4, 3, 1) in the ratio 3 : 4 internally are, given by :, (a), , 2 20 10, ,, ,, 7 7 7, , (b), , 10 15 2, , ,, 7 7 7, , (c), , 20 5 15, , ,, 7 7 7, , (d), , 15 20 3, ,, ,, 7 7 7, , 17. Let L, M, N be the feet of the perpendiculars drawn from a, , 22. What is the locus of a point which is equidistant from the, points (1, 2, 3) and (3, 2, – 1) ?, (a) x + z = 0, , (b) x – 3z = 0, , (c) x – z = 0, , (d) x – 2z = 0, , 23. The points (0, 7, 10), (– 1, 6, 6) and (– 4, 9, 6) form, (a) a right angled isosceles triangle, (b) a scalene triangle, , point P(7, 9, 4) on the x, y and z-axes respectively. The, , (c) a right angled triangle, , coordinates of L, M and N respectively are, , (d) an equilateral triangle, , (a) (7, 0, 0), (0, 9, 0), (0, 0, 4) (b) (7, 0, 0), (0, 0, 9), (0, 4, 0), (c) (0, 7, 0), (0, 0, 9), (4, 0, 0) (d) (0, 0, 7), (0, 9, 0), (4, 0, 0), 18. The points A(4, – 2, 1), B(7, – 4, 7), C (2, – 5, 10) and, D (– 1, – 3, 4) are the vertices of a, (a) tetrahedron, , (b) parallelogram, , (c) rhombus, , (d) square, , 19. If L, M and N are the feet of perpendiculars drawn from the, point P(3, 4, 5) on the XY, YZ and ZX-planes respectively,, then, (a) distance of the point L from the point P is 5 units., (b) distance of the point M from the point P is 3 units., , 24. The point in YZ-plane which is equidistant from three points, A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1) is, (a) (0, 3, 1) (b) (0, 1, 3), , (c) (1, 3, 0) (d) (3, 1, 0), , 25. Perpendicular distance of the point P(3, 5, 6) from y-axis is, (a), , 41, , (c) 7, , (b) 6, (d) None of these, , 26. The coordinates of the point R, which divides the line, segment joining P(x1, y1, z1) and Q(x2, y2, z2) in the ratio, k : 1, are, (a), , æ kx 2 - x1 ky2 - y1 kz2 - z1 ö, ,, ,, çè, ÷, 1- k, 1- k, 1- k ø, , (b), , æ kx 2 + x1 ky2 + y1 kz2 + z1 ö, ,, ,, çè, ÷, 1+ k, 1+ k, 1+ k ø, , (c), , æ kx 2 + x1 ky2 + y1 kz2 + z1 ö, ,, ,, çè, ÷, 1- k, 1- k, 1- k ø, , (c) distance of the point N from the point P is 4 units., (d) All of the above., 20. Consider the following statements, I., , The x-axis and y-axis together determine a plane known, as xy-plane., , II., , Coordinates of points in xy-plane are of the form (x1, y1, 0)., , Choose the correct option., , (d) None of these

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EBD_7762, 154, , 27., , 28., , MATHEMATICS, , The ratio, in which YZ-plane divides the line segment, joining the points (4, 8, 10) and (6, 10, – 8), is, , yz-plane is, , (a) 2 : 3 (externally), , (b) 2 : 3 (internally), , (a) 8, , (b) 7, , (c) 1 : 2 (externally), , (d) 1 : 2 (internally), , (c) 6, , (d) None of these, , Assertion : Points (– 4, 6, 10), (2, 4, 6) and (14, 0, – 2) are, , collinear. Ratio in which B divides AC is 1 : m. The value of, , Reason : Point (14, 0, – 2) divides the line segment joining, , m is, , by other two given points in the ratio 3 : 2 internally., , (a) 2, , (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, , (d) 5, , 35. Assertion : If three vertices of a parallelogram ABCD are, is (1, –2, 8)., , (c) Assertion is correct, reason is incorrect, , mid-point of AC and BD coincide., , (d) Assertion is incorrect, reason is correct., , (a) Assertion is correct, reason is correct; reason is a, , The ratio in which YZ-plane divides the line segment formed, (a) 2 : 3 (externally), , (b) 2 : 3 (internally), , (c) 1 : 3 (externally), , (d) 1 : 3 (internally), , If the origin is the centroid of a DABC having vertices, (b) b = 8, , (c) c = – 2, , (d) None of these, , The (0, 7, – 10), (1, 6, – 6) and (4, 9, –6) are the vertices, of an isosceles triangle., , II., , (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., A (2a, 2, 6), B (– 4, 3b, –10) and C (8, 14, 2c), then the sum of, , (a) a = – 2, , I., , correct explanation for assertion., , 36. If the origin is the centroid of the triangle with vertices, , A(a, 1, 3), B(– 2, b, – 5) and C(4, 7, c), then, , Centroid of the triangle whose vertices are (x1, y1, z1),, (x2, y2, z2) and (x3, y3, z3) is, æ x1 + x2 + x3 y1 + y2 + y3 z1 + z2 + z3 ö, ,, ,, çè, ÷ø, 3, 3, 3, , value of a and c is, (a) 0, , (a) Only I is true., , (b) Only II is true., , (c) Both are true., , (d) Both are false., , (b) 1, , (c) 2, , (d) 3, , 37. What is the locus of a point which is equidistant from the, points (–1, –2, –3) and (3, –2, – 1) ?, (a) x + z = 0, , (b) x – 3z = 0, , (c) x – z = 0, , (d) 2x + z = 0, , 38. What is the shortest distance of the point (1, 2, 3) from, x- axis ?, (a) 1, , Choose the correct option., , 32., , (c) 4, , Reason : Diagonals of a parallelogram bisect each other and, , by joining the points (– 2, 4, 7) and (3, – 5, 8), is, , 31., , (b) 3, , A(3, –1, 2), B (1, 2, –4) and C (–1, 1, 2), then the fourth vertex, , correct explanation for assertion., , 30., , 34. Given that A(3, 2, – 4), B(5, 4 – 6) and C(9, 8, – 10) are, , collinear., , (a) Assertion is correct, reason is correct; reason is a, , 29., , 33. The perpendicular distance of the point P(6, 7, 8) from, , (b), , 6, , (c), , 13, , (d), , 14, , 39. The equation of locus of a point whose distance from the, y-axis is equal to its distance from the point (2, 1, –1) is, , Distance between the points (2, 3, 5) and (4, 3, 1) is a 5 ., , (a) x2 + y2 + z2 = 6, , (b) x2 – 4x2 + 2z2 + 6 = 0, , The value of ‘a’ is, , (c) y2 – 2y – 4x2 + 2z + 6 = 0, , (d) x2 + y2 – z2 = 0, , (a) 2, , (b) 3, , (c) 9, , (d) 5

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INTRODUCTION TO THREE DIMENSIONAL GEOMETRY, , 40. If the perpendicular distance of the point (6, 5, 8) from the, Y-axis is 5l unit, then l is equal to, (a) 5, , (b) 3, , (c) 4, , (d) 2, , 41. Find the coordinates of the point which is three fifth of the, way from (3, 4, 5) to (– 2, – 1, 0)., (a) (1, 0, 2) (b) (2, 0, 1), , (c) (0, 2, 1) (d) (0, 1, 2), , 42. The coordinates of the points which trisect the line segment, joining the points P(4, 2, – 6) and Q(10, – 16, 6) are, , 155, , (c), , æ 5 7 17 ö, ç 7 ,- 3 , 3 ÷, è, ø, , (d), , æ 5 7 17 ö, ç - 3 , 3 ,- 3 ÷, è, ø, , Topic 2 : Centroid of a Tetrahedron, Area of a Triangle, Condition of Collinearity of Three Points., 45. If the middle point of the sides of a triangle ABC are (0, 0);, (1, 2) and (–3, 4), then the area of triangle is, (a) 40, , (b) 20, , (c) 10, , 46. Points ( –2, 4, 7), (3, –6, –8) and (1, –2, –2) are, , (a) (6, – 4, – 2) and (8, 10, – 2), , (a) Collinear, , (b) (6, – 4, – 2) and (8, – 10, 2), , (b) Vertices of an equilateral triangle, , (c) (–6, 4, 2) and (– 8, 10, 2), , (c) Vertices of an isosceles triangle, (d) None of these, , (d) None of these, 43. Assertion : Coordinates (–1, 2, 1), (1, –2, 5), (4, –7, 8) and, (2, –3, 4) are the vertices of a parallelogram., Reason : Opposite sides of a parallelogram are equal and, (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 44. The coordinates of the foot of the perpendicular drawn from, the point A(1, 0, 3) to the join of the points B(4, 7, 1) and, C(3, 5, 3) are, , æ 5 7 17 ö, ç 3, 3 , 3 ÷, è, ø, , 47. The point A divides the join of the points (–5,1) and (3,5) in, the ratio k : 1 and coordinates of points B and C are, (1, 5) and (7, –2) respectively. If the area of DABC be 2 units,, then k equals, (a) 7, 9, , diagonals are not equal., , (a), , (d) 60, , (b) (5, 7, 17), , (b) 6, 7, , (c) 7, 31/9, , (d) 9, 31/9, , 48. The ordered pair (l, m) such that the points (l,m, – 6),, (3, 2, – 4) and (9, 8, – 10) become collinear is, (a) (3, 4), , (b) (5, 4), , (c) (4, 5), , (d) (4, 3), , 49. Which of the following set of points are non-collinear?, (a) (1, –1, 1), (–1, 1, 1), (0, 0, 1), (b) (1, 2, 3), (3, 2, 1), (2, 2, 2), (c) (–2, 4, –3), (4, –3, –2), (–3, –2, 4), (d) (2, 0, –1), (3, 2, –2), (5, 6, –4), 50. The points (5, 2, 4), (6, –1, 2) and (8, –7, k) are collinear if k is, equal to, (a) –2, , (b) 2, , (c) 3, , (d) – 1

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EBD_7762, 156, , MATHEMATICS, , Exercise 2 : Exemplar & Past Year MCQs, 8., , NCERT Exemplar MCQs, 1., 2., , The distance of point P(3, 4, 5) from the YZ-plane is, (a) 3 units (b) 4 units, (c) 5 units (d) 550, What is the length of foot of perpendicular drawn from the, point P (3, 4, 5) on Y-axis?, (a), , 3., , 34, , (c) 5, (d) None of these, Distance of the point (3, 4, 5) from the origin (0, 0, 0) is, (a), , 4., , (b), , 41, , 50, , (b) 3, , (c) 4, , (d) 5, , If the distance between the points (a, 0, 1) and (0, 1, 2) is, 27 , then value of a is, , 5., , 6., , 7., , (a) 5, (b) ±5, (c) – 5, (d) None of these, x-axis is the intersection of two planes are, (a) xy and xz, (b) yz and zx, (c) xy and yz, (d) None of these, Equation of Y-axis is considered as, (a) x = 0, y = 0, (b) y = 0, z = 0, (c) z = 0, x = 0, (d) None of these, The point (– 2, – 3, – 4) lies in the, (a) first octant, (b) seventh octant, (c) second octant, (d) eighth octant, , 9., , A plane is parallel to yz-plane so it is perpendicular to :, (a) x-axis, (b) y-axis, (c) z-axis, (d) None of these, The locus of a point for which y = 0 and z = 0, is, (a) equation of X-axis, , (b) equation of Y-axis, , (c) equation of Z-axis, , (d) None of these, , 10. The locus of a point for which x = 0 is, (a) xy-plane, (b) yz-plane, (c) zx-plane, (d) None of these, 11. If a parallelopiped is formed by planes drawn through the, points (5, 8, 10) and (3, 6, 8) parallel to the coordinate plane,, then the length of diagonal of the parallelopiped is, (a), , 2 3, , (b), , 3 2, , (c), , 2, , (d), , 3, , 12. L is the foot of the perpendicular drawn from a point P(3,4,5), on the XY-plane. The coordinates of point L are, (a) (3, 0, 0), , (b) (0, 4, 5), , (c) (3, 0, 5), , (d) None of these, , 13. L is the foot of the perpendicular drawn from a point, (3, 4, 5) on X-axis. The coordinates of L are, (a) (3, 0, 0), , (b) (0, 4, 0), , (c) (0, 0, 5), , (d) None of these, , Exercise 3 : Try If You Can, 1., , The number of points on x-axis which are at a distance, c (c < 3) from the point (2, 3) is, (a) 2, (b) 1, (c) infinite (d) no point, , 2., , ABC is a triangle and AD is the median. If the coordinates of, A are ( 4, 7, –8) and the coordinates of centroid of the triangle, ABC are (1, 1, 1), what are the coordinates of D?, (a), , æ 1, ö, çè – , 2,11÷ø, 2, , (c) (–1, 2, 11), , (b), , 11ö, æ 1, çè – , –2, ÷ø, 2, 2, , (d) (–5, –11, 19), , 3., , Find the centre and radius of the sphere passing through, the points O(0, 0, 0), A(a, 0, 0), B(0, b, 0) and C(0,0, c)., 1 2 2 2, a +b +c, 2, , (a), , (a , b, c ) and, , (b), , 1 2, æa b cö, a + b2 + c2, ç , , ÷ and, 2, è2 2 2ø, , (c), , æa b cö, 2, 2, 2, ç 2 , 2 , 2 ÷ and a + b + c, è, ø, , (d), , (a , b, c ) and a 2 + b2 + c 2

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INTRODUCTION TO THREE DIMENSIONAL GEOMETRY, , 4., , 5., , 6., , 7., , A cube of side 3 units has one vertex at point (1, 1, 1) and the, three edges from this vertex are respectively parallel to positive, x-axis and negative y-axis and z-axis. The co-ordinates of, other vertices of the cube can not be:, (a) (1, –2, 1), , (b) (4, –2, –2), , (c) (1, 1, –2), , (d) None of these, , The mid-points of the sides of a triangle are (5, 7, 11),, (0, 8, 5) and (2, 3, – 1), then the vertices are, (a) (7, 2, 5), (3, 12, 17), (– 3, 4, – 7), (b) (7, 2, 5), (3, 12, 17), (3, 4, 7), (c) (7, 2, 5), (– 3, 11, 15), (3, 4, 8), (d) None of the above, Find the ratio in which the line segment joining the points, (4, 8, 10) and (6, 10, – 8) is divided by the yz-plane., (a) 2 : 3 externally, , (b) 3 : 2 externally, , (c) 4 : 5 internally, , (d) None of these, , The ends of the rod of length l moves on two mutually, perpendicular lines, find the locus of the point on the rod, which divides it in the ratio m1: m2, (a) m12 x2 + m22 y2 =, , 8., , l2, , ( m1 + m2 ) 2, , æ mm l ö, (b) (m2 x)2 + (m1y)2 = ç 1 2 ÷, è m1 + m2 ø, , 2, , æ mm l ö, (c) (m1x)2 + (m2 y)2 = ç 1 2 ÷, è m1 + m2 ø, (d) None of these, , 2, , A(3, 2, 0), B(5, 3, 2) and C(–9, 6, – 3) are the vertices of a, triangle ABC. If the bisector of ÐABC meets BC at D, then, coordinates of D are, (a), , (c), , æ 19 57 17 ö, ç 8 , 16 ,16 ÷, è, ø, , (b), , æ 19 57 17 ö, ç 8 , - 16 ,16 ÷, è, ø, , (d) None of these, , æ 19 57 17 ö, ç - 8 , 16 ,16 ÷, è, ø, , 157, , 9., , If the origin is shifted (1, 2 – 3) without changing the, directions of the axis, then find the new coordinates of the, point (0, 4, 5) with respect to new frame., (a) (–1, 2, 8), , (b) (4, 5, 1), , (c) (3, –2, 4), , (d) (6, 0, 8), , 10. If the origin is the centroid of a DABC having vertices, A(a, 1, 3), B(– 2, b, – 5) and C(4, 7, c), then, , 11., , (a) a = – 2, , (b) b = 8, , (c) c = – 2, , (d) None of these, , If the co-ordinates of the point in which the line joining the, points (3, 5, –7) and (–2, 1, 8) is intersected by the plane yz, , é 13 ù, is êa, ,c ú , then (a + b – c) =, ë b û, (a) 2, , (b) 3, , (c) 4, , (d) 5, , 12. A parallelopiped is formed by planes drawn through the, points (2, 3, 5) and (5, 9, 7) parallel to the coordinate planes., The length of the diagonal of the parallelopiped is, (a) 5, , (b) 7, , (c) 11, , (d) 13, , 13. In DABC the mid-point of the sides AB, BC and CA are, respectively (l, 0, 0), (0, m, 0) and (0, 0, n). Then,, AB2 + BC2 + CA 2, l 2 + m2 + n 2, , is equal to, , (a) 8, , (b) 16, , (c) 9, , (d) 25, , 14. In three dimensional space the path of a point whose distance, from the x-axis is 3 times its distance from the yz-plane is:, (a) y2 + z2 = 9x2, , (b) x2 + y2 =3z2, , (c) x2 + z2 = 3y2, , (d) y2 – z2 = 9x2, , 15. The ratio in which the YZ-plane divide the line segment, formed by joining the points (–2, 4, 7) and (3, –5, 8) is 2 : m., The value of m is, (a) 2, , (b) 3, , (c) 4, , (d) 1

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EBD_7762, 158, , MATHEMATICS, , 1, 2, 3, 4, 5, , (c), (c), (a), (b), (b), , 6, 7, 8, 9, 10, , (c), (d), (a), (b), (c), , 11, 12, 13, 14, 15, , (a), (b), (b), (b), (d), , 1, 2, , (a), (b), , 3, 4, , (a), (b), , 5, 6, , (a), (c), , 1, 2, , (d), (b), , 3, 4, , (b), (d), , 5, 6, , (a), (a), , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (c) 21, (d), (b) 31, 16, 26, (a) 22, (d), (a) 32, 17, 27, (a), (c) 33, 18 (b) 23, 28, (b), (b) 34, 19 (d) 24, 29, (c) 25, (d), (a) 35, 20, 30, Exercise 2 : Exemplar & Past Year MCQs, (b), (a), (a) 13, 7, 9, 11, (a) 10, (b), (d), 8, 12, Exercise 3 : Try If You Can, (c), (a), (b) 13, 7, 9, 11, (a) 10, (a), (b) 14, 8, 12, , (c), (a), (c), (a), (a), , 36, 37, 38, 39, 40, , (a), (d), (c), (c), (d), , 15, , (b), , (a), , (a), (a), , 41, 42, 43, 44, 45, , (d), (b), (a), (a), (b), , 46, 47, 48, 49, 50, , (a), (c), (b), (c), (a)

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13, , LIMITS AND DERIVATIVES, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , 3, , Number of Questions, , JEE MAIN, BITSAT, 2, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 6, 3, , Critical Concepts, , Algebra of limits, Limits of, Trigonometric Functions, Sandwich, Theorem, L Hospital's Rule, Limit of a, Series, Derivative by First Principle,, Algebra of Derivative of Functions,, Product Rule & Quotient Rule of, Derivatives., , Rating of, Difficulty Level, , CUS, (chapter utility score), Out of 10, , 4/5, , 8.5

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LIMITS AND DERIVATIVES, , 161

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EBD_7762, 162, , MATHEMATICS, , Topic 1 : Limits, Existance of Limits, Standard Limits,, Sandwitch Theorem, Limits By Factorisation,, Substitution & Rationalisation., 1., , 2., , 3., , (b) –2, , 1+ x + 1- x, The value of lim, is, 1+ x, x ®0, (a) 2, (b) –2, (c) 1, , 1+ x - 1 - x, (b) 2, (c) –1, , (c), , 5., , 1, 2, , 1, 2, , -, , 1, 2, , 1, 2, , equals, (a) 1, 8., , x ®a, , é 1 + x -1 ù 1, lim ê, ú=, x ®0, x, ë, û 2, (a) Both I and II are true, (c) Only II is true, , (b) Only I is true, (d) Both I and II are false, , a sin x - 1, is, x ® 0 sin x, (a) log a, (c) log (sin x), , (b) sin x, (d) cos x, , 13. Value of lim, , (c) 0, , (d) ¥, 14., , lim, , 2sin 2 3x, , x ®0, , (a) 12, , x2, , is equal to :, , (b) 18, , (c) 0, 1/ x, , 15. Assertion: lim (1 + 3 x), x®0, , (d) 6, , 3, , =e ., , Reason: Since lim (1 + x)1/x = e ., x® 0, , (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., , 1, 2+x - 2, is equal to, then ‘a’, a 2, x, , (b) 2, , If value of lim, , é x15 - 1 ù 3, lim ê 10 ú =, x ®1 x - 1, ë, û 2, , II., , x ®2, , x ®0, , x ®0, , I., , lim h(x) = 4., , If value of lim, , x, is, tan x, , (d) –2, , Which of the following is/are true?, (a) Both I and II are true (b) Only I is true, (c) Only II is true, (d) Both I and II are false, 7., , (b) –1, (d) does not exist, , (a) 0, (b) 1, (c) 4, (d) not defined, 12. Which of the following is/are true?, , x2 - 4, ,x¹2, Consider the function h (x) =, x-2, Then,, I. h (2) is not defined., , II., , lim, , is, , 11., , (d) does not exist, , (b), , x2, , (d) –1, , (d) –1, , 1+ x + x2 -1, lim, =, x, x ®0, , (a), 6., , -, , (1 + x 2 ) - 1 - x 2, , (a) 1, (c) 0, , 1- x - 4, Value of lim, is, x-5, x ®5, , (b), , (c) 1, , x®0, , (d) –1, , x ®0, , (a) 0, , (b) –2, , 10. The value of lim, , x, , Evaluate lim, (a) 1, , 4., , (c) 0, , ïì x 2 + 1, x ³ 1, , then the value of lim f (x) is, x ®1, ïî 3x - 1, x < 1, , If f (x) = í, (a) 2, , é x2 - 1 ù, ú is, The value of lim ê 2, x ® -1 ëê x + 3x + 2 ûú, (a) 2, , 9., , (c) 3, , (d) 4, , a + 2x - 3x, 2 3, is equal to, , where m, m, 3a + x - 2 x, , is equal to, (a) 2, (b) 8, , (c) 9, , (d) 3, , 16., , sin m 2q, is equal to :, q, q® 0, (a) 0, (b) 1, lim, , (c) m, , (d) m2

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LIMITS AND DERIVATIVES, , 163, , é f (x) ù, 17. If lim ê, ú exists, then which one of the following correct ?, x ®a ë g( x ) û, , x®0, , ‘a’ is, (a) 0, , (a) Both lim f (x ) and lim g( x) must exist, x®a, , (b), , x® a, , lim f (x ) need not exist but lim g( x) must exist, , x®a, , x ®0, , (a) 1, (c) 4, , x® a, , lim f (x ) must exist but lim g( x) need not exist, , x®a, , x® a, , 18. The value of lim, , 1+, , x® 0, , (a), 19., , 2, 3, , 1, 3, , (b) – p, , 2, 5, , (d), , 1, 5, , (c), , (d) cos x, , (d), , -, , 1, p, , (b) Only I is true, (d) Both I and II are false, , (d) 0, , x, , x(e - 1), is, 1 - cos x, (b) 2, (d) does not exist, , 31. The value of lim, , x ®0, , (a) 0, (c) –2, 32., , 1 - cos 2 x, , lim, , is, , 2x, , x®0, , (a) 1, (c) zero, 33. Let f (x) =, , (b) –1, (d) does not exist, , {, , x + 2 , x £ -1, cx 2 , x > -1, , If lim f (x) exists, then c is equal to, , (c) 1, , x ®-1, , (d) 3, , x ®1, , (a + h) 2 sin (a + h) - a 2 sin a, h®0, h, I. a2 sin a + 2a cos a, II., (a) Both I and II are true (b), (c) Only II is true, (d), , (a) 1, , a cos a + 2a sin a, Only I is true, Both I and II are false, 2, , (b) 0, , 34. If value of limp, x®, , 4, , (c) 2, , 4 2 - (cos x + sin x), 1 - sin 2x, , (d) 3, 5, , is a 2 , then the, , value of ‘a’ is, (a) 2, (b) 3, (c) 4, (d) 5, x n - 2n, 35. If lim, = 80 and n Î N, then the value of ‘n’ is, x®2 x - 2, (a) 2, (b) 3, (c) 4, (d) 5, 1 - cos x + 2 sin x - sin 3 x - x 2 + 3x 4, is, x® 0, tan 3 x - 6 sin 2 x + x - 5x 3, (b) 2, (c) – 1, (d) – 2, , 36. The value of lim, (a) 1, , is equal to, , 25. If f (x) = | x | – 5, then the value of lim f (x) is, , 37., , lim, , sin(p cos 2 x), , x®0, , (a) -p, , x2, , 38. The value of, , x ®5, , (a) 9, (c) 0, , (c) sin x, , ., , x cot x, is, x ®0 1 - cos x, (b) –2, (c) 2, , (a) 1, , (b) 2, (c) 4, (d) 8, sin (2 + x) - sin(2 - x), 23. If lim, is equal to p cos q, then sum, x ®0, x, of p and q is, (a) 2, (b) 1, (c) 3, (d) 4, lim, , (d) 3, , 30. The value of lim, , ìïa + bx, x < 1, x =1, 22. Suppose f (x) = í4,, ïîb - ax, x > 1, and if lim f (x) = f (1) then the value of a + b is, , 24., , (c) 2, , 3, , lim (sec x – tan x) is equal to, , (a) 0, , x2, (b) – 4, , (a) 4, , x ®p /2, , (b) 2, , sin 2x, , x®0, , ax 2 + bx + c, (where a + b + c ¹ 0) is 1., x ®1 cx 2 + bx + a, , (a) 0, , (b) 6, , 29. Evaluate lim, , lim, , 1 1, +, 1, II. lim x 2 is, x ® -2 x + 2, 4, (a) Both I and II are true, (c) Only II is true, , {, , +3 , x £ 0, 28. If f (x) = 32x, f (x) is, (x + 1) , x > 0 then the value of lim, x ®0, 2, , 1, p, Which of the following is/are true?, , I., , 21., , (c), , (d) 3, , (b) 2, (d) None of these, , (a) 0, , cos x, The value of lim, is, x® 0 p - x, , (a) p, , 20., , (b), , x, x, -1+, 3, 3 is, x, , (c) 2, , sin 4x, is, sin 2x, , 27. Value of lim, , (c) Both lim f (x ) and lim g( x) need not exist, (d), , (b) 1, , x® a, , x®a, , a, sin x, is equal to then the value of, x (1 + cos x), 2, , 26. If value of lim, , (b) 1, (d) None of these, (a), , p, 4, , equals, , (b) p, lim, , p, q® 4, , (b), , (c) p/2, , (d) 1, , cos q + sin q, is, p, q+, 4, , -p, 4, , (c), , – 2, , (d), , 2

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EBD_7762, 164, , 39., , MATHEMATICS, , If f (x) =, , x+ | x |, , then the value of lim f (x) is, x, x ®0, , (a) 0, (c) does not exist, 40., , Let a and b be the distinct roots of ax2 + bx = c = 0, then, lim, , 1 - cos(ax 2 + bx + c), ( x - a )2, , x®a, , (a), , (c), , 41., , 47., , (b) 2, (d) None of these, , 2, , (b) 0, , -a 2, (a - b) 2, 2, , (d), , ((a - n)nx - tan x )sin nx, x2, , x ®0, , (a) 2, , (b) 4, (c) 6, 2, 2sin x + sin x -1, 48. Evaluate: xlim, ®p /6 2sin 2 x - 3sin x + 1, (a) 3, (b) – 3, (c) 1, , is equal to, , a2, (a - b ) 2, 2, , If lim, , p, ì k cos x, ïï p - 2x , when x ¹ 2, ., Let f (x) = í, ï3, when x = p, ïî, 2, æpö, If lim f (x) = f ç ÷ , then k is equal to, p, è2ø, x®, , (a) 0, , 42., , 1, (a - b) 2, 2, , (a) a ¹ 1, , = 0, where n is non-zero, , lim, x ®0, , 1, 2, , (b), , -1, 2, , (d) – 2, , -5, (a + 2)2 / 3, 3, , (b), , 5, (a - 2)2 / 3, 3, , (c), , 5, (a + 2) -2 / 3, 3, , (d), , 5, (a + 2)2/ 3, 3, , x - sin x, , lim, , x + sin 2 x, , x ®0, , (b) c2, a2 + b2, (d) c2, a2 – b2, , 52. What is the value of lim, (a) 0, , 53. If lim, (b) 2, (d) does not exist, , tan x - sin x, is equal to, x ®0, sin 3 x, , (c) 1, lim, x®4, , 5, 4, , (b), a x – xa, xa – aa, , x ®0, , x sin 5x, , (a) –1, , (b) 0, (d) Not defined, , (c), , (b) 0, x ®2, , (a), , 5, 16, , (d), , 25, 4, , (d) 2, , 1+ 2+ x - 3, is, x -2, (b), , 8 3, , (c) 0, , 1, 4 3, , (d) None of these, , x - 2a + x - 2a, x 2 - 4a 2, , x ®2a, , (a), , ?, , (c) 1, , 1, , 55. The value of lim, (b) 0, (d) does not exist, , sin 2 4 x, , = –1, then a is equal to:, , 54. The value of lim, , x-4, is equal to, x-4, , (a) 1, (c) – 1, , (b) 0, (d) None of these, x® 0, , lim, , 1, 2, , is equal to, , (a) 1, (c) ¥, , x ®1, , (a), , 46., , 51., (c) 1, , (d) a ¹ 2, , (a), , lim [x – 1], where [.] is greatest integer function, is equal, to, (a) 1, (c) 0, , 45., , (d), , (c) a ¹ –1, , ( x + 2)5/3 - ( a + 2)5/3, ., x-a, x®a, , 1, n+, n, , cosax - cos bx, m, , where m and n are, is equal to, cos cx -1, n, , respectively, (a) a2 + b2, c2, (c) a2 – b2, c2, 44., , (c) n, , (b) a ¹ 0, , 50. Evaluate : lim, , 2(2 x - 3) ù, é 1, lim ê, - 3, is equal to, x®2 x - 2, x - 3x 2 + 2x úû, ë, (a), , 43., , (b), , (d) – 1, , ì| x | +1 , x < 0, ï, , x = 0 then lim f (x) exists for all, 49. If f (x) = í0, x®a, ïî| x | -1 , x > 0, , real number, then a is equal to, n +1, n, , (d) 8, , 1, a, , (b), , 1, 2 a, , (c), , a, 2, , is, (d) 2 a

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LIMITS AND DERIVATIVES, , 165, , é, æ xö ù, ê1 - tan çè 2 ÷ø ú [1 - sin x], ë, û, is, 56. lim, p é, x ® 1 + tan æ x ö ù [ p - 2 x ]3, çè ÷ø ú, 2 ê, 2 û, ë, 1, (a) ¥, (c) 0, (b), 8, , Topic 2 : Derivative, Derivative by First Principle,, Algebra of Derivatives; Addition, Subtraction,, Product and Quotient Rule of Differentiation, 1, (d), 32, , æ 1 - cos{2( x - 2)} ö, 57. lim ç, ÷ is equal to, x®2 è, x-2, ø, (a) equals 2, (b) equals –, (c) equals, , 1, , (a), 2, , (c), , (d) does not exist, , 2, , 58. Let f : R ® [0, ¥) be such that lim f(x) exists and, x® 5, , lim, , x® 5, , ( f ( x ) )2 - 9 = 0, x -5, , (a) 0, , . Then lim f(x) equals :, x® 5, , (b) 1, , 59. The value of lim, , (c) 2, , tan 2 x - 2 tan x - 3, , x ®0 tan 2, , (a) 0, 60., , lim, , (a), (c), 61., , x - 4 tan x + 3, , (b) 1, , ö, ç 8xz - 4x + 8xz ÷, è, ø, 2, , 3, , 4, , z, 2, , 2, , 21 / 3, , 23 / 3, , .z, , (d) None of these, , z, , 1, 1 ö, æ, equals to, lim ç, 3, 2, h ÷ø, h ®0 è h 8 + h, , (a), , -, , 1, 8, , (b), , 62. The value of lim, , 1, 8, , (c), , 1, 48, , (d), , -, , 1, 48, , cos ( sin x ) - cos x, , is equal to, x4, (a) 1/5, (b) 1/6, (c) 1/4, (d) 1/2, 63. A function f is said to be a rational function, if, x®0, , g(x), , where g (x) and h (x) are polynomials, f (x) =, h (x), such that h (x) ¹ 0, then, , g (a), h (a), (b) h (a) = 0 and g (a) ¹ 0 Þ lim f (x) does not exist, , (a) h (a) ¹ 0 Þ lim f (x) =, x® a, , x® a, , (c) Both (a) and (b) are true, (d) Both (a) and (b) are false., , (t + 1)2, 2t 2, , (d), , (t - 1)2, , 2t, , ( t + 1)2, -2t 2, , ( t - 1)2, , -1, (1 - x) 2, , (b), , -2, (1 - x) 2, , (d), , 3, (1 - x) 2, , 1, 2, (b) 2 x, (c), (d), x, x, x, 67. Which of the following is/are true?, I. The derivative of f (x) = sin 2x is 2(cos2 x – sin2 x)., II. The derivative of g (x) = cot x is – cosec2 x., (a) Both I and II are true (b) Only I is true, (c) Only II is true, (d) Both I and II are false, 68. If a and b are fixed non-zero constants, then the derivative, of (ax + b)n is, (a) n(ax + b)n – 1, (b) na(ax + b)n – 1, n–1, (c) nb(ax + b), (d) nab(ax + b)n – 1, n, 69. The derivative of sin x is, (a) n sinn–1 x, (b) n cosn – 1 x, n–1, (c) n sin x cos x, (d) n cosn – 1 x sin x, 2, 70. The derivative of (x + 1) cos x is, (a) – x2 sin x – sin x – 2x cos x, (b) – x2 sin x – sin x + 2 cos x, (c) – x2 sin x – x sin x + 2 cos x, (d) – x2 sin x – sin x + 2 x cos x, , (a), , 1, 2, , (b), , 66. The derivative of 4 x - 2 is, , (d) 3, , is equal to, , (b), , 11 / 3, , -2t 2, , 1, x is, 65. The derivative of, 1, 1x, 2, (a), (1 + x) 2, , (c), , tan x = 3, is, , 1- t, , then the value of f ' (1/t) is, 1+ t, , 1+, , (d) 3, , (c) 2, , x 3 z 2 - (z - x) 2, , x ®0 æ 3, , 64. If f (t) =, , æ pö, 71. If f (x) = x sin x, then f ¢ ç ÷ is equal to, è 2ø, (a) 0, , (b) 1, , (c) – 1, , (d), , 1, 2, , 72. Which of the following is/are true?, I. The derivative of x2 – 2at x = 10 is 18., II. The derivative of 99x at x = 100 is 99., III. The derivative of x at x = 1 is 1., (a) I, II and III are true, (b) I and II are true, (c) II and III are true, (d) I and III are true

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EBD_7762, 166, , 73., , 74., , MATHEMATICS, , The derivative of function 6x100 – x55 + x is, (a) 600x100 – 55x55 + x, (b) 600x99 – 55x54 + 1, 99, 54, (c) 99x – 54x + 1, (d) 99x99 – 54x54, 1, Derivative of x + sin x + 2 is, x, (a) 2x + cos x, (b) 2x + cos x + (–2) x–3, –3, (c) 2x – 2x, (d) None of these, , 1, , 1, (b) 1 - 2, x, x, If f (x) = axn, then a =, , 78., , 1, (d) 1 + 2, x, , f ¢ (1), , (c) n · f ¢(1) (d), n, Derivative of x sin x, (a) x cos x, (b) x sin x, (c) x cos x + sin x, (d) sin x, Assertion. For the function, f (x) =, , 79., , (b), , n, f ¢ (1), , sin a, , (b), , sin 2 (a + y), sin a, , x100 x 99, x2, +, + ... +, + x + 1, f ¢(1) = 100f ¢ (0)., 100 99, 2, , 81., , x +1 1, is 2 ., x, x, (a) Both I and II are true (b) Only I is true, (c) Only II is true, (d) Both I and II are false, Derivative of the function f (x) = (x – 1) (x – 2) is, (a) 2x + 3, (b) 3x – 2, (c) 3x + 2, (d) 2x – 3, Let 3f (x) – 2 f (1/x) = x, then f '(2) is equal to, , 82., , 2, 1, (a), (b), 7, 2, What is the derivative of, , The derivative of f (x) =, , f (x) =, (a), (c), , -, , (c) 2, , (d), , 7, 2, , (d), , cos 2 (a + y), sin a, , ( x + 3), 3, , ( x + 3), , 2, , 2, , +, , -, , 2, (2x - 1), 1, , ( 2 x - 1), , 2, , x, dy, , then value of (1 + cos x), – sin x is, 2, dx, (b) x2, (c) x, (d) None, , 85. Differential coefficient of, , 2, , (b), (d), , -, , 3, ( x + 3), 3, , ( x + 3), , 2, , 2, , +, , -, , x sin x, is, 1 + cos x, , (a), , - x - sin x, 1 + cos x, , (b), , x - sin x, 1 + cos x, , (c), , x + sin x, 1 - cos x, , (d), , x + sin x, 1 + cos x, , (a), , x - y2, x2 - y, , (b), , 2, , x-y, x 2 - y2, (d) 2, x - y2, x-y, , é 2 + 2x - 2x 2 ù, ê, ú, x 2 + 1 úû, êë, , (a), , (b), , 2, æ 2x - 1ö é 2 + 2x - 2x 2 ù, sin ç, ê, ú, è x 2 - 1÷ø ëê, x 2 + 1 ûú, , (c), , 2, æ 2x - 1ö é 2 + 2x - 2x 2 ù, sin ç, ê, ú, ÷, è x 2 + 1ø ëê, x 2 + 1 ûú, , (d), , æ 2x + 1ö, sin ç, è x 2 - 1÷ø, , 2, , é 2 + 2x - 2x 2 ù, ê, ú, x 2 + 1 úû, êë, , 88. If a, b are fixed non-zero constant, then the derivative of, a, b, - 2 + cos x is ma + nb – p, where, 4, x x, -2, , p = sin x, x3, , (b) m =, , -4, 2, , n = 3 , p = sin x, 5, x, x, , (c) m =, , -4, -2, , n = 3 , p = - sin x, 5, x, x, , 2, (2 x - 1) 2, , x - y2, , (c), , æ 2x + 1ö, sin ç 2 ÷, è x + 1ø, , 1, ( 2 x - 1) 2, , x2 - y, , dy, is, dx, , dy, æ 2x - 1ö, 87. If y = f ç, and f ' (x) = sin x2, then value of, is, ÷, 2, dx, è x + 1ø, , (a) m = 4x3, n =, , 7x, ?, (2x - 1) (x + 3), 3, , (c) sin2 (a + y), , 86. If x3 + y3 = 3xy, then value of, , d n, (x ) = n . x n -1 ., Reason:, dx, (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., Which of the following is/are true?, I. The derivative of, f(x) = 1 + x + x2 + ... + x50 at x = 1 is 1250., , II., , 80., , sin ( a + y ), , (a) – x, , (c) 1, , 2, , (a) f ¢(1), , 77., , (a), , 84. If y = x tan, , 1 ö, æ, Derivative of ç x +, ÷ is, xø, è, , (a), , 76., , If sin y = x sin (a + y), then value of dy/dx is, , 2, , 2, , 75., , 83., , (d) m = 4x3, n =, , 2, , p = – sin x, x3

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LIMITS AND DERIVATIVES, , 167, , 89. If a is a fixed non-zero constant, then the derivative of, sin(x + a), is, cos x, , (a), , cosa, (b), cos 2 x, , - cosa, cos 2 x, , (c), , sin a, cos 2 x, , (d), , - sin a, cos 2 x, , æpö, 90. If f (x) = |cos x – sin x|, then f ¢ ç ÷ is equal to, è4ø, (b) - 2, (d) None of these, , (a), 2, (c) 0, , æ x+y ö, ÷ for all x, y Î R (xy ¹ 1) and, 91. If f(x) + f (y) = f ç, è 1 - xy ø, , f (x), æ 1 ö, = 2. Then, f ¢ ç, ÷ is, x ®0 x, è 3ø, , x100 x 99, x2, +, + ... + x + 1,, 100 99, 2, f ¢(1) = mf ¢ (0), where m is equal to, (a) 50, (b) 0, (c) 100, (d) 200, The function u = ex sin x, v = ex cos x satisfy the equation, , (b), , d2u, = 2v, dx 2, , 2, , (c), , d v, = -2u, dx 2, , (d) All of these, , Topic 3 : Evaluation of Limits Using L-Hospital’s Rule,, Evaluation of Limits when x ® ¥, Limits by Expansion, Method., , p, 95. The value of lim (1 - x) tan æç xö÷ is :, è, 2 ø, x ®1, p/4, , (b), , 2/p, , (c), , (b) 15, , 99. Let f ( x ) = lim, , 2p / 3, , 1 + x 2n, , n ®¥, , lim f ( x ) ¹ lim f ( x ) (b), , (a), , (d) 3p / 4, , x ®0+, , æ x2 + 5 x + 3 ö, 97. Lim ç 2, ÷, x ®¥ çè x + x + 2 ÷ø, (a) e4, (b) e2, , (b) 2, (d) None of these, x, , (c), , e3, , (d) 25, , log(2 + x ) - x 2 n sin x, , x ®1+, , x ®1-, , . Then, , lim f ( x ) = sin1, , x ®1+, , lim f ( x) doesn’tt exist (d) None of these, , (c), , x ®1-, , (cos x)1/ 2 - (cos x)1/ 3, , 101. If lim, , is, , sin 2 x, (b) –1/12, , (c) 2/3, , (sin nx ) [(a - n )nx - tan x], x2, , x® 0, , (d) 1/3, , = 0 , then the value, , of a, (a), 102. lim, , 1, n, , (b) n -, , 1, n, , (c), , n+, , 1, n, , ( x + 1)10 + ( x + 2)10 + .... + ( x + 100)10, x10 + 1010, , x®¥, , (d) None, is equal to, , (a) 0, (b) 1, (c) 10, (d) 100, 103. Let f (x) = x – [x], where [x] denotes the greatest integer £ x, and g ( x) = lim, , { f ( x )}2 n - 1, , n®¥ { f ( x )}2n, , (a) 0, (c) –1, , , then g(x) is equal to, +1, (b) 1, (d) None of these, , ra, ra, 104. If z r = cos, + i sin 2 , where r = 1, 2, 3, ...., n, then, 2, n, n, lim z1 z2 z3 ...zn is equal to, , (a), , cos a + i sin a, , (b), , cos(a / 2) - i sin(a / 2), , (c), , eia / 2, , (d), , 3 ia, , (d) 1, , e, , 105. Evaluate lim éê a 2 x 2 + ax + 1 - a 2 x 2 + 1 ùú ., x ®¥ ë, û, (a) 1, , (b), , 96. Evaluate lim x x ., (a) 1, (c) 0, , (c) 5, , n®¥, , BEYOND NCERT, , (a), , (a) 10, , (a) 1/6, , f (x) =, , (a), , lim bn ., , n®¥, , x ®0, , 3, 4, 3, 3, (a), (b), (c), (d), 4, 3, 6, 2, 2, 92. If f be a function given by f (x) = 2x + 3x – 5. Then,, f ¢(0) = mf ¢(–1), where m is equal to, (a) – 1, (b) – 2, (c) – 3, (d) – 4, 93. For the function, , du, dv, v - u = u 2 + v2, dx, dx, , 1æ, 125 ö, recurrence relation bn +1 = ç 2bn +, ÷ , bn ¹ 0. Then find, ç, 3è, bn2 ÷ø, , 100. lim, , lim, , 94., , 98. Let the sequence <bn > of real numbers satisfies the, , 106. The value of lim, , x ®¥, , 1, 2, 1, xn e x, (2 ), , (c), , 1, 4, , 1, xn e x, - (3 ), , xn, , (d) 2, , (where n Î N ) is, , (a), , æ2ö, log n ç ÷, è3ø, , (b) 0, , (c), , æ2ö, n log n ç ÷, è3ø, , (d) not defined

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EBD_7762, 168, , MATHEMATICS, , æ 100, x2 ö, 2, x, æ, ö, 107. lim ç, + ç cos ÷ ÷ =, è, xø ÷, x ®¥ ç e x, è–1, ø, , (a) e, (c) (1 + e–2), , 108. If lim, , n®¥, , 1 - (10), , n, , =-, , 1 + (10)n +1, , (a) 2, , (b) –1, , n p sin 2 (n!), , 0 < p < 1 is equal to, n +1, n ®¥, (a) 0, (b) ¥, (c) 1, , 117. Lim, (b) e –4, (d) e–2, , 118. lim log tan x (tan 2 x) is equal to, +, x ®0, , a, , then the value of a is –, 10, (c) 1, , (d) 0, , 1, (1 + x )1/ x - e + ex, 2 ., 109. Evaluate lim, x ®0, x2, , (a), , 11e, 24, , (b) e, , (c), , 7e, 19, , (b) –, , 1, 2, , (c), , (d) 1, , 1, 2, , (b) –1, , 120. If lim, , (b) 2a, ax 2 + bx + c, , x®1, , ( x - 1), , (a) 2, , (b), , 1, 3, , (b), , 2, 3, , (c) 0, , (d), , -¥, , (c), , -2, 3, , (d), , 2, 9, , (c) 4, , 2, 7, , (b), , (d), , 5, 2, , (c), , 19, 52, , (d) None of these, , 1, 12, , (c), , ¥, , (d) does not exist, , (c), , log(3 + x ) - log(3 - x), = k , then the value of k is, x, x®0, , 114. If lim, , (c), , -, , æ f (1 + x) ö, lim ç, ÷, x ® 0 è f (1) ø, , 1, 3, , (d), , 2, 3, , 1/ n, 115. If f ( x) = lim n( x - 1), then for x > 0, y > 0, f(xy) is, n®¥, , (b) f (x) + f (y), (d) None of these, , æ, tan p x, 1 ö, + lim ç 1 + 2 ÷, x ®- 2 x + 2, x®¥ è, x ø, , 1/ x, , equals, , (b) e1/2, , (c) e2, , (a) 0, , (b) 0, , x, , 116. Value of the lim, , (a) p + 1 (b) p – 1, , 3, 2, , (d) e3, , x®0, , (a), , equal to, (a) f(x) f(y), (c) f(x) – f(y), , ( x - a)( x - b)( x - c), =, ( x + 1), x®1, , 123. lim (cosec x )1/ log x is equal to :, , 1, p, , 0, , (d) 4a, , = 2 then lim, , (a), , (a) 1, , æ 1 ö, x tan ç 2 ÷ + 3 | x |2 +7, è px ø, 113. lim, is equal to, x ®-¥, | x |3 +7 | x | +8, , (b), , a, 2, , 122. Let f : R ® R be such that f (1) = 3 and f '(1) = 6. Then, , 5, , 2, 3, , 2, , (c), , n r3 - 8, is equal to, n ® ¥ r =3 r 3 - 8, , æ 2, ö, 112. lim ç x - x ÷ =, ç, x ®¥ è 3x - 2 3 ÷ø, , -, , æ a ö, x -1, 119. Evaluate lim 2 tan ç x ÷ ., x ®¥, è2 ø, , (d) None of these, , (a) 1, , (a), , (b) –1, (d) None of these, , 121. The limit lim P, , é3, 2 3, 2ù, 111. The value of lim ê (n + 1) - (n - 1) ú is, û, n®¥ ë, , (a), , (a) 1, (c) 0, , (a) a, , 2, n ü, ì 1, +, + .... +, ý is equal to, 110. lim í, 2, 2, n®¥ î1 - n, 1- n, 1 - n2 þ, , (a) 0, , (d) None, , (c) p/2 + 1, , is, (d) p/2 – 1, , (b) 1, , 1, e, , (d) None of these, , æ x 2 - 1ö, 124. The value of the limit lim ç, ÷, x®¥ è x 2 + 1ø, (a) e2, , (b) e–2, , 125. If Lim æçç1 + l + m ö÷÷, x ®¥ è, x x2 ø, (a) l = -1, m = 2, , (c) e, 2x, , = e 2 then, , (b), , l = 2, m = 1, , (c), , l = 1, m = any real number, , (d) l = m = any real number, , x2, , equals to, (d) e–1

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LIMITS AND DERIVATIVES, , 169, , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, 1., , integer function, then lim f (x) is equal to, x®0, , sin x, is equal to, x®p x - p, lim, , (a) 1, , (b) 2, , (c) – 1, , (d) – 2, , 2, , 2., , x cos x, is equal to, x ® 0 1 - cos x, lim, , (a) 2, 3., , 4., , (b), , lim, , xm -1, , x ®1 x n, , -1, , 9., , 3, 2, , (d) 1, , (b), , m, n, , (c), , -, , m, n, , 4, 9, , (b), , 1, 2, , lim, , x®0, , (b) 1, sin x, , x +1 - 1- x, , (b) 0, , sec 2 x - 2, is, x ® p / 4 tan x - 1, (a) 3, (b) 1, , x ®1, , ), , x - 1 ( 2 x - 3), 2x + x - 3, , 1, 10, (c) 1, , (a), , 2, , (b) – 1, (d) None of these, , x ® 2-, , (d), , m, , 2, , 13., , -1, 2, , n2, , (d) – 1, , (c), , 1, 2, , (b), , 3, 2, , (a), , (b) 1, , x+, , 15. If y =, (d) 0, (a) 1, , (d) – 1, , 1, x, (b), , 16. If f (x) =, (a), , (c) 0, , (d) 2, , x-4, 2 x, , 5, 4, , (b), 1+, , is equal to, -1, 10, (d) None of these, , (b), , ì sin[ x], , [ x] ¹ 0, ï, 10. If f (x) = í [ x], , where [.] denotes the greatest, ï 0,, [ x] = 0, î, , 1, 2, , (c), , -1, 2, , (d), , 1, 4, , æ1ö, 14. If f (x) = x – [x] , x Î R, then f ' ç ÷ is equal to, è 2ø, , is equal to, (c) 1, , (b) x2 – 7x + 8 = 0, (d) x2 – 10x + 21 = 0, , tan 2 x - x, is equal to, 3x - sin x, , lim, , x®0, , (a) 2, , (c), , x ® 2+, , (a) x2– 6x + 9 = 0, (c) x2 – 14x + 49 = 0, , lim, , (, lim, , | sin x |, is equal to, x, (a) 1, (c) Does not exist, lim, , x ®0, , whose roots are lim f (x) and lim f ( x ) is, (d) 0, , cosec x - cot x, lim, is equal to, x, x®0, -1, 2, , (b) 0, (d) Does not exist, , ìï x 2 - 1, 0 < x < 2, 12. If f (x) = í, , then the quadratic equation, ïî2 x + 3, 2 £ x < 3, , is equal to, , (a) 2, 8., , -, , 1 - cos 4 q, lim, is equal to, q ® 0 1 - cos 6 q, , (a), 7., , (c), , (1 + x )n - 1, is equal to, x, x ®0, (a) n, (b) 1, (c) – n, , (a), 6., , 3, 2, , lim, , (a) 1, 5., , 11., , (a) 1, (c) – 1, , , then, 1, 2, , (c) 0, , (d) – 1, , dy, at x = 1 is equal to, dx, , 1, , (c), , 2, , , then f ' (1) is equal to, 4, 5, , (c) 1, , (d) 0, , 1, , dy, x2, ' then, is equal to, 1, dx, 1- 2, x, - 4x, - 4x, (b), 2, 2, x2 - 1, ( x - 1), , 17. If y =, , (a), , (c), , 1 - x2, 4x, , (d) 0, , (d), , 4x, 2, , x -1

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EBD_7762, 170, , 18., , MATHEMATICS, , dy, sin x + cos x, at x = 0 is equal to, , then, dx, sin x - cos x, , If y =, , (a) – 2, , 19., , If y =, , sin( x + 9), dy, , then, at x = 0 is equal to, cos x, dx, , (b) sin 9, , (c) 0, , 1, 100, (c) 0, , 29., , (b) 0, , (, , x®0, , (a), 25., , x, -p, , (a), , (b), , ) is equal to:, p, , (c), , (c), , 1, 16, , (d), , 1, 8, , a1 + a2 + ........... + an, , e a1 +a2 +¼an, a1 + a 2 + ..... + a n, (c), n, (d) a1a2 a3 .......an, 30. For each t Î R , let [t] be the greatest integer less than or, equal to t. Then, [JEE MAIN 2018, S], æé 1 ù é 2 ù, é15 ù ö, lim x ç ê ú + ê ú + ... + ê ú ÷, è, x, x, ë x ûø, x ®0+ ë û ë û, , (a) is equal to 15., (c) does not exist (in R)., [JEE MAIN 2014, A], , p, 2, , 31. The value of lim, , n®¥, , (b) is equal to 120., (d) is equal to 0., , 1 + 2 + 3 +¼ n, n 2 + 100, , is equal to :, [BITSAT 2018, A, BN], , (d) 1, (a), , lim (cosec x )1/ log x is equal to : [BITSAT 2014, A, BN], , x®0, , (a) 0, 1, (c), e, 26., , 2, , [JEE MAIN 2017, A, BN], , (b), , 1, (d) Does not exist, 2, If f(x) = x100 + x99 + ... + x + 1, then f '(1) is equal to, (a) 5050, (b) 5049, (c) 5051, (d) 50051, If f (x) = 1 – x + x2 – x3 + ... – x99 + x100, then f '(1) is, equal to, (a) 150, (b) – 50, (c) – 150, (d) 50, , lim, , equals :, , ( p - 2x )3, , 1, 1, (b), 4, 24, The value of, , (d) Does not exist, , sin p cos 2 x, , cot x - cos x, , p, x®, 2, , x, 1/ x ö, æ a1/ x + a1/, 2 + .......... + an, lim ç 1, ÷, n, x®¥ è, ø, ai > 0, i = 1, 2, ...... n, is, [BITSAT 2017, A, BN], , xn – a n, for some constant a, then f '(a) is equal to, x–a, , If f (x) =, , lim, , (b) 1, (d) None of these, , (a), , Past Year MCQs, 24., , , where a > b > 1, is equal to, , nx, , (c), , 23., , 28., , (b) 100, , (a) 1, , 22., , - bn, , (a) –1, (c) 0, , (d) 1, , x2, x100, + ... +, , then f ' (1) is equal to, If f (x) = 1 + x +, 2, 100, , (a), , 21., , n ®¥ a n, , [BITSAT 2016, C, BN], , (d) Does not exist, , (a) cos 9, 20., , a n + bn, , lim, , (b) 0, , 1, 2, , (c), , 27., , ¥, , (c) 2, , (b) 1, (d) None of these, , æ1ö, æ1ö, If lim x sin ç ÷ = A and lim x sin ç ÷ = B , then which, x ®0, x ®¥, èxø, èxø, one of the following is correct? [BITSAT 2015, C, BN], (a) A =1 and B = 0 (b) A =0 and B = 1, (c) A =0 and B = 0 (d) A =1 and B = 1, , (b), , 32., , lim, , x ®0, , 1, 2, , (d) 0, , x , sin x, , x ∗ sin 2 x, , is equal to, , [BITSAT 2018, A], , (a) 1, , (b) 0, , (c) ¥, , (d) None of these

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LIMITS AND DERIVATIVES, , 171, , Exercise 3 : Try If You Can, 1., , lim, , 1, , n®¥ n3, , (a), , 2., , {[12 x ] + [2 2 x] + [32 x] + ... + [ n 2 x]} equals, , x/2, , If lim, , (b), , (c), , x/3, , æ 1 ö, x 2 tan ç 2 ÷ + 3 | x |2 +7, è px ø, | x |3 +7 | x | +8, , x ®-¥, , (a) 1, 3., , (b) –1, , (d) 0, , x/6, , 8., , -C, =, , then c =, p, , (c) 2, , (d) –2, , lim lim (1 + cos 2 m n!px) is equal to, , (b) 1, (d) None of these, , If k Î I such that lim çæ cos k p ÷ö, n ®¥ è, 4 ø, , 2n, , kp ö, æ, - ç cos ÷, 6 ø, è, , 9., , 2n, , = 0,, , then, (a) k must not be divisible by 24, (b) k is divisible by 24 or k is divisible neither by 4 nor by, 6, (c) k must be divisible by 12 but not necessarily by 24, (d) None of these, n, , 5., , Given that, , lim, , n®¥, , å, , r=1, , log ( n + r ) - log n, 1ö, æ, = 2 ç log 2 - ÷ ,, è, 2ø, n, , 1, k, k, k 1/ n, then lim k [(n + 1) (n + 2) ...(n + n) ], is equal to, n®¥ n, (a), 6., , 4k, e, , æ 4ö, (b) ç ÷, è eø, , 1/ k, , (c), , æ 4ö, çè ÷ø, e, , k, , æ eö, (d) ç ÷, è 4ø, , k, , Let f(x) be a polynomial function satisfying, æ1ö, æ1ö, f ( x ) × f ç ÷ = f ( x ) + f ç ÷ . If f(4) = 65 and l , l , l are, 1 2 3, x, è ø, è xø, in GP, then f '(l1 ), f '(l2 ), f '(l3 ) are in, (a) AP, (c) HP, , 7., , (b) G P, (d) None of these, , sin x ù, é, (where [×] denotes the, lim ê min( y 2 - 4 y + 11), x ®0 ë, x úû, greatest integer function) is, , (b) 6, (d) does not exist, , ì, ü, ï, ï, ï, ï, ïï, ïï, x, The value of lim í, ý is, 3, x ®¥ ï, x, ï, x+, 3, ï, ï, x, ï, x+, ....¥ ï, 3, ïî, ïþ, x+ x, , (a) 0, (c) ¥, , m ®¥ n ®¥, , (a) –2, (c) 0, 4., , (a) 5, (c) 7, , If [x] denotes the greatest integer £ x , then, , (b) 1, (d) None of these, , ìï, æ [12 (sin x ) x ] + [22 (sin x) x ] + .... + [ n2 (sin x ) x, lim í lim ç, ç, x ®0 + ï, n3, în ®¥ è, , öüï, ÷ý ,, ÷, øïþ, , (where [.] denotes the greatest integer function) is equal to, (a) 0, (b) 2/3, (c) 4/3, (d) 1/3, 10., , lim cos[ p n 2 + n ], n Î I is equal to, , n ®¥, , (a) 0, (c) –1, , (b) 1, (d) None of these, , 11. If Sn denotes the sum of n terms of a GP whose common, ratio is r, then ( r - 1), , dS n, is equal to, dr, , (a) (n - 1)Sn + n Sn-1, , (b) (n - 1)Sn - n Sn-1, , (c) (n - 1) Sn, , (d) None of these, , 12. If l = lim (sin x + 1 - sin x ) and, x ®¥, , m = lim [sin x + 1 - sin x ]; where [.] denotes the, x ®-¥, , greatest integer function, then :, (a), , l=m=0, , (b), , l = 0; m is undefined, , (c), , l, m both do not exist, , (d) l = 0, m ¹ 0 (although m exist)

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EBD_7762, 172, , 13., , MATHEMATICS, , The limit lim, , 2 x + 23- x - 6, , x® 2, , (a) 1, , 2, , -x, , 1- x, , is equal to, , (c), , -2, , (b) 2, , (c) 4, , Let a = min{x + 2 x + 3, x Î R} and b = lim, , 14., , q®0, , n, , The value of, , åa, , r, , .b, , n -r, , is, , (d) None of these, , 3 ´ 2n, , 8, , (d) 8, , 2, , 4 n +1 - 1, , 1 - cos q, q2, , 15., ., , -1, ìé 1/ 2 1/ 2 -1, ü, ù, log a, 2(ax )1/ 4, ï æa +x ö, ú - 2 4 ïý, lim íêçç 1/ 4 1/ 4 ÷÷ - 3/ 4 1/ 4 1/ 2 1/ 2 1/ 4, x® a ê a, x - a x + a x - a3/ 4 ú, -x ø, ï ëè, ï, û, î, þ, , is, (a) a, (c) a2, , (b) a3/4, (d) None of these, , r=0, , (a), , 2 n +1 - 1, , (b), , 3 ´ 2n, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, , (b), (a), (a), (c), (a), (a), (b), (c), (a), (a), (b), (a), (a), , 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, , (b), (a), (d), (a), (a), (c), (b), (a), (c), (d), (c), (c), (b), , 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, , (b), (d), (a), (c), (b), (d), (a), (d), (d), (b), (b), (d), (c), , 1, 2, 3, 4, , (c), (a), (a), (b), , 5, 6, 7, 8, , (a), (c), (c), (d), , 9, 10, 11, 12, , (b), (a), (c), (d), , 1, 2, , (b), (a), , 3, 4, , (d), (b), , 5, 6, , (c), (b), , 2 n +1 + 1, 3 ´ 2n, , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (a) 53, (c), (c) 79, 40, 66, (a), (a) 80, 41 (d) 54, 67, (b), (b) 81, 42 (b) 55, 68, (c) 56, (d), (b) 82, 43, 69, (d), (d) 83, 44 (d) 57, 70, (a) 58, (d), (b) 84, 45, 71, (c), (c) 85, 46 (d) 59, 72, (c) 60, (b), (b) 86, 47, 73, (d), (b) 87, 48 (b) 61, 74, (b), (b) 88, 49 (b) 62, 75, (c), (b) 89, 50 (d) 63, 76, (a), (c) 90, 51 (b) 64, 77, (c) 65, (b), (a) 91, 52, 78, Exercise 2 : Exemplar & Past Year MCQs, (a), (d) 25, 13 (b) 17, 21, (a) 26, 14 (b) 18 (a), 22, (a), (d) 27, 15 (d) 19, 23, (a) 20, (b), (b) 28, 16, 24, Exercise 3 : Try If You Can, (b), (d), (b) 13, 7, 9, 11, (b), (a), (b), 8, 10, 12, 14, , (d), (d), (b), (a), (b), (c), (d), (b), (c), (b), (a), (d), (d), , 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, , (c), (c), (d), (b), (c), (a), (c), (a), (b), (c), (d), (c), (c), , (c), (a), (b), (c), , 29, 30, 31, 32, , (d), (b), (b), (b), , (d), (c), , 15, , (c), , 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, , (b), (b), (d), (c), (a), (b), (c), (d), (a), (d), (b), (a), (a), , 118, 119, 120, 121, 122, 123, 124, 125, , (a), (c), (d), (a), (c), (c), (b), (c)

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14, , MATHEMATICAL REASONING, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 2, , JEE MAIN, BITSAT, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, JEE Main, BITSAT, , Weightage, 3, 0, , Critical Concepts, Statements: Negation of a Statement,, Compound Statement, Quantifiers,, Implications: Contrapositive, & Converse, Truth Value., , Rating of, Difficulty Level, , CUS, (Chapter Utility Score), Out of 10, , 2/5, , 5.3

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MATHEMATICAL REASONING, , 175

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EBD_7762, 176, , MATHEMATICS, , Topic 1 : Statement, Truth Value of a Statement,, Logical Connectives, 1., , 2., , 3., , 4., , Which of the following is a statement?, (a) May you live long!, (b) May God bless you!, (c) The sun is a star., (d) Hurrah! we have won the match., Which of the following is not a statement?, (a) Please do me a favour. (b) 2 is an even integer., (c) 2 + 1 = 3., (d) The number 17 is prime., Which of the following is not a statement?, (a) 2 is an even integer., (b) 2 + 1= 3., (c) The number 17 is prime., (d) x + 3 = 10, x Î R., Which of the following is not a proposition, (a), , 11., , 12., , 13., , 14., , 3 is a prime, , 2 is irrational, (c) Mathematics is interesting, (d) 5 is an even integer, Which of the following is not a statement?, (a) Roses are red, (b) New Delhi is in India, (c) Every square is a rectangle, (d) Alas! I have failed, The sentence “There are 35 days in a month” is, (a) a statement, (b) not a statement, (c) may be statement or not (d) None of these, Which of the following is a statement?, (a) Everyone in this room is bold, (b) She is an engineering student, (c) sin2q is greater than 1/2, (d) Three plus three equals six, The sentence “New Delhi is in India”, is, (a) a statement, (b) not a statement, (c) may be statement or not (d) None of theese, Which of the following is/are connectives?, (a) Today, (b) Yesterday, (c) Tomorrow, (d) “And”, “or”, Consider the following statements, I. “Every rectangle is a square” is a statement., II. “Close the door” is not a statement., (b), , 5., , 6., , 7., , 8., , 9., , 10., , 15., , 16., , 17., , 18., , Choose the correct option., (a) Only I is false., (b) Only II is false., (c) Both are true., (d) Both are false., Which of the following is not a statement?, (a) Every set is a finite set, (b) 8 is less than 6, (c) Where are you going?, (d) The sum of interior angles of a triangle is 180 degrees, Which of the following is an open statement?, (a) Good morning to all (b) Please do me a favour, (c) Give me a glass of water (d) x is a natural number, A compound statement p or q is false only when, (a) p is false, (b) q is false, (c) both p and q are false, (d) depends on p and q, If p, q, r are statement with truth vales F, T, F respectively,, then the truth value of p®(q ® r) is, (a) false, (b) true, (c) true if p is true, (d) none, A compound statement p and q is true only when, (a) p is true, (b) q is true, (c) both p and q are true (d) none of p and q is true, A compound statement p ® q is false only when, (a) p is true and q is false, (b) p is false but q is true, (c) at least one of p or q is false, (d) both p and q are false, Assertion: The sentence “8 is less than 6” is a statement., Reason: A sentence is called a statement, if it is either true, or false but not both., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., If p : Pappu passes the exam,, q : Papa will give him a bicycle., Then, the statement ‘Pappu passing the exam, implies, that his papa will give him a bicycle’ can be symbolically, written as, (a) p ® q (b) p « q, (c) p Ù q (d) p Ú q

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MATHEMATICAL REASONING, , 19. “Paris is in England” is a _________, (a) statement, (b) sentence, (c) both ‘a’ and ‘b’, (d) neither ‘a’ nor ‘b’, 20. Consider the following, I. New Delhi is in Nepal., II. Every relation is a function., III. Do your homework., Choose the correct option., (a) I and II are statements., (b) I and III are statements., (c) II and III are statements., (d) I, II and III are statements., 21. Which of the following is the conditional p ® q?, (a) q is sufficient for p, (b) p is necessary for q, (c) p only if q, (d) if q, then p, 22. Which of the following statement is false?, (a) A quadratic equation has always a real root, (b) The number of ways of seating 2 persons in two chairs, out of n persons is P(n, 2), (c) The cube roots of unity are in GP, (d) None of the above, 23. For the statement “17 is a real number or a positive, integer”, the “or” is, (a) inclusive, (b) exclusive, (c) Only (a), (d) None of these, 24. Let p : A quadrilateral is a parallelogram, q : The opposite side are parallel, Then the compound proposition, 'A quadrilateral is a parallelogram if and only if the opposite, sides are parallel' is represented by, (a) p Ú q (b) p ® q, (c) p Ù q, (d) p « q, 25. Truth value of the statement ‘It is false that 3 + 3 = 33, or 1 + 2 = 12’ is, (a) T, (b) F, (c) both T and F, (d) 54, 26. Consider the following statements, p : A tumbler is half empty., q : A tumbler is half full., Then, the combination form of “p if and only if q” is, (a) a tumbler is half empty and half full, (b) a tumbler is half empty if and only if it is half full, (c) Both (a) and (b), (d) None of the these, Topic 2 : Converse, Inverse and Contrapositive of the, Conditional Statement, Negative of, Compound Statement, 27. Which of the following is the converse of the statement?, “If Billu secure good marks, then he will get a bicycle.”, (a) If Billu will not get bicycle, then he will not secure, good marks., , 177, , (b) If Billu will get a bicycle, then he will secure good, marks., (c) If Billu will get a bicycle, then he will not secure good, marks., (d) If Billu will not get a bicycle, then he will secure good, marks., 28. ~ (p Ú (~ p)) is equal to, (a) ~ p Ú q, (b) (~ p) Ù q, (c) ~ p Ú ~ p, (d) ~ p Ù ~ p, 29. If ( p Ù ~ r ) Þ (q Ú r) is false and q and r are both false,, then p is, (a) True, (c) May be true or false, 30., , (b) False, (d) Data insufficient, , ~ ( (~ p) Ù q ) is equal to, (a), , p Ú (~ q), , (b), , pÚq, , (c), , p Ù (~ q), , (d), , ~pÙ~q, , 31. The inverse of the statement (p Ù ~ q) ® r is, (a) ~ (p Ú ~q) ® ~ r, (b) (~p Ù q) ® ~ r, (c) (~p Ú q) ® ~ r, (d) None of these, 32. Negation of the statement (p Ù r) ® (r Ú q) is, (a) ~ (p Ù r) ® ~ (r Ú q) (b) (~p Ú ~r) Ú (r Ú q), (c) (p Ù r) Ù (r Ù q), (d) (p Ù r) Ù (~ r Ù ~q), 33. Read the following statements., Statement: If x is a prime number, then x is odd., I. Contrapositive form : If a number x is not odd, then, x is not a prime number., II. Converse form : If a number x is odd, then it is a, prime number., Choose the correct option., (a) Both I and II are true. (b) Only I is true., (c) Only II is true., (d) Neither I nor II true., 34. The negation of the statement “ 2 is not a complex, number” is, (a), , 2 is a rational number, , (b), , 2 is an irrational number, , (c), , 2 is a complex number, (d) None of the above, 35. The contrapositive of the statement, 'If I do not secure good, marks then I cannot go for engineering', is, (a) If I secure good marks, then I go for engineering., (b) If I go for engineering then I secure good marks., (c) If I cannot go for engineering then I donot secure good, marks., (d) None.

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EBD_7762, 178, , 36., , MATHEMATICS, , If p Þ (~ p Ú q) is false, the truth values of p and q are, , respectively, (a) F, T, (b) F, F, (c) T, T, (d) T, F, 37. p Þ q can also be written as, (a) p Þ ~ q, (b) ~ p Ú q, (c) ~ q Þ ~ p, (d) None of these, 38. Assertion: The contrapositive of (p Ú q) ® r is ~ r ® ~ p Ù ~ q., Reason: If (p Ù ~ q) ® (~ p Ú r) is a false statement, then, respective truth values of p, q and r are F, T, T., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 39. If p Þ (q Ú r) is false, then the truth values of p, q, r are, respectively, (a) T, F, F (b) F, F, F, (c) F, T, T (d) T, T, F, 40. If Ram secures 100 marks in maths, then he will get a, mobile. The converse is, (a) If Ram gets a mobile, then he will not secure, 100 marks, (b) If Ram does not get a mobile, then he will secure, 100 marks, (c) If Ram will get a mobile, then he secures 100 marks, in maths, (d) None of these, 41. Consider the following statements, Statement-I: The negation of the statement “The number, 2 is greater than 7” is “The number 2 is not greater than 7”., Statement-II: The negation of the statement “Every, natural number is an integer” is “every natural number is, not an integer”., Choose the correct option., (a) Only Statement I is true, (b) Only Statement II is true, (c) Both Statement are true, (d) Both Statement are false, 42. Assertion : The negation of (p Ú ~ q) Ù q is (~ p Ù q) Ú ~ q., Reason : ~ (p ® q) º p Ù ~ q, (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 43. The statement p: For any real numbers x, y if x = y, then, 2x + a = 2y + a when a Î Z., (a) is true, (b) is false, (c) its contrapositive is not true, (d) None of these, 44. Let p : I am brave,, q : I will climb the Mount Everest., The symbolic form of a statement,, 'I am neither brave nor I will climb the mount Everest' is, (a) p Ù q (b) ~ (p Ù q) (c) ~ p Ù ~ q (d) ~ pÙ q, 45. Consider the following statements., , I. If a number is divisible by 10, then it is divisible by 5., II. If a number is divisible by 5, then it is divisible by 10., Choose the correct option., (a) I is converse of II., (b) II is converse of I., (c) I is not converse of II., (d) Both ‘a’ and ‘b’ are true., 46. Let p: Kiran passed the examination,, q: Kiran is sad, The symbolic form of a statement “It is not true that Kiran, passed therefore he is said” is, (a) (~ p® q), (b) (p® q), (c) ~ (p® ~ q), (d) ~ ( p«q), 47. Consider the following statements, P : Suman is brilliant, Q : Suman is rich, R : Suman is honest, The negation of the statement “Suman is brilliant and, dishonest if and only if Suman is rich” can be expressed as, (a) ~ Q « ~ P Ú R, (b) ~ Q « ~ P Ù R, (c) ~ ( P Ù ~ R) « Q, (d) ~ P Ù (Q «~ R ), 48. Let S be a non-empty subset of R. Consider the following, statement :, P : There is a rational number x Î S such that x > 0., Which of the following statement is the negation of the, statement P ?, (a) There is no rational number x Î S such than x < 0., (b) Every rational number x Î S satisfies x < 0., (c) x Î S and x < 0 Þ x is not rational., (d) There is a rational number x Î S such that x < 0., 49. Consider the following statements, p : x, y Î Z such that x and y are odd., q : xy is odd. Then,, (a) p Þ q is true, (b) : q Þ p is true, (c) Both (a) and (b), (d) None of these, 50. Assertion: ~ (p ® q) º p Ù ~ q, Reason: ~ (p « q) º (p Ú ~ q) Ù (q Ù ~ p), (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., BEYOND NCERT, Topic : 3 Truth Table, Logical Equivalence, Tautology and, Contradiction, Duality, Algebra of Statement, 51. Which of the following is true?, (a) p Þ q º ~ p Þ ~ q, (b), , ~ (p Þ ~ q) º ~ p Ù q, , (c), , ~ (~ p Þ ~ q) º ~ p Ù q, , (d) ~ (~ p Û q) º [~ (p Þ q)Ù ~ (q Þ p)], 52. Consider the following statement., “If a triangle is equiangular, then it is an obtuse angled, triangle.”, This is equivalent to

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MATHEMATICAL REASONING, , a triangle is equiangular implies that it is an obtuse, angled triangle., II. for a triangle to be obtuse angled triangle it is sufficient, that it is equiangular., Choose the correct option., (a) Both are correct., (b) Both are incorrect., (c) Only I is correct., (d) Only II is correct., 53. Which of the following is true?, (a) p Þ q º ~ p Þ ~ q, , 179, , I., , (b), , ~ (p Þ ~ q) º ~ p Ù q, , (c), , ~ (~ p Þ ~ q) º ~ p Ù q, , (d), , ~ (~ p Û q) º [~ (p Þ q)Ù ~ (q Þ p)], , 54. The conditional ( p Ù q) Þ p is, (a) A tautology, (b) A fallacy i.e., contradiction, (c) Neither tautology nor fallacy, (d) None of these, 55. If ~q Ú p is F, then which of the following is correct?, (a) p « q is T, (b) p ® q is T, (c) q ® p is T, (d) p ® q is F, 56. Consider the following statements., I. If a statement is always true, then the statement is, called “tautology”., II. If a statement is always false, then the statement is, called “contradiction”., , 57., , 58., 59., , 60., , Choose the correct option., (a) Both the statements are false., (b) Only I is false., (c) Only II is false., (d) Both the statements are true., If p, q are true and r is false statement, then which of the, following is true statement?, (a) (p Ù q) Ú r is F, (b) (p Ù q) ® r is T, (c) (p Ú q) Ù (p Ú r) is T, (d) (p ® q) « (p ® r) is T, Which of the following is true?, (a) p Ù ~p º T, (b) p Ú ~p º F, (c) p ® q º q ® p, (d) p ® q º (~q) ® (~p), The dual of statement [pÚq) Ù ~ q]Ú ~ q is, (a) [(p Ú q) Ù ~ q] Ú ~ q, (b) [(p Ù q) Ù ~ q] Ú ~ q, (c) [(p Ù q) Ú ~ p] Ú (~ q), (d) [(p Ù q) Ú ~ p] Ù (~ q), Assertion: ~ ( p «~ q) is equivalent to p « q ., Reason: ~ ( p «~ q) is a tautology, (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, 1., , 2., , 3., , 4., , 5., , Which of the following is a statement?, (a) x is a real number, (b) Switch off the fan, (c) 6 is a natural number (d) Let me go, Which of the following is not a statement., (a) Smoking is injurious to health, (b) 2 + 2 = 4, (c) 2 is the only even prime number, (d) Come here, The connective in the statement :, “2 + 7 > 9 or 2 + 7 < 9” is, (a) and, (b) or, (c) >, (d) <, The connective in the statement :, “Earth revolves round the Sun and Moon is a satellite of, earth” is, (a) or, (b) Earth, (c) Sun, (d) and, The negation of the statement, “A circle is an ellipse” is, (a) An ellipse is a circle., (b) An ellipse is not a circle., (c) A circle is not an ellipse., , 6., , 7., , 8., , 9., , (d) A circle is an ellipse., The negation of the statement “7 is greater than 8” is, (a) 7 is greater than 8, (b) 7 is not greater than 8, (c) 8 is less than 7, (d) none of these, The negation of the statement “72 is divisible by 2 and 3” is, (a) 72 is not divisible by 2 or 72 is not divisible by 3., (b) 72 is not divisible by 2 and 72 is not divisible by 3., (c) 72 is divisible by 2 and 72 is not divisible by 3., (d) 72 is not divisible by 2 and 72 is divisible by 3., The negation of the statement “Plants take in CO2 and, give out O2” is, (a) Plants do not take in CO2 and do not given out O2, (b) Plants do not take in CO2 or not give out O2, (c) Plants take is CO2 and do not give out O2, (d) Plants take in CO2 or do not give out O2, The negation of the statement :, “Rajesh or Rajni lived in Bangalore” is, (a) Rajesh did not live in Bangalore or Rajni lives in, Bangalore., (b) Rajesh lives in Bangalore and Rajni did not live in, Bangalore.

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EBD_7762, 180, , 10., , 11., , 12., , 13., , 14., , 15., , 16., , 17., , MATHEMATICS, , (c) Rajesh did not live in Bangalore and Rajni did not, live in Bangalore., (d) Rajesh did not live in Bangalore or Rajni did not live, in Bangalore., The negation of the statement “101 is not a multiple of 3”, is, (a) 101 is a multiple of 3 (b) 101 is a multiple of 2, (c) 101 is an odd number (d) 101 is an even number, The contrapositive of the statement, “If 7 is greater than 5, then 8 is greater than 6” is, (a) If 8 is greater than 6, then 7 is greater than 5., (b) If 8 is not greater than 6, then 7 is greater than 5., (c) If 8 is not greater than 6, then 7 is not greater than 5., (d) If 8 is greater than 6, then 7 is not greater than 5., The converse of the statement, “If x > y, then x + a > y + a” is, (a) If x < y, then x + a < y + a, (b) If x + a > y + a, then x > y, (c) If x < y, then x + a > y + a, (d) If x > y, then x + a < y + a, The converse of the statement “If sun is not shining, then, sky is filled with clouds” is, (a) If sky is filled with clouds, then the Sun is not shining, (b) If Sun is shining, then sky is filled with clouds, (c) If sky is clear, then Sun is shining, (d) If Sun is not shining, then sky is not filled with clouds., The contrapositive of the statement, “If p, then q”, is, (a) If q, then p, (b) If p, then ~ q, (c) If ~ q, then ~ p, (d) If ~ p, then ~ q, 2, The statement “If x is not even, then x is not even” is, converse of the statement, (a) If x2 is odd, then x is even, (b) If x is not even, then x2 is not even, (c) If x is even, then x2 is even, (d) If x is odd, then x2 is even, The contrapositive of statement “If Chandigarh is capital, of Punjab, then Chandigarh is in India” is, (a) “If Chandigarh is not in India, then Chandigarh is, not the capital of Punjab”., (b) “If Chandigarh is in India, then Chandigarh is capital, of Punjab”., (c) “If Chandigarh is not capital of Punjab, then, Chandigarh is not the capital of India”., (d) “If Chandigarh is capital of Punjab, then Chandigarh, is not in India”., Which of the following is the conditional p ® q?, (a) q is sufficient for p., (b) p is necessary for q., (c) p only if q., (d) if q, then p., , 18. The negation of the statement “The product of 3 and 4 is, 9” is, (a) it is false that the product of 3 and 4 is 9, (b) the product of 3 and 4 is 12, (c) the product of 3 and 4 is not 12, (d) it is false that the product of 3 and 4 is not 9., 19. Which of the following is not a negation of the statement, “A natural number is greater than zero”., (a) A natural number is not greater than zero., (b) It is false that a natural number is greater than zero., (c) It is false that a natural number is not greater than, zero., (d) None of the above, 20. Which of the following statement is a conjunction?, (a) Ram and Shyam are friends., (b) Both Ram and Shyam are tall., (c) Both Ram and Shyam are enemies., (d) None of the above., Past Year MCQs, 21. In the truth table for the statement (p Ù q) ®, (q Ú ~ p), the last column has the truth value in the, following order is, [BITSAT 2013, A, BN], (a) TTFF, , (b), , FTTT, , (c) TFTT, , (d), , TTTT, , 22. The statement : ( p « : q ) is: [JEE MAIN 2014, A, BN], (a) a tautology, (b) a fallacy, (c) equavalent to p « q, (d) equivalent to : p « q, 23. The negation of ~ s Ú (~ r Ù s) is equivalent to :, [JEE MAIN 2015, A, BN], (b) s Ù r, (d) s Ù (r Ù ~ s), , (a) s Ú (r Ú ~ s), (c) s Ù ~ r, 24. The Boolean Expression, (p Ù : q) Ú qÚ (: p Ù q) is equivalent to:, (a), , pÚ q, , (c) : p Ù q, 25. The following statement, (p ® q) ® [(~p ® q) ® q], (a) a fallacy, (c) equivalent to ~ p ® q, 26. The Boolean expression, , [JEE MAIN 2016, A, BN], (b) p Ú : q, (d) p Ù q, [JEE MAIN 2017, S, BN], is :, (b) a tautology, (d) equivalent to p ® ~q, , ~ (pÚ q) Ú (~ p Ù q) is equivalent to :, (a) p, , (b) q, , [JEE MAIN 2018, A, BN], (c) ~q, (d) ~p

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MATHEMATICAL REASONING, , 181, , Exercise 3 : Try If You Can, 1., , 2., , 3., , 4., , If S*(p, q, r) is the dual of the compound statement S(p,q,r), and S (p,q,r) = ~ p Ù [~ (q Ú r)] then S*(~p, ~q, ~r) is, equivalent to –, (a) S (p, q, r), (b) ~ S (~p, ~q, ~r), (c) ~ S (p, q, r), (d) S*(p, q, r), The false statement of the following is, (a), , p Ù (~ p) is a contradiction, , (b), , ( p Þ q) Û (~ q Þ~ p) is a contradiction, , (c), , ~ (~ p) Û p is a tautology, , (d), , p Ú (~ p) Û p is a tautology, , If each of the following statements is true, then, p Þ ~q, ~r Þ q, (a) r is false, (b) r is true, (c) q is true, (d) None of these, For integers m and n, both greater than 1, consider the, following three statements :, P : m divides n, Q : m divides n2, R : m is prime,, then, (a) Q Ù R ® P, (c), , 5., , 6., , 7., , 8., , Q®R, , (b), , PÙQ ® R, , (d) Q ® P, , Let f be a function from a set X to a set Y. Consider the, following statements:, P : For each x Î X, there exists unique y Î Y such that f(x) = y, Q : For each y Î Y, there exists x Î X such that f(x) = y., R : There exist x1, x2 Î X such that x1 ¹ x2 and f(x1) = f(x2)., The negation of the statement “f is one-to-one and onto” is, (a) P or not R, (b) R or not P, (c) R or not Q, (d) P and not R, If p and q are true statement and r, s are false statements,, then the truth value of ~ [(pÙ~r) Ú (~q Ú s)] is, (a) true, (b) false, (c) false if p is true, (d) None of these, If p is any statement, t is tautology and c is a contradiction,, then which of the following is not correct?, (a) pÚ (~ p) = c, (b) pÚ t = t, (c) p Ù t = p, (d) p Ù c = c, Dual of following statement are given which one is not, correct?, (a) (p Ú q) Ù (r Ú s), (pÙq) Ú (rÙ s), (b) [p Ú (~q) Ù (~ p)], [ p Ù (~ q)] Ú (~p), , 9., , (c) (p Ù q) Ú r, (p Ú q) Ù r, (d) (p Ú q) Ú s, (p Ù q) Ú s, If p : 4 is an even prime number, q : 6 is a divisor of 12 and, r: the HCF of 4 and 6 is 2, then which one of the following, is true?, (a), , ( p Ù q), , (b), , ( p Ú q)Ù ~ r, , (c) ~ (q Ù r ) p, (d) ~ p Ú (q Ù r ), 10. Negation of the compound proposition., If the examination is difficult, then I shall pass if I study, hard., (a) The examination is difficult and I study hard but I, shall not pass., (b) The examination is difficult and I study hard and I, shall pass., (c) The examination is not difficult and I study hard and, I shall pass., (d) None of these., 11. Identify the false statements, (a) ~ [p Ú (~ q)] º (~ p) Ú q, (b) [p Ú q] Ú (~ p) is a tautology, (c) [p Ù q) Ù (~ p) is a contradiction, (d) ~ [p Ú q] º (~ p) Ú (~ q), 12. The converse of the statement if x< y then x2 < y2 is, (a) If x is not less then y then x2 is not less than y2, (b) If x2 < y2 then x < y, (c) If x2 ³ y2 then x ³ y, (d) None, 13. If p : It rains today, q : I go to school, r : I shall meet any, friends and s : I shall go for a movie, then which of the, following is the proposition :, “If it does not rain or if I do not go to school, then I shall, meet my friend and go for a movie”., (a), , ~ ( p Ù q) Þ (r Ù s), , (b), , ~ ( p Ù ~ q) Þ (r Ù s), , (c), , ~ ( p Ù q) Þ (r Ú s), , (d) None of these, , 14. In the truth table for the statement ( ~ p Ú q)®[p Ù ( q Ú ~ q)],, the last column has the truth value in the following order is, (a) TTFF (b) FTFT, (c) TFFT, (d) TTTT, 15. The propositions (p Þ ~ p) Ù (~ p Þ p) is, (a) Tautology and contradiction, (b) Neither tautology nor contradiction, (c) Contradiction, (d) Tautology

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EBD_7762, 182, , MATHEMATICS, , 1, 2, 3, 4, 5, 6, , (c), (a), (d), (c), (d), (a), , 7, 8, 9, 10, 11, 12, , (d), (a), (d), (c), (c), (d), , 13, 14, 15, 16, 17, 18, , (c), (b), (c), (a), (a), (a), , 1, 2, 3, , (c), (d), (b), , 4, 5, 6, , (d), (c), (b), , 7, 8, 9, , (b), (b), (c), , 1, 2, , (c), (b), , 3, 4, , (b), (a), , 5, 6, , (c), (b), , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (b), 19 (a) 25 (a) 31 (c) 37, (c), 20 (a) 26 (b) 32 (d) 38, (a), 21 (c) 27 (b) 33 (a) 39, (c), 22 (a) 28 (b) 34 (c) 40, (c), 23 (a) 29 (a) 35 (b) 41, (b), 24 (d) 30 (a) 36 (d) 42, Exercise 2 : Exemplar & Past Year MCQs, (c), 10 (a) 13 (a) 16 (a) 19, (d), 11 (c) 14 (c) 17 (c) 20, (d), 12 (b) 15 (b) 18 (a) 21, Exercise 3 : Try If You Can, (a), (d) 11 (d) 13, (a), 7, 9, (d) 10 (a) 12 (b) 14, (d), 8, , 43, 44, 45, 46, 47, 48, , (a), (c), (d), (b), (a), (b), , 49, 50, 51, 52, 53, 54, , (a), (c), (c), (a), (c), (a), , 22, 23, 24, , (c), (b), (a), , 25, 26, , (b), (d), , 15, , (c), , 55, 56, 57, 58, 59, 60, , (b), (d), (c), (d), (c), (c)

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15, , STATISTICS, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 3, , 2, , JEE MAIN, BITSAT, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 3, 2, , Critical Concepts, , Measures of Dispersion, Range,, Quartiles, Mean Deviation, Variance, and Standard deviation, Analysis of, Frequency Distributions, , Rating of, Difficulty Level, , 3.5/5, , CUS, (Chapter Utility Score), Out of 10, 7

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STATISTICS, , 185

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EBD_7762, 186, , MATHEMATICS, , Topic 1 : Mean, Median and Mode, 1., , 2., , 3., 4., , 5., , The observation which occur most frequently is known as :, (a) mode, (b) median, (c) weighted mean, (d) mean, The reciprocal of the mean of the reciprocals of n observation, is the :, (a) geometric mean, (b) median, (c) harmonic mean, (d) average, The median of 18, 35, 10, 42, 21 is, (a) 20, (b) 19, (c) 21, (d) 22, While dividing each entry in a data by a non-zero number a,, the arithmetic mean of the new data:, (a) is multiplied by a, (b) does not change, (c) is divided by a, (d) is diminished by a, In the following frequency distribution, class limits of some, of the class intervals and mid-value of a class are missing., However, the mean of the distribution is known to be 46.5, , Class, intervals, x -x, x -x, x -x, x -x, x - 100, 1, , 2, , 2, , 3, , 3, , 4, , 4, , 5, , 5, , Mid-values, , Frequency, , 15, 30, M, 75, 90, , 10, 40, 30, 10, 10, , the values of x1, x 2 , x 3 , x 4 , x 5 respectively will be, , 6., 7., , 8., , 9., , 10., , (a) (0, 20, 40, 60, 80), (b) (40, 50, 60, 70, 80), (c) (10, 20, 40, 70, 80), (d) (0, 19.5, 39.5, 69.5, 80), The mode of the following series 3, 4, 2, 1, 7, 6, 7, 6, 8, 6, 5 is, (a) 5, (b) 6, (c) 7, (d) 8, If you want to measure the intelligence of a group of, students,which one of the following measures will be more, suitable?, (a) Arithmetic mean, (b) Mode, (c) Median, (d) Geometric mean, If mean of the n observations x1, x2, x3,... xn be x , then the, mean of n observations 2x1 + 3, 2x2 + 3, 2x3 + 3, ...., 2xn + 3 is, (a) 3x + 2 (b) 2x + 3, (c) x + 3, (d) 2x, In computing a measure of the central tendency for any set, of 51 numbers, which one of the following measures is welldefined but uses only very few of the numbers of the set?, (a) Arithmetic mean, (b) Geometric mean, (c) Median, (d) Mode, A set of numbers consists of three 4’s, five 5’s, six 6’s, eight, 8’s and seven 10’s. The mode of this set of numbers is, (a) 6, (b) 7, (c) 8, (d) 10, , 11., , 12., , 13., , 14., , 15., , 16., 17., , 18., , Consider the following data which represents the runs, scored by two batsmen in their last ten matches as, Batsman A : 30, 91, 0, 64, 42, 80, 30, 5, 117, 71, Batsman B : 53, 46, 48, 50, 53, 53, 58, 60, 57, 52, Which of the following is/are true about the data?, I. Mean of batsman A runs is 53., II. Median of batsman A runs is 42., III. Mean of batsman B runs is 53., IV. Median of batsman B runs is 53., (a) Only I is true, (b) I and III are true, (c) I, III and IV are true (d) All are true, The value which represents the measure of central, tendency, is/are, (a) mean, (b) median, (c) mode, (d) All of these, The mean of 13 observations is 14. If the mean of the first, 7 observations is 12 and that of the last 7 observations is, 16, what is the value of the 7th observation ?, (a) 12, (b) 13, (c) 14, (d) 15, We can grouped data into ....... ways., (a) three, (b) four, (c) two, (d) None of these, The average of 5 quantities is 6, the average of three of, them is 4, then the average of remaining two numbers is :, (a) 9, (b) 6, (c) 10, (d) 5, The range of set of observations 2, 3, 5, 9, 8, 7, 6,5, 7, 4, 3 is, (a) 6, (b) 7, (c) 4, (d) 5, The mean of six numbers is 30. If one number is excluded,, the mean of the remaining numbers is 29. The excluded, number is, (a) 29, (b) 30, (c) 35, (d) 45, Mean of 100 items is 49. It was discovered that three items, which should have been 60, 70, 80 were wrongly read as, 40, 20, 50 respectively. The correct mean is, 1, (c) 50, (d) 80, 2, Mean of 20 observations is 15.5 Later it was found that the, observation 24 was misread as 42. The corrected mean is:, (a) 14.2 (b) 14.8, (c) 14.0, (d) 14.6, The mean of a set of 20 observation is 19.3. The mean is, reduced by 0.5 when a new observation is added to the set., The new observation is, (a) 19.8 (b) 8.8, (c) 9.5, (d) 30.8, The observations 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95, are arranged in ascending order. What is the value of x if, the median of the data is 63?, (a) 61, (b) 62, (c) 62.5, (d) 63, The median of a set of 9 distinct observation is 20.5. If, each of the largest 4 observations of the set is increased by, , (a) 48, 19., , 20., , 21., , 22., , (b) 82

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STATISTICS, , 187, , 2, then the median of the new set is, (a) increased by 2, (b) decreased by 2, (c) two times the original median, (d) remains the same as that of original set, 23. Given (i) 85 observations which are not sorted and (ii) 150, observations which are sorted and arranged in an increasing, order. The median values of (i) & (ii) respectively can be found, as, 43rd, , (a) (i), observation (ii) A.M. of, observation, , 75 th, , and 76th, , (b) (i) 43rd observation (ii) 76th observation, (c) (i) can not be found (ii) can not be found, (d) None of these, 24. If mean of the n observations x1, x2, x3,... xn be x , then the, mean of n observations 2x1 + 3, 2x2 + 3, 2x3 + 3, ...., 2xn + 3, is, (a) 3x + 2 (b), , 2x + 3, , (c), , x+3, , (d), , 25. If the mean of n observations 12, 22, 32,...., n2 is, , 2x, 46n, , then, 11, , n is equal to, (a) 11, (b) 12, (c) 23, (d) 22, 26. If the mean of four observations is 20 and when a constant, c is added to each observation, the mean becomes 22. The, value of c is :, (a) – 2, (b) 2, (c) 4, (d) 6, 27. The arithmetic mean of a set of observations is x . If each, observation is divided by a then it is increased by 10, then, the mean of the new series is:, x, x + 10, x + 10a, (b), (c), (d) a x + 10, a, a, a, 28. The average weight of students in a class of 35 students is, 40 kg. If the weight of the teacher be included, then average, , (a), , 1, kg, the weight of the teacher is, 2, (a) 40.5 kg (b) 50 kg (c) 41 kg, (d) 58 kg, , rises by, , 29. The mean of n items is X . If the first item is increased by, 1, second by 2 and so on, the new mean is :, (a), , X+, , x, 2, , (b) X + x, , n +1, (d) None of these, 2, 30. In a batch of 15 students, if the marks of 10 students who, passed are 70, 50, 95, 40, 60, 70, 80, 90, 75, 80 then the, median marks of all the 15 students is:, (a) 40, (b) 50, (c) 60, (d) 70, , (c) X +, , Topic 2 : Measures of Dispersion, Mean Deviation,, Variance and Standard Deviation, 31. The measure of dispersion is:, (a) mean deviation, (b) standard deviation, (c) quartile deviation, (d) all (a) (b) and (c), 32. The coefficient of variation is computed by:, (a), , mean, standard deviation, , (b), , standard deviation, mean, , (c), , mean, ×100, standard deviation, , (d), , standard deviation, ´100, mean, , 33. Find the mean deviation about the mean for the data, 4, 7, 8, 9, 10, 12, 13, 17, (a) 3, (b) 24, (c) 10, (d) 8, 34. Find the mean deviation about the mean for the data., xi, fi, , 5, 7, , 10, 4, , 15, 6, , 20, 3, , 25, 5, , (a) 6, (b) 7.3, (c) 8, (d) 6.32, 35. The mean and SD of 63 children on an arithmetic test are, respectively 27, 6 and 7.1. To them are added a new group, of 26 who had less training and whose mean is 19.2 and, SD 6.2. The values of the combined group differ from the, original as to (i) the mean and (ii) the SD is, (a) 25.1, 7.8, (b) 2.3, 0.8, (c) 1.5, 0.9, (d) None of these, 36 Find the mean and variance for the following data, 6, 7, 10, 12, 13, 4, 8, 12, (a) mean = 9, variance = 9.25, (b) mean = 3, variance = 7.5, (c) mean = 7, variance = 12, (d) mean = 9, variance = 12.5, 37. The method used in Statistics to find a representative value, for the given data is called, (a) measure of skewness, (b) measure of central tendency, (c) measure of dispersion, (d) None of these, 38. Consider the following data, Size, 20 21 22 23 24, Frequency 6 4 5 1 4, I., II., III., IV., (a), (c), , Mean of the data is 22.65., Mean deviation of the data is 1.25., Mean of the data is 21.65., Mean deviation of the data is 2.25., I and II are true, (b) II and III are true, I and IV are true, (d) III and IV are true

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EBD_7762, 188, , MATHEMATICS, , 39. The number which indicates variability of data or, observations, is called, (a) measure of central tendency, (b) mean, (c) median, (d) measure of dispersion, 40. Number which is mean of the squares of deviations from, mean, is called ...... ., (a) standard deviation, (b) variance, (c) median, (d) None of these, 41. The standard deviation of the wages of 85 employees is, ` 15.40. After one year each of them is given an increment, of ` 25. The standard deviation of new wages (in `) is, (a) 15.40 (b) 40.40, , (c) 20.40, , (d) 10.40, , 42. The variance of n observations x1, x2, ...., xn is given by, 1 n, å xi - x, n i =1, , (, , (a), , s2 =, , (c), , 1 n, s2 = å x i + x, n i =1, , (, , ), ), , (b), , s2 =, , 2, (d) s =, , 1 n, å xi - x, n i =1, , (, , n, , (, , 1, å xi + x, n i =1, , ), ), , 2, , 2, , 43. The measure of variability which is independent of units,, is called, (a) mean deviation, (b) variance, (c) standard deviation, (d) coefficient of variation, 44., , 45., , 1 n, 2, å ( xi - x ) be the S.D., n - 1 i =1, of a set of observations x1, x2, ....xn, then, n, n, (a) S £ r, (b) S = r, n -1, n -1, n, (c) S ³ r, (d) None of these, n -1, If X and Y are two variates connected by the relation, , Let r be the range and S 2 =, , aX + b, and Var (X) = s2, then write the expression, c, for the standard deviation of Y., a, a, s, (a), (b), (c) | a · c | (d) | a · c | s, c, c, 46. Consider the following data, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, I. The variance of the data is 33., II. The standard deviation of the data is 4.74., (a) Only Statement I is true, (b) Only Statement II is true, (c) Both statements are true, (d) Both statements are false, 47. Variance of the numbers 3, 7, 10,18, 22 is equal to, Y=, , (a) 12, , (b) 6.4, , (c), , 49.2, , (d) 49.2, , 48. The mean deviation from the mean of the following data :, Marks, , 0-10 10-20 20-30 30-40 40-50, , No. of Students, , 5, , 8, , 15, , 16, , 6, , is, , (a) 10, (b) 10.22, (c) 9.86, (d) 9.44, 49. The first of two samples has 100 items with mean 15 and, SD 3. If the whole group has 250 items with mean 15.6, and SD = 13.44 the SD of the second group is, (a) 5, (b) 4, (c) 6, (d) 3.52, 50. The mean deviation from the median of the following data is, Class interval 0-6 6-12 12-18 18-24 24-30, 4, 5, 3, 6, 2, Frequency, (a) 14, (b) 10, (c) 5, (d) 7, 51. Consider the following frequency distribution, x, , A 2A 3A 4A 5A 6A, , f, , 2, , 1, , 1, , 1, , 1, , 1, , where, A is a positive integer and has variance 160. Then, the value of A is., (a) 5, (b) 6, (c) 7, (d) 8, 52. Coefficient of variation of two distribution are 50% and, 60% and their standard deviation are 10 and 15,, respectively. Then, difference of their arithmetic means is, (a) 3, (b) 4, (c) 5, (d) 6, 53. Assertion : The mean deviation of the data 2, 9, 9, 3, 6, 9,, 4 from the mean is 2.57, Reason : For individual observation,, Mean deviation (X) =, , å xi - x, , n, (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 54 The mean of 5 observation is 4.4 and their variance is 8.24., If three of the observations are 1, 2 and 6, then difference, of the other two observations is, (a) 5, (b) 4, (c) 6, (d) 9, 55. Consider the following data., 36, 72, 46, 42, 60, 45, 53, 46, 51, 49, (a) 6, (b) 8, (c) 7, (d) None of these, 2, 56. Given N = 10, Sx = 60 and Sx = 1000. The standard deviation, is, (a) 6, (b) 7, (c) 8, (d) 9, 57. The standard deviation of 5 scores 1, 2, 3, 4, 5 is a . The, value of ‘a’ is, (a) 2, (b) 3, (c) 5, (d) 1

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STATISTICS, , 58. Let x1 , x2,...., xn be n observations, and let x be their, arithmetic mean and s2 be the variance., Assertion : Variance of 2x1, 2x2, ..., 2xn is 4s2., Reason : Sum of the deviations from mean ( x ) is 1., , 59., , 60., , 61., , 62., , (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., The variance of the data 2, 4, 6, 8, 10 is, (a) 8, (b) 7, (c) 6, (d) None of these, The mean deviation from the mean for the set of, observations –1, 0, 4 is, (a) 3, (b) 2, (c) 1, (d) None of these, The S. D of 15 items is 6 and if each item is decreased or, increased by 1, then standard deviation will be, (a) 5, (b) 6, (c) 7, (d) None of these, The mean and variance for first n natural numbers are, respectively, n +1, n2 - 1, , variance =, 2, 12, n -1, n2 + 1, (b) mean =, , variance =, 2, 12, 2, n, +1, n -1, (c) mean =, , variance =, 2, 12, , (a) mean =, , n -1, n2 + 1, , variance =, 2, 2, 63. Statement-I : The mean and variance for first n natural, , (d) mean =, , n +1, n2 + 1, and, , respectively.., 2, 12, Statement-II : The mean and variance for first 10 positive, multiples of 3 are 16.5 and 74.25. respectively., (a) Only Statement I is true, (b) Only Statement II is true, (c) Both statements are true, (d) Both statements are false, Find the mean and standard deviation for the following, data :, , numbers are, , 64., , xi, fi, , 6 10 14 18 24 28 30, 2 4 7 12 8 4 3, , (a) mean = 6.59, S.D = 19 (b) mean = 8, S.D = 19, (c) mean = 19, S.D = 6.59 (d) mean = 19, S.D = 6, 65. The mean deviation from the mean of the set of, observations – 1, 0 and 4 is, (a) 3, (b) 1, (c) – 2, (d) 2, , 189, , 66. Variance of the data 2, 4, 5, 6, 8, 17 is 23.33. Then, variance, of 4, 8, 10, 12, 16, 34 will be, (a) 23.33 (b) 25.33, (c) 93.32, (d) 98.32, 67. If n = 10, x = 12 and å x i2 = 1530 , then the coefficient of, variation is, (a) 35% (b) 42%, (c) 30%, (d) 25%, 68. The variance of 20 observations is 5. If each observation is, multiplied by 2, then the new variance of the resulting, observation is, (a) 23 × 5 (b) 22 × 5, (c) 2 × 5, (d) 24 × 5, 69. For two data sets, each of size 5, the variances are given to, be 4 and 5 and the corresponding means are given to be 2, and 4, respectively. The variance of the combined data set is, 11, 13, 5, (b) 6, (c), (d), 2, 2, 2, 70. All the students of a class performed poorly in Mathematics., The teacher decided to give grace marks of 10 to each of, the students. Which of the following statistical measures, will not change even after the grace marks were given ?, (a) mean (b) median, (c) mode, (d) variance, 71. The coefficient of variation from the given data, Class interval 0-10 10-20 20-30 30-40, 40-50, Frequency, 2, 10, 8, 4, 6, is :, (a) 50, (b) 51.9, (c) 48, (d) 51.8, 72. Coefficient of variation of two distribution are 60 and 70,, and their standard deviations are 21 and 16, respectively., What are their arithmetic means?, (a) 35, 22.85, (b) 22.85, 35.28, (c) 36, 22.85, (d) 35.28, 23.85, BEYOND NCERT, , (a), , Topic 3 : Quartile, Quartile Deviation, 73. Which of the following is/are used for the measures of, dispersion?, (a) Range, (b) Quartile deviation, (c) Standard deviation, (d) All of these, 74. 25% of the items of a data are less than 35 and 25% of the, items are more than 75. Quartile Deviation of the data is, (a) 55, (b) 20, (c) 35, (d) 75, 75. The quartile deviation of the following items :, 12, 7, 15, 10, 16, 17, 25 is, (a) 4.5, (b) 13.5, (c) 9, (d) 3.5, 76. Consider the following statements :, I. Measures of dispersion Range, Quartile deviation,, mean deviation, variance, standard deviation are, measures of dispersion, Range = Maximum value – minimum values, II. Mean deviation for ungrouped data, M.D. ( x ) =, , Ρ | xi – x |, n, , M.D. (M) =, , Ρ | xi – M |, n

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EBD_7762, 190, , MATHEMATICS, , III. Mean deviation for grouped data, S fi | xi – x |, M.D. ( x ) =, N, Ρ fi | xi – M |, M.D. (M) =, N, where N = Ρ fi, , 77., , Which of the above statements are true?, (a) Only (I), (b), Only (II), (c) Only (III), (d) All of the abvoe, Which of the following, in case of a discrete data, is not, equal to the median?, (a) 50th percentile, (b) 5th decile, (c) 2nd quartile, (d) Lower quartile, , NCERT Exemplar MCQs, 1., 2., , The mean deviation of the data 3, 10, 10, 4, 7, 10, 5 from the, mean is, (a) 2, (b) 2.57, (c) 3, (d) 3.75, Mean deviation for h observations x1, x2, ..., xn from their, mean x is given by, , å ( xi – x ), , 9., , (d), , 1 n, å ( xi – x )2, n i =1, , 10., , n, , å ( xi – x )2, , i =1, , 3., , 4., , 5., , When tested, the lives (in hours) of 5 bulbs were noted as, follows, 1357, 1090, 1666, 1494, 1623, The mean deviations (in hours) from their mean is, (a) 178, (b) 179, (c) 220, (d) 356, Following are the marks obtained by 9 students in a, mathematics test, 50, 69, 20, 33, 52, 39, 40, 65, 59, The mean deviation from the median is, (a) 9, (b) 10.5, (c) 12.67, (d) 14.76, The standard deviation of data 6, 5, 9, 13, 12, 8 and 10 is, 52, 52, (b), (c), (d) 6, 6, 7, 7, If x1, x2, ..., xn be n observations and x be their arithmetic, mean. Then, formula for the standard deviation is given, by, , (a), , 6., , (a), (c), , å ( xi – x ), , 2, , å( xi – x )2, n, , 8., , 1 n, å | xi – x |, n i =1, , i =1, , (c), , 7., , (b), , n, , (a), , 78. For a symmetrical distribution Q1 = 25 and Q3 = 45, the, median is, (a) 20, (b) 25, (c) 35, (d) None of these, 79. If 25% of the item are less than 20 and 25% are more than, 40, the quartile deviation is, (a) 20, (b) 30, (c) 40, (d) 10, 80. For a series the value of mean deviation is 15. The most, likely value of its quartile deviation is, (a) 12.5 (b) 11.6, (c) 13, (d) 9.7, , (b), (d), , å ( xi – x )2, n, å ( x 2i, +x –2, n, , 11., 12., , 13., , 14., 15., , The mean of 100 observations is 50 and their standard, deviation is 5. The sum of squares of all observation is, (a) 50000 (b) 250000 (c) 252500 (d) 255000, Let a, b, c, d and e be the observations with mean m and, standard deviation s. The standard deviation of the, observations a + k, b + k, c + k, d + k and e + k is, (a) s, (b) ks, (c) s + k, (d) s/k, Let x1, x2, x3, x4 and x5 be the observations with mean m and, standard deviations. Then, standard deviation of the, observations kx1, kx2, kx3, kx4 and kx5 is, (a) k + s (b) s / k, (c) ks, (d) s, Let x1, x2, ... xn be n observations. Let wi = lxi + k for i = 1, 2,, ..., n, where l and k are constants. If the mean of xi's is 48 and, their standard deviation is 12, the mean of wi's is 55 and, standard deviation of wi's is 15, then the value of l and k, should be, (a) l = 1.25, k = – 5, (b) l = – 1.25, k = 5, (c) l = 2.5, k = – 5, (d) l = 2.5, k = 5, Standard deviation for first 10 natural numbers is, (a) 5.5, (b) 3.87, (c) 2.97, (d) 2.87, Consider the following data, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, If 1 is added to each number, then variance of the numbers, so obtained is, (a) 6.5, (b) 2.87, (c) 3.87, (d) 8.25, Consider the first 10 positive integers. If we multiply each, number by – 1 and then add 1 to each number, the variance, of the numbers so obtained is, (a) 8.25, (b) 6.5, (c) 3.87, (d) 2.87, The following information relates to a sample of size 60, åx2, = 18000, and åx = 960. Then, the variance is, (a) 6.63, (b) 16, (c) 22, (d) 44, Coefficient of variation of two distributions are 50 and 60, and their arithmetic means are 30 and 25, respectively., Then, difference of their standard deviations is, (a) 0, (b) 1, (c) 1.5, (d) 2.5

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STATISTICS, , 191, , 16. The standard deviation of some temperature data in °C is, 5. If the data were converted into °F, then the variance would, be, (a) 81, (b) 57, (c) 36, (d) 25, 17., , 18., 19., , 20., , Past Year MCQs, The variance of first 50 even natural numbers is, [JEE MAIN 2014, C], 437, 833, (a) 437 (b), (c), (d) 833, 4, 4, If M. D. is 12, the value of S.D. will be [BITSAT 2014, C, BN], (a) 15, (b) 12, (c) 24, (d) None of these, The mean of the data set comprising of 16 observations is 16., If one of the observation valued 16 is deleted and three new, observations valued 3, 4 and 5 are added to the data, then the, mean of the resultant data, is:, [JEE MAIN 2015, A], (a) 15.8 (b) 14.0, (c) 16.8, (d) 16.0, The mean square deviation of a set of n observation x 1, x2,, 1 n, 2, .... xn about a point c is defined as n å ( xi - c ) ., i =1, , The mean square deviations about – 2 and 2 are 18 and 10, respectively, the standard deviation of this set of, observations is, [BITSAT 2015, A], (a) 3, (b) 2, (c) 1, (d) None of these, 21. The arithmetic mean of the data 0, 1, 2, ......, n with frequencies, 1, nC1, nC2,....., nCn is, [BITSAT 2015, S], , (a) 3a2 – 34a + 91 = 0, (b) 3a2 – 23a + 44 = 0, 2, (c) 3a – 26a + 55 = 0, (d) 3a2 – 32a + 84 = 0, 23. The arithmetic mean of numbers a, b, c, d, e is M. What is, the value of (a – M) + (b – M) + (c – M) + (d – M) + (e – M)?, [BITSAT 2016, C], (a) M, (b) a + b + c + d + e, (c) 0, (d) 5 M, 24. The marks obtained by 60 students in a certain test are, given below :, No. of, No. of, Marks, Marks, students, students, 10 - 20, 2, 60 - 70, 12, 20 - 30, 3, 70 - 80, 14, 30 - 40, 4, 80 - 90, 10, 40 - 50, 5, 90 - 100, 4, 50 - 60, 6, Median of the above data is, [BITSAT 2017, A], (a) 68.33, (b) 70, (c) 68.11, (d) None of these, 9, , 25. If, , i =1, , 9, , å (x i - 5)2 = 45 , then the standard, i =1, , deviation of the 9 items x1, x2, ..., x9 is :, [JEE MAIN 2018, A], (a) 4, (b) 2, (c) 3, (d) 9, 26. Number of solutions of the equation, , n, 2n, (c) n + 1, (d), 2, n, 22. If the standard deviation of the numbers 2, 3, a and 11 is 3.5,, then which of the following is true?[JEE MAIN 2016, C], , (a) n, , å (xi - 5) = 9 and, , tan -1 (1 + x ) + tan -1 (1 - x ) =, , (b), , (a) 3, , (b) 2, , p, are, 2, (c) 1, , [BITSAT 2018, A], (d) 0, , Exercise 3 : Try If You Can, 1., , 2., , In a set of 2n distinct observations, each of the observations, below the median of all the observations is increased by 5, and each of the remaining observations is decreased by 3., Then the mean of the new set of observations:, (a) increases by 1, , (b) decreases by 1, , (c) decreases by 2, , (d) increases by 2, , The frequency distribution of daily working expenditure of, families in a locality is as follows:, Expenditure 0-50 50-100 100-150 150-200 200-250, in ` (x ):, 24, 33, 37, 25, No. of, b, families (f ):, , If the mode of the distribution is ` 140, then the value of b, is, , 3., , 4., , 5., , (a) 34, (b) 31, (c) 26, (d) 36, The mean of the numbers a, b, 8, 5, 10 is 6 and the variance, is 6.80. Then which one of the following gives possible, values of a and b?, (a) a = 0, b = 7, (b) a = 5, b = 2, (c) a = 1, b = 6, (d) a = 3, b = 4, The mean of five numbers is 0 and their variance is 2. If, three of those numbers are –1, 1 and 2, then the other two, numbers are, (a) –5 and 3, (b) –4 and 2, (c) –3 and 1, (d) –2 and 0, If the median and the range of four numbers, {x, y, 2x + y, x – y}, where 0 < y < x < 2y, are 10, and 28 respectively, then the mean of the numbers is :, (a) 18, (b) 10, (c) 5, (d) 14

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EBD_7762, 192, , 6., , MATHEMATICS, , The marks obtained by 60 students in a certain test are, given below :, No. of, No. of, Marks, Marks, students, students, 10 - 20, 2, 60 - 70, 12, 20 - 30, 3, 70 - 80, 14, 30 - 40, 4, 80 - 90, 10, 40 - 50, 5, 90 - 100, 4, 50 - 60, 6, Mean, Median and Mode of the above data are respectively, (a) 64.33, 68.33, 76.33, (b) 60, 70, 80, (c) 66.11, 71.11, 79.11, (d) None of these, Consider the following frequency distribution, , 7., , x, , A 2A 3A 4A 5A 6A, , f, , 2, , 1, , 1, , 1, , 1, , 1, , where, A is a positive integer and has variance 160. Then, the value of A is., (a) 5, (b) 6, (c) 7, (d) 8, Suppose a population A has 100 observations 101, 102,, ............., 200 and another population B has 100 observations, 151, 152, ................ 250. If VA and VB represent the variances, , 8., , of the two populations, respectively then, , VA, is, VB, , 4, 9, (d), (c), 9, 4, The harmonic mean of the following distribution, , (a) 1, 9., , (b), , (a), , (b), , 10 + 20, , 10 + 10, (c), (d) None of these, 10, 11. The mean of five observations is 4 and their variance is, 5×2. If three of these observations are 2, 4 and 6, then the, other two observations are, (a) 3 and 5, (b) 2 and 6, (c) 5 and 8, (d) 1 and 7, 12. In an experiment with 15 observations on x, the following, results were available:, , å x2 = 2830, å x = 170, , One observation that was 20 was found to be wrong and, was replaced by the correct value 30. The corrected, variance is, (a) 8.33, (b) 78.00, (c) 188.66 (d) 177.33, 13. Coefficient of variation of two distribution are 60 and 70,, and their standard deviations are 21 and 16, respectively., What are their arithmetic means?, (a) 35, 22.85, (b) 22.85, 35.28, (c) 36, 22.85, (d) 35.28, 23.85, 14. If the mean deviation of the numbers 1, 1 + d, 1 + 2d, ...., 1 + 100d from their mean is 255, then d is equal to :, (a) 20.0, (b) 10.1, (c) 20.2, (d) 10.0, 15. Following are the marks obtained, out of 100 by two students, Raju and Sita in 10 tests., Raju 25 50 45 30 70 42 36 48 35 60, , 2, 3, , C.I., 4.5-5.5 5.5-6.5 6.5-7.5 7.5-8.5 8.5-9.5, is, f, 8, 10, 18, 6, 4, (a) 7, (b) 6.54, (c) 6.32, (d) 6.89, 10. If the standard deviation of the observations –5, –4, –3,, , Sita, I., II., III., IV., (a), (c), , 10 70 50 20 95 55 42 60 48 80, , Raju is more intelligent., Sita is more intelligent., Raju is more consistent., Sita is more consistent., I and IV are true, (b) II and III are true, I and III are true, (d) II and IV are true, , –2, –1, 0, 1, 2, 3, 4, 5 is 10. The standard deviation of, observations 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 will be, , 1, 2, 3, 4, 5, 6, 7, 8, , (a), (c), (c), (c), (c), (b), (b), (b), , 9, 10, 11, 12, 13, 14, 15, 16, , (d), (c), (c), (d), (c), (c), (a), (b), , 17, 18, 19, 20, 21, 22, 23, 24, , (c), (c), (d), (b), (b), (d), (d), (b), , 1, 2, 3, , (b), (b), (a), , 4, 5, 6, , (c), (a), (c), , 7, 8, 9, , (c), (a), (c), , 1, 2, , (a), (d), , 3, 4, , (d), (d), , 5, 6, , (d), (a), , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (a) 33, (a), (a) 49, 25, 41, (d), (b) 50, 26 (b) 34, 42, (c) 35, (a), (d) 51, 27, 43, (a) 52, 28 (d) 36 (a), 44, (c) 37, (b), (b) 53, 29, 45, (c) 38, (b), (a) 54, 30, 46, (d), (d) 55, 31 (d) 39, 47, (b), (d) 56, 32 (d) 40, 48, Exercise 2 : Exemplar & Past Year MCQs, (a) 13, (a), (a) 19, 10, 16, (d), (d) 20, 11 (d) 14, 17, (a), (a) 21, 12 (d) 15, 18, Exercise 3 : Try If You Can, (c), (b), (d) 13, 7, 9, 11, (a) 10, (c), (b) 14, 8, 12, , (b), (d), (c), (c), (a), (a), (c), (c), , 57, 58, 59, 60, 61, 62, 63, 64, , (a), (c), (a), (b), (b), (a), (b), (c), , 65, 66, 67, 68, 69, 70, 71, 72, , (d), (c), (d), (b), (a), (d), (c), (a), , (b), (a), (d), , 22, 23, 24, , (d), (c), (a), , 25, 26, , (b), (c), , (a), (b), , 15, , (b), , 73, 74, 75, 76, 77, 78, 79, 80, , (d), (b), (d), (d), (d), (c), (d), (a)

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16, , PROBABILITY-1, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 3, , 2, , JEE MAIN, BITSAT, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 3, 2, , Critical Concepts, , Random Experiment, Mutually, Exclusive Event, Exhaustive Event,, Probability of the event 'A or B',, Addition Theorem, , Rating of, Difficulty Level, , CUS, (chapter utility score), Out of 10, , 4/5, , 7.9

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PROBABILITY-1, , 195

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EBD_7762, 196, , MATHEMATICS, , Topic 1 : Events, Types of Events, Algebra of Events,, Probability of an Event., 1., , 2., , In a simultaneous throw of 2 coins, the probability of having, 2 heads is:, 1, 1, 1, 1, (a), (b), (c), (d), 8, 6, 4, 2, If 10 objects are distributed at random among 10 persons,, then the probability that at least one of them will not get, anything., 1010 - 10!, 1010 - 1, (b) 1010 (c) 1, (d), 10, 10, 1010, If the letters of the word ATTEMPT are written down at, random, the chance that all T's are consecutive, is, 6, 1, (a), (b), 7, 42, , (a), 3., , 1, (d) None of these, 7, In a game called 'odd man out', m (m > 2) persons toss a, coin to determine who will buy refreshments for the entire, group. A person who gets an outcome different from that of, the rest of the members of the group is called the odd man, out. The probability that there is a loser in any game is, (a) 1/2m, (b) m/2m – 1, (c) 2/m, (d) None of these, The probability of getting sum more than 7 when a pair of, dice are thrown is:, 7, 5, (a), (b), 36, 12, 7, (c), (d) None of these, 12, The probability of raining on day 1 is 0.2 and on day 2 is, 0.3. The probability of raining on both the days is, (a) 0.2, (b) 0.1, (c) 0.06, (d) 0.25, In four schools B1, B2, B3 , B4 the percentage of girls, students is 12, 20, 13, 17 respectively. From a school, selected at random, one student is picked up at random, and it is found that the student is a girl. The probability, that the school selected is B2, is, , (c), , 4., , 5., , 6., 7., , 10, 13, 17, 6, (b), (c), (d), 31, 62, 62, 31, Find the probability of getting the sum as a perfect square, number when two dice are thrown together., (a) 5/12, (b) 7/18, (c) 7/36, (d) None of these, 10 apples are distributed at random among 6 persons. The, probability that at least one of them will receive none, is, , (a), , 9., , (b), , 14, , C4, , 15, , C5, , 137, (d) None of these, 143, A bag contains 10 balls, out of which 4 balls are white, and the others are non-white. The probability of getting a, non-white ball is, , (c), , 10., , 2, 3, 1, 2, (b), (c), (d), 5, 5, 2, 3, The dice are thrown together. The probability of getting, the sum of digits as a multiple of 4 is:, , (a), 11., , 1, 1, 1, 5, (b), (c), (d), 9, 3, 4, 9, 12. A die is loaded so that the probability of a face i is, proportional to i, i = 1, 2, ......, 6. Then the probability of, an even number occurring when the die is rolled., , (a), , (a), , 2, 7, , (b), , 1, 7, , (c), , 3, 7, , (d), , 4, 7, , 13. The probability that in the toss of two dice we obtain the, sum 7 or 11 is :, (a), , 1, 6, , (b), , 1, 18, , (c), , 2, 9, , (d), , 23, 108, , 2, is the probability of an event, then the probability, 11, of the event 'not A', is, , 14. If, , 11, 9, 11, 2, (b), (c), (d), 9, 11, 2, 11, 15. An experiment is called random experiment, if it, (a) has more than one possible outcome, (b) is not possible to predict the outcome in advance, (c) Both (a) and (b), (d) None of the above, 16. Seven people seat themselves indiscriminately at round table., The probability that two distinguished persons will be next, to each other is, , (a), , (a), 8., , 6, 143, , (a), , 1, 3, , (b), , 1, 2, , (c), , 1, 4, , (d), , 2, 3, , 17. Cards are drawn one-by-one at random from a well-shuffled, pack of 52 playing cards until 2 aces are obtained from the, first time. The probability that 18 draws are required for, this is, (a) 3/34, (b) 17/455, (c) 561/15925, (d) None of these

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PROBABILITY-1, , 197, , 18. An event can be classified into various types on the basis, of the, (a) experiment, (b) sample space, (c) elements, (d) None of the above, 19. An event which has only ...... sample point of a sample, space, is called simple event., (a) two, (b) three, (c) one, (d) zero, 20. Four persons are selected at random out of 3 men, 2 women, and 4 children. Find the probability that there are exactly, 2 children in the selection., 11, 8, (b), 21, 21, 21. If an event has more than, called a/an, (a) simple event, (c) compound event, 22. A letter is chosen at, ‘ASSASSINATION’., , (a), , 10, 7, (d), 21, 21, one sample point, then it is, , (c), , (b) elementary event, (d) None of these, random from the word, 6, ., 13, , I., , The probability that letter is a vowel is, , II., , The probability that letter is a consonant is, , 7, ., 13, , (a) Only I is correct., (b) Both I and II are correct., (c) Only II is correct., (d) Both are incorrect., 23. The probability that in the random arrangement of the, letters of the word ‘UNIVERSITY’, the two I’s does not, come together is, (a), , 4, 5, , (b) 1/ 5, , (c) 1/10, , (d) 9/10, , 24. When three cards are drawn at random from a well shuffled, pack of cards, then what is the probability that all of them, are from the different suits?, 13, , (a), , (c), , 52, , C3, , (b), , C3, , 3×13 C3 13C1 13C1, 52, , C3, , 4 × (13 C1 ×13 C1 13C1 ), 52, , C3, , (d) None of these, , 25. Let A be a set containing n elements. A subset P of the set, A is chosen at random. The set A is reconstructed by, replacing the elements of P, and another subset Q of A is, chosen at random. The probability that P Ç Q contains, exactly m (m < n) elements is, (a), , (c), , 3n – m, 4n, n, , C m 3n – m, 4n, , (b), , n, , Cm 3 m, 4n, , (d) None of these, , 26. Two dice are thrown simultaneously. The probability of, 1, . The value of ‘m’ is, m, (a) 3, (b) 2, (c) 6, (d) 9, 27. A coin is tossed 3 times, the probability of getting exactly, , obtaining a total score of seven is, , m, . The value of ‘m’ is, 8, , two heads is, , (a) 1, (b) 2, (c) 3, (d) 4, 28. The probability that the two digit number formed by digits, 1, 2, 3, 4, 5 is divisible by 4 is, (a), , 1, 30, , (b), , 1, 20, , 1, (d) None of these, 5, If 12 identical balls are to be placed in 3 identical boxes, then, the probability that one of the boxes contains exactly 3 balls is, , (c), 29., , (a), , æ 1ö, 220 ç ÷, è 3ø, , (c), , 55 æ 2 ö, ç ÷, 3 è 3ø, , 12, , 11, , æ 1ö, 22 ç ÷, è 3ø, , 11, , æ 2ö, (d) 55 ç ÷, è 3ø, , 10, , (b), , 30. In a knock out chess tournament, eight players P1, P2, …P8, participated. It is known that whenever the players Pi and, Pj play, the players Pi will win j if i < j. Assuming that the, players are paired at random in each round, what is the, probability that the player P4 reaches the final?, (a) 31/35, (b) 4/35, (c) 8/35, (d) None of these, 31. In a simultaneous toss of two coins, the probability of, getting exactly 2 tails is, (a) 1, (b) 4, 32. A die is thrown., , m, . The value of m + n is, n, (c) 5, (d) 2, , 1, ., 2, II. The probability of a number more than 6 will appear is 1., (a) Only I is correct., (b) Only II is correct., (c) Both I and II are correct., (d) Both I and II are incorrect., 33. If three distinct numbers are chosen randomly from the, first 100 natural numbers, then the probability that all, three of them are divisible by both 2 and 3 is, (a) 4 /25 (b) 4/35, (c) 4/33, (d) 4/1155, 34. Fifteen coupons are numbered 1, 2, 3, ......, 15. Seven, coupons are selected at random one at a time with, replacement. The probability that the largest number, appearing on selected coupon is 9 is, (a) (9/16)6, (b) (8/15)7, (c) (3/5)7, (d) None of these, , I., , The probability of a prime number will appear is

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EBD_7762, 198, , 35., , 36., 37., , 38., , 39., , MATHEMATICS, , Five different games are to be distributed among 4 children, randomly. The probability that each child get at least one, game is, (a) 1/4, (b) 15/64, (c) 21/64, (d) None of these, A die is thrown. The probability of getting a number less, than or equal to 6 is, (a) 6, (b) 1, (c) 2, (d) 5, In a school there are 40% science students and the, remaining 60% are arts students. It is known that 5% of, the science students are girls and 10% of the arts students, are girls. One student selected at random is a girl. What is, the probability that she is an arts student?, 3, 1, 3, 1, (a), (b), (c), (d), 4, 5, 5, 3, A card is selected from a pack of 52 cards., I., , The probability that card is an ace of spades, is, , II., , The probability that the card is black card, is, , (c) 1 -, , (n - 1)!, n, , (b), , n -1, , 1, ., 2, , (n - 1)!, , 41., , 43., , 2, , 2, , 1 3n, 1 3n, ., . 6 n (d), 2 C3, 2 6 n C3, C3, C3, A bag contains 10 balls, out of which 4 balls are white and, the others are non-white. The probability of getting a, non-white ball is, , (b), , 3n, , 6n, , (c), , 2, 3, 1, 2, (a), (b), (c), (d), 5, 5, 2, 3, In a leap year the probability of having 53 Sundays or 53, Mondays is, , 2, 4, 5, 3, (b), (c), (d), 7, 7, 7, 7, A fair die is thrown once. The probability of getting a, composite number less than 5 is, , (a), 44., , (a), , 1, 3, , (b), , 14, 15, , (c), , 1, 5, , (d), , 4, 5, , 1, 3, , (b), , 1, 6, , (c), , 1, 2, , (d), , 1, 4, , 47. 10 different books and 2 different pens are given to 3 boys, so that each gets equal number of things. The probability, that the same boy does not receive both the pens, is, (a), , 5, 11, , (b), , 7, 11, , (c), , 2, 3, , (d), , 6, 11, , 2, 4, 1, 4, (b), (c), (d), 15, 15, 15, 90, 49. Three identical dice are rolled. The probability that the, same number will appear on each of them is:, , (a), , 1, 1, 1, 3, (b), (c), (d), 6, 36, 18, 28, The number 1, 2, 3, ......., n are arranged in a random, order. The probabiltiy that the digits 1, 2, 3, .....k (k < n), appears as neighbours in that order is, , (a), , 1, 1, 1, (a), (b), (c), (d) None of these, 3, 4, 2, If 6n tickets numbered 0, 1, 2, ....., (6n – 1) are placed in a, bag and three are drawn at random, then the probability, that the sum of numbers on the ticket is 6n, is, , 42., , (a), , (n - 1)!, , 3n, , (b), , 46. There are four machines and it is known that exactly two, of them are faulty. They are tested, one by one, in a random, order till both the faulty machines are identified. Then, the probability that only two testes are needed is, , nn, , 40., , 6n, , 1, 15, , 48. The probability that a two digit number selected at random, will be a multiple of ‘3’ and not a multiple of ‘5’ is, , (d) None of these, nn, Let x = 33n. The index n is given a positive integral value, at random. The probability that the value of x will have 3, in the unit's place, is, , (a), , (a), , 2, ., 52, , (a) Only I is false., (b) Only II is false., (c) Both I and II are false. (d) Both I and II are true., If n objects are distributed at random among n persons, the, probability that at least one of them will not get anything is, (a) 1 -, , 45. Two numbers are selected randomly from the set S = {1, 2,, 3, 4, 5, 6} without replacement one by one. The probability, that minimum of the two numbers is less than 4 is, , 1, 6, , (c), , 2, 3, , (d) 0, , 50., , 51., , (a), , 1, n!, , (b), , k!, n!, , (c), , (n – k )!, n!, , (d), , (n – k +1)!, n!, , x 1, x 2 , x3 , ......., x50 are fifty real numbers such that, xr < xr +1 for r = 1, 2, 3, ...., 49. Five numbers out of these, are picked up at random. The probability that the five, numbers have x20 as the middle numbers, is, 20, , (a), , 50, 19, , (c), , C2 ´ 30 C2, C5, , C2 ´ 31C2, 50, , C5, , 30, , (b), , C2 ´ 19 C2, 50, , C5, , (d) None of these, , 52. A coin is tossed repeatedly until a tail comes up for the, first time. Then, the sample space for this experiment is, (a) {T, HT, HTT}, (b) {TT, TTT, HTT, THH}, (c) {T, HT, HHT, HHHT, HHHHT, ...}, (d) None of the above

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PROBABILITY-1, , 199, , 53. The probability that a randomly chosen two-digit positive, integer is a multiple of 3, is, 1, 1, 1, 1, (b), (c), (d), 3, 5, 2, 4, 54. In a convex hexagon two diagonals are drawn at random., The probability that the diagonals intersect at an interior, point of the hexagon, is, , (a), , (a), , 5, 12, , (b), , 7, 12, , 2, (d) None of these, 5, 55. The chance that the vowels are separated in an arrangement, of the letters of the word "HORROR" is, (a) 1/2, (b) 2/3, (c) 3/4, (d) None of these, 56. A coin is tossed twice. Then, the probability that atleast, one tail occurs is, , (c), , 1, 1, 1, 3, (b), (c), (d), 3, 4, 2, 4, 57. In a leap year, the probability of having 53 Sundays or 53, Mondays is, , (a), , (a), , 2, 7, , (b), , 3, 7, , (c), , 4, 7, , (d), , 5, 7, , Topic 2 : Mutually Exclusive and Mutually Exhaustive, Events, Venn Diagrams, Addition Theorem of Probability., 1+ 4p 1- p, 1- 2p, ,, and, are the probabilities of three, 4, 2, 2, mutually exclusive events, then value of p is, , 58. If, , 1, 1, 1, (c), (b), 3, 2, 4, 59. If A and B are two events, such that, , (a), , (d), , 2, 3, , 3, 1, 2, , P(A Ç B) = , P(Ac ) =, 4, 4, 3, where Ac stands for the complementary event of A, then, P(B) is given by:, P(A È B) =, , 1, 2, 1, 2, (b), (c), (d), 3, 3, 9, 9, 60. In the following Venn diagram circles A and B represent, two events:, B, A, , (a), , The probability of the union of shaded region will be, (a) P(A) + P(B) – 2P(A Ç B), (b) P(A) + P(B) – P(A Ç B), (c) P(A) + P(B), (d) 2P(A) +2 P(B) – P(A Ç B), , 61. For the three events A, B and C, P (exactly one of the events, A or B occurs) = P (exactly one of the two events B or C, occurs) = P (exactly one of the events C or A occurs) = p, and P (all the three events occur simultaneously) = p2,, where 0 < p < 1/2. Then the probability of at least one of, the three events A, B and C occurring is, (a), , 3 p + 2 p2, 2, , (b), , p + 3 p2, 4, , p + 3 p2, 3 p + 2 p2, (d), 2, 4, 62. The probability of choosing at random a number that is, divisible by 6 or 8 from among 1 to 90 is equal to :, (c), , (a), , 1, 6, , (b), , 1, 30, , (c), , 11, 80, , (d), , 23, 90, , 63. A die is rolled. Let E be the event “die shows 4” and F be, the event “die shows even number”, Then, E and F are, (a) mutually exclusive, (b) exhaustive, (c) mutually exclusive and exhaustive, (d) None of the above, 64. If A and B are two events, then which of the following is, true?, (a), , P ( A È B) = P ( A ) + P ( B), , (b), , P ( A È B) = P ( A ) + P ( B ) - å P ( wi ), "wi Î A Ç B, , (c), , P ( A È B) = P ( A ) + P ( B ) - P ( A Ç B ), , (d) Both (b) and (c), 65. A bag contains 8 red and 7 black balls. Two balls are drawn, at random. The probability that both the balls are of the, same colour is :, (a), , 14, 15, , (b), , 11, 15, , (c), , 7, 15, , (d), , 4, 15, , 66. The probability that a company executive will travel by, 2, 1, and that he will travel by plane is . The prob3, 5, ability of his journey by train or plane is, , train is, , 2, 13, 15, 15, (b), (c), (d), 15, 15, 13, 2, 67. A die is rolled, let E be the event “die shows 4” and F, be the event “die shows even number”. Then, I. E and F are mutually exclusive., II. E and F are not mutually exclusive., (a) Only I is true., (b) Only II is true., (c) Neither I nor II is true.(d) Both I and II are true., 68. A die is thrown. Let A be the event that the number obtained, is greater than 3. Let B be the event that the number, obtained is less than 5. Then P(AÈB) is, , (a), , (a), , 3, 5, , (b) 0, , (c) 1, , (d), , 2, 5

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EBD_7762, 200, , 69., , 70., , 71., , MATHEMATICS, , The probability that a police inspector Ravi will catch a, thief in a day is 1/4 and the probability he will catch a, robber in that day is 1/5 and the probability that he will, catch both a thief and a robber in a day is 1/15 then what is, the probability that Ravi will catch at least 1 mischief?, (a) 23/60, (b) 19/60, (c) 7/20, (d) None of these, If A and B are mutually exclusive events and if P(B) =, P(A È B) =, , 13, , then P(A) is equal to, 21, , (a) 1/7, , (b) 4/7, , (c) 2/7, , 1, ,, 3, , (d) 5/7, , The probability that a card drawn from a pack of 52 cards, will be a diamond or king is:, 4, 1, 2, 1, (b), (c), (d), 13, 13, 13, 52, Events A, B, C are mutually exclusive events such, , (a), 72., , (b) Only II and III are correct., (c) Only I and III are true., (d) All three statements are correct., 76. If A, B and C are three mutually exclusive and exhaustive, events of an experiment such that 3P(A) = 2P(B) = P(C),, then P(A) is equal to ..., 1, 2, 5, 6, (b), (c), (d), 11, 11, 11, 11, 77. Two events A and B have probabilities 0.25 and 0.50, respectively. The probability that both A and B occur, simultaneously is 0.14. Then the probability that neither A, nor B occurs is, (a) 0.39, (b) 0.25, (c) 0.11, (d) None of these, 78. If P(A) = 1/4, P(B) = 2/5 then find the range of P(A È B), (a) (1/5, 13/20), (b) (1/4, 13/20), (c) (2/5, 13/20), (d) None of these, 79. The probabilities of three event A, B, and C are P(A) = 0.6,, , (a), , P(B) = 0.4, and P(C) = 0.5. If P ( A È B ) = 0.8,, P ( A Ç C ) = 0.3, P ( A Ç B Ç C ) = 0.2, and, , 1 - 2x, 3x + 1, 1- x, , P( B) =, that P ( A) =, and P (C ) =, 2, 3, 4, The set of possible values of x are in the interval is, é1 1 ù, , (a) [0 , 1] (b) ê 3 , 2 ú, û, ë, 73., , (c), , é1 2 ù, ê3 , 3ú, û, ë, , 80. If, P( B) =, , 3, 1, , P(A Ç B Ç C ) =, 4, 3, , 1, and P( A Ç B Ç C ) = , then P (B Ç C) is, 3, , 4 2, 1, Assertion : P(A È B) = = and P(A Ç B) =, 6 3, 6, , (a), , 2, 1, =, 3, 3, , 1, 52, , (b), , 2, 13, , (c), , 4, 13, , (d), , 1, 12, , (b), , 1, 1, (c), 6, 15, BEYOND NCERT, , (d), , 1, 9, , Topic 3 : Odds Against and odd in Favour of an event,, Boole’s Inequality, Demorgans Law., , (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., The probability that a card drawn from a pack of 52 cards, will be a diamond or king is, (a), , 75., , é 1 13 ù, ê3 , 3 ú, û, ë, , Consider a single throw of die and two events., A = the number is even = {2, 4, 6}, B = the number is a multiple of 3 = {3, 6}, , Reason : P(A Ç B) = P(AÈ B) = 1 –, , 74., , (d), , P ( A È B È C ) ³ 0.85, then the range of P ( B Ç C ) is, (a) (0.2, 0.35), (b) [0.2, 0.35], (c) (0.2, 0.35], (d) [0.2, 0.35], , 1, 13, , If A and B are events such that P(A) = 0.42, P(B) = 0.48, and P(A and B) = 0.16. then,, I. P(not A) = 0.58, II. P(not B) = 0.52, III. P(A or B) = 0.47, (a) Only I and II are correct., , 81. A card is drawn from a pack of 52 cards. A gambler bets, that it is a spade or an ace. What are the odds against his, winning this bet?, (a) 17 : 52 (b) 52 : 17, (c) 9 : 4, (d) 4 : 9, 1, 82. Let A and B are any two events such that P ( A) = and, 2, 1, c, c, c, c, P ( B ) = then the value of P ( A Ç B ) + P ( A È B c )c, 3, has the value equal to, 4, 3, (c), 5, 4, If A and B are arbitrary events, then, (a) P (A Ç B) ³ P (A) + P (B), (b) P (A È B) £ P (A) + P (B), (c) P (A Ç B) = P (A) + P(B), (d) None of the these, , (a), 83., , 2, 3, , (b), , (d), , 5, 6

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PROBABILITY-1, , 201, , 84. A bag contains 25 tickets, numbered from 1 to 25, two, tickets are drawn from the bag without replacement, what, is the probability that the 2nd ticket is a perfect square if it, is know that 1st ticket was a perfect square number?, (a) 1/8, (b) 1/6, (c) 1/4, (d) None of these, 85. If the odds in favour of an event be 3 : 5, then the probability of non-occurrence of the event is, (a), , 3, 5, , (b), , 5, 3, , (c), , 3, 8, , (d), , 86. If E and F are events such that P(E) =, P(E and F) =, I., , (a), , 1, 1, , P(F) = and, 2, 4, , (a), , 37, 60, , (b), , (c), , 1, 5, , 1, 8, , (d), , 37, 60, , 47, 60, , (b), , (c), , 1, 4, , 3, 4, , (d), , 89. If A and B are two events such that P (A) =, , 1, , then,, 8, , P(E or F) =, , 7, 60, , 88. In a horse race the odds in favour of three horses are 1 : 2,, 1 : 3 and 1 : 4. The probability that one of the horse will, win the race is, , 5, 8, , P( B) =, , 5, 8, , 3, 8, (a) Only I is true., (b) Only II is true., (c) Both I and II are true. (d) Neither I nor II is true., , II., , 87. In a given race the odds in favour of three horses A, B, C, are 1 : 3; 1 : 4; 1 : 5 respectively. Assuming that dead head, is impossible the probability that one of them wins is, , 2, , then which of the following is not correct?, 3, , 2, 3, , (a), , P ( A È B) ³, , (c), , 1, 1, £ P( A Ç B) £, 6, 2, , P(not E and not F) =, , 1, and, 2, , 1, 3, , (b), , P( A Ç B' ) ³, , (d), , 1, 1, £ P( A 'Ç B) £, 6, 2, , 90. If P (A) = P (B) = x and P(A Ç B) = P(A'ÇB' ) = 1/3, then, x=?, (a) 1/2, , (b) 1/3, , (c) 1/4, , (d) 1/6, , Exercise 2 : Exemplar & Past Year MCQs, 4., , NCERT Exemplar MCQs, 1., , In a non-leap year, the probability of having 53 Tuesday or, 53 Wednesday is, (a), , 1, 7, , (b), , 5., , 3, (d) None of these, 7, Three numbers are chosen from 1 to 20. Find the probability, that they are not consecutive, , 3., , (a), , 186, 190, , (b), , (c), , 188, 190, , (d), , (a), , 29, 52, , (b), , 1, 2, , (c), , C3, , 26, 51, , ( ), , P (A) £ P B, , (c) P(A) < P(B), , 18, 20, , 1, 4, 1, 5, (b), (c), (d), 5, 5, 30, 9, If A and B are mutually exclusive events, then, , (a), 6., , 187, 190, , 7., , While shuffling a pack of 52 playing cards, 2 are, accidentally dropped. The probability that the missing, cards to be of different colours is, (a), , 2, 1, 1, 1, (b), (c), (d), 7, 2, 3, 6, If without repetition of the numbers, four-digit numbers, are formed with the numbers 0, 2, 3 and 5, then the, probability of such a number divisible by 5 is, , (a), , 2, 7, , (c), 2., , If seven persons are to be seated in a row. Then, the, probability that two particular persons sit next to each, other is, , (d), , 27, 51, , (b), , P ( A ) ³ P ( B), , (d) None of these, , If P ( A È B) = P ( A Ç B) for any two events A and B, then, (a) P(A) = P(B), (c) P(A) < P(B), , (b) P(A) > P(B), (d) None of these

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EBD_7762, 202, , 8., , MATHEMATICS, , If 6 boys and 6 girls sit in a row at random, then the, probability that all the girls sit together is, (a), , 1, 432, , (b), , then the set of all possible values of common difference is, (±1, ±2, ±3, ±4, ±5) ., , 12, 431, , (a) Statement -1 is true, Statement -2 is true ; Statement -2, is not a correct explanation for Statement -1, , 1, (d) None of these, 132, A single letter is selected at random from the word, “PROBABILITY”. The probability that the selected letter, is a vowel is, , (b) Statement -1 is true, Statment -2 is false, , (c), 9., , (c) Statement -1 is false, Statment -2 is true., (d) Statement -1 is true, Statement -2 is true ; Statement -2, is a correct explanation for Statement -1., , 1, 4, 2, (b), (c), (d) 0, 3, 11, 11, If the probabilities for A to fail in an examination is 0.2, and that for B is 0.3, then the probability that either A or B, fails as, (a) > . 5 (b) 0.5, (c) £ .5, (d) 0, The probability that atleast one of the events A and B occurs, is 0.6. If A and B occur simultaneously with probability, , 14. A bag contains 5 brown and 4 white socks. A man pulls, out 2 socks. Find the probability that they are of the same, colour., [BITSAT 2014, A], , 0.2, then P( A ) + P( B ) is equal to, (a) 0.4, (b) 0.8, (c) 1.2, (d) 1.6, 12. If M and N are any two events, the probability that atleast, one of them occurs is ..., , [BITSAT 2016, A], , (a), , 10., , 11., , (a), , (c) P(M) + P(N) + P ( M Ç N ), (d) P(M) + P(N) + 2P ( M Ç N ), Past Year MCQs, Four numbers are chosen at random (without replacement), from the set {1, 2, 3, ...20}., [JEE MAIN 2010, S], Statement -1: The probability that the chosen numbers, 1, when arranged in some order will form an AP is, ., 85, , 2, 9, , (c), , 5, 9, , (d), , 7, 9, , 1, 1, 1, 1, (b), (c), (d), 3, 5, 10, 2, For three events A, B and C, P(Exactly one of A or B, occurs), [JEE MAIN 2017,A], = P(Exactly one of B or C occurs), 1, = P(Exactly one of C or A occurs) = and, 4, 1, P(All the three events occur simultaneously) =, ., 16, Then the probability that at least one of the events, occurs, is :, , (a), , (b) P(M) + P(N) – P ( M Ç N ), , 13., , (b), , 15. If three vertices of a regular hexagon are chosen at random,, then the chance that they form an equilateral triangle is :, , 16., , (a) P(M) + P(N) – 2P ( M Ç N ), , 4, 9, , (a), , 3, 16, , (b), , 7, 32, , (c), , 7, 16, , (d), , 7, 64, , 17. The probability of getting 10 in a single throw of three fair, dice is :, [BITSAT 2018, A], (a), , Statement -2 : If the four chosen numbers form an AP,, , 1, 6, , (b), , 1, 8, , (c), , 1, 9, , (b), , æ2 3ö, ç , ÷, è3 4ø, , (d), , 1, 5, , (c), , æ1 3ö, ç , ÷, è3 4ø, , Exercise 3 : Try If You Can, 1., , A die is rolled three times, then find the probability of, getting a large number than the previous number., (a), , (b), , 5, 54, , 1, (d) None of these, 17, A and B are two events. Odds against A are 2 to 1, odds in, favour of A È B are 3 to 1. If x £ P (B) £ y. then the ordered, pair (x, y) is :, (c), , 2., , 7, 54, , æ 5 3ö, (a) ç , ÷, è 12 4 ø, , 3., , (d) None, If the integers m and n are chosen at random between 1, and 100, then the probability that a number of the form, 7m + 7n is divisible by 5 equals, (a), , 1, 4, , (b), , 1, 7, , (c), , 1, 8, , (d), , 1, 49

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PROBABILITY-1, , 4., , 5., , 6., , 203, , Let X be a universal set such that n(X) = k. The probability, of selecting two subsets A and B of the set X such that, B = A is :, (a) 1/2, (b) 1/(2k – 1) (c) 1/2k, (d) 1/3k, Let A, B, C be three events. If the probability of occurring, exactly one event out of A and B is 1 – a, out of B and C, and A is 1 – a and that of occurring three events, simultaneously is a2, then the probability that at least one, out of A, B, C will occur is, (a) 1/2, (b) Greater than 1/2, (c) Less than 1/2, (d) Greater than 3/4, If A, B and C are three events, then which of the following, is/are not correct?, (a) P (Exactly two of A, B and C occur), , £ P( A Ç B ) + P ( B Ç C ) + P (C Ç A), (b), , 10., , (a), 11., , 12., , P ( A È B È C ) £ P( A) + P( B) + P(C ), , (c) P (Exactly one of A, B and C occur), £ P( A) + P( B) + P(C ) - P ( B Ç C ) - P(C Ç A) - P( A Ç B ), , 7., , (d) None of these, A student has a collection of blue and red marbles. The, numbers of red marbles belong to the set (20, 21, 22, 23,, ...., 38). If two marbles are chosen simultaneously at, random from his collection, the probability that they have, 1, . Which of the following is not, 2, possible number for blue marbles, (a) 21, (b) 36, (c) 38, (d) 15, Let w be a complex cube root of unity with w ¹ 1. A fair, die is thrown three times. If r1, r2 and r3 are the numbers, obtained on th e die, then the probability that, r1, , 9., , r2, , 13., , (a), (c), , 170, 22, , C5, , (b) 1 -, , 150, 22, , C5, , (d) None of these, , (b), , 64, 127, , (c), , 63, 128, , (d), , 31, 128, , Given that n is odd, the number of ways in which three, numbers in A.P. can be selected from 1, 2, 3, ..., n is, (a), , 3(n - 1), n ( n - 2), , (b), , 3(n + 1) 2, 2n (n - 1) (n - 2), , (c), , n-2, n (n - 1), , (d), , 3(n - 1), 2n ( n - 2), , Two distinct numbers a and b are chosen randomly from, the set {2, 22, 23, .... 225}. Then the probability that logab is, an integer is, 21, 62, 31, 131, (b), (c), (d), 200, 300, 300, 300, Two integers x and y are chosen at random from the set, , {x : 0 £ x £ 10, x ÎI} then probability for | x – y | £ 5 is, 87, 91, 87, 91, (c), (b), (d), 100, 121, 100, 121, Three natural numbers are taken at random from the set, , (a), 14., , A = {x | 1 £ x £ 100, x Î N }. The probability that the AM, of the numbers taken is 25, is, 77, , (a), , 15., , 25, , C2, , 100, , 74, , (c), , r3, , w + w + w = 0 is, (a) 1/18 (b) 1/9, (c) 2/9, (d) 1/36, A box contains 2 fifty paise coins, 5 twenty five paise, coins and 15 ten paise coins. Five coins are taken out of, the box at random. Probability that the value of these five, coins is less than one rupee and fifity paise is, , 1, 2, , (a), , different colours is, , 8., , A set S contains 7 elements. A non-empty subset A of S, and an element x of S are chosen at random. Then the, probability that x Î A is:, , (b), , C3, , C72, , 100, , C2, , 100, , C3, , (d) None of these, , C97, , If a and b are chosen randomly from the set consisting of, numbers 1, 2, 3, 4, 5, 6 with replacement. Then the, probability that lim [(a x + b x ) / 2]2 / x = 6 is, x ®0, , (a) 1/3, , (b) 1/4, , (c) 1/9, , (d) 2/9

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EBD_7762, 204, , MATHEMATICS, , 1, 2, 3, 4, 5, 6, 7, 8, 9, , (a), (a), (c), (b), (b), (d), (b), (c), (c), , 10, 11, 12, 13, 14, 15, 16, 17, 18, , (b), (c), (d), (c), (a), (c), (a), (c), (c), , 19, 20, 21, 22, 23, 24, 25, 26, 27, , (c), (c), (c), (b), (a), (b), (c), (c), (c), , 1, 2, , (a), (b), , 3, 4, , (c), (c), , 5, 6, , (d), (a), , 1, 2, , (b), (a), , 3, 4, , (c), (b), , 5, 6, , (b), (d), , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (b), 28 (c) 37 (b) 46 (b) 55, (d), 29 (c) 38 (a) 47 (a) 56, (b), 30 (b) 39 (a) 48 (b) 57, (a), 31 (c) 40 (a) 49 (b) 58, (b), 32 (a) 41 (b) 50 (d) 59, (b), 33 (d) 42 (b) 51 (b) 60, (a), 34 (d) 43 (b) 52 (c) 61, (d), 35 (b) 44 (b) 53 (b) 62, (d), 36 (b) 45 (d) 54 (a) 63, Exercise 2 : Exemplar & Past Year MCQs, (a) 9 (c) 11 (c) 13, (b), 7, (c) 10 (c) 12 (b) 14, (a), 8, Exercise 3 : Try If You Can, (a), 7 (c), 9 (c) 11 (d) 13, (c) 10 (b) 12 (b) 14, (c), 8, , 64, 65, 66, 67, 68, 69, 70, 71, 72, , (d), (c), (b), (b), (c), (a), (c), (c), (b), , 73, 74, 75, 76, 77, 78, 79, 80, 81, , (b), (c), (a), (b), (a), (c), (d), (a), (c), , 15, 16, , (c), (c), , 17, , (b), , 15, , (c), , 82, 83, 84, 85, 86, 87, 88, 89, 90, , (d), (b), (b), (d), (c), (b), (b), (b), (a)

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17, , RELATIONS AND, FUNCTIONS-2, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 2, , JEE MAIN, BITSAT, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , ., , Weightage, , 3, 4, , Critical Concepts, , Types of Relation, Kinds of Mapping, of Functions, Composition of, Functions, Invertible Functions,, Inverse of a Function., , Rating of, Difficulty Level, , CUS, (chapter utility score), Out of 10, , 3/5, , 6

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RELATIONS AND FUNCTIONS-2, , 207

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EBD_7762, 208, , MATHEMATICS, , Topic 1 : Types of Relations., 1., , 2., , 3., , 4., , 5., , 6., , 7., 8., , 9., 10., , 11., , Let P = {(x, y) | x2 + y2 = 1, x, y Î R}. Then, P is, (a) Reflexive, (b) Symmetric, (c) Transitive, (d) Anti-symmetric, For real numbers x and y, we write x R y Û x – y + 2 is, an irrational number. Then, the relation R is, (a) Reflexive, (b) Symmetric, (c) Transitive, (d) None of these, Let L denote the set of all straight lines in a plane. Let a, relation R be defined by a R b Û a ^ b, a, b Î L . Then, R, is, (a) Reflexive, (b) Symmetric, (c) Transitive, (d) None of these, Let S be the set of all real numbers. Then, the relation, R = {(a, b) : 1 + ab > 0} on S is, (a) Reflexive and symmetric but not transitive, (b) Reflexive and transitive but not symmetric, (c) Symmetric, transitive but not reflexive, (d) Reflexive, transitive and symmetric, Let R be a relation on the set N be defined by, {(x, y) | x, y Î N, 2x + y = 41}. Then, R is, (a) Reflexive, (b) Symmetric, (c) Transitive, (d) None of these, Let A = {1, 2, 3} and B = {2, 4, 6, 8}., Consider the rule f : A ® B, f(x) = 2x " x Î A. The, domain, codomain and range of f respectively are, (a) {1, 2, 3}, {2, 4, 6}, {2, 4, 6, 8}, (b) {1, 2, 3}, {2, 4, 6, 8}, {2, 4, 6}, (c) {2, 4, 6, 8}, {2, 4, 6, 7}, {1, 2, 3}, (d) {2, 4, 6}, {2, 4, 6, 8}, {1, 2, 3}, The relation "less than" in the set of natural numbers is :, (a) only symmetric, (b) only transitive, (c) only reflexive, (d) equivalence relation, Let R and S be two non-void relations in a set A. Which of, the following statements is not true., (a) R and S transitive, Þ R È S is transitive, (b) R and S transitive, Þ R Ç S is transitive, (c) R and S symmetric, Þ R È S is symmetric, (d) R and S reflexive, Þ R Ç S is reflexive, The relation R = { (1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} is :, (a) symmetric only, (b) reflexive only, (c) an equivalence relation (d) transitive only, Let A be the non-empty set of children in a family. The, relation 'x is brother of y' in A is:, (a) reflexive, (b) symmetric, (c) transitive, (d) None of these, Let A = {1, 2, 3, 4} and let R = {(2, 2), (3, 3), (4, 4), (1,, 2)} be a relation on A. Then R is:, (a) reflexive, (b) symmetric, (c) transitive, (d) None of these, , 12. If R is a relation in a set A such that (a, a) Î R for every, a Î A, then the relation R is called, (a) symmetric, (b) reflexive, (c) transitive, (d) symmetric or transitive, 13. A relation R in a set A is called empty relation, if, (a) no element of A is related to any element of A, (b) every element of A is related to every element of A, (c) some elements of A are related to some elements of A, (d) None of the above, 14. A relation R in a set A is called universal relation, if, (a) each element of A is not related to every element of A, (b) no element of A is related to any element of A, (c) each element of A is related to every element of A, (d) None of the above, 15. A relation R in a set A is said to be an equivalence relation,, if R is, (a) symmetric only, (b) reflexive only, (c) transitive only, (d) All of these, 16. Let R = {(3, 3) (5, 5), (9, 9), (12, 12), (5, 12), (3, 9),, (3, 12), (3, 5)} be a relation on the set A = {3, 5, 9, 12}., Then, R is:, (a) reflexive, symmetric but not transitive., (b) symmetric, transitive but not reflexive., (c) an equivalence relation., (d) reflexive, transitive but not symmetric., 17. A relation R in a set A is called transitive, if for all a 1, a2,, a3 Î A, (a1, a2) Î R and (a2, a3) Î R implies, (a), , ( a 2 ,a1 ) Î R, ( a3 , a1 ) Î R, , (b), , ( a1 , a3 ) Î R, ( a3 , a 2 ) Î R, , (c), (d), 18. If R = {(x, y) : x is father of y}, then R is, (a) reflexive but not symmetric, (b) symmetric and transitive, (c) neither reflexive nor symmetric nor transitive, (d) Symmetric but not reflexive, 19. If R = {(x, y) : x is exactly 7 cm taller than y}, then R is, (a) not symmetric, (b) reflexive, (c) symmetric but not transitive, (d) an equivalence relation, 20. If R = {(x, y) : x is wife of y}, then R is, (a) reflexive, (b) symmetric, (c) transitive, (d) an equivalence relation, 21. Let R be the relation in the set Z of all integers defined by, R = {(x, y) : x – y is an integer}. Then R is, (a) reflexive, (b) symmetric, (c) transitive, (d) an equivalence relation, 22. Let R be the relation in the set {1, 2, 3, 4} given by, R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}., (a) R is reflexive and symmetric but not transitive, (b) R is reflexive and transitive but not symmetric, (c) R is symmetric and transitive but not reflexive, (d) R is equivalence relation

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RELATIONS AND FUNCTIONS-2, , 23. Let A = (1, 2, 3). We define, R1 = {(1, 2), (3, 2), (1, 3)}, R2 = {(1, 3), (3, 6), (2, 1), (1, 2)}. Then, the relation on A, (a) R1 is relation and R2 is not, (b) R1 and R2 are both is relation, (c) R1 and R2 are both non-relation, (d) None of these, 24. Let R be a relation on the set A of ordered pairs of positive, integers defined by (x, y) R (u, v), if and only if xv = yu., Then, R is, (a) reflexive, (b) symmetric, (c) transitive, (d) an equivalence relation, 25. The relation R on the set Z defined by R = {(a, b) : (a –, b) is divisible by 5} divides the set Z into how many, disjoint equivalence classes ?, (a) 5, (b) 2, (c) 3, (d) 4, 26. Let R be the relation on the set of all real numbers defined, by a R b iff |a – b| £ 1. Then, R is, (a) Reflexive and symmetric, (b) Symmetric only, (c) Transitive only, (d) Anti-symmetric only, 27. Which one of the following relations on the set of real, numbers R is an equivalence relation ?, (a) aR1b Û | a | = | b |, (b) aR 2 b Û a ³ b, 28., , 29., , 30., , 31., , (c) aR 3b Û a divides b (d) aR 4 b Û a < b, Let R be the relation defined in the set A of all triangles, as R = {(T1, T2) : T1 is similar to T2}. If R is an equivalence, relation and there are three right angled triangles T1 with, sides 3, 4, 5; T2 with sides 5, 12, 13 and T3 with sides 6, 8,, 10. Then,, (a) T1 is related to T2, (b) T2 is related to T3, (c) T1 is related to T3, (d) None of these, For the set A = {1, 2, 3}, define a relation R in the set A as, follows, R = {(1, 1), (2, 2), (3, 3), (1, 3)}, Then, the ordered pair to be added to R to make it the, smallest equivalence relation is, (a) (1, 3) (b) (3, 1), (c) (2, 1), (d) (1, 2), On the set N of all natural numbers, define the relation R, by a R b, iff GCD of a and b is 2. Then, R is, (a) reflexive, but not symmetric, (b) symmetric only, (c) reflexive and transitive, (d) not reflexive, not symmetric, not transitive, A relation R is defined over the set of non-negative integers, 2, , 209, , 33. Let A = {1, 2, 3}. Then, the number of relations containing, (1, 2) and (1, 3), which are reflexive and symmetric but, not transitive, is, (a) 1, (b) 2, (c) 3, (d) 4, , Topic 2 : Mappings, Mapping of Functions, Kinds of, Mapping of functions., 34. Which of the following functions from I to itself is a, bijection?, (a) f(x) = x3, (b) f(x) = x + 2, (c) f (x) = 2x + 1, (d) f (x) = x2 + x, 35. Which of the following function is an odd function ?, , (b) {(6,0)( 11,5), (3,3, 3), (c) {(6, 0)(0, 6)}, (d) ( 11,5), (2, 4 2), (5 11), (4 2, 2)}, 32. Let A = {1, 2, 3}and R = {(1, 2), (2, 3)} be a relation in A., Then, the minimum number of ordered pairs may be added,, so that R becomes an equivalence relation, is, (a) 7, (b) 5, (c) 1, (d) 4, , f (x) = 1 + x + x 2 - 1 - x + x 2, , (b), , æ a x +1 ö, ÷, f ( x ) = xç, ç a x -1 ÷, è, ø, , æ 1- x 2 ö, ÷, f ( x) = logç, ç 1+ x 2 ÷, è, ø, (d) f(x) = k, k is a constant, 36. A function f from the set of natural numbers to integers, (c), , ì n -1, , when n is odd, ï, is, defined by f (n) = í 2, n, ï - , when n is even, î 2, , 37., , 38., , 39., 40., , 41., , 2, , as xRy Þ x + y = 36 what is R?, (a) {(0, 6)}, , (a), , 42., , 43., , (a) neither one-one nor onto (b) one-one but not onto, (c) onto but not one-one, (d) one-one and onto both, Let X = {– 1, 0, 1}, Y = {0, 2} and a function f : X ® Y, defined by y = 2x4, is, (a) one-one onto, (b) one-one into, (c) many-one onto, (d) many-one into, Let X = {0, 1, 2, 3} and Y = {–1, 0, 1, 4, 9} and a function, f :X ® Y defined by y = x2, is, (a) one-one onto, (b) one-one into, (c) many-one onto, (d) many-one into, Let g(x) = x2 – 4x – 5, then, (a) g is one-one on R, (b) g is not one-one on R, (c) g is bijective on R, (d) None of these, The mapping f : N ® N given by f(n) = 1 + n2, n Î N, when N is the set of natural numbers, is, (a) one-one and onto, (b) onto but not one-one, (c) one-one but not onto (d) neither one-one nor onto, The function f: R ® R given by f(x) = x3 – 1 is, (a) a one-one function, (b) an onto function, (c) a bijection, (d) neither one-one nor onto, If N be the set of all natural numbers, consider f : N ® N, such that f(x) = 2x, " x Î N, then f is, (a) one-one onto, (b) one-one into, (c) many-one onto, (d) None of these, The function f : A ® B defined by f(x) = 4x + 7, x Î R is, (a) one-one, (b) many-one, (c) odd, (d) even

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EBD_7762, 210, , MATHEMATICS, , 44. The smallest integer function f(x) = [x] is, (a) one-one, (b) many-one, (c) Both (a) & (b), (d) None of these, 45. The signum function, f : R ® R is given by, , 46., , ì 1 if x > 0, ï, f ( x ) = í 0, if x = 0 is, ï-1 if x < 0, î, (a) one-one, (b) onto, (c) many-one, (d) None of these, Which of the following functions is NOT one-one ?, (a) f : R ® R defined by f ( x) = 6 x - 1., (b) f : R ® R defined by f ( x) = x 2 + 7., (c) f : R ® R defined by f ( x) = x3., 2x + 1, ., x-7, Let f : R ® R be defined as f (x) = 2x3. then, (a) f is one-one onto, (b) f is one-one but not onto, (c) f is onto but not one-one, (d) f is neither one-one nor onto, , (d), , 47., , 48., , 53., , 56., , x2 +1, , then, 2, , (a) f is one-one onto, (b) f is one-one but not onto, (c) f is onto but not one-one, (d) f is neither one-one nor onto, 49. A function f : X ® Y is said to be onto, if for every y Î Y,,, there exists an element x in X such that, (a) f(x) = y, (b) f(y) = x, (c) f(x) + y = 0, (d) f(y) + x = 0, 50. f : X ®Y is onto, if and only if, (a) range of f = Y, (b) range of f ¹ Y, (c) range of f < Y, (d) range of f ³ Y, 51. Let f : {x, y, z} ® {1, 2, 3} be a one-one mapping such, that only one of the following three statements is true and, remaining two are false : f (x) ¹ 2, f (y) = 2, f (z) ¹ 1, then, (a) f (x) > f (y) > f (z), (b) f (x) < f (y) < f (z), (c) f (y) < f (x) < f (z), (d) f (y) < f (z) < f (x), 52. Consider the four functions f1, f2, f3 and f4 as follows, f2, , f1, , a, b, c, d, e, f, , 3, 4, X1, , (i), , f3, , X2, , 1, 2, 3, 4, X1, , 1, , a, , 1, , 2, 3, 4, , b, c, , 2, , X1, , 55., , f : R - {7} ® R defined by f ( x) =, , Let f : R ® R be defined as f (x) =, , 1, 2, , 54., , (iii), , X3, , (ii), , f4, , X2, , a, b, c, , 3, 4, X1, , a, b, c, d, e, f, , d, (iv), , X4, , 57., 58., , 59., , (a) f1 and f2 are onto, (b) f2 and f4 are onto, (c) f2 and f3 are onto, (d) f3 and f4 are onto, Let f : R ® R be defined as f(x) = x4, then, (a) f is one-one onto, (b) f is many-one onto, (c) f is one-one but not onto, (d) f is neither one-one nor onto, The function f : R ® R defined by f(x) = x2 + x is., (a) one-one, (b) onto, (c) many-one, (d) None of the above, Assertion : Let A = {–1, 1, 2, 3} and B = {1, 4, 9}, where, f : A ® B given by f(x) = x2, then f is a many-one function., Reason : If x 1 ¹ x 2 Þ f(x 1 ) ¹ f(x 2), for every, x1, x2 Î domain, then f is one-one or else many-one., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., Consider the following statements, Statement - I : An onto function f : {1, 2, 3} ® {1, 2, 3}, is always one-one., Statement - II : A one-one function f :{1, 2, 3} ® {1, 2, 3}, must be onto., (a) Only I is true, (b) Only II is true, (c) Both I and II are true (d) Neither I nor II is true, Let A = {1, 2, 3} and B = {a, b, c}, then the number of, bijective functions from A to B are, (a) 2, (b) 8, (c) 6, (d) 4, Let A = {1, 2, 3} and B = {a, b, c}, and let f = {(1, a), (2,, b), (P, c)}be a function from A to B. For the function f, to be one-one and onto, the value of P =, (a) 1, (b) 2, (c) 3, (d) 4, The function f : R ® R defined by f (x) = sin x is :, (a) into, (b) onto, (c) one-one, (d) many one, , ìï x | x | -4, x Î Q, 60. If f : R ® R, f (x) = í, , then f (x) is, ïî x | x | - 3 x Ï Q, (a) one to one and onto, (b) many to one and onto, (c) one to one and into, (d) many to one and into, 61. A function f : R ® [–1, 1] defined by, f(x) = sin x, "x Î R, where R is the subset of real numbers, is one-one and onto if R is the interval:, , (a) [0, 2p], , (b), , é p pù, êë - 2 , 2 úû, , (c) [ -p, p ], (d) [ 0, p ], 62. Let f : R ® R be function defined by, f (x) = sin (2x –3), then f is, (a) injective, (b) surjective, (c) bijective, (d) None of these, 63. Let f : R ® R be a function defined by, f(x) = x3 + 4, then f is, (a) injective, (b) surjective, (c) bijective, (d) None of these

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RELATIONS AND FUNCTIONS-2, , 211, , 64. The number of all one-one functions from set A = {1, 2,, 3} to itself is, (a) 2, (b) 6, (c) 3, (d) 1, 65. lf A = {l,3,5,7} and B = {l,2,3,4,5,6, 7,8}, then the number, of one–to–one functions from A into B is, (a) 1340 (b) 1680, (c) 1430, (d) 1880, 66. If the function gof is defined and is one-one then, (a) neither f nor g is one-one, (b) f and g both are necessarily one-one, (c) g must be one-one, (d) None of the above, 67. The number of equivalence relations in the set {1, 2, 3}, containing (1, 2) and (2, 1) is, (a) 2, (b) 3, (c) 1, (d) 4, 68. The function f : [0, p] ® R, f (x) = cos x is, (a) one-one function, (b) onto function, (c) a many one function (d) None of these, 69. The function f : R ® R defined by, f (x) = (x – 1) (x – 2) (x – 3) is, (a) one-one but not onto (b) onto but not one-one, (c) both one-one and onto (d) neither one-one nor onto, 70. The number of surjection from, A = {1, 2, ......., n}, n ³ 2 onto B = {a, b} is, (a) nP2, (b) 2n – 2, n, (c) 2 – 1, (d) None of these, 71. The number of surjective functions from A to B where, A = {1, 2, 3, 4} and B = {a, b} is, (a) 14, (b) 12, (c) 2, (d) 15, 72. If f : R ® S , defined by f ( x) = sin x - 3 cos x + 1, is, onto, then the interval of S is, (a) [ –1, 3] (b) [–1, 1], (c) [ 0, 1] (d) [0, 3], 73. Let function f : R ® R be defined by f (x) = 2x + sin x for, x Î R , then f is, (a) one-one and onto, (b) one-one but NOT onto, (c) onto but NOT one-one, (d) neither one-one nor onto, , f ( x) = 5 x ( x - 4) , then f–1 (x) is, , (a), (c), , 74. If f : R ® R and g : R ® R defined by f(x) = 2x + 3 and, g(x) = x2 + 7, then the value of x for which f(g(x)) = 25 is, (a) ± 1, (b) ± 2, (c) ± 3, (d) ± 4, 75. If f: R ® R is given by, ì -1, when x is rational, f (x) = í, î1, when x is irrational,, , (c), , 2 + 4 + log 5 x, , (d) None of these, , x, x, 1- x, 1, (b), (c), (d), 1+ x, 1- x, x, x, 80. Let f : R ® R, g : R ® R be two functions such that, f(x) = 2x – 3, g(x) = x3 + 5. The function (fog)–1 (x) is, equal to, , (a), , 1/ 3, , 7ö, æ, (b) ç x - ÷, 2ø, è, , 1/ 3, , æ x-7ö, (d) ç, ÷, è 2 ø, , (a), , æ x+7ö, ç, ÷, è 2 ø, , (c), , æ x-2ö, ç, ÷, è 7 ø, , 1/ 3, , 1/ 3, , 81. If f : R ® R defined by f ( x ) =, function, then f –1 is equal to, ( 4x + 5), (a), (b), 2, 3x + 2, (c), (d), 2, , 2x - 7, is an invertible, 4, , ( 4x + 7), 2, 9x + 3, 5, , 3 + 4x, 6x - 4, 3 - 4x, 9 + 2x, (b), (c), (d), 6x - 4, 3 + 4x, 6x - 4, 6x - 4, 83. If the binary operation * on the set of integers Z, is defined, by a * b = a + 3b2, then the value of 8 * 3 is, (a) 32, (b) 40, (c) 36, (d) 35, , (a), , 3, , (d) 0, , 3x + x 3, 1+ x ö, g, x, =, (, ), 76. Given f ( x ) = log æç, an, d, , t hen, ÷, 1 + 3x 2, è 1- x ø, fog(x) equals, (a) – f (x), (b) 3f(x), (c) [f (x)]3, (d) None of these, , x, , 84. If f(x) =, , (a), , ), , (b) – 1, , æ1ö, ç ÷, è5ø, , (b), , x (x - 4), , 79. If f(x) = x – x2 + x3 – x4 + ... to ¥ for |x| < 1, then f–1(x) =, , then (fof) 1 - 3 is equal to, (a) 1, , 2 - 4 + log 5 x, , 2ü, 82. Consider the function f in A = R - ì, í ý defined as, î3þ, 4x + 3, f (x) =, , then f –1 is equal to, 6x - 4, , Topic 3 : Composition of Functions, Inverse of a Function,, Binary Operations., , (, , 77. If f : R ® R, g : R ® R and h : R ® R are such that, f (x) = x2, g(x) = tan x and h(x) = log x, then the value of, (go (foh)) (x), if x = 1 will be, (a) 0, (b) 1, (c) –1, (d) p, 78. Let f : [4, ¥) ® [1, ¥) be a function defined by, , 1+ x2, 3x, , 1+ x, , 3x, , 2, , , then (fof of) (x) is, , (b), , x, 1 + 3x 2, , (d) None of these, 1 - x2, 85. For binary operation * defined on R – {1} such that, a, a*b =, is, b +1, (a) not associative, (b) not commutative, (c) commutative, (d) both (a) and (b), (c)

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EBD_7762, 212, , MATHEMATICS, , 86. The binary operation * defined on N by a * b = a + b + ab, for all a, b Î N is, (a) commutative only, (b) associative only, (c) both commutative and associative, (d) None of these, 87. If a binary operation * is defined by a * b = a 2 + b2 + ab + 1,, then (2 * 3) * 2 is equal to, (a) 20, (b) 40, (c) 400, (d) 445, 88. If a * b denote the bigger among a and b and if, a . b = (a * b) + 3, then 4.7 =, (a) 14, (b) 31, (c) 10, (d) 8, 89. Let A = {1, 2, 3, 4, 5} and functions, f : A ® A and g : A ® A be defined by, f (1) = 3, f (2) = 5, f (3) = 3, f (4) = 1, f (5) = 2; g (1) = 4,, g (2) = 1, g (3) = 1, g (4) = 2, g (5) = 3. Then, (a) fog = {(1,1), (2, 3), (3, 2), (4, 5)}, (b) fog = {(1,1), (2,3), (3,3), (4,5), (5,3)}, (c) gof = {(1, 1), (2, 3), (3, 3), (4, 4), (5, 5)}, (d) gof = {(2, 2), (2, 3), (3, 1), (4, 1), (5, 1)}, 90. Let f : (4, 6) ® (6, 8) be a function defined by, éx ù, f ( x ) = x + ê ú (where [.] denotes the greatest integer, ë2û, , function), then f -1 ( x) is equal to, (a), , éxù, x-ê ú, ë2û, , (c) x – 2, , (b), (d), , -x - 2, 1, éx ù, x+ê ú, ë2û, , If f : R ® R is given by f (x) = 1 - x 2 , then fof is, (b) x2, (c) x, (d) 1– x2, (a), x, 92. If f is an even function and g is an odd function, then the, function fog is, (a) an even function, (b) an odd function, (c) neither even nor odd (d) a periodic function, 93. If f : X ® Y is a function such that there exists a function, g : Y ® X such that gof = IX and fog = IY, then f must be, (a) one-one, (b) onto, (c) one-one and onto, (d) None of these, 94. Which of the following option is correct?, (a) gof is one-one Þ g is one-one, (b) gof is one-one Þ f is one-one, (c) gof is onto Þ g is not onto, (d) gof is onto Þ I is onto, 91., , 95., , The inverse of f (x) =, (a), , (b), , 2 + 3x, 1, log10, 2 - 3x, 2, , 1, 2 + 3x, 1, 2 - 3x, log10, log10, (d), 3, 2 - 3x, 6, 2 + 3x, The domain of the function, f ( x ) = 24- x C3x -1 + 40-6 x C8x -10 is,, (a) {2, 3}, (b) {1, 2, 3}, (c) {1, 2, 3, 4}, (d) None of these, , (c), , 96., , 1, 1+ x, log10, 3, 1- x, , 2 10 x - 10 - x, is, 3 10 x + 10- x, , 97. Let f (x) = – 1 + | x – 1 |, –1 £ x £ 3 and, g (x) = 2 – | x + 1 |, –2 £ x £ 2, then (fog) (x) is equal to, ìx - 1 - 2 £ x £ 0, í, îx + 1 0 < x £ 2, , (a), , ìx + 1 - 2 £ x £ 0, í, îx - 1 0 < x £ 2, , (b), , (c), , ì - 1 - x -2 £ x £ 0, í, 0<x£2, î x -1, , (d) None of these, , 98. Which of the following is not a binary operation on the, indicated set?, (a) On Z+, * defined by a * b = a – b, (b) On Z+, * defined by a * b = ab, (c) On R, * defined by a * b = ab2, (d) None of the above, 99. Consider a binary operation * on N defined as, a * b = a3 + b3, (a) * is both associative and commutative, (b) * is commutative but not associative, (c) * is associative but not commutative, (d) * is neither commutative nor associative, 100. Assertion : f : R ® R is a function defined by, 2x + 1, 3x - 1, f(x) =, . Then f–1(x) =, ., 3, 2, Reason : f(x) is not a bijection., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., ax + b, . Then fof (x) = x provided that, cx + d, (a) d = – a, (b) d = a, (c) a = b = c = d = 1, (d) a = b = 1, , 101. Let f ( x ) =, , ìï2x + a ; x ³ -1, 102. If f (x) = í 2, and, ïîbx + 3 ; x < -1, , ìx + 4 ; 0 £ x £ 4, g (x) = í, î-3x - 2 ; - 2 < x < 0, If domain of g (f (x)) is [–1, 4], then –, (a) a = 0, b > 5, (b) a = 2, b > 7, (c) a = 2, b > 10, (d) a = 0, b Î R, 103. Let f : (2, 3) ® (0, 1) be defined by f(x) = x – [x]. Then,, f–1(x) equals to, (a) x – 2 (b) x + 1, (c) x – 1, (d) x + 2, 104. Let A = R – {3} and B = R – {1}. If f : A ® B defined by, f (x) =, , x-2, is invertible, then the inverse of f is, x -3, , (a), , 3y + 2, y -1, , (b), , 3y - 2, y +1, , (c), , 3y - 2, y -1, , (d) None of these

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RELATIONS AND FUNCTIONS-2, , 213, , 105. If f : R ® R, f (x) = x3 + 2, then f –1 (x) is, (a), , 1/ 2, , (x - 1), , (b), , x -2, , (d) (x - 2)1/ 2, (x - 2)1/ 3, 106. Consider the following statements, I. The operation * defined on Z+ by a * b = |a – b| is a, binary operation., II. The operation * defined on Z+ by a * b = a is not a, binary operation., (a) Only I is true, (b) Only II is true, (c) Both I and II are true (d) Neither I nor II is true, 107. The range of the function f (x) = 7 – x Px – 3 is, (a) {1, 2, 3}, (b) {1, 2, 3, 4, 5, 6}, (c) {1, 2, 3, 4}, (d) {1, 2, 3, 4, 5}, 108. A binary operation * on the set {0, 1, 2, 3, 4, 5} is defined, as, (c), , 109., , 110., 111., , 112., , , if a + b < 6, ì a+b, a*b = í, îa + b - 6 , if a + b ³ 6, the identity element is, (a) 0, (b) 1, (c) 2, (d) 3, Let * be a binary operation on set Q of rational numbers, ab, defined as a * b =, . The identity for * is, 5, (a) 5, (b) 3, (c) 1, (d) 6, Let * be the binary operation on N given by a * b = HCF, (a, b) where, a, b Î N. The value of 22 * 4 is, (a) 1, (b) 2, (c) 3, (d) 4, Assertion : If f : R ® R and g : R ® R be two mappings, such that f(x) = sin x and g(x) = x2, then fog ¹ gof., Reason : (fog) x = f(x) g(x) = (gof) x, (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., Let f be a function with domain X and range Y. Let A, B, Í X and C, D Í Y. Which of the following is not true?, (a) f (A È B) = f (A) È f (B), (b) f (A Ç B) = f (A) Ç f (B), (c) f–1 (C È D) = f–1 (C) È f–1 (D), (d) f–1 (C Ç D) = f–1 (C) Ç f–1 (D), , ìï x 3 - 1, x < 2, . Then, 113. Let f (x) = í 2, ïîx + 3, x ³ 2, ìï (x + 1)1/3 , x < 2, (a) f – 1 (x) = í, 1/3, ïî(x - 3) , x ³ 2, 1/3, ïì (x + 1) , x < 7, (b) f – 1 (x) = í, 1/2, ïî(x - 3) , x ³ 7, 1/3, ïì (x + 1) , x < 1, (c) f – 1 (x) = í, 1/2, ïî(x - 3) , x ³ 7, (d) f – 1 (x) does not exist, , 114. If f : Q ® Q, f (x) = 2x ; g : Q ® Q, g (x) = x + 2, then, value of (fog)–1 (20) is, (a) 5, (b) – 8, (c) 4, (d) 8, 115. In the set N of natural numbers, define the binary operation, * by m * n = GCD (m, n), m, n Î N. Then, which of the, following is true?, I. * is not a binary operation, II. * is a binary operation, III. Inverse of each element of N exist, IV. Inverse of each element of N does not exist, (a) I and IV are true, (b) II and III are true, (c) Only I is true, (d) II and IV are true, 116. Let f(x) = 2x2, g(x) = 3x + 2 and fog (x) = 18 x2 + 24x + c,, then c =, (a) 2, (b) 8, (c) 6, (d) 4, 117. Let f (x) =, , g(x) =, , 2, x -3, , 2 + ax, x, , , x ¹ 3 The inverse of f (x) is, , , x ¹ 0 . Then a =, , (a) 5, , (b) 2, (c) 3, (d) 4, x, 118. If f (x) =, , then (fofo........of )(x) is equal to :, x -1, 19 times, (a), , x, (b), x -1, , æ x ö, çè x - 1÷ø, , 19, , (c), , 19x, x -1, , (d) x, , pö, æ, pö, æ, 119. If f (x) = sin2 x + sin2 ç x + ÷ + cos x cos ç x + ÷ and, 3ø, è, 3ø, è, æ5ö, g ç ÷ = 1, then gof (x) is equal to, è4ø, (a) 1, (b) 48, (c) – 48, (d) – 1, 120. If g(x) = x – 2 is the inverse of the function f(x) = x + 2,, then graph of g (x) is the image of graph of f(x) about, the line y = kx. Here k =, (a) 1, (b) 2, (c) 3, (d) 4, 121. Let A = N × N and * be the binary operation on A defined, by (a, b) * (c, d) = (a + c, b + d). Then * is, (a) commutative, (b) associative, (c) Both (a) and (b), (d) None of these, 122. If the binary operation * is defined on the set Q+ of all, positive rational numbers by a * b =, , ab, . Then 3 *, 4, , æ1 1ö, ç * ÷, è5 2ø, , is equal to, 3, 5, 3, 3, (b), (c), (d), 160, 160, 10, 40, 123. Let S be a finite set containing n elements. Then the total, number of binary operations on S is:, , (a), , (a), , nn, , 2, , (b) nn, , (c), , 2n, , 2, , (d) n2

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EBD_7762, 214, , MATHEMATICS, , 124. If f : B ® A is defined by f ( x ) =, defined by g ( x ) =, , 3x + 4, and g : A ® B is, 5x - 7, , 7x + 4, ì3ü, , where A = R – í ý and, 5x - 3, î5þ, , ì7 ü, B = R – í ý and IA is an identity function on A and IB is, î5þ, identity function on B, then, (a) fog = IA and gof = IA (b) fog = IA and gof = IB, (c) fog = IB and gof = IB (d) fog = IB and gof = IA, , (, , 125. If f : R ® R be given by f ( x ) = 3 - x 3, , ), , 1, 3, , , then fof (x) is, , 1, , (b) x3, (c) x, (d) (3 – x3), x3, 126. If f(x) = |x| and g(x) = |5x – 2|, then, (a) gof (x) = |5x – 2|, (b) gof (x) = |5| x | – 2|, (c) fog (x) = |5| x | – 2|, (d) fog (x) = |5x + 2|, 127. If f(x) = ex and g(x) = logex, then which of the following, is true?, (a), , (a), , f {g ( x )} ¹ g {f ( x )}, , f {g ( x )} = g {f ( x )}, , (b), , (c) f {g ( x )} + g {f ( x )} = 0 (d) f {g ( x )} - g {f ( x )} = 1, 128. For a binary operation * on the set {1, 2, 3, 4, 5}, consider, the following multiplication table., *, , 1, , 2, , 3, , 4, , 5, , 1, , 1, , 1, , 1, , 1, , 1, , 2, , 1, , 2, , 1, , 2, , 1, , 3, , 1, , 1, , 3, , 1, , 1, , 4, , 1, , 2, , 1, , 4, , 1, , 5, , 1, , 1, , 1, , 1, , 5, , Which of the following is correct?, (a) (2 * 3) * 4 = 1, (b) 2 * (3 * 4) = 2, (c) * is not commutative, (d) (2 * 3) * (4 * 5) = 2, 129. If f (x) = sin x + cos x, g (x) = x2 – 1, then g (f (x)) is, invertible in the domain, é pù, é p pù, (a) ê0, ú (b) ê- , ú (c), ë 2û, ë 4 4û, 130. The inverse of the function, , f (x) =, , e x - e- x, ex + e- x, , é p pù, ê- 2 , 2 ú (d) [0, p], ë, û, , + 2 is, , æ x - 3ö, loge ç, è x - 1 ÷ø, , (c), , æ x + 2ö, loge ç, è x - 3 ÷ø, , (b), 1/ 2, , Topic 4: Composition of Relations, Inverse of a Relation., 131. The relation R is defined on the set of natural numbers as, {(a, b) : a = 2b}. Then, R–1 is given by, (a) {(2, 1), (4, 2), (6, 3),...} (b) {(1, 2), (2, 4), (3, 6),...}, (c) R–1 is not defined, (d) None of these, 132. Let R = {(1, 3), (2, 2), (3, 2)} and S = {(2, 1), (3, 2), (2, 3)}, be two relations on set A = {1, 2, 3}. Then RoS =, (a) {(1, 3), (2, 2), (3, 2), (2, 1), (2, 3)}, (b) {(3, 2), (1, 3)}, (c) {(2, 3), (3, 2), (2, 2)}, (d) {(2, 3), (3, 2)}, 133. If R is an equivalence relation on a set A, then R–1 is, (a) Reflexive only, (b) Symmetric but not transitive, (c) Equivalence, (d) None of these, 134. The relation R defined in A = {1, 2, 3} by aRb, if | a 2 – b2 | £, 5. Which of the following is false?, (a) R = {(1, 1), (2, 2), (3, 3), (2, 1), (1, 2), (2, 3), (3,2)}, (b) R – 1 = R, (c) Domain of R = {1, 2, 3}, (d) Range of R = {5}, 135. If R be a relation < from A = {1, 2, 3, 4} to B = {1, 3, 5}, , 136., , 137., , 138., , 139., , 1/ 2, , (a), , BEYOND NCERT, , æ x - 1ö, loge ç, è 3 - x ÷ø, , 1/2, , æ x +1ö, (d) loge çè, ÷, x - 2ø, , 1/ 2, , 140., , i.e., (a, b) Î R Û a < b, then RoR -1 is, (a) {(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}, (b) {(3, 1), (5, 1), (3, 2), (5, 2), (5, 3), (5, 4)}, (c) {(3, 3), (3, 5), (5, 3), (5, 5)}, (d) {(3, 3), (3, 4), (4, 5)}, Assertion : If the relation R defined in A = {1, 2, 3} by, aRb, if |a2 – b2| £ 5, then R– 1 = R, Reason : For above relation, domain of R–1 = Range of R., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., Let R be a relation on a set A such that R = R–1, then R is, (a) Reflexive, (b) Symmetric, (c) Transitive, (d) None of these, R is a relation from {11, 12, 13} to {8, 10, 12} defined by, y = x – 3. The relation R–1 is, (a) {(11, 8), (13, 10)}, (b) {(8, 11), (10, 13)}, (c) {(8, 11), (9, 12), (10, 13)}, (d) None of these, If R Í A × B and S Í B × C be two relations, then, (SoR)–1 is equal to, (a) S –1 oR –1, (b) RoS, (c) R–1 oS–1, (d) None of these, If f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then fog is, (a) {(2, 2), (3, 3)}, (b) {(5, 3), (6, 2)}, (c) {(2, 2), (5, 5)}, (d) {(6, 6), (3, 3)}

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RELATIONS AND FUNCTIONS-2, , 215, , (a) nP2, (c) 2n – 1, , NCERT Exemplar MCQs, 1., , 2., , 3., , 4., , 5., , 6., , 7., , Let T be the set of all triangles in the Euclidean planen, and let a relation R on T be defined as aRb, if a is congruent, to b, " a, b Î T. Then, R is, (a) reflexive but not transitive, (b) transitive but not symmetric, (c) equivalence, (d) None of these, Consider the non-empty set consisting of children in a family, and a relation R defined as a R b if a is brother of b. Then R, is, (a) symmetric but not transitive, (b) transitive but not symmetric, (c) neither symmetric nor transitive, (d) both symmetric and transitive, The maximum number of equivalence relations on the set, A = {1, 2, 3} are, (a) 1, (b) 2, (c) 3, (d) 5, If a relation R on set {1, 2, 3} be defined by R = {(1, 2)},, then R is, (a) reflexive, (b) transitive, (c) symmetric, (d) None of these, Let us define a relation R in R as aRb if a ³ b. Then R is, (a) an equivalence relation, (b) reflexive, transitive but not symmetric, (c) symmetric, transitive but not reflexive, (d) neither transitive nor reflexive but symmetric, The relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}, on set A = {1, 2, 3} is, (a) Reflexive but not symmetric, (b) Reflexive but not transitive, (c) Symmetric and transitive, (d) Neither symmetric nor transitive, The identity element for the binary operation * defined on, ab, , " a, b Î Q – {0} is, 2, (a) 1, (b) 0, (c) 2, (d) None of these, If the set A contains 5 elements and the set B contains 6, elements, then the number of one-one and onto mapping, from A to B is, (a) 720, (b) 120, (c) 0, (d) None of these, If A = {1, 2, 3, ....., n} and B = {a, b}. Then, the number of, surjections from A into B is, , Q – {0} as a * b =, , 8., , 9., , (b) 2n – 2, (d) None of these, , 1, "x Î R . Then f is, x, (a) one-one, (b) onto, (c) bijective, (d) f is not defined, Let f : R ® R be defined by f(x) = 3x2 – 5 and g : R ® R, x, by g ( x ) = 2, . Then gof is, x +1, , 10. Let f : R ® R be defined by f ( x ) =, , 11., , (a), , (c), , 3x 2 - 5, 9x 4 - 30x 2 + 26, 3x 2, x 4 + 2x 2 - 4, , (b), , (d), , 3x 2 - 5, 9x 4 - 6x 2 + 26, 3x 2, 9x 4 + 30x 2 - 2, , 12. Which of the following functions from Z into Z are bijective?, (a) f(x) = x3, (b) f(x) = x + 2, (c) f(x) = 2x + 1, (d) f(x) = x2 + 1, 13. If f : R ® R be the function defin ed by, f(x) = x 3 + 5, then f –1 (x) is equal to:, (a) (x + 5)1/3, (b) (x – 5)1/3, (c) (5 – x)1/3, (d) (5 – x), 14. If f : A ® B and g : B ® C be the bijective functions, then, (gof )–1 is, (a) f –1og–1, (b) fog, (c) g–1of –1, (d) gof, , ì3ü, 3x + 2, 15. Let f : R – í ý ® R be defined by f ( x ) =, . Then, 5x - 3, î5þ, (a) f –1(x) = f(x), , (b) f –1(x) = – f(x), , (c) (fof )x = – x, , (d) f –1(x) =, , 1, f(x), 19, , 16. If f (x) is defined on [0, 1] by the rule, ì x : x is rational, f (x) = í, î1 - x : x is irrational, , then for all x Î R, f (f(x)) is, (a) constant, (b) 1 + x, (c) x, (d) None of these, 17. If f : [2, ¥) ® R be the function defined by f (x) = x2 – 4x, + 5, then the range of f is, (a) R, (b) [1, ¥), (c) [4, ¥), (d) [5, ¥), 18. Let f : N ® R be the function defined by

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EBD_7762, 216, , MATHEMATICS, , 2x - 1, and g : Q ® R be another function defined, 2, 3, by g(x) = x + 2. Then (gof), is, 2, f (x) =, , (a) 1, , (b) 0, , (c), , 7, 2, , (d) 3, , 19. Let f : R ® R be defined by, , (a) f is one-one onto, , (c) f is many-one onto, (d) f is many-one into, 23. If r = {(x, y) |x2 + y2 = 1; x, y Î R}. Then, r is, [BITSAT 2015, A], (a) reflexive, (b) symmetric, (c) transitive, (d) anti-symmetric, 24. Let f (x) =, , x>3, ì 2x :, ï 2, f ( x ) = íx : 1 < x £ 3, ï 3x :, x £1, î, , (a), , p, 4, , (b), , (c) does not exist, , p, ì, ü, ínp + : n Î Z ý, 4, î, þ, , (d) None of these, , If g is the inverse of a function f and f ' ( x ) =, , g ¢ ( x ) is equal to:, (a), , 1, , [JEE MAIN 2017, A], (a) neither injective nor surjective, (b) invertible, (c) injective but not surjective, (d) surjective but not injective, 26. Let f and g be functions from R to R defined as, , 1 + x5, , ì 7 x 2 + x - 8, x £ 1, ì | x |, x < -3, ï, ï, f ( x) = í 4 x + 5, 1 < x £ 7 , g ( x) = í0, - 3 £ x < 2, ï 8 x + 3, x > 7, ï 2, î, î x + 4, x ³ 2, , , then, , [JEE MAIN 2014, C], , Then, , (b) 1 + { g ( x )}, , 5, , 1 + { g ( x )}, , 5, , (c) 1 + x5, 22., , 1, , [BITSAT, 2016, A], (b) d = a, (d) a = b = c = d = 1, , x, é 1 1ù, 25. The function f : R ® ê - , ú defined as f(x) =, , is :, 2, 2, 1 + x2, ë, û, , Past Year MCQs, 21., , ax + b, , then fof (x) = x, provided that :, cx + d, , (a) d = – a, (c) a = b = 1, , Then f (– 1) + f (2) + f (4) is, (a) 9, (b) 14, (c) 5, (d) None of these, 20. Let f : R ® R be given by f(x) = tan x. Then f –1(1) is, , (b) f is one-one into, , [BITSAT 2017, S], , (a) (fog) (–3) = 8, , (b) (fog) (9) = 683, , (c) (gof) (0) = – 8, , (d) (gof) (6) = 427, , (d) 5x4, , Let f : R ® R be a function defined by f (x) =, where m ¹ n , then, , x-m, ,, x-n, , [BITSAT 2014, A], , Exercise 3 : Try If You Can, 1., , ì, æp, öü, If f ( x ) = (1 + tan x) í1 + tan ç - x ÷ý and let g(x) be, è4, øþ, î, defined for all real x, then which of the following statements, is/are true for gof(x)?, (a) domain of gof(x) is R, , (b) gof(x) is constant " x Î D f, , (c) gof(x) is surjective function, (d) None of these, 2., , If the function f : R ® R defined by f ( x ) =, , ax 2 + 6 x - 8, a + 6 x - 8x2, , is onto for a Î [m, n], then mn - 3 is equal to., (a) 4, (b) 19, (c) 5, (d) None of these

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RELATIONS AND FUNCTIONS-2, , 217, , 9., , 3., , Let f : R + ® {-1, 0, 1} defined by, , 4., , f ( x ) = sgn( x - x 4 + x 7 - x8 - 1) where sgn denotes, signum function. Then f(x) is, (a) many-one and onto, (b) many-one and into, (c) one-one and onto, (d) one-one and into, Let f (x) = x2 + ax + b cos x, a being an integer and b is a, real number. Find the number of ordered pairs (a, b) for, which the equations f (x) = 0 and f ( f (x)) = 0 have the, same (nonempty) set of real roots, (a) 1, (b) 2, (c) 3, (d) 4, , 5., , æ 2 log10 x + 2 ö, If f ( x) = log100 x ç, ÷ ; g ( x ) = { x}; where {x}, -x, è, ø, denotes the fractional part of x. If the function (fog)(x), exists, then the maximum possible range of g(x) is, (a) (0, 10–1), (b) (0, 10–2), , (c), 6., , 7., , -2, , -2, , -1, , (0, 10 ) È (10 , 10 ) (d) None of these, , Let f(x) = max {1 + sinx, 1, 1 – cosx}, x Î [0, 2p] and, g(x) = max{1, |x – 1|}, xÎR, then which of the following, is incorrect., (a) g(f(0)) = 1, (b) g(f(1)) = 1, (c) f(g(1)) = 1, (d) f(g(0)) = 1 + sin1, Let X and Y be two non-empty sets. Let f : X ® Y be a, function., , For, , and, , AÌ X, , B Ì Y,, , define, , f ( A)= { f ( x) : x Î A}; f -1 ( B) = {x Î X : f ( x) Î B}, then, , (a), (c), 8., , f(f, , -1, , ( B )) = B, , f -1 ( f ( A)) = A, , (b), (d), , f ( f -1 ( B)) Ì B, f -1 ( f ( A)) Ì A, , 3 | x | - x - 2 and g (x) = sin x, then domain of, definition of fog (x) is, If f (x) =, , (a), , (b), , (c), , pü, ì, í 2 np + ý, 2 þ n ÎI, î, , U, , n ÎI, , 7p, 11p ö, æ, , 2 np +, çè 2np +, ÷, 6, 6 ø, , 7p ü, ì, í 2 np + ý, 6 þ n ÎI, î, , (d) {(4m + 1), , p, 7p, 11p ù, é, : m Î I } U ê 2n p +, , 2 np +, 2, 6, 6 úû, ë, nÎI, , Let f (x) = sin x and g (x) = loge | x |. If the ranges of the, composition functions fog and gof ar e R1 and R 2 ,, respectively, then, (a) R1 = {u: – 1 £ u < 1}, R2 = {v: – ¥ < v < 0}, (b) R1 = {u: – ¥ < u < 0}, R2 = {v: – ¥ < v < 0}, (c) R1 = {u: – 1 < u < 1}, R2 = {v: – ¥ < v < 0}, (d) R1 = {u: – 1 £ u £ 1}, R2 = {v: – ¥ < v £ 0}, 10. A relation R on the set of complex numbers is defined, z –z, by z1 R z2 if and only if 1 2 is real, then R is, z1 + z2, (a) Not reflexive, (b) Symmetric, (c) Not transitive, (d) None of these, 11., , log y, For positive real numbers x and y, let f (x, y) = x 2 . If, the sum of the solutions of the equation, , 4096 f ( f (x, x), x) = x13 can be expressed as, , m, ( where m,, n, , n are coprime numbers), then (m – 10n) is, (a) 1, (b) 2, (c) –1, (d) None of these, 12. If two roots of the equation, , ( p - 1)( x 2 + x + 1)2 - ( p + 1)( x 4 + x 2 + 1) = 0 are real and, æ æ 1 öö, 1- x, , then f ( f ( x)) + f ç f ç ÷ ÷ is, distinct and f ( x ) =, 1+ x, è è x øø, equal to, (a) p, (b) –p, (c) 2p, (d) –2p, 13. Let f (z) = sin z and g(z) = cos z. If * denotes a composition, of functions, then the value of (f + ig) * (f – ig) is :, –e, (a) i e, , (c), , –iz, , – i e– e, , –iz, , -e, (b) ie, , iz, , (d) None of these, , 2, p - 2ü, ì, 14. If A = íx : - £ x £, ý , B = {y : – 1 £ y £ 1|} and f, 5, 5 þ, î, (x) = cos (5x + 2), then the mapping f : A ® B is, (a) one-one but not onto (b) onto but not one-one, (c) both one-one and onto (d) neither one-one nor onto, 15. Let f and g be functions from R to R defined as, ì7 x 2 + x - 8, x £ 1, ì | x |, x < -3, ïï, ï, f ( x ) = í4x + 5, 1 < x £ 7 g ( x ) = í0, - 3 £ x < 2, ï 8x + 3, x > 7, ï x 2 + 4, x ³ 2, î, îï, , Then, (a) (fog) (–3) = 8, (c) (gof) (0) = – 8, , (b) (fog) (9) = 683, (d) (gof) (6) = 427

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EBD_7762, 218, , MATHEMATICS, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, , (b), (a), (b), (a), (d), (b), (b), (a), (b), (c), (c), (b), (a), (c), , 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, , (d), (d), (b), (c), (a), (c), (d), (b), (a), (d), (a), (a), (a), (c), , 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, , (b), (b), (c), (a), (a), (b), (a), (d), (c), (b), (b), (c), (c), (b), , 1, 2, 3, , (c), (b), (d), , 4, 5, 6, , (d), (b), (a), , 7, 8, 9, , (c), (c), (d), , 1, 2, , (b), (c), , 3, 4, , (b ), (d), , 5, 6, , (c), (c), , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (a) 85 (d), 43 (a) 57 (c), 71, (a) 86 (c), 44 (b) 58 (c), 72, (a), 45 (c) 59 (d), 73, 87 (d), (b) 88 (c), 46 (b) 60 (d), 74, (b) 89 (b), 47 (a) 61 (b), 75, (b) 90 (c), 48 (d) 62 (d), 76, (a) 91 (c), 49 (a) 63 (c), 77, (b) 92 (a), 50 (a) 64 (b), 78, (b) 93 (c), 51 (c) 65 (b), 79, (d) 94 (b), 52 (d) 66 (d), 80, (b) 95 (b), 53 (d) 67 (a), 81, (a) 96 (a), 54 (c) 68 (a), 82, (d) 97 (d), 55 (a) 69 (b), 83, (b) 98 (a), 56 (c) 70 (b), 84, Exercise 2 : Exemplar & Past Year MCQs, (c) 19 (a), 10 (d) 13 (b), 16, (b) 20 (b), 11 (a) 14 (a), 17, (d) 21 (b), 12 (b) 15 (a), 18, Exercise 3 : Try If You Can, (b), (d), (a) 13 (b), 7, 9, 11, (d) 10 (b), (a) 14 (c), 8, 12, , 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, , (b), (c), (a), (a), (d), (c), (c), (a), (a), (a), (a), (b), (c), (b), , 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, , (b), (d), (d), (b), (c), (a), (a), (a), (c), (a), (a), (b), (c), (b), , 22, 23, 24, , (b), (b), (a), , 25, 26, , (d), (b), , 15, , (b), , 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, , (b), (a), (b), (b), (b), (c), (c), (d), (c), (b), (b), (b), (c), (a)

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18, , INVERSE TRIGONOMETRIC, FUNCTIONS, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 2, , JEE MAIN, BITSAT, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , % Weightage, , JEE Main, , 3, , BITSAT, , 3, , Critical Concepts, , Rating of Difficulty, , CUS (chapter utility score), out of 10, , Domain and Range of, Inverse Trigonometric, functions, Properties of, Inverse Trigonometric, Functions, , 4/5, , 4.7

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INVERSE TRIGONOMETRIC FUNCTIONS, , 221

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EBD_7762, 222, , MATHEMATICS, , Topic 1 : Trigonometric Functions and Its Inverse, Domain, and Range of Inverse Trigonometric Functions, 1., , æ1, 5 ö÷, is, The value of tan ç cos -1, ç2, 3 ÷ø, è, 3- 5, (a) 3 + 5, (b), 2, 2, , (c), 2., , 4., 5., , 7., 8., 9., , 10., , (d) None of these, , 6, (b), 17, , (c), , p, 3, , (b), , 3, 17, , 4, 17, , (d), , p, 4, , (b) [1, 2), , (c), , pü, ì, ílog ý (d) {– sin 1}, 2þ, î, , (a), , æ p 3p ö, çè 4 , 4 ÷ø, , (b), , é p 3p ù, ê4, 4 ú, ë, û, , (c), , ì p 3p ü, í , ý, î4 4 þ, , (d) None of these, 2, , æ cos -1 (3x - 1) ö, + 1÷ is, 13. The range of the function y = ç, π, è, ø, (a) [1, 4], , (b) [0, p], , (c) [1, p], , (d) [0, p2], , 14. If a and b are the roots of the equation x2 – 4x + 1 = 0 (a > b), then the value of, f ( a, b ) =, , is –, (a) 56, , æ1, b3, b ö a3, aö, æ1, cosec2 ç tan -1 ÷ +, sec 2 ç tan -1 ÷, è2, aø 2, bø, 2, è2, , (b) 66, , (c) 40, , (d) 18, , Topic 2 : Principle Value for Inverse Trigonometric, , p, (c), (d) None of these, 6, The domain of the function, cos–1 log2 (x2 + 5x + 8) is(a) [2, 3] (b) [–3, –2], (c) [–2, 2] (d) [–3, 1], If 6 sin–1(x2 – 6x + 8.5) = p, then x is equal to, (a) 1, (b) 2, (c) 3, (d) 8, , If sin–1 x = tan –1 y, what is the value of, , f (x) = sin -1 (log[ x ]) + log(sin -1[ x]); (where [.] denotes, the greatest integer function) is, , -1, -1, -1, 12. Range of f(x) = sin x + tan x + sec x is, , The domain of the function defined by f ( x ) = sin -1 x - 1, is, (a) [1, 2], (b) [–1, 1], (c) [0, 1], (d) None of these, –1, 2, If sin (x – 7x + 12) = np, " nÎI, then x =, (a) –2, (b) 4, (c) –3, (d) 5, If tan–1(cot q) = 2q, then q is equal to, (a), , 6., , 5, 17, , The range of the function, , (a) R, , 2ö, æ, -1 5, + tan -1 ÷ is, The value of cot ç cosec, 3, 3ø, è, , (a), 3., , 5, 6, , 11., , 1, x2, , -, , 1, , ?, , y2, (a) 1, (b) –1, (c) 0, (d) 2, Assertion: The domain of the function sec–1 x is the set, of all real numbers., Reason: For the function sec–1 x, x can take all real values, except in the interval (–1, 1)., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., Domain of cos–1[x] is, (a) [–1, 2], (b) [–1, 2), (c) (–1, 2], (d) None of these, , Functions, Intervals for Inverse Function, -1 æ -1ö, 15. Principal value of sin çè ÷ø is equal to, 2, , (a), , p, 3, , (b), , -, , p, 3, , (c), , p, 6, , (d), , -, , p, 6, , -, , p, 2, , æ -2 ö, 16. Principal value of cosec -1 ç, ÷ is equal to, è 3ø, , (a), , -, , p, 3, , (b), , p, 3, , (c), , p, 2, , (d), , 17. What is the principle value of cosec–1 (– 2) ?, p, p, p, (b), (c) –, 4, 2, 4, 18. Principal value of sec–1(2) is equal to, , (a), , (a), , p, 6, , (b), , p, 3, , (c), , 2p, 3, , (d) 0, , (d), , 5p, 3, , (d), , 5p, 3, , 19. Prinicpal value of tan –1 ( 3 ) is equal to, (a), , p, 6, , (b), , p, 3, , (c), , 2p, 3

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EBD_7762, 224, , MATHEMATICS, , p, is, 4, (d) infinite, , 39., , -1, -1, The number of roots of equation tan 2 x + tan 3x =, , 40., , (a) 0, (b) 1, (c) 2, The value of x satisfying the equation, 1, , 3 tan -1, , 2+ 3, , - tan -1, , 1, 1, = tan -1 is, x, 3, , (a) x = 2, (c), , 41., , 1, , x=, , (, , 2- 3, The maximum value of, -1, , -1, , 2, , (a) 2, , 48., 1, 2, , (b), , x=, , d), , None of these, , 43., , 4, 3, + cos -1 , then x, 5, 5, , 45., , 46., , 7, 3, , (c), , (d), , 14 3, , (, , 6 7, , ), , -, , 2 6, 5, , (b), , æ 1ö, æ 2ö, tan -1 ç ÷ + tan -1 ç ÷ is equal to, è 4ø, è 9ø, , (a), , 1, æ 3ö, cos -1 ç ÷, è 5ø, 2, , (c), , 1, æ 3ö, tan -1 ç ÷, è 5ø, 2, , (c) tan–1(15), , 1, 2, , (b), , 0,, , 1, 2, , (c), , (b), , 1 -1 æ 3 ö, sin ç ÷, è 5ø, 2, , (d), , æ 1ö, tan -1, , (b), (d), , 0, -, , 1, 2, , (d), , 0, ±, , 1, 2, , (b), , 2, 5, , (c), , çè ÷ø, 2, , cot–1 (15), æ 25 ö, tan -1 ç ÷, è 24 ø, , é, x +1ö, -1 æ x ö ù is equal to, The limit lim x ê tan -1 æç, ÷ø - tan çè, ÷, è, x, +, 2, x + 2 ø úû, x®¥ ë, , 1, 3, , (d), , 1, 5, , 1 ì p / K , if x > 0., =í, , then the value of, x î -p / K , if x < 0, , (b), , p, 2, , 5p, 4, , (d) None of these, , -1 æ a ö, -1 æ b ö p, 52. If tan çè ÷ø + tan çè ÷ø = , then x is equal to, x, x, 2, , (a), 53., , ab, , (b), , (c) 2ab, , 2ab, , (d) ab, , cos-1 (cos(2 cot -1 ( 2 - 1))) is equal to, (a), , (b), , 2 -1, , p, 4, , 3p, (d) None of these, 4, If tan–1x – tan–1y= tan–1A, then A is equal to, , (c), 54., , -1 æ 1 ö, -1 æ 1 ö, -1 æ 7 ö, The value of tan èç ø÷ + tan çè ÷ø + tan çè ÷ø is, 2, 3, 8, , æ 7ö, tan -1 ç ÷, è 8ø, , (c), , -2 6, , 6, (c) (d) None of these, 5, The solution set of the equation sin–1x = 2 tan–1x is, (a) {1, 2}, (b) {–1, 2}, (c) {–1, 1, 0}, (d) {1, 1/2, 0}, , (a), , 47., , 4, , ±, , (a) p, , 1, The value of cos 2 cos -1 x + sin -1 x at x = is, 5, , (a), , 44., , (b), , (a), , K is, (a) 3, (b) 4, (c) 2, (d) 1, 51. The value of tan–1(1) + tan–1(0) + tan–1(2) + tan–1 (3) is, equal to, , has the value :, (a), , -, , 50. If tan -1 x + tan -1, , 5p2, (b), 4, (d) None of these, , If tan -1 (x + 1) + cot -1 ( x - 1) = sin -1, , 3, 4, 7, , 1, (d) None of these, 3, If tan–1 (x – 1) + tan –1 x + tan–1 (x + 1) = tan–1 3x, then the, value of x are, , (c), , (a) –1, , (sec x) + (cosec x) is equal to, , 42., , 1, 2, , 1, æ, ö, 49. If sin ç sin -1 + cos -1 x ÷ = 1, then the value of x is, è, ø, 5, , 2, , p2, (a), 2, (c) p2, , (b), , (a) x – y, , (b) x + y, , (c), , x-y, 1 + xy, , (b), , x+y, 1 - xy, , -1 æ x - 1 ö + tan -1 æ x + 1 ö = p ,, ÷, ç, ÷, then x is equal to, 55. If tan ç, è x+2ø, è x+2ø 4, , (a), , 1, 2, , (b), , -, , 1, , (c), , 2, , ±, , 5, 2, , (d), , é, , æ, , ë, , è, , 56. Assertion: The value of sin ê tan -1 ( - 3 ) + cos -1 ç -, , ±, , 1, 2, , 3 öù, ÷ ú is 1., 2 øû, , Reason: tan–1(–x) = tan x and cos–1(–x) = cos–1x., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct.

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INVERSE TRIGONOMETRIC FUNCTIONS, , 57., , 225, , æ ab + 1ö, æ bc + 1ö, æ ca + 1ö, cot -1 ç, + cot -1 ç, + cot -1 ç, è a - b ÷ø, è b - c ÷ø, è c - a ÷ø is equal, , 67., , tan -1, , to …, (a) 0, 58., , (b), , p, 4, , (c) 1, , 6, 17, , (c), , 16, 7, , 59. If u = cot, , (b), (d), -1, , tan a - tan, , equal to, (a), tan a, (c) tan a, 60., , 7, 16, , 69., , None of these, , æp uö, tan a , then tan ç 4 - 2 ÷ is, è, ø, , (b), , (c), , p, 2, , (d), , 3p, 4, , -1 æ 2k - k ö, -1 æ x 3 ö, If A = tan çè, , then the, ÷ø and B = tan ç, è k 3 ÷ø, 2k - x, , value of A–B is, (a) 10°, (b) 45°, (c) 60°, (d) 30°, 2, x, xy, y2, x, y p, +, 62. If cos -1 + cos -1 = , then value of, is, 4 2 3 9, 2, 3 6, 3, 1, (a), (b), 4, 2, 1, (c), (d) None of these, 4, 63. If 4 cos–1x + sin–1x = p, then the value of x is, (a), , 3, 2, , (b), , æ 2a, 64. If sin -1 ç, è 1 + a2, , (a) a/b, , 1, 2, , (c), , 3, 2, , (d), , 2, 3, , ö, -1 æ 2b ö, = 2 tan -1 x, then x =, ÷ + sin ç, 2÷, ø, è1+ b ø, , (b) b/a, , (c), , a+b, 1 + ab, , (d), , a+b, 1 - ab, , 1/ x, , æπ, ö, equals, lim ç - tan -1 x ÷, ø, x ®¥ è 2, (a) 0, (b) 1, (c) ¥, 66. If tan–13 + tan–1 x = tan –18, then x =, , 65., , (a) 5, , (b), , 1, 5, , (c), , 5, 14, , -, , 1, 2, , (c), , ±, , 1, 2, , (d) 0, , 11p, has, 6, (a) No solution, (b) Only one solution, (c) Two solutions, (d) Three solutions, If 2 tan–1 (cos x) = tan –1 (2 cosec x) then value of x is, , (b), , p, 3, , (c), , p, 4, , (d), , p, 2, , æ 1 ö, –1, 70. If cos -1 ç, ÷ = q , then the value of cosec ( 5) is, è 5ø, , p, æ pö, æpö, (d) - q, ç 2 ÷ +q (b) ç 2 ÷ - q (c) 2, è ø, è ø, 71. What is the value of, tan (tan–1x + tan–1y + tan–1z) – cot (cot–1x + cot–1y + cot–1z) ?, (a) 0, (b) 2 (x + y + z), (a), , (b), cot a, (d) cot a, , p, 4, , 2, , (a) 0, , 1, 1, 1, 1, tan -1 + tan -1 + tan -1 + tan -1 =, 3, 5, 7, 8, , (a) p, 61., , -1, , (b), , –1, –1, 68. The equation 2 cos x + sin x =, , 4, 2ö, æ, The value of tan ç cos -1 + tan -1 ÷ =, è, 5, 3ø, , (a), , 1, , (a), , (d) 5, , x -1, x +1 p, + tan -1, =, x-2, x+2 4, , (d) –1, (d), , 14, 5, , (c), , 3p, 2, , (d), , 3p, +x+y+z, 2, , 72. If k £ sin -1 x + cos -1 x + tan -1 x £ K, then, (a), , k = – p, K = p, , p, 3p, , K=, 4, 4, 73. Assertion:, , (c), , k=, , (b), , k = 0, K =, , p, 2, , (d) k = 0, K = p, , æ3ö, æ 2ö, æ1ö, cosec-1 ç ÷ + cos -1 ç ÷ - 2 cot -1 ç ÷ - cot -1 ( 7 ), 2, 3, è ø, è ø, è7ø, is equal to cot–1 7., Reason: sin -1 x + cos -1 x =, , p, p, -1, -1, , tan x + cot x = ,, 2, 2, , æ1ö, æ1ö, cosec -1x = sin -1 ç ÷ , cot -1 ( x ) = tan -1 ç ÷, èxø, èxø, (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., p, 74. If sin–1 (1– x ) – 2 sin–1 x = , then x equals, 2, (a), , 0, -, , (c) 0, , 1, 2, , 1, 2, (d) None of these, , (b), , 0,

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EBD_7762, 226, , 75., , 76., , MATHEMATICS, , æp, ö, cot ç - 2 cot -1 3÷ =, è4, ø, , 85. The value of sec2(tan–12) + cosec2(cot–13) is, (a) 12, (b) 5, (c) 15, (d) 9, , (a) 7, (b) 6, (c) 5, (d) None of these, The + ve integral solution of, , 86., , -1, , tan x + cos, , y, , -1, , 1 + y2, , = sin, , -1, , (a), , x = 1, y = 2; x = 2, y = 7, , (b), , x = 1, y = 3; x = 2, y = 4, , 3 is, 10, , (a), , The value of, (a) 1, , (a), (b), (c), (d), , 78., , 79., 80., , æ 6x, If sin -1 ç, è 1 + 9x 2, (a) 3, (b), , ö, -1, ÷ = 2 tan (ax) , then a =, ø, 8, (c) 6, , -1, If tan–1k – tan–1 3 = tan, , (b) 2, , 1, , then k =, 13, (c) 4, , -1 æ 1 ö, , (d) 5, (a), , If x, a Î R, where, , (c) 4, , -a, 3, , <x<, , 83., , 84., , æ 3a 2 x – x 3, , then tan – 1 çç 3, 2, 3, è a – 3ax, , a, , ö, ÷ is, ÷, ø, 91., , p, If sin ç ÷ + cosec ç ÷ = , then the value of x is, 5, 4, è ø, è ø 2, (a) 4, (b) 5, (c) 1, (d) 3, -1 æ 5 ö, , æ xö, æ x - yö, is equal to (Where x > y > 0), tan -1 ç ÷ - tan -1 ç, è yø, è x + y ø÷, , (c), , 3p, 4, , (b), , 6, 13, , 5 2, , (c), , 51, ,, 50, , 2 2, , (d), , 4, 5, , (d) –, , 5 2, , ì, æ -3 ö ü, sin í2 cos -1 ç ÷ ý is equal to, è 5 øþ, î, , (c), , 24, 25, , é 1 - sin x + 1 + sin x ù, x, cot -1 ê, ú is p - ., 2, ë 1 - sin x - 1 + sin x û, , (d) 5, , (d) None of these, , p, 4, , (d), , p, 2, (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not a, correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., If xy + yz + zx = 1, then :, (a) tan–1 x + tan –1 y + tan –1 z = 0, (b) tan–1 x + tan–1 y + tan –1 z = p, p, (c) tan–1 x + tan–1 y + tan –1 z =, 4, , Reason: tan–1 x + cos–1 x =, , æaö, 3 tan – 1 ç ÷, èxø, , -, , 1, , (a), , (d) 5, , æxö, (a) 3 tan – 1 ç ÷, èaø, , (a), , 2, , (b), , 6, 24, (b), 25, 25, 90. Assertion: The value of, , æxö, (b) – 3 tan – 1 ç ÷, èaø, , -1 æ x ö, , 1, , (d) 0.96, , equal to, , (c), , 29, 3, , (c), , unique solution, no solution, infinitely many solutions, None of these, , 89., , -1, , (b) 2, , 4, 31, , (d) 9, , çè ÷ø + 2 tan ( 3) = kp, then k =, 2, , (a) 1, , (b), , 88. Solve for x : {x cos(cot–1x) + sin(cot–1 x)}2 =, , -1 æ 1 ö, -1, -1 æ 1 ö, -1, If sin ç ÷ + sec (2) + 2tan ç ÷ + sec (5), è 5ø, è 3ø, , + sin, , 82., , (c) 4, , The value of sin [2 sin–1 (0.8)] is equal to, (a) sin 1.2° (b) sin 1.6°, (c) 0.48, , (a) 1, 81., , sin (tan -1 x + cot –1 x), is, sin (sin -1 x + cos –1 x), , (b) 2, , 2, 23, , æ 3ö, 87. The equation sin –1x – cos–1x = cos–1 çè ÷ø has, 2, , (c) x = 0, y = 0; x = 3, y = 4, (d) None of these, 77., , é, 1, æ 5 öù, tan êcos -1, - sin -1 ç, ÷ ú is equal to, 82, è 26 ø û, ë, , p, 4, , (d) None of these, , (d) tan–1 x + tan–1 y + tan –1 z =, , p, 2, , æ 1 ö, -1, 92. The value of sin –1 ç, ÷ + cot (3) is, è 5ø, p, p, p, (b), (c), (a), 6, 3, 4, , 93. The solution of sin–1 x – sin –12x = ±, (a), , ±, , 1, 3, , (b), , ±, , 1, 4, , (c), , ±, , (d), , p, 2, , (d), , ±, , p, is, 3, , 3, 2, , 1, 2

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INVERSE TRIGONOMETRIC FUNCTIONS, , 229, , Exercise 3 : Try If You Can, 1., , 2., , 3., , 4., , 1, + tan– 1 x = pk. The value of, x, k for different values of x are k1 and k2, satisfy the equation, x2 y 2, +, = 1 and y2 = 12(x – k2), then the equation of their, 4k1 5, common tangent is, (a) y = 2x – 5, (b) 3y = x + 3, (c) y = x + 3, (d) y = 3x + 1, The complete set of values of a for which the function, f(x) = tan–1(x2 – 18x + a) > 0 " x Î R is, (a) (81, ¥), (b) [81, ¥), (c) (–¥, 81), (d) (–¥, 81], If [sin–1 cos–1 sin–1 tan–1 x] = 1, where [.] denotes the, greatest integer function, then x belongs to the interval, (a) [tan sin cos 1, tan sin cos sin 1], (b) [tan sin cos 1, tan sin cos sin 2], (c) [–1, 1], (d) [sin cos tan 1, sin cos sin tan 1], p, In triangle ABC, ÐC = and, 2, æ bx ö, -1, -1 æ ax ö, sin ( x) = sin ç ÷ + sin -1 ç ÷ , where a, b, c are the, è c ø, è c ø, sides of triangle, then total number of different values of x is, (a) 2, (b) 3, (c) 1, (d) None of these, , 5., , cos -1[cos{2 cot -1 ( 2 - 1)}] is equal to, , 6., , p, (d), 4, The set of values of x for which the identity, , (a), , 2 -1, , cos, , -1, , (c), , -1, , 3p, 4, , 9., , 10., , (b), , é, ê 0,, ë, , 1ù, 2 úû, , (c), , é1 ù, ê 2 , 1ú, ë, û, , p, p, p, <A<, (b) p, if 0 < A <, 4, 2, 4, (c) both (a) and (b), (d) None of these, 11. cot–1(2.12) + cot–1(2.22) + cot–1(2.32) +... is equal to, p, p, p, p, (a), (b), (c), (d), 3, 5, 4, 2, 12. If the equation x3 + bx2 + cx + 1 = 0 (b < c) has only one real, root a., Then the value of 2 tan –1 (cosec a) + tan–1 (2 sin a sec2 a), is:, , (a) 0 if, , p, p, (b) –p, (c), 2, 2, 13. The number of solutions of the equation, , ( n2 + 1)( n2 - 2n + 2), 2, ( n2 - n + 1) is, , (a) tan–1 1, (b) tan–1 n, –1, (c) tan (n + 1), (d) tan–1 (n – 1), If x1, x2, x3, x4 are roots of the equation x4 – x3 sin 2b + x2 cos, 2b – x cos b – sin b = 0 then, , 4, , å tan -1 xi, , i =1, , is equal to, , -, , (a), , tan -1 x =, , ( x2 + 1), , 2, , k, , 14. If, , (b) 3, (d) none of these, , å cos-1 br =, r =1, k, , kp, for any k ³ 1 where br ³ 0"r and, 2, , A = å (br )r , Then lim, , 15., , 1, 2, , (1 + x 2 )1/ 3 - (1 - 2 x)1/ 4, x + x2, , x® A, , r =1, , (a), , (d) p, , - 4 x 2 is, , (a) 2, (c) 4, , (d) {–1, 0, 1}, , 10, 50, + sec -1, +...+, 3, 7, , p, p, -b, - 2b, (d), 2, 2, The minimum integral value of a for which the quadratic, equation (cot–1 a)x2 – (tan–1 a)3/2 x + 2(cot–1 a)2 = 0 has, both positive roots, (a) 1, (b) 2, (c) 3, (d) 4, The value of, , æ1, ö, tan -1 ç tan 2A) + tan -1 (cot A) + tan -1 (cot 3 A) ÷ is, è2, ø, , x 1, ö p, x + cos ç +, 3 - 3 x 2 ÷ = holds good is, 2, 2, è, ø 3, , -1, 2 + sec-1, The sum of series sec, , sec, , 8., , (b) 1 - 2, , (b) p – 2b, , (c), , -1 æ, , (a) [0, 1], 7., , (a) p – b, , If sin– 1 x + cos– 1 x + tan– 1, , (b) 0, , (c), , A, 2, , (d), , =, , p, 2, , 1, æ, ö, 1, ö, -1 æ, S = tan -1 ç, ÷ + tan ç 2, ÷ + ..., 2, è n + n +1 ø, è n + 3n + 3 ø, æ, ö, 1, + tan -1 ç, ÷ , then tan S is equal to :, è 1 + ( n + 19)( n + 20) ø, 20, n, (a), (b), 2, 401 + 20 n, n + 20n + 1, n, 20, (c), (d), 2, 401, +, 20 n, n + 20n + 1

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EBD_7762, 230, , MATHEMATICS, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, , (b), (b), (a), (b), (c), (b), (b), (a), (d), (b), , 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, , (c), (d), (a), (a), (d), (a), (c), (b), (b), (d), , 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, , (b), (a), (b), (a), (c), (a), (a), (c), (b), (d), , 1, 2, 3, , (c), (d), (b), , 4, 5, 6, , (d), (a), (a), , 7, 8, 9, , (b), (c), (a), , 1, 2, , (c), (a), , 3, 4, , (a), (b), , 5, 6, , (d), (c), , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (d), 31 (d) 41 (b) 51 (a) 61, (c), 32 (b) 42 (a) 52 (a) 62, (c), 33 (d) 43 (a) 53 (c) 63, (d), 34 (b) 44 (c) 54 (c) 64, (b), 35 (b) 45 (d) 55 (c) 65, (b), 36 (a) 46 (c) 56 (c) 66, (c), 37 (b) 47 (b) 57 (a) 67, (a), 38 (d) 48 (d) 58 (d) 68, (c), 39 (b) 49 (d) 59 (a) 69, (b), 40 (a) 50 (c) 60 (b) 70, Exercise 2 : Exemplar & Past Year MCQs, (b), 10 (b) 13 (d) 16 (c) 19, (b), 11 (a) 14 (b) 17 (a) 20, (c), 12 (d) 15 (a) 18 (c) 21, Exercise 3 : Try If You Can, (b) 9, (b) 11 (a) 13, (c), 7, (c) 10 (c) 12 (b) 14, (a), 8, , 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, , (a), (c), (d), (b), (a), (a), (a), (a), (d), (c), , 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, , (b), (a), (d), (b), (c), (a), (a), (b), (d), (b), , 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, , (d), (b), (d), (c), (c), (b), (b), (b), (d), (a), , 22, 23, 24, , (c), (b), (d), , 25, 26, 27, , (b), (c), (d), , 28, , (c), , 15, , (c)

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19, , MATRICES, , Chapter, , Trend, Analysis, , Number of Questions, , off JEE Main and BITSAT (Year 2010-2018), , JEE MAIN, BITSAT, , 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 4, 3, , Critical Concepts, , Operations on Matrices, Properties, of Multiplication of Matrices,, Transpose of a Matrix, Symmetric &, Skew Symmetric matrices, Elementary, operation of a Matrix, Inverse of, a Matrix by Elementary operations, , Rating of, Difficulty Level, , CUS, (chapter utility score), Out of 10, , 4/5, , 6.3

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MATRICES, , 233

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EBD_7762, 234, , MATHEMATICS, , Topic 1: Order of Matrices, Types of Matrices, Addition, and subtraction of Matrices, , 1., , 2., , é x + y 2x + z ù é4 7 ù, If ê, ú=ê, ú , then the values of x, y, z, ë x - y 2z + w û ë0 10 û, and w respectively are, (a) 2, 2, 3, 4, (b) 2, 3, 1, 2, (c) 3, 3, 0, 1, (d) None of these, , If A = [aij]2×2, where aij =, , ( i + 2 j)2, 2, , , then A is equal to, , é9 25ù, é9 / 2 25 / 2 ù, (b) ê 8, ê8 18 ú, 18 úû, ë, û, ë, é9 25ù, é9 / 2 15 / 2 ù, (c) ê 4 9 ú, (d) ê 4, 9 úû, ë, û, ë, In a upper triangular matrix n × n, minimum number of, zeros is, (a) n(n – 1)/2, (b) n(n + 1)/2, (c) 2n (n – 1)/2, (d) None of these, A square matrix A = [aij]n × n is called a lower triangular, matrix if aij = 0 for, (a) i = j, (b) i < j, (c) i > j, (d) None of these, For what values of x and y are the following matrices equal, , (a), , 3., , 4., 5., , 3y ù, é 2x + 1, é x + 3 y2 + 2ù, A=ê, B, =, ú, ê, ú, ,, y2 - 5y ûú, -6 ûú, ëê 0, ëê 0, 6., , 7., , 8., , (a) 2, 3, (b) 3, 4, (c) 2, 2, (d) 3, 3, A square matrix A = [aij]n × n is called a diagonal matrix if, aij = 0 for, (a) i = j, (b) i < j, (c) i > j, (d) i ¹ j, 6 3y - 2ù, é x + 3 z + 4 2y - 7 ù, é 0, ê 4x + 6 a - 1, ú, ê, 0 ú, 2x, -3 2c + 2 úú, If ê, = ê, êë b - 3, êë 2b + 4 -21, 3b z + 2c úû, 0 úû, then, the values of a, b, c, x, y and z respectively are, (a) – 2, – 7, – 1, – 3, – 5, – 2 (b), 2, 7, 1, 3, 5, – 2, (c) 1, 3, 4, 2, 8, 9, (d), – 1, 3, –2, –7, 4, 5, The construction of 3 × 4 matrix A whose elements aij is, , given by, , (i + j) 2, is, 2, , (a), , é2 9 / 2 8 25ù, ú, ê, 5 18 ú, ê9 4, êë8 25 18 49úû, , (b), , é 2 9 / 2 25 / 2 9 ù, ú, ê, 5 45 / 2ú, ê9 / 2 5 / 2, êë 25 18, 25 9 / 2 úû, , (c), , 9/2, 8, 25 / 2ù, é 2, ú, ê, 8, 25 / 2 18 ú, ê9 / 2, êë 8 25 / 2 18, 49 / 2úû, , (d), , None of these, , é x + y + zù é9ù, If ê x + y ú = ê 5 ú then the value of (x, y, z) is:, êë y + z úû êë 7 úû, (a) (4, 3, 2), (b) (3, 2, 4), (c) (2, 3, 4), (d) None of these, 10. If A is a 3 × 2 matrix, B is a 3 × 3 matrix and C is a 2 × 3, matrix, then the elements in A, B and C are respectively, (a) 6, 9, 8 (b) 6, 9, 6 (c) 9, 6, 6 (d) 6, 6, 9, 11. Consider the following statements., I. If a matrix has 24 elements, then all the possible orders, it can have are 24 × 1, 1 × 24, 2 × 4, 4 × 2, 2 × 12, 12, × 2, 3 × 8, 8 ×3, 4 ×6 and 6 × 4., II. For a matrix having 13 elements, its all possible orders, are 1 × 13 and 13 × 1., III. For a matrix having 18 elements, its all possible orders, are 18 × 1, 1 × 18, 2 × 9, 9 × 2, 3 × 6, 6 × 3., IV. For a matrix having 5 elements, its all possible orders, are 1 × 5 and 5 × 1., Choose the correct option, (a) Only I is false, (b) Only II is a false, (c) Only III is false, (d) All are true, 12. If A = [aij]3 × 4 is matrix given by, , 9., , 3 ù, é 4 -2 1, ê5 7 9, 6 úú, A= ê, êë 21 15 18 -25úû, Then, a23 + a24 will be equal to the element, (a) a14, (b) a44, (c) a13, (d) a32, 13. A square matrix B = [bij] m × m is said to be a diagonal matrix,, if, (a) all its non-diagonal elements are non-zero i.e., bji ¹, 0, ;, i ¹j, (b) all its diagonal elements are zero, i.e., bij = 0, i = j, (c) all its non-diagonal elements are zero i.e, bij = 0 when i ¹ j, (d) None of the above, 14. Choose the incorrect statement., (a) A matrix A = [3] is a scalar matrix of order 1, é -1 0 ù, (b) A matrix B = ê 0 -1ú is a scalar matrix of order 2, ë, û, é 3 0, ê, 0, 3, (c) A matrix C = êê, 0, ë0, matrix, (d) None of the above, , 0ù, ú, 0ú, ú of order 3 is not a scalar, 3û

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EBD_7762, 244, , MATHEMATICS, , If Z is an idempotent matrix, then (I + Z)n, (a) I + 2n Z, (b) I + (2n – 1) Z, (c) I – (2n – 1) Z, (d) None of these, , 13., , é 0 – tan q / 2 ù, Let A = ê, (q ¹ np), 0 úû, ë tan q / 2, , 14., , é cos q – sin q ù, é1 0 ù, ,I =ê ú, B = ê, ú, ësin q cos q û, ë0 1û, Then the matrix I + A is equal to, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, , (a), (b), (a), (b), (c), (d), (a), (c), (c), (b), (a), (d), , 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, , (c), (c), (a), (b), (b), (c), (d), (a), (b), (a), (d), (c), , 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, , (c), (c), (b), (d), (b), (d), (a), (b), (c), (a), (a), (c), , 1, 2, 3, , (a), (d), (b), , 4, 5, 6, , (d), (d), (d), , 7, 8, 9, , (a), (b), (c), , 1, 2, , (a), (c), , 3, 4, , (b), (a), , 5, 6, , (a), (a), , (a) (I – A)B, (c) (I + A)2B, écos a - sin a, 15. Let F(a) = êê sin a cos a, êë 0, 0, , (b) (I – A)2B, (d) (I – A)2, 0ù, ú, 0ú where a Î R . Then, 1úû, , [F(a)]-1 is equal to, , (a), , F(-a), , (c), , F(2a), , ANSW ER KEYS, Exerci se 1 : Topi c-wi se MCQs, (d), (d), (a), (b), 37, 49, 61, 73, (a), (a), (a), (b), 38, 50, 62, 74, (d), (b), (a), (a), 39, 51, 63, 75, (b), (b), (b), (c), 40, 52, 64, 76, (b), (c), (b), (b), 41, 53, 65, 77, (c), (c), (d), (a), 42, 54, 66, 78, (a), (c), (b), 43, 55, 67, 79 (b), (b), (b), (d), (b), 44, 56, 68, 80, (b), (c), (c), (a), 45, 57, 69, 81, (d), (a), (c), (c), 46, 58, 70, 82, (d), (d), (d), (c), 47, 59, 71, 83, (a), (a), (a), (c), 48, 60, 72, 84, Exerci se 2 : Exempl ar & Past Year MCQs, (d), (d), (d), (c), 10, 13, 16, 19, (a), (d), (b), (a), 11, 14, 17, 20, (a), (a), (d), (a), 12, 15, 18, 21, Exercise 3 : Try If You Can, (a), (b), (a), (b), 7, 9, 11, 13, (b), (a), (c), (a), 8, 10, 12, 14, , (b) F(a -1 ), (d) None of these, , 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, , (b), (b), (b), (b), (b), (b), (b), (a), (b), (a), (c), (c), , 15, , (a), , 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, , (a), (c), (d), (c), (b), (d), (b), (a), (a), (c), (b), (a), , 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, , (b), (a), (b), (c), (a), (d), (d), (b), (c), (b), (b), (b)

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20, , DETERMINANTS, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 4, , 3, JEE MAIN, BITSAT, , 2, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 4, 6, , Critical Concepts, , Properties of Determinants, Area of, a Triangle, Adjoint of a matrix,, Solution of system of linear equations, using inverse of a matrix, , Rating of, Difficulty Level, , 4.5/5, , CUS, (chapter utility score), Out of 10, 7.3

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DETERMINANTS, , 247

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EBD_7762, 248, , MATHEMATICS, , Topic 1 : Minor and Co-factor of an element of a, Determinant, Value of a Determinant Singular and, Non-Singular Matrices, 1., , a + ib, , c + id, , -c + id a - ib, , 9., , (a) 2013 (b) 2014, (c) (2013)2 (d) (2014)2, 10. Find the cofactors of elements a12, a22, a32, respectively of, , =, , sin q, 1 ù, é 1, ê - sin q, 1, sin, qúú, the matrix ê, êë -1, - sin q, 1 úû, , (a) (a + b)2 (b) (a + b + c + d)2, (c) (a2 + b2 – c2 – d2), (d) a2 + b2 + c2 + d2, 2., , 3., , 4., , 5., , cos15°, , sin15°, , sin 75° cos 75°, , =, , (a) 0, (b) 5, (c) 3, (d) 7, If the area of a triangle ABC, with vertices A(1, 3), B(0,, 0) and C(k, 0) is 3 sq. units, then the value of k is, (a) 2, (b) 3, (c) 4, (d) 5, é3 5ù, é1 17 ù, If A = ê, and B = ê, ú, ú , then | AB | is equal to :, ë2 0û, ë0 -10 û, (a) 80, (b) 100, (c) – 110, (d) 92, If Ai j denotes the cofactor of the element a ij of the, , 2 -3, , 6., , 7., , 5, , determinant 6 0 4 , then value of a111A31 + a13A32 +, 1 5 -7, a13A33 is, (a) 0, (b) 5, (c) 10, (d) – 5, If ci j is the cofactor of the element ai j of the determinant, 2 -3 5, 6, , 0, , 4, , 1, , 5, , -7, , , then write the value of a32.c32, , (a) 110, (b) 22, (c) – 110, (d) – 22, If A, B, C are the angles of a triangle, then the value of, -1, , cos C, , D = cos C, , -1, , cos B, cos A is, , cos B cos A, , 8., , -1, , (a) cosA cosB cosC, (c) 0, The roots of the equation, , (b) sinA sinB sinC, (d) None of these, , 0 x 16, x, , 5, , 7, , 0 9, , x, , = 0 are :, , (a) 0, 12 and 12, (c) 0, 12 and 16, , r – 1ö, æ r, ÷ , r Î N. The, Matrix M r is defined as M r = çç, r, –, 1, r ÷ø, è, value of det( M1 ) + det( M 2 ) + det( M 3 ) + .... + det(M2014) is, , (b) 0 and ±12, (d) 0, 9 and 16, , 11., , (a) 0, 2, –2 sin q, (b) 2, 0, 2sin q, (c) 2, 0, –2 sin q, (d) – 2sin q, 2, 0, Let A = [aij] be a matrix of order 3 × 3., Assertion: Expansion of determinant of A along second, row and first column gives the same value., Reason: Expanding a determinant along any row or, column gives the same value., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., x 3 7, , 12. If (x + 9) = 0 is a factor of 2, 7, factor is:, (a) (x – 2) (x – 7), (b), (c) (x + 9) (x – a), (d), 13. The value of, a2, , a, , x 2 =0, then the other, 6 x, , (x – 2) (x – a), (x + 2) (x + a), , 1, , cos(nx) cos(n + 1)x cos(n + 2)x, sin(nx), , sin(n + 1)x, , sin(n + 2)x, , is independent of :, (a) n, (b) a, (c) x, (d) None of these, 14. Consider the following statements, I. To every rectangular matrix A = [aij] of order n, we, can associate a number (real or complex) called, determinant of A., II. Determinant is a function which associates each, square matrix with a unique number (real or complex)., (a) Only I is true, (b) Only II is true, (c) Both I and II are true (d) Neither I nor II is true

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DETERMINANTS, , 249, , 15. The parameter on which the value of the determinant, 2, , 1, a, a, cos(p - d)x cos px cos(p + d)x d oe s n ot d e p en d, sin(p - d)x sin px sin(p + d)x, , upon is, (a) a, , (b) p, , (c) d, , (d) x, , 2 3ù, é1, , then | A | is, 16. If A = ê, 4, -5 6 úû, ë, (a) 2, (b) 0, (c) –2, (d) Does not exist, , 17. The maximum and minimum value of (3 × 3) determinant, whose elements belongs to {0, 1} is, (a) 1, – 1, (b) 2, – 2, (c) 4, – 4, (d) None of these, x, sin q cos q, 18. The determinant - sin q - x, 1, is, cos q, 1, x, , 23. The value of, (a) –1, 24., , (c) ± 3, , cos80°, , (b) 1, , is, , (c) 2, , (d) 0, , 1 4, The value of the determinant D = 0 12, 1 2, , 3, 9 is, 2, , 2 6 9, 1 7 8 is, 1 4 5, , (a) 0, (b) 3, (c) 5, (d) 7, 2, 27. If 1, w, w the cube roots of unity, then the value of, , éa 2 ù, 3, 20. If A = ê, ú and | A | = 125, then the value of a is, ë 2 aû, , (b) ± 2, , sin 80°, , (a) 2, (b) 4, (c) 6, (d) 8, 25. The area of the triangle formed by the points (1, 2), (k,, 5) and (7, 11) is zero then the value of k is, (a) 0, (b) 3, (c) 5, (d) 7, 26. The minor of the element a11 in the determinant, , (a) independent of q only, (b) independent of x only, (c) independent of both q and x, (d) None of the above, 19. If area of triangle is 4 sq units with vertices (–2, 0), (0, 4), and (0, k), then k is equal to, (a) 0, –8 (b) 8, (c) – 8, (d) 0, 8, , (a) ± 1, , sin10° - cos10°, , w2n, , w2n, , 1, , wn, , wn, , w2n, , 1, , is equal to, , (b) w, , (c) w2, , (d) 0, , Topic 2 : Properties of Determinant, 28. If x, y, z are all distinct and, x x 2 1 + x3, y y 2 1 + y 3 = 0 , then the value of xyz is, z z 2 1 + z3, , é 1 0 xù, é 1 0 aù, ê, ú, ê, ú, 21. Let A = ê 2 3 b ú , B = ê 2 3 y ú, êë -3 1 z úû, êë-3 1 c úû, , Assertion: det A + det B = det C., Reason: A + B = C., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 22. In how many ways, the determinant of order 3 can be, expanded?, (a) 5, (b) 4, (c) 3, (d) 6, , wn, , (a) 1, , (d) ± 5, , é 1 0 a + xù, and C = êê 2 3 b + y úú, êë -3 1 c + z úû, , 1, , (a) –2, (c) –3, , 29., , (b) –1, (d) None of these, , 2xy, , x2, , y2, , x2, , y2, , 2xy =, , y2, , 2xy, , x2, , (a) (x3 + y3)2, (c) – (x2 + y2)3, , (b) (x2 + y2)3, (d) –(x3 + y3)2, , 30. The value of the determinant of nth order,, , being given by, , x, 1, , 1, x, , 1 ..., 1 ..., , 1 1 x ..., ... ... ... ..., , , is, , (a) (x – 1)n–1 (x + n – 1), , (b) (x – 1)n–1 (x + n – 1), , (c) (1 – x)n–1 (x + n – 1), , (d) None of these

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EBD_7762, 250, , MATHEMATICS, , 31. If x is positive integer, then, x!, ( x + 1)! ( x + 2 )!, ( x + 1)! ( x + 2 )! ( x + 3)!, ( x + 2 )! ( x + 3)! ( x + 4 )!, , 32., , is equal to, , then the value of k is :, (a) 2, (b) 4, , (a) 2x! (x + 1)!, (b) 2x! (x + 1)! (x + 2)!, (c) 2x! (x + 3)!, (d) 2(x + 1)! (x + 2)! (x + 3)!, The solution set of the equation, , 1, , 4, , 1 2x, , x+ y, , 2 z, , (c) {1, 5}, , (d) {2, – 1}, , yz + 2x, , z, , 2z ;, , y + xz, , yz, , z, , (ei b - e - i b ) 2, , 4 is, , (b), , (c), , x ( 2 y - z y), , (d) None of these, , y ( 2 z - y z), , x a b, The factors of a x b are:, a b x, , ), , ig, , (e - e, , -i g 2, , ), , 4, , (c) independent of a , b only, , 1 + sin x cos x, log(1 + x), 2 , -1 < x £ 1, is, 1 + x2, , 0, , (a) 1, (b) – 2, (c) – 1, (d) 0, 41. If rows and columns of the determinant are interchanged,, then its value, (a) remains unchanged, (b) becomes change, (c) is doubled, (d) is zero, 42. If b2 – ac < 0, a < 0 then the value of, a, b, ax + by, b, c, bx + cy is, ax + by bx + cy, 0, , x – a, x – b and x + a + b, x + a, x + b and x + a + b, x + a, x + b and x – a – b, x – a, x – b and x – a – b, , 1 1 ∗ ac 1 ∗ bc, 1 1 ∗ ad 1 ∗ bc is equal to:, 1 1 ∗ ac 1 ∗ bc, , (b) 1, (d) 3, , If a, b, g are the roots of px3 + qx2 + r = 0, then the value, ga, , of the determinant bg ga ab is, ga ab bg, (a) p, (b) q, (c) 0, , (d) r, , x 10 5, 3 5 6, 7, 8, 9, If D =, , then 5 3 6 equal to:, 8 7 9, 10 x 5, , (b) –D, , -i g 2, , (b) dependent on a , b and g, , x2, , z ( 2 y - z y), , (a) D, , D = (ei b + e- i b ) 2, , x, f (x) = 1, , (a), , ab bg, , 37., , 4, , (d) independent of a , g only, 40. The coefficient of x in, , z, , (a) a + b + c, (c) 0, 36., , (d) 8, , (a) independent of a , b and g, , 5x 2, , where x, y, z are positive real numbers, is, , 35., , (ei a - e - i a )2, , (e + e, , 5 = 0 is:, , The value of the determinant, , (a), (b), (c), (d), , (ei a + e- i a ) 2, ig, , (a) {0, 1} (b) {1, 2}, , 34., , (c) 6, , 39. If a , b, g Î R, then the determinant, , 20, , 1 ,2, , 33., , y+z x-z x -y, 38. If y - z z + x y - x = kxyz,, z-y z-x x+y, , (c) Dx, , (d) 0, , (a) Zero, (b) Positive, (c) Negative, (d) b2 + ac, 43. If any two rows (or columns) of a determinant are identical, (all corresponding elements are same), then the value of, determinant is, (a) 1, (b) –1, (c) 0, (d) 2, 44. Area of the triangle whose vertices are (a, b + c), (b, c + a), and (c, a + b), is, (a) 2 sq units, (b) 3 sq units, (c) 0 sq unit, (d) None of these, 45. Consider the following statements, I. If any two rows (or columns) of a determinant are, interchanged, then sign of determinant changes., II. If any two rows (or columns) of a determinant are, interchanged, then the value of the determinant, remains same., (a) Only I is true, (b) Only II is true, (c) Both I and II are true (d) Neither I nor II is true

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DETERMINANTS, , 251, , (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., , 1 1, 1, 1 . for x ¹ 0, y ¹ 0 , then D is, 46. If D = 1 1 + x, 1 1 1+ y, , (a) divisible by x but not y, (b) divisible by y but not x, (c) divisible by neither x nor y, (d) divisible by both x and y, 47. The value of x obtained from the equation, x+a, , b, , g, , g, , x +b, , a, , a, , b, , x+g, , 54., = 0 will be, , (b) 0 and a + b + g, , (c) 1 and (a - b - g), , (d) 0 and a 2 + b 2 + g 2, , p, q-y r-z, p q r, 48. If p - x, q, r - z = 0 , then the value of + + is, x y z, p-x q-y, r, , (a) 0, (b) 1, (c) 2, (d) 4pqr, 49. For positive numbers x, y, z the numerical value of the, , determinant, , 3, , log y z, , log z x, , log z y, , 5, , (a) 0, (c) 1, 50. If, , 2a, , 2a, , 2b, , b-c-a, , 2b, , 2c, , 2c, , c-a-b, , (a) 0, , is, , (b) log x log y log z, (d) 8, , a -b-c, , (b) 1, , 3, = k ( a + b + c ) , then k is, , (c) 2, , 51. If a, b, c are cube roots of unity, then, , (b) – 39, , sin 2 x, , cos 2 x 1, , cos 2 x, , sin 2 x, , ,10, , 12, , (d) 3, ea, , e 2a, , e 3a - 1, , eb, , e 2b, , e 3b - 1 =, , ec, , e 2c, , e 3c - 1, , (a) 0, (b) e, (c) e2, (d) e3, 52. Assertion: If a, b, c are even natural numbers, then, a -1 a a +1, D = b - 1 b b + 1 is an even natural number.., c -1 c c +1, , Reason: Sum and product of two even natural number is, also an even natural number., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, , (c) 96, , (d) 0, , 1 is equal to:, 2, , (a) 0, (b) 12 cos2 x – 10 sin2 x, (c) 12 cos2 x – 10 sin2 x – 2 (d) 10 sin 2x, 55. If a, b, c are in A. P., then the value of, x ∗1, x∗2, x∗3, , x∗2, x ∗3, x∗4, , x∗a, x∗b, is:, x∗c, , (a) 3, (b) – 3, (c) 0, (d) None of these, 56. For positive numbers x, y, z, the numerical value of the, 1, , determinant log y x, log z x, , log x y log x z, , log y x, , 14 17 20 is equal to:, 15 18 21, , (a) 57, , (a) 0 and -(a + b + g ), , 1, , 13 16 19, , 53., , log x y log x z, 1, , log y z is, log z y, 1, , (a) 0, (b) 1, (c) 2, (d) None of these, 57. If a, b, c are in A.P. then the determinant, 2 y + 4 5y + 7 8 y + a, 3 y + 5 6 y + 8 9 y + b is equal to, 4 y + 6 7 y + 9 10 y + c, (a) 0, (b) y2 + 10, , (c) 4(y2 + 5), , (d) y3, , 1+ a, 1, 1, 58. If 1 + b 1 + 2b, 1 = 0 where, 1 + c 1 + c 1 + 3c, , a ¹ 0, b ¹ 0, c ¹ 0 , then a -1 + b -1 + c -1 is, (a) 4, (b) –3, (c) –2, (d) –1, 59. If each of third order determinant of value D is multiplied, by 4, then value of the new determinant is:, (a) D, (b) 21D, (c) 64D, (d) 128D, 60. The value of determinant, sin 2 13° sin 2 77° tan135°, sin 2 77° tan135° sin 2 13° is, tan135° sin 2 13° sin 2 77°, , (a) –1, , (b) 0, , (c) 1, , (d) 2

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EBD_7762, 252, , 61., , 62., , MATHEMATICS, , Let A be a matrix of order 3 and let D denotes the value of, determinant A. Then determinant (– 2A) =, (a) – 8 D (b) – 2D, (c) 2 D, (d) 8 D, –1, –1, –1, If a + b + c = 0 such that, 1∗ a, , 1, , 1, , 1, , 1∗ b, , 1, , 1, , 1, , 1∗ c, , 64., , (a) f(x) = 0, , (b) f(x) = 1, , (c) f(x) = 2, , (d) None of these, ab, - b2, bc, , (b) abc, (d) None of these, x, , 2, , y y, , 2, , x, , If x ¹ y ¹ z and, , z, , 66., , 67., , z2, , x, , 3, 3, , y = 0, then xyz is, z3, , equal to:, (a) 1, (b) – 1, (c) 0, (d) x + y + z, If a1, a2, a3, ................. are positive numbers in G.P. then, , the value of, , log an, , log an +1, , log an +2, , log an +1, , log an+ 2, , log an +3, , log an+ 2, , log an +3, , log an +4, , (a) 1, (b) 4, (c) 3, The value of the determinant, cos a, , - sin a, , 1, , sin a, , cos a, , 1, , cos(a + b) - sin(a + b) 1, , 1+ a x, 2, , x + 11 x + r, , is, , sin 2 a sin a cos a cos 2 a, D = sin 2 b, sin 2 g, , sin b cos b, sin g cos g, , cos 2 b then D cannot exceed, cos 2 g, , (a) 1, , (b) 0, , 1, (d) None of these, 2, 71. If x, y Î R, then the determinant, , (c), , -, , cos x, - sin x, 1, sin x, cos x, 1 lies in the interval, cos(x + y) - sin(x + y) 0, , (a), , é - 2, 2 ù, ë, û, , (b) [–1, 1], , (c), , é - 2, 1ù, ë, û, , (d), , a, , b c, , é -1, - 2 ù, ë, û, , ka kb, 72. If D = x y z , then value of kx ky, p q r, kp kq, 2, 3, (a) k D, (b) k D, (c) kD, 32 + k, , 42, , kc, kz is, kr, (d) k4D, , 32 + 3 + k, , 73. If 42 + k 52, 52 + k 6 2, , 5, , (d) 0, , 42 + 4 + k = 0, then the value of k is, 52 + 5 + k, , 4, , 3, , x51 y 41 z 31 is, x, , y, , z, , (a) x + y + z, (c) 0, , is, , 2, 2, 2, If a + b + c = – 2 and, , 2, , x+p, , (a) 0, (b) –1, (c) 2, (d) 1, 74. If x, y, z are integers in AP, lying between 1 & 9 and x51,, y41 & z31 are three digit numbers, then the value of, , (a) independent of a, (b) independent of b, (c) independent of a and b (d) None of the above, 68., , x+6, , ac, bc is :, -c 2, , (a) 0, (c) 4a2b2c2, , 65., , x +9, , x + 5 x + 10 x + q, , (b) – abc, (d) None of these, , cos2 x cos x.sin x - sin x, 2, cos x , then for all x, If f(x) = cos x sin x sin x, sin x, - cos x, 0, , -a 2, The value of ab, ac, , x+4, , (a) x + 15, (b) x + 20, (c) x + p + q + r, (d) None of these, 70. Suppose a, b, gÎR are such that sin a, sin b, sin g ¹ 0 and, , =l, , then the value of l is :, (a) 0, (c) abc, , 63., , 69. If p, q, r are in A.P., then the value of, , x, , 75., , y, z, , 2, , 2, , 2, , 2, , (1 + b ) x (1 + c ) x, , f (x) = (1 + a ) x 1 + b x (1 + c ) x ,, (1 + a2 ) x (1 + b 2 ) x 1 + c 2 x, then f (x) is a polynomial of degree, (a) 1, (b) 0, (c) 3, , C1, C1, C1, , x, y, z, , C2, C2, C2, , (b) x – y + z, (d) None of these, x, y, z, , C3, C3, , =, , C3, , (a) xyz (x – y) (y – z) (z – x), , (d) 2, , (b), , xyz, (x – y)(y – z)(z – x), 6, , (c), , xyz, (x – y)(y – z)(z – x) (d) None of these, 12

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EBD_7762, 254, , MATHEMATICS, , 90. Assertion : D = a11A11 + a12A12 + a13A13 where, Aij is, cofactor of aij., Reason : D = Sum of the products of elements of any row, (or column) with their corresponding cofactors., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 91., , é cos a - sin a 0ù, ê, ú, If M (a) = ê sin a cos a 0ú ;, êë 0, 0, 1 úû, é cos b 0 sin b ù, M (b) = ê 0, 1, 0 ú,, ê, ú, êë - sin b 0 cos b úû, then value of [M(a) M (b)]–1 is, , 92., , 93., , 94., , (a), , M ( b ) .M ( a ), , (b), , M ( -a ) .M ( b ), , (c), , M ( -b ) M ( a ), , (d), , M ( -b ) M ( -a ), , é1 3 l + 2ù, ú, If the matrix ê2 4, 8 ú is singular, then l =, ê, êë3 5 10 úû, (a) –2, (b) 4, (c) 2, (d) – 4, 1 1 1, If A = 1 1 1 Then adj A =, 1 1 1, , é 5 5a a ù, ê, ú, 95. Let A = ê 0 a 5a ú . If |A2| =25, then | a | equals to, êë 0 0, 5 úû, , 96., , 97., , (b) 1, , (c), , 98. If the system of equations x + ly + 2 = 0, lx + y – 2 = 0,, lx + ly + 3 = 0 is consistent, then, (a) l = ±1, (b) l = ± 2, (c) l = 1, – 2, (d) l = –1, 2, 99. The equations 2x + 3y + 4 = 0; 3x + 4y + 6 = 0 and, 4x + 5y + 8 = 0 are, (a) consistent with unique solution, (b) inconsistent, (c) consistent with infinitely many solutions, (d) None of the above, 100. If [a] denotes the integral part of a and x = a3y + a2z,, y = a1z + a3z and z = a2x + a1y, where x, y, z are not all, zero. If a1 = m – [m], m being a non-integral constant,, then a1a2a3 is, (a) > 1, (b) > – 1, (c) < 1, (d) < – 1, 101. Given : 2x – y – 4z = 2, x – 2y – z = – 4, x + y + lz = 4,, then the value of l such that the given system of equation, has NO solution, is, (a) 3, (b) 1, (c) 0, (d) – 3, 102. If the equations x + ay – z = 0, 2x – y + az = 0, ax + y + 2z, = 0 have non-trivial solutions, then a =, (a) 2, (b) – 2, (c) 3, (d) - 3, 103. The value (s) of m does the system of equations 3x + my =, m and 2x – 5y = 20 has a solution satisfying the conditions, x > 0, y > 0., (a) m Î (0 ¥), , 15 ö, æ, (b) mÎ ç - ¥, - ÷ È (30, ¥), 2ø, è, , (a) 0, (b) 3, (c) 5, (d) 7, If A is a non-singular matrix of order 3, then |adj A| =, |A|n. Here the value of n is, (a) 2, (b) 4, (c) 6, (d) 8, , (a) 52, , Topic 4 : Solution of System of Linear Equations, , 1, 5, , (d) 5, , é1 0 3ù, If A = êê 2 1 1 úú , then the value of | adj (adj A) | is, ëê 0 0 2úû, (a) 14, (b) 16, (c) 15, (d) 12, é1 0 ù, If A = ê1 1 ú , then value of A–n is, ë, û, , (a), , é -1 0 ù, ê n 1ú, ë, û, , é 0 -1ù, ê2 n ú, ë, û, , (b), , (c), , é 1 0ù, ê - n 1ú, ë, û, , (d) None of these, , æ 15 ö, (c) m Î ç - , ¥ ÷, è 2, ø, (d) None of these, 104. Consider the system of linear equations;, x1 + 2x2 + x3 = 3, 2x1 + 3x2 + x3 = 3, 3x1 + 5x2 + 2x3 = 1, The system has, (a) exactly 3 solutions, (b) a unique solution, (c) no solution, (d) infinite solutions, 105. The system of linear equations : x + y + z = 0, 2x + y – z =, 0, 3x + 2y = 0 has :, (a) no solution, (b) a unique solution, (c) an infinitely many solution, (d) None of these, 106. The system of simultaneous linear equations kx + 2y – z = 1,, (k – 1) y – 2z = 2 and (k + 2) z = 3 have a unique solution, if k equals:, (a) – 1, (b) – 2, (c) 0, (d) 1, æ 0 0 -1ö, 107. Let A = ç 0 -1 0 ÷ . The only correct, ç -1 0 0 ÷, è, ø, statement about the matrix A is

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EBD_7762, 256, , MATHEMATICS, , Reason: If f(q) = c, then f(q) is independent of q., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., , ±, , (c), , ±, , 1, 2, 1, 2, , 1, , ,±, , 6, 1, , ,±, , 6, , ,±, ,±, , 1, 2, 1, 3, , (b), , ±, , (d), , ±, , 1, 3, 1, 2, , ,±, ,±, , 1, 2, 1, 2, , x, 2, , 123. If f (x) = 2sin x x, tan x, x, (a) 2, , 1, , é f ¢ (x) ù, 2x , then lim ê, is, x ®0 ë x ú, û, 1, , (b) – 2, , (c) 1, é- 1, , 124. The rank of the matrix êê 2, , é 0 2b c ù, 122. The value of a, b, c when ê a b - c ú is, ê, ú, ëê a - b c úû, orthogonal, are, , (a), , cos x, , êë 1, , (a) 1 if a = 6, (c) 3 if a = 2, , ,±, ,±, , (b) 2 if a = –1, (d) 1 if a = –6, , sin q sin f cos q, cos q sin f - sin q , then, - sin q sin f sin q cos f, 0, , 125. Let D =, , 1, 2, , 2, 5 ù, ú, - 4 a - 4ú is, - 2 a + 1 úû, , sin q cos f, cos q cos f, , 1, 6, , (d) – 1, , (a) D is independent of q (b), (c) D is a constant, , dD ù, =0, dq úû q=p /2, , (d) None of these, , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, 1., , 2 x 5 6 -2, If 8 x = 7 3 , then the value of x is, , (a) 3, , (b) ± 3, , (c) ± 6, , (d) 6, , 6., , -1, , a -b b +c a, , 2., , 3., , 4., , The value of b - a c + a b is, c-a a +b c, (a) a3 + b3 + c3, (b) 3bc, (c) a3 + b3 + c3 – 3abc, (d) None of these, If the area of a triangle with vertices (– 3, 0), (3, 0) and (0,, k) is 9 sq. units. Then, the value of k will be, (a) 9, (b) 3, (c) – 9, (d) 6, The determinant, b2 - ab b - c, , a - b b - ab, , bc - ac, , 2, , ab - a, , (a), (b), (c), (d), , -1, , cos A is equal to, cos B cos A, -1, , (a) 0, (c) 1, , 7., , (b) – 1, (d) None of these, , cos t t 1, f (t ), If f ( t ) = 2sin t t 2t , then lim, is equal to, t ®0 t 2, sin t t t, , (a) 0, , (b) –1, , 2, , c - a ab - a, , equals to, , 8., , abc (b – c) (c – a) (a – b), (b – c) (c – a) (a – b), (a + b + c) (b – c) (c – a) (a – b), None of these, , The number of distinct real roots of cos x sin x cos x = 0, cos x cos x sin x, , The maximum value of, , (c) 2, , (d) 3, , 1, 1, 1, 1, 1 + sin q 1 is (q is real, 1 + cos q, 1, 1, , number), (a), , sin x cos x cos x, , 5., , cos C cos B, , cos C, , bc - ac, , 2, , p, p, £ x £ is, 4, 4, (a) 0, (b) 2, (c) 1, (d) 3, If A, B and C are angles of a triangle, then the determinannt, , in the interval -, , 9., , 1, 2, , (b), , 3, 2, , (c), , 2, , 0, x -a x -b, 0, x - c , then, If f ( x ) = x + a, x+b x +c, 0, , (d), , 2 3, 4

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DETERMINANTS, , 257, , (a) f (a) = 0, (c) f (0) = 0, , (a) abc, (b) a + b + c, (c) –(a + b + c), (d) –abc, 18. The value of l , for which the lines, , (b) f (b) = 0, (d) f (1) = 0, , 2 l -3, , 10. If A = 0 2, 1 1, , 3x - 4 y = 13, 8 x - 11y = 33 and 2 x - 3 y + l = 0 are, concurrent is, [BITSAT 2014, A], , 5 , then A–1 exists, if, 3, , (b) l ¹ 2, , (a) l = 2, (c) l ¹ –2, 11. If A and B are invertible, following is not correct?, (a) adj A = |A| . A–1, (c) (AB)–1 = B–1 A–1, 12. If x, y and z are all, 1+ x, , 1, , 1, , 1, , 1+ y, , 1, , 1, , 1, , 1+ z, , (d) None of these, matrices, then which of the, , 19., , (b) det (A)–1 = [det (A)]–1, (d) (A + B)–1 = B–1 + A–1, different from zero and, , = 0 , then the value of x–1 + y–1 + z–1 is, , 20., , (b) x–1y–1z–1, (d) – 1, , (a) xyz, (c) – x – y – z, x+ y, , x, , 13. The value of x + 2 y, x+ y, , x + 2y, , x, x + 2y, , x + y is, x, , 21., , (a) 9x2(x + y), (b) 9y2 (x + y), 2, (c) 3y (x + y), (d) 7x2 (x + y), 14. If there are two values of a which makes determinant,, 1 -2, D= 2, , a, , 0, (a) 4, , 4, , 5, -1 = 86 , then the sum of these numbers is, 2a, (b) 5, , (c) – 4, , 23., , 15. If a, b ¹ 0, and f ( n ) = a n + b n and, 1 + f (1) 1 + f ( 2 ), , 1 + f (1) 1 + f ( 2 ) 1 + f ( 3) = K (1 - a ) (1 - b ) ( a - b), ,, 1 + f ( 2 ) 1 + f ( 3) 1 + f ( 4 ), 2, , 2, , 2, , then K is equal to:, , [JEE MAIN 2014, S], 1, (a) 1, (b) –1, (c) ab, (d), ab, 16. If A is an 3 × 3 non-singular matrix such that AA' = A'A, and B = A–1A', then BB' equals [JEE MAIN 2014, A], (a) B –1, , (b), , ( B )¢, -1, , (c), , 22., , (d) 9, , Past Year MCQs, , 3, , I +B, , 1, (d) 9, 7, The set of all values of l for which the system of linear, equations : 2x1 – 2x2 + x3 = lx1 [JEE MAIN 2015, A], 2x1 – 3x2 + 2x3 = lx2; –x1 + 2x2 = lx3, has a non-trivial solution,, (a) contains two elements., (b) contains more than teo elements, (c) is an empty set., (d) is a singleton, If the lines p1x + q1y = 1, p2x + q2y = 1 and p3x + q3y = 1, be concurrent, then the points (p1, q1), (p2, q2) and (p3, q3), [BITSAT 2015, A], (a) are collinear, (b) form an equilateral triangle, (c) form a scalene triangle, (d) form a right angled triangle, The system of linear equations, [JEE MAIN 2016, A], x + ly – z = 0; lx – y – z = 0, x + y – lz = 0; has a non-trivial solution for:, (a) exactly two values of l., (b) exactly three values of l., (c) infinitely many values of l., (d) exactly one value of l., Let M be a 3 × 3 non-singular matrix with det (M) = a. If, [M–1 adj (adj (M)] = KI, then the value of K is, [BITSAT 2016, A], (a) 1, (b) a, (c) a2, (d) a3, If [ ] denotes the greatest integer less than or equal to the, real number under consideration and –1 < x < 0; 0 < y < 1;, 1 < z < 2 , then the value of the determinant, [BITSAT 2016, A], , (a) –1, , (d) I, , 17. One of the roots of, b, , c, , a, , x+b, , c, , a, , b, , x+c, , = 0 is :, , [BITSAT 2014, A], , (c), , [ x] + 1 [ y], [z], [ x ] [ y ] + 1 [ z ] is, [x], [ y ] [ z] + 1, , (a) [z], (b) [y], (c) [x], (d) None of these, 24. Let a1, a2 and b1, b2 be the roots of ax2 + bx + c = 0 and, px2 + qx + r = 0 respectively. If the system of equations, a1y + a2z = 0 and b1y + b2z = 0 has a non-trivial solution,, then, [BITSAT 2016, A], (a), , x+a, , (b) –7, , (c), , b2, q, , 2, , a2, p, , 2, , c2, , =, , ac, pr, , (b), , =, , bc, qr, , (d) None of these, , r, , 2, , =, , ab, pq

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EBD_7762, 258, , 25., , MATHEMATICS, , 1, 1, 1, +, +, =, 1+ a 1+ b 1+ c, (a) 0, (b) 1, , é1 3 l + 2ù, ú, If the matrix ê2 4, 8 ú is singular, then l =, ê, êë3 5 10 úû, , [BITSAT 2016, A], (a) –2, (b) 4, (c) 2, (d) – 4, 26. Let k be an integer such that triangle with vertices, (k, –3k), (5, k) and (–k, 2) has area 28 sq. units. Then the, orthocentre of this triangle is at the point :, [JEE MAIN 2017, A, BN], (a), , æ 1ö, çè 2, ÷ø, 2, , (b), , (c), , æ 3ö, çè1, ÷ø, 4, , 3ö, æ, (d) ç1, - ÷, è, 4ø, , 1ö, æ, çè 2, - ÷ø, 2, , 27. If S is the set of distinct values of ‘b’ for which the, following system of linear equations x + y + z = 1, x + ay + z = 1; ax + by + z = 0[JEE MAIN 2017, A], has no solution, then S is :, (a) a singleton, (b) an empty set, (c) an infinite set, (d) a finite set containing two or more elements, 28. Let w be a complex number such that 2w + 1 = z where, 1, , z=, , 2, -3 . If 1 -w - 1 w = 3k, then k is equal to :, , (a) 1, , w2, , w7, , (b) –z, , (c) z, , [JEE MAIN 2017, A], (d) – 1, , q-y r-z, , p, , 30., , 1, , 2, , 1, , 29., , 1, , p q r, q, r - z = 0 , then the value of + +, If p - x, x y z, p-x q-y, r, is, [BITSAT 2017, A], (a) 0, (b) 1, (c) 2, (d) 4pqr, If a system of equation – ax + y + z = 0, x – by + z = 0; x + y – cz = 0 (a, b, c ¹ –1), has a non-zero solution then, , x - 4 2x, , 31. If, , [BITSAT 2017, A], (c) 2, , (d) 3, , 2x, , 2x, , x - 4 2x, , 2x, , 2x, , = (A+ Bx)(x - A) 2 , then the, , x-4, , ordered pair (A, B) is equal to : [JEE MAIN 2018, A], (a) (– 4, 3) (b) (– 4, 5), (c) (4, 5), (d) (– 4, – 5), 32. If the system of linear equations, x + ky + 3z = 0, 3x + ky – 2z = 0, 2x + 4y – 3z = 0, xz, has a non-zero solution (x, y, z), then 2 is equal to :, y, [JEE MAIN 2018, A], (a) 10, (b) – 30, (c) 30, (d) – 10, 33. The points represented by the complex numbers, 5, i on the argand plane are, 3, [BITSAT 2018, C], vertices of an equilateral triangle, vertices of an isosceles triangle, collinear, None of these, , 1 + i, – 2 + 3i,, (a), (b), (c), (d), , é3 - 2 4 ù, 1, 34. If matrix A = êê1 2 - 1úú and A -1 = adj (A) , then k, k, êë0 1, 1 úû, is, [BITSAT 2018, A], (a) 7, (b) – 7, (c) 15, (d) – 11, 35. If x, y, z are complex numbers, and, , 0 -y -z, D= y, , 0, , z, , x, , - x then D is, , [BITSAT 2018, A], , 0, , (a) purely real, (c) complex, , (b) purely imaginary, (d) 0, , Exercise 3 : Try If You Can, 1., , Let curve f (x, y) is locus of point (x, y) which satisfy, 2x, , 2y, , 6, , x, , a, , 2b, , 3l + 2 4l, , 2a, , 3b +, , –1, , 2, , 1, , –1, , 2, , 2, , y, , 1, , 11, , – 2l, , 72, , 2a, , – 81b, , = 0 " a, b, g Î R. If minimum distance between curve, f (x, y) and line 4x – 3y + 12 = 0 is d, then the value of d is, (a) 1, (b) 2, (c) 3, (d) 4

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EBD_7762, 260, , MATHEMATICS, , 14. If A, B, C are the angles of a triangle, then the value of, , x y 2 z3, , D2 = x 4 y 5 z 6 . Then D1D2 is equal to, x7 y8 z9, , (a), , D32, , (b), , (c), , D 42, , (d) None of these, , D 22, , |A – B| ¹ 0, A4 = B4, C3A = C3B, B3A = A3B, then, |A3 + B3 + C3| =, (a) 0, (b) 1, (c) 3| A |3, (d) 6, , 13., , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, , (d), (a), (a), (b), (a), (a), (c), (b), (d), (a), (a), (a), (a), , 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, , (b), (b), (d), (b), (a), (d), (c), (c), (d), (b), (c), (b), (b), , 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, , (d), (b), (d), (a), (b), (d), (a), (a), (c), (c), (b), (d), (a), , 1, 2, 3, 4, , (c), (d), (b), (d), , 5, 6, 7, 8, , (c), (a), (a), (a), , 9, 10, 11, 12, , (c), (d), (d), (d), , 1, 2, , (c), (a), , 3, 4, , (b), (a), , 5, 6, , (a), (d), , sin 2A, , sin C, , sin B, , determinant sin C, sin B, (a) p, (c) 2 p, , sin 2B, , sin A is, , sin A sin 2C, (b) 0, (d) None of these, , é3 7 ù, 15. If A = ê, ú , then the value of the determinant, ëê1 2 úû, |A2013 – 3A2012| is equal to, (a) 8, (b) – 8, (c) 9, (d) –7, , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (b), 40 (b) 53 (d) 66 (d) 79, (d), 41 (a) 54 (a) 67 (a) 80, (c), 42 (b) 55 (c) 68 (d) 81, (c), 43 (c) 56 (a) 69 (d) 82, (b), 44 (c) 57 (a) 70 (a) 83, (b), 45 (a) 58 (b) 71 (a) 84, (a), 46 (d) 59 (c) 72 (b) 85, (d), 47 (a) 60 (b) 73 (d) 86, (c), 48 (c) 61 (a) 74 (c) 87, (d), 49 (d) 62 (c) 75 (c) 88, (b), 50 (b) 63 (b) 76 (c) 89, (a), 51 (a) 64 (c) 77 (c) 90, (d), 52 (d) 65 (c) 78 (c) 91, Exercise 2 : Exemplar & Past Year MCQs, (b), 13 (b) 17 (c) 21 (b) 25, (a), 14 (c) 18 (b) 22 (b) 26, (a), 15 (a) 19 (a) 23 (a) 27, (b), 16 (d) 20 (a) 24 (a) 28, Exercise 3 : Try If You Can, (a), (c) 11 (d) 13, (a), 7, 9, (b), (d), (a), (b), 8, 10, 12, 14, , 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, , (b), (a), (a), (c), (b), (c), (a), (a), (b), (d), (b), (b), (c), , 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, , (c), (a), (a), (c), (b), (b), (c), (d), (c), (c), (d), (a), (c), , 29, 30, 31, 32, , (c), (b), (b), (a), , 33, 34, 35, , (c), (c), (b), , 15, , (d), , 118, 119, 120, 121, 122, 123, 124, 125, , (b), (b), (b), (a), (c), (b), (d), (b)

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21, , CONTINUITY AND, DIFFERENTIABILITY, , Chapter, , Trend, Analysis, , off JEE Main and BITSAT (Year 2010-2018), , 5, , Number of Questions, , 4, , 3, , JEE MAIN, BITSAT, , 2, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 3, 4, , Critical Concepts, , Continuous Functions,, Discontinuous Functions,, Differentiability of a Function,, Derivatives of Composite Function,, Chain Rule, Derivatives of Different, functions, Successive Differentiation,, Mean Value Theorems, , Rating of, Difficulty Level, , CUS, (chapter utility score), Out of 10, , 4/5, , 8.3

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EBD_7762, 264, , MATHEMATICS, , Topic 1: Continuous Functions, Continuity of a Function, at a Point, Continuity of a Function in an Interval,, Discontinuouns Functions., 1., , 1, f (x) =, 1 + tan x, (a) is a continuous, real-valued function for all, x Î (– ¥, ¥), , 3p, (b) is discontinuous only at x =, 4, (c) has only finitely many discontinuities on, (– ¥, ¥), (d) has infinitely many discontinuities on (– ¥, ¥), , 2., , 3., , 5., , (e x - 1) 2, for x ¹ 0, and f(0) = 12. If f is, æ xö, æ xö, sin ç ÷ log ç1 + ÷, è aø, è 4ø, continuous at x = 0, then the value of a is equal to, (a) 1, (b) –1, (c) 2, (d) 3, , Let f (x) =, , ì, - x 2 , when x £ 0, ïï 5 x - 4, when 0 < x £ 1, í, If f (x) = ï4 x 2 - 3x, when1 < x < 2 , then, îï 3x + 4, when x ³ 2, , (a), (b), (c), (d), 6., , If f ( x ) =, (a), , 7., , 8., , 9., , ì3, if 0 £ x £ 1, ï, f ( x ) = í4, if 1 < x < 3 are, ï5, if 3 £ x £ 10, î, (a) 1,3, (b) 3,10, (c) 1,3,10 (d) 0,1,3, The relationship between a and b, so that the function, , 1, 2, , (c) 2, , ì1 - sin 3 x, p, ,x <, ï, 2, 2, ï 3 cos x, p, ïï, Let f (x ) = íp, x =, 2, ï, p, ï q(1 - sin x ), ,x >, ï, 2, 2, ïî (p - 2x ), , 3, (d), 2, , (a), , a = b+, , 2, 3, , (b) a - b =, , (c), , a +b =, , 2, 3, , (d) a + b = 2, , 3, 2, , ì x-4, ï x - 4 + a, x < 4, ï, Then f (x) is continuous at, 10. Let f (x) = ía + b, x = 4., ï x-4, + b, x > 4, ï, î x-4, , x = 4 when, (a) a = 0, b = 0, (c) a = –1, b = 1, , 12., , 4+x -2, , x ¹ 0 be continuous at x = 0, then f(0) =, x, 1, (b), 4, , ì ax + 1, if x £ 3, is continuous at, f defined by f ( x ) = í, î bx + 3, if x > 3, x = 3 , is, , 11., , f (x) is continuous at x = 0, f (x) is continuous at x = 2, f (x) is discontinuous at x = 1, None of these, , p, , (p, q) =, 2, ö, æ1, (b) ç , 2 ÷, ø, è2, , (a) (1, 4), æ1, ö, (d) None of these., (c) ç , 4 ÷, è2, ø, All the points of discontinuity of the function f defined by, , ì3x - 4, 0 £ x £ 2, Let f (x) = í, î2x + l, 2 < x £ 9, If f is continuous at x = 2, then what is the value of l ?, (a) 0, (b) 2, (c) – 2, (d) – 1, The value of l, for which the function, , ìïl( x 2 - 2x ) if x £ 0, is continuous at x = 0, is :, f (x) = í, ïî 4x + 1, if x > 0, (a) 1, (b) – 1, (c) 0, (d) None of these, 4., , If f(x) is continuous at x =, , (b) a = 1, b = 1, (d) a = 1, b = –1, , ì 1 + kx - 1 - kx, , for - 1 £ x < 0, ï, If f ( x ) = í, is, x, ï, 2, î 2x + 3x - 2 , for 0 £ x £ 1, continuous at x = 0, then k is equal to, (a) – 4, (b) – 3, (c) – 2, (d) – 1, The number of points at which the function, f (x) =, , 1, ,, x - [ x ] [.] denotes the greatest integer function is, , not continuous is, (a) 1, (c) 3, , (b) 2, (d) None of these, , ì1 - 2 sin x, , if x ¹, ïï, 13. If f ( x ) = í p - 4x, ï, a, , if x =, ïî, then a is equal to, , (a) 4, , (b) 2, , p, 4 is continuous at p ,, 4, p, 4, , (c) 1, , (d), , 1, 4

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CONTINUITY AND DIFFERENTIABILITY, , 14., , 265, , ,, x£2, ì 1, ï, If the function f (x) = íax + b , 2 < x < 4, ï 7, ,, x³4, î, , is continuous at x = 2 and 4, then the values of a and b are., (a) a = 3, b = –5, (b) a = –5, b = 3, (c) a = –3, b = 5, (d) a = 5, b = –3, 15. If we can draw the graph of the function around a point, without lifting the pen from the plane of the paper, then the, function is said to be, (a) not continuous, (b) continuous, (c) not defined, (d) None of these, 16. If f(x) = x2 – sin x + 5, then, (a) f(x) is continuous at all points, (b) f(x) is discontinuous at x = p., (c) It is discontinuous at x =, , p, 2, , (d) None of the above, ìï51/ x , x < 0, 17. Let f (x) = í, and l Î R, then at x = 0, ïîl[x], x ³ 0, (a), (b), (c), (d), , f is discontinuous, f is continuous only, if l = 0, f is continuous only, whatever l may be, None of these, , ì 8x - 4 x - 2 x + 1, ,, x>0, ï, 18. If f ( x ) = í, is continuous at, x2, ï x, îe sin x + px + l ln 4, x £ 0, , x = 0. Then, the value of l is, (a) 4 loge2, (b) 2 loge2, (c) loge2, (d) None of these, ìïx k cos(1/ x), x ¹ 0, 19. If f (x) = í, is continuous at x = 0,, ,, x=0, ïî0, then, (a) k < 0, , (b) k > 0, , (c) k = 0, , (d) k ³ 0, , 1, , then the points of discontinuity of the, 1- x, function f [ f {f(x)}] are, , 20. If f(x) =, , (a) {0, –1} (b) {0,1}, , (c) {1, –1} (d) None, , ìx + l , x < 3, ï, 21. If f (x) = í4, , x = 3 is continuous at x = 3, then the, ï3x - 5 , x > 3, î, , value of l is equal to :, (a) 1, (c) 0, , (b) – 1, (d) does not exist, , 22. If f (x) = x1/x – 1 for all positive x ¹ 1 and if f is continuous at, 1, then x equals:, 1, (c) e, e, The value of p for which the function, , (a) 0, 23., , ì, ï, ï, f ( x ) = í sin, ï, ï, î, , (b), , (d) e 2, , (4 x - 1)3, ,x¹0, é x2 ù, x, log ê1 + ú, p, 3 úû, ëê, 12(log 4)3 , x = 0, , may be continuous at x = 0, is, (a) 1, (b) 2, (c) 3, (d) None of these, 24. Find the value of a and b if, ì 1/(|x + 2|), -1, ï ae, ; -3 < x < - 2, +, 1/(|, 2|), x, ï 2-e, ï, f ( x) = í, b;, x = -2, ï, 4, ïsin æ x - 16 ö ; -2 < x < 0, ÷, ï ç 5, î è x + 32 ø, is continuous at x = – 2., , æ 2ö, (a) sin ç ÷, 0, è 5ø, (c), , æ3ö, æ 3ö, sin ç ÷, - sin ç ÷, è5ø, è5ø, , æ -2 ö, (b) sin ç ÷,1, è 5 ø, , 2, 2, (d) sin æç ö÷ - sin æç ö÷, 5, è ø, è5ø, , ì x3 + x 2 - 16x + 20, ,x ¹ 2, ï, 25. Let f ( x ) = í, ( x - 2 )2, ï, k, ,x = 2, î, , If f(x) is continuous for all x, then k =, (a) 3, (b) 5, (c) 7, (d) 9, p, ì k cos x, ïï p - 2x , if x ¹ 2, 26. The function f ( x ) = í, is continuous at, p, ï 3,, if x =, ïî, 2, , p, , when k equals, 2, (a) – 6, (b) 6, 27. If f : R ® R is defined by, x=, , (c) 5, , (d) – 5, , ì 2sin x - sin 2x, , if x ¹ 0, ï, f ( x ) = í 2x cos x, then the value of a, so, ïî, a,, if x = 0, , that f is continuous at 0, is, (a) 2, (b) 1, (c) – 1, , (d) 0

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EBD_7762, 266, , 28., , MATHEMATICS, , ì sin 5x, , x¹0, ïï 2, is continuous at x = 0, then, If f ( x ) = í x + 2x, ï k+1,, x=0, ïî, 2, , 35. The set of points of discontinuity of the function, f(x) = lim, , n ®¥ 3 n, , (b) – 2, , (, , (c) 2, , (d), , 1, 2, , (a), , 30., , 31., , (c), , ), , 33., , ), , lim f ( x ) = a, , x®a, , (b) f (x) is continuous at x = a, (c) f (x) is discontinuous at x = a, (d) None of these, The function f(x) = [x]2 – [x2] (where [y] is the greatest, integer function less than or equal to y), is discontinuous at :, (a) all integers, (b) all integers except 0 and 1, (c) all integers except 0, (d) all integers except 1, , x, are not equal at x = 0., x, (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., In the interval [7, 9] the function f(x) = [x] is, discontinuous at _______, where [x] denotes the greatest, integer function, (a) 2, (b) 4, (c) 6, (d) 8, The no. of points of discontinuity of the function, f (x) = x – [x] in the interval (0, 7) are, (a) 2, (b) 4, (c) 6, (d) 8, , ïìx , if x £ 4, If the function f (x) = í, ïîax, if x > 4, , (a), , a = 2n + (3/ 2); b Î R; n Î I, , (b), , a = 4n + 2; b Î R; n Î I, , (c), , a = 4n + (3/ 2); b Î R +1 ; n Î I, , a = 4n + 1; b Î R + ; n Î I, 37. The number of points at which the function, , f (x) =, , 1, is discontinuous is, log | x |, , (a) 1, (b) 2, 38. If the function,, , x, is continuous at x = 0., x, Reason : The left hand limit and right hand limit of the, , is continuous at x = 4, then a =, (a) 2, (b) 4, (c) 6, , (d) None of these, , (d), , (c) 3, , (d) 4, , ì x + a 2 2 sin x , 0 £ x £ p / 4, ï, f (x) = í, x cot x + b, , p/ 4 £ x £ p/ 2, ïb sin 2x - a cos 2x , p / 2 £ x £ p, î, , Assertion : The function f (x) =, , 2, , 34., , p, ì, ü, ínp ± , n Î Iý, 6, î, þ, , = sin(p( x + a)) for x < –1, where [x] denotes the integral part of x, then for what values, of a, b, the function is continuous at x = –1?, , function f (x) =, , 32., , p, ì, ü, (b) ínp ± , n Î Iý, 3, î, þ, , 36. Given f ( x ) = b([ x ]2 + [ x ]) + 1 for x ³ -1, , ì x 2 / a - a, when x < a, ïï, 0, when x = a, then, 29. If f ( x ) = í, ï a - x 2 / a , when x > a, ïî, , (, , is given by, , - (2 cos x ) 2 n, , (a) R, , the value of k is, (a) 1, , (2 sin x ) 2 n, , is continuous in the interval [0, p] then the value of (a, b), are :, (a) (– 1, – 1), (b) (0, 0), (c) (– 1, 1), (d) (1, 0), , æ px ö, 39. The point of discontinuity of f (x) = tan ç, ÷ other, è x + 1ø, than x = –1 are :, (a) x = 0, (b) x = p, (c), , 40., , x=, , 2m + 1, 1 - 2m, , ì ee/x - e -e/ x, ,, ï 1/x, If f (x) = í e + e -1/ x, ï, k, ,, î, , (d) x =, x¹0, , 2m - 1, 2m + 1, , then, , x=0, , (a) f is continuous at x, when k = 0, (b) f is not continuous at x = 0 for any real k., (d) 8, , (c), , lim f (x) exist infinitely, , x ®0, , (d) None of these

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CONTINUITY AND DIFFERENTIABILITY, , 267, , 41. If f(x y) = f(x). f(y) for all x, y and f(x) is continuous at x = 2,, then f(x) is not necessarily continuous in :, (a) (– ¥, ¥) (b) (0, ¥), (c) (–¥, 0) (d) (2, ¥), 42. The number of discontinuous functions y(x) on [– 2, 2], satisfying x2 + y2 = 4 is, (a) 0, (b) 1, (c) 2, (d) > 2, 43., , ì, æ1ö, ï x cos ç ÷ , if x ¹ 0, f (x) = í, is, èxø, ï, 0, , if x = 0, î, , (a) discontinuous at x = 0 (b) continuous at x = 0, (c) Does not exist, (d) None of the above, 44. If f : R ® R is defined by, ì x+2, ,if, ï 2, ïï x + 3x + 2, -1, f (x) = í, ,if, ï, 0, ,if, ï, îï, , x Î R - {-1, -2}, x = -2, x = -1, , then f is continuous on the set, (a) R, (b) R – {– 2}, (c) R – {– 1}, (d) R – {– 1, – 2}, 45. If f (x) = (x + 1)cot x be continuous at x = 0 then f (0) is, equal to:, (a) 0, (b) – e, (c) e, (d) None, Topic 2: Differentiability, Differentiability of a Function at, a Point, Differentiability of a Function in an Interval,, Relation between Continuity and Differentiability., , (a) f(x) is continuous at x = 0 but not differentiable at, x=0, (b) f(x) is continuous as well as differentiable at x = 0, (c) f(x) is discontinuous at x = 0, (d) None of these., 47. Suppose f is a real function and c is a point in its domain., The derivative of f at c is defined by (if limit exist), , (c), , lim, , h ®0, , lim, , h ®0, , h, f ( c + h ) + f (c), h, , discontinuous every where, continuous as well as differentiable for all x, continuous for all x but not differentiable at x = 0, neither differentiable nor continuous at x = 0, , 49. If f ( x ) = 3, (a), (b), (c), (d), , x4, , x ¹ 0 and f (0) = 0 is :, |x|, , continuous for all x but not differentiable for any x, continuous and differentiable for all x, continuous for all x and differentiable for all x ¹ 0, continuous and differentiable for all x ¹ 0, , 50. f(x) = maximum {2 sin x, 1 – cos x} is not differentiable, when x is equal to, (a) 1, (b) –1, -1 æ 3 ö, (c) 0, (d) p - cos ç 5 ÷, è ø, ì [x] - 1, ,x ¹1, ï, , 51. If f (x) = í x - 1, ïî, , (a), (b), (c), (d), , then f(x) is, , 0 , x =1, , continuous as well as differentiable at x = 1, differentiable but not continuous at x = 1, continuous but not differentiable at x = 1, neither continuous nor differentiable at x = 1, , æ1ö, 52. Assertion : f (x) = xn sin ç ÷ is differentiable for all real, è xø, values of x (n ³ 2)., x ®0, , ì x, , x¹0, ï, f (x) = í x 2, then :, ï, ïî 0, x = 0, , (a), , (a), (b), (c), (d), , then f(x) is, , Reason : For n ³ 2, lim f ( x) = 0, , 46. If a function f(x) is defined as, , f (c - h ) - f (c), , 48., , ì æ 1 1ö, ï -ç + ÷, If f ( x ) = í xe è x x ø , x ¹ 0, ï0, ,x = 0, î, , (b) lim, , h ®0, , (d) lim, , h ®0, , f ( c + h ) - f (c), h, f ( c - h ) + f (c), h, , (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 53. Which of the following functions is differentiable at x = 0?, (a), (c), , ( ), sin ( x ) + x, , (b) cos ( x ) - x, , cos x + x, , (d) sin ( x ) - x, tan[ x ]p, , , where [.] denotes the, [1+ | log(sin 2 x + 1) |], greatest integer function and |.| stands for the modulus of, the function, then f(x) is, (a) discontinuous "x Î I, (b) continuous " x, (c) non differentiable " x Î I, (d) a periodic function with fundamental period 1., , 54. If f (x) =

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EBD_7762, 268, , 55., , 56., , MATHEMATICS, , Which of the following function is not differentiable at x = 0?, (i), , f ( x ) = ( x - 1) ( x - 1)( x - 2), , (ii), , f ( x ) = sin(| x - 1|)- | x - 1|, , (iii), , f ( x) = tan(| x - 1|)+ | x - 1|, , (a) only (i) (b) only (iii), (c) Both (ii) and (i), (d) Both (ii) and (iii), At how many points between the interval (–¥, ¥) is the, function f (x) = sin x is not differentiable., (a) 0, (b) 7, (c) 9, (d) 3, , 58., , Then which one of the following is true?, (a) f is neither differentiable at x = 0 nor at x =1, (b) f is differentiable at x = 0 and at x =1, (c) f is differentiable at x = 0 but not at x = 1, (d) f is differentiable at x = 1 but not at x = 0, Which of the following functions is not differentiable at, x = 1?, (a) f(x) = (x2 – 1) |(x – 1) (x – 2)|, (b) f(x) = sin (|x – 1|) – |x – 1|, (c) f(x) = tan(|x – 1|) + |x – 1|, (d) None of these, , 60., , tan px 2 + ( x + 1) n sin x, , n ®¥, , x 2 + (x + 1) n, , , then, , (a) f is continuous at x = 0, (b) f is differentiable at x = 0, (c) f is continuous but not differentiable at x = 0, (d) None of these, Assertion : f (x) = | x | sin x, is differentiable at x = 0., Reason : If f (x) is not differentiable and g (x) is differentiable, at x = a, then f (x) . g (x) can still be differentiable at x = a., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., , ìï x [ x ] ,, 61. The function f ( x ) = í, ïî( x - 1) x,, (a) differentiable at x = 2 (b), (c) continuous at x = 2, (d), , 62., , if 0 £ x < 2, is, if 2 £ x < 3, , not differentiable at x = 2, None of these, , Given, f ( x) = x 2 e2( x -1) , 0 £ x £ 1, = a Sgn ( x + 1)cos (2 x - 2) + bx 2 , 1 < x £ 2,, , f (x) is differentiable at x = 1 provided, (a), , a = -1, b = 2, , (d) a = 3, b = -4, , 64. If f ( x ) = ae x + b x 2 , a b Î R and f(x) is differentiable at, x = 0. Then, a and b are, (a) a = 0, b Î R (b), (c) b = 0, a Î R (d), , a = 1, b = 2, a = 4, b = 5, , 65. Let f : R ® R be a function defined by, f (x) = min {x + 1, x + 1} ,Then which of the following is true ?, , 57., , If f ( x ) = lim, , a = -3, b = 4, , 63. The number of points in (1, 3), where f (x) = a [x ], a > 1, is not, differentiable, where [x] denotes the integral part of x., (a) 5, (b) 7, (c) 9, (d) 11, , 1, ì, if x ¹ 1, ï( x –1) sin, Let f ( x) = í, x –1, ïî, 0, if x = 1, , 59., , (c), , 2, , 2, , (b) a = 1, b = -2, , (a) f (x) is differentiable everywhere, (b) f (x) is not differentiable at x = 0, (c) f (x) ³ 1 for all x Î R, (d) f (x) is not differentiable at x = 1, 66. If f ( x ) = x +, (a), (b), (c), (d), , x, x, +, + ...to ¥ , then at x = 0, f(x), 1 + x (1 + x ) 2, , has no limit, is discontinuous, is continuous but not differentiable, is differentiable, , x, x, x, +, +, + ... to, 1 + x (x + 1)(2x + 1) (2x + 1)(3x + 1), ¥ , then at x = 0, f(x), (a) has no limit, (b) is discontinuous, (c) is continuous but not differentiable, (d) is differentiable, (a) 1, (b) 2, (c) 3, (d) 4, 68. Let f : R ® R be a function defined by f (x) = max {x, x3}., The set of all points where f (x) is NOT differentiable is, (a) {-1, 1} (b) {-1, 0} (c) {0, 1} (d) {-1, 0, 1}, , 67. If f ( x ) =, , ì x sin1 / x , x ¹ 0, 69. f(x) = í, 0, , x = 0 at x = 0 is, î, , (a), (b), (c), (d), , continuous as well as differentiable, differentiable but not continuous, continuous but not differentiable, neither continuous nor differentiable, , 70. Let f : R ® R be a function defined by, f (x) = min {x + 1, x + 1} ,Then which of the following is, true?, (a) f (x) is differentiable everywhere, (b) f (x) is not differentiable at x = 0, (c) f (x) ³ 1 for all x Î R, (d) f (x) is not differentiable at x = 1, 71. If both limh ®0, , f ( c + h ) - f (c), h, , and lim+, h ®0, , f (c + h ) - f (c), h

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CONTINUITY AND DIFFERENTIABILITY, , 269, , are finite and equal, then, (a) f is continuous at a point c, (b) f is not continuous at c, (c) f is differentiable at a point c in its domain, (d) None of the above, , (c) – 2x sin 2x2, 79. If 2x + 2y = 2 x + y, then, , 2x - y, , (a), Topic 3: Chain Rule of Differentiation, Differentiation of, Inverse Functions, Implicit Functions, Parametric Functions,, Logarithmic Differentiation, Second Order Derivative., 72. If y = 2–x, then, , dy, is equal to :, dx, , x, , (a), , -, , (c), , 2–x, , 2, , log 2, , (d), , 73. If y = sec x°, then, (a) sec x tan x, (c), , log, 2x, , 75., , 2x, 1+ x, , 4, , (b), , 2x, 1 + 4x, , (c), , dy, is :, dx, , -2x, 1 + x4, , 76. If y = log tan x then the value of, , (a), , (c), , 1, , (b), , 2 x, 2sec 2 x, , (1+log e, 77. If y = e, , (a) e, (c) 0, , (d), x), , , then, , (d), , dy, is :, dx, , sec, , 2, , x, , sec, , 1 - 2x, , (d) None of these, , 2x - 2 y, , æ 1 + cos 2q ö, 3p, dy, at q =, is :, ç 1 - cos 2q ÷ , then, è, ø, 4, dq, , p, dy, , the value of, is equal to :, 4, dx, , (b) 2, , then, , x, , 2 x tan x, , dy, is equal to :, dx, (b) 1, (d) loge x . x, , dy, If y = (cos x2)2, then, is equal to :, dx, (a) – 4x sin 2x2, (b) – x sin x2, , (c), , 1, 2, , (d) -, , 1, 2, , -2x, , d2y, , is, dx 2, (a) 2, (b) 1, (c) 0, 83. If f(x) = |loge |x||, then f '(x) equals, , (d) –1, , (a), , 1, , xp0, |x|, , (b), , 1, 1, for | x | > 1 and - for | x | < 1, x, x, , (c), , -, , (d), , 1, 1, for x > 0 and - for x < 0, x, x, , 1 + x2, , 1, 1, for | x | > 1 and, for | x | < 1, x, x, , x tan x, 2, , 2y - 1, , æ 3 + 2 log e x ö, æ log (e / x 2 ) ö, 82. If y = tan –1 ç e, + tan–1 ç 1– 6 log x ÷ ,, 2 ÷, è, e ø, è log e (ex ) ø, , 1, 1, (c) (log x) (d) (x log x), , 1, (b), x, , x- y, (b) 2, , 2x - 1, , (a) – 2, , dy, If y = cot –1 (x2), then the value of, is equal to:, dx, , (a), , 78., , t=, , (d) None of these, , 74. If y = log (log x), then the value of, (a) ey, , 1, 2, , ey, , 2y - 1, , (a) – 2, (b) 2, (c) ± 2, (d) None of these, 81. If x = sin t cos 2t and y = cos t sin 2t, then at, , dy, is equal to :, dx, (b) sec x° tan x°, , p, sec x ° tan x °, 180, , dy, =, dx, , 2x + 2y, , (c), , 80. If y =, , (b) 2x log 2, , x +1, , (d) – x cos 2x2, , 84. The value of, (a) –, (c), 85. If x =, , d é -1 æ a - x ö ù, tan ç, ÷ is :, dx êë, è 1 + ax ø úû, , 1, , (b), , 1 + x2, 1, , 1+ t, , 2, , 1+ a, , 2, , -, , 1, 1+ x2, , (d) None of these, , 1 + x2, 1 - t2, , 1, , and y =, , 2t, 1+ t, , 2, , , then, , dy, is equal to :, dx

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EBD_7762, 270, , MATHEMATICS, , (a), 86., , -, , y, x, , y, x, , (b), , (c) -, , x, y, , x, y, , (d), , 1, , (d), , The derivative of, æ 1 – x2, cos –1 ç, ç 1 + x2, è, (a), , 3, 2, , ö, æ 1 – 3x 2 ö, ÷ w.r.t. cot –1 ç, ÷ is:, ÷, ç 3x – x 3 ÷, ø, è, ø, (b) 1, , 1, 2, , (c), , (a) sin y, (c) e, , 2, 3, , (d), , ( ), , p, 6, , (a), , p, (b) 6, , (c), , (d), , 6, , x +1, x, , 89. If y = tan, , (c), , (1+ x)2, , (c) – 1, , (d), , x, 1+ x, , 1, (d) 4, , a2 - x2, , 1, , (c), , 2, , a - x2, , 92., , -, , y, x, , x, y, , (b), , 1, 2 x (1 - x ), , -, , 96., , If y =, , (a), , d2y, dx2, , (c), , y, , 1, , (c), , 1 - x2, , -, , 1- x2, , 1, 2 x (1 - x ), , x =0, , is, (d) 3e7, , 1 d2 y, y dx 2, , (, , log 1+ tan 2 x, , ) , then, , 1, sec 2 x, 2, , dy, is equal to, dx, , (b) sec2x, (d), , (, , 1, log 1+ tan 2 x, 2, e, , -1, 98. If f(x) = (logcot x tan x)(logtanxcot x)–1 + tan, , 1, , (, , dy, dx, , (d) 1, , (d) None of these., , (c) sec x tan x, , x, (c) - y, , {, , d2y, dx 2, , (d) None of these, , 1 - x 1 - x - x 1 - x2, , 1, 2, , (c) – 1, , (b), , (a), , (d), , y, x, , )}, , ), 4x, , 4 - x2, , , then, , f¢(2) is equal to, (a), , 1, 2, , (b) -, , 1, 2, , (c) 1, , (d) – 1, , 99. If y = logax + logxa + logxx + logaa, then, , dy, is equal to, dx, , log a, x, +, x, log a, , (a), , 1, + x log a, x, , (b), , (c), , 1, + x log a, x log a, , 1, log a, (d) x log a 2, x ( log x ), , 1, , (b), , (c) –, , f ( x), f '( x ), f '' f '' 2( y - z ), and z =, , then, - +, (f ')2 =, f( x ), f '( x ), f, f, ff, , (b) 2 a 2 - x 2, , d é -1 æ, ù, sin ç x 1 - x - x 1 - x 2 ö÷ ú is equal to, ê, è, øû, dx ë, (a), , 1, 2, , (b) 0, , 97. If y = e 2, , dy, æx-yö, 91. If sec ç, ÷ = a , then dx is, èx+yø, , (a), , (b), , 1, , d æ, 2, 2, 2, -1 æ x ö ö, ç x a - x + a sin ç ÷ ÷ is equal to, dx è, è a øø, , (a), , (a) –1, , (a) 1, , x -x ö, çç, 3 / 2 ÷÷ , then y¢(1) is equal to, è 1+ x, ø, , (a) 0, , ì 1 - x ïü ù, d é 2, -1 ï, êsin cot í, ý ú equals, dx êë, îï 1 + x ïþ úû, , 6, , -1 æ, , 1, (b), 2, , dy, at (1, p) is equal to, dx, (b) – x cos y, (d) sin y – x cos y, , 95. If , y = e3x + 7 , then the value of, , -1, , 1, 1+ x, , (b), , 94., , p, , dy, =, If x 1 + y + y 1 + x = 0 , then, dx, , (a), , 90., , 1, , 2, , 93. If sin y + e - x cos y = e , then, , æ pö, 87. If f ( x ) = 1 + cos2 x 2 , then the value of f ¢ çç, ÷÷ is, è 2 ø, , 88., , x (1 - x )(1 - x ), , 2, , 100. Let y = t10 + 1 and x = t8 + 1, then, , d2 y, dx 2, , is equal to

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EBD_7762, 272, , MATHEMATICS, , 112. If y = e, (a), , 121. The 2nd derivative of a sin3t with respect to a cos3t at, , dy, , then ×, =, dx, , xx, , t=, x, (b) yx (1 + log e x ), , y(1 + log e x ), , x, (d) None of these, (c) ye (1 + log e x), 113. The value of the derivative of |x – 1| + |x – 3|, at x = 2 is :, (a) –2, (b) 0, (c) 2, (d) not defined, æ 2x ö, 114. The derivative of sin –1 ç, ÷ with respect to, è 1+ x2 ø, , é1 - x 2 ù, cos -1 ê, ú is equal to :, ë1 + x 2 û, (a) 1, (b) – 1, (c) 2, (d) None of these, 115. Let f (x) = ex, g(x) = sin–1x and h (x) = f (g(x)), then, h'(x)/h(x) =, , (a), , -1, , (b) 1/ 1 – x 2, (c) sin–1 x, (d) 1/(1 – x2), –1, 116. If y = cos (cos x), then y ' (x) is equal to, (a) 1 for all x, (b) –1 for all x, (c) 1 in 2nd and 3rd quadrant, (d) –1 in 3rd and 4th quadrant, 117. Let g(x) be the inverse of an invertible function f(x), which, is differentiable for all real x, then g''(f(x)) equals, f ''( x), f '( x) f ''( x) - ( f '( x))3, (a), (b), f '( x ), ( f '( x ))3, (c), , esin, , f '( x) f ''( x ) - ( f '( x)) 2, ( f '( x))2, , If F(x) = (hogof) (x), then F¢¢ ( x ) is equal to, , (a) a cosec3x, (c) 2x cot x2, , dy, is equal to :, dx, , (a) 12, (b) 32, 124. If x2 + y2 = 1, then, , 3, , x3, , (a), , e, , (c), , 3x 3 e x, , (b) 3x 2 2e x 3, 3, , (d) 3x 3e x 3 + 3x 2, , (c) 36, , 2, yy¢¢ - ( 2y¢ ) + 1 = 0, , (b) yy¢¢ - ( y¢ )2 + 1 = 0, , (c), , 2, yy¢¢ - ( y¢ ) - 1 = 0, , (d) yy¢¢ - 2 ( y¢ )2 + 1 = 0, , (c) 3 x - 2 x 2 (d) 1 - 3 x + 2 x 2, , (b) 2 - 4x, , d é ì x æ x - 2 öü, 126. dx êêlog íe çè x + 2 ÷ø ý, þ, ë î, , 3/ 4 ù, , (a) 1, , (b), , 127. If yx = ey – x, then, , (a), , 1 + log y, y log y, , ú is equal to, úû, , x2 +1, x2 - 4, , (c), , 128., , ( log y ), , 2, , x2 -1, x2 - 4, , x, (d) e, , x2 -1, x2 - 4, , dy, is equal to, dx, , (b), , 1 + log y, , (c), , for x ³ 0, ì sin x,, 119. Let f ( x ) = í, and g(x) = ex. Then the, î1 - cos x, for x £ 0, , 120. The derivative of e x with respect to log x is, , is equal to, ds2, (d) 10, , (a), , (a) 1 - 2x, , sin (a + y), sin a, , (b) – 1, (d) None of these, , d2 u, , 2, 125. If y = x - x 2 , then the derivative of y 2 with respect to x is, , (d) None of these, , value of (gof)¢ (0) is, (a) 1, (c) 0, , (b) 2 cot x2 – 4x2 cosec2x2, (d) – 2 cosec2x, , 123. If u = x2 + y2 and x = s + 3t, y = 2s – t, then, , (d) None of these, , (b), , (c) sin (a + y), , (b) 2, , 1, (d) None of these, 12a, 122. Let f(x) = sinx, g(x) = x2 and h(x) = logex., , 2, , sin a, sin (a + y), , 4 2, 3a, , (c), , x, , 118. If sin y = x sin (a + y), then, (a), , (a), , p, is, 4, , (d), , (1 + log y )2, y log y, , (1 + log y )2, log y, , d æ -1 1 + x2 - 1ö, ç tan, ÷ is equal to :, ÷ø, dx çè, x, 1, (a), 1 + x2, , (b), , 2, 1 + x2, , (d), , (c), , x2, 2 1 + x 2 ( 1 + x 2 - 1), 1, 2(1 + x 2 )

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CONTINUITY AND DIFFERENTIABILITY, , 273, , 129. If f '' (x) = – f (x) and g (x) = f ' (x) and, æ, F( x ) = ç, è, , 2, , 2, , æ xö ö æ æ xö ö, f ç ÷ ÷ + ç g ç ÷ ÷ and given that, è 2ø ø è è 2 ø ø, , F (5) =5, then F (10) is equal to –, (a) 5, (b) 10, (c) 0, , (a) n2y, , (b) – n2y, , 131. If y = sin x, , (c), , x, , é 6 log 2 - 3p ù, û, 6 ë, , 2, , -, , p, 6, , dx, , 2, , +x, , (c) –y, , (b), , dy, is, dx, , (d) 2x2y, , dy, p, at x = - is, dx, 6, , , then the value of, , p, 2 6, , d2 y, , 1 £ x £ 2 at the point, , p, 26, , é6 log 2 + 3p ù, û, 6 ë, , é6 log 2 + 3p ù, û, 6 ë, , (d) None of these, , 2, 7, , (b), , 1, 2, , (c) 2, , (d), , 7, 2, , d, f ( x ) is, dx, (b) 2, , 134. If 2f (sin x ) + f (cos x ) = x , then, (a) sin x + cos x, 1, , (c), , (d) None of these, , 1- x 2, , 135. Let f (x) be a twice differentiable function and f '' (0) = 5, then, the value of lim, , 3f (x) - 4f (3x) + f (9x), , is, x2, (a) 0, (b) 120, (c) –120, (d) does not exist, 136. The set of the points where f(x) = x | x | is twice, differentiable, will be, (a) R, (b) R0, (c) R+, (d) R–, x ®0, , Topic 4: Rolle’s Theorem, Langrange’s Mean Value, Theorem, 137. In [0, 1] Lagranges Mean Value theorem is NOT applicable to, , (a), , (c), , (a) b = 8, c = – 5, (c) b = 5, c = – 8, , 4, , the value of b and c are, 3, (b) b = – 5, c = 8, (d) b = – 5, c = – 8, , f (b) - f (a), = f ¢ (c) , if a = 0,, b-a, b = 1/2 and f (x) = x (x – 1) (x – 2), the value of c is, , 142. In the mean value theorem, , 132. If y = 3 cos (log x) + 4 sin (log x), then, (a) xy2 + y1 + y = 0, (b) xy2 + y1 – y = 0, (c) x2y2 + xy1 + y = 0, (d) None of these, 133. Let 3f(x) – 2f(1/x) = x, then f '(2) is equal to, (a), , 1, log e 3 (c) log3e (d) loge3, 2, 139. If f(x) = xa log x and f(0) = 0, then the value of a for, which Rolle’s theorem can be applied in [0, 1] is, (a) – 2, (b) – 1, (c) 0, (d) 1/2, 140. The equation ex–8 + 2x – 17 = 0 has, (a) two real roots, (b) one real root, (c) eight real roots, (d) four real roots, 141. Rolle’s Theorem holds for the function x3 + bx2 + cx,, , (a) 2 log3e (b), , (d) 15, , 130. If y = (x + 1 + x 2 )n, then (1 + x2), , (a), , 138. A value of c for which the Mean Value Theorem holds for, the function f(x) = logex on the interval [1, 3] is, , ì 1, ï 2-x, ï, f (x) = í, 2, ïæç 1 - x ö÷, ïîè 2, ø, , f ( x) = x x, , 1, ì sin x, 2, (b) f ( x ) = ïí x , x ¹ 0, 1, ïî 1,, x =0, x³, 2, x<, , (d) f ( x) = x, , 15, 21, (b) 1 + 15 (c) 1 (d) 1 + 21, 6, 6, 143. Assertion : Rolle's theorem can not be verified for the, function f (x) = |x| in the interval [–1, 1]., Reason : The function f (x) = |x| is differentiable in the, interval (–1, 1) everywhere., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 144. The value of c in Lagrange's theorem for the function, (a) 1 -, , ì, æ1ö, ï x cos ç ÷ , x ¹ 0, f ( x) = í, è xø, in the interval [–1, 1] is, ï, 0,, x= 0, î, 1, (a) 0, (b), 2, 1, (c), (d) non-existent in the interval, 2, 145. The value of c in Rolle’s Theorem for the function, f(x) = ex sinx, x Î [0, p] is, p, p, p, 3p, (b), (c), (d), 6, 4, 2, 4, 146. Let f(x) satisfy the requirements of Lagrange’s mean value, , (a), , theorem in [0, 2]. If f(0) = 0 and f ¢ ( x ) £, 2], then, (a), , f (x) £ 2, , (b) f(x) £ 1, (c) f(x) = 2x, (d) f(x) = 3 for atleast one x in [0, 2], , 1, for all x in [0,, 2

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EBD_7762, 274, , MATHEMATICS, , dn, (xn log x), then In– nIn –1=, 152. If I n =, n, dx, (a) n, (b) n –1, (c) n!, (d) (n –1)!, , BEYOND NCERT, Topic 5: Removable Discontinuity,, , y + ...¥, , 153. If x = e y + e, p, ì -1, ïïcos {cot x}, x < 2, 147. f ( x ) = í, p, ï p[ x] - 1,, x³, ïî, 2, , (a), , where [×] represents the greatest function and {}, × represents, the fractional part function. The jump of discontinuity is, p, p, - 1 (b), (a), (c) 1, (d) p, 2, 2, 148. Which of the following function(s) has/have removable, discontinuity at x = 1?, , (a), , f ( x) =, , 1, ln | x |, , (b) f ( x ) =, , 1, , (c), , f ( x) = 2, , 21- x, , (d) f ( x ) =, , 149. Which of the following is true about, , (a), , 2, , 151. If f ( x ) = x tan( x ) - x ln(1 + x ), then the value of, d 4 ( f ( x )), dx 4, (a) 0, , at x = 0 is, (b) 6, , y2, y -1, , x, 1+ x, , (d), , dy, =, dx, , y, (c) 1 - y, , dn y, dx n, , -y, (d) 1 - y, , = -a cos x + bsin x , then, , (c) 6, , (d) 8, , (b), , (c), , cos x, 2 y -1, , (d) None of these, 1, 1, , x+, , (a), , 1, x + ..., , y, x, (b), 2y - x, x- y, , then, , dy, =, dx, , cos x, 2 y +1, , y cos x, 2y -1, , dy, ., dx, , 2y, (c) x - y, , (d) 1, , 158. If y = log x + log x + log x + ....¥ , then, (a), , x, 2 y -1, , (b), , x, 2 y +1, , (c), , 1, x (2 y - 1), , (d), , 1, x (1 - 2 y ), , dy, =, dx, , 159. If y = log (1 + sin x), then y4 + y3y1 + y22 is, (a) 1, (b) –1, (c) 2, (d) 0, 160., , dn, dx n, , (a), (c) 12, , , then, , 1- x, x, , (a), , x+, , BEYOND NCERT, , 3, , (b), , (c), , (b) 4, , 157. If y = x +, , (a) f(x) is continuous at x = 2, (b) f(x) has removable discontinuity at x = 2, (c) f(x) has non-removable discontinuity at x = 2, (d) Discontinuity at x = 2 can be removed by redefining, function at x = 2., , 4, , y2, 1- y, , dy, is, dx, , 156. If y = sin x + sin x + sin x + ...........¥ then, , x2 - x, , Topic 6: Successive Differentiation, nth Derivative, Differentiation of Infinite Series., , x + e x +...to ¥, , n =, (a) 2, , x -1, , 150. The function f ( x ) = sin(loge | x |), x ¹ 0, and 1 if x = 0, (a) is continuous at x = 0, (b) has removable discontinuity at x = 0, (c) has jump discontinuity at x = 0, (d) has oscillating discontinuity at x = 0, , 1, x, , 155. If y = a cos x – b sin x and, , 1, , ì ( x - 2) æ x 2 - 1 ö, ï, ç, ÷; x ¹ 2, ï | x - 2 | çè x 2 + 1 ÷ø, f ( x) = í, ï, 3, x=2, ;, ï, 5, î, , (b), , 154. If y = e x + e, , 3, , x + 1 - 2x, , 1+ x, x, , , x > 0, then, , (d) 24, (c), , (log x) =, , (n - 1)!, x, , n, , (n - 2)!, x, , n, , (b), , n!, xn, , n -1, (d) (-1), , (n - 1)!, xn

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CONTINUITY AND DIFFERENTIABILITY, , 275, , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, 1., , x2, + 1 , then which of the following, 2, can be a discontinuous function?, (a) f(x) + g(x), (b) f(x) – g(x), , If f(x) = 2x and g ( x ) =, , g ( x), (d) f x, ( ), , (c) f(x).g(x), 2., , 3., , The function f ( x ) =, , is, 4 x - x3, (a) discontinuous at only one point, (b) discontinuous at exactly two points, (c) discontinuous at exactly three points, (d) None of the above, The set of points wher e the function f given by, f (x) = |2x – 1| sin x is differentiable is, , æ1ö, (b) R - ç ÷, è2ø, (c) (0, ¥), (d) None of these, The function f(x) = cot x is discontinuous on the set, (a), , 5., , 6., , 7., , {x = np, n Î Z}, , (b), , 1, , where x ¹ 0, then the value of the, x, function f at x = 0, so that the function is continuous at, x = 0, is, (a) 0, (b) – 1, (c) 1, (d) None of these, , 2, If f ( x ) = x sin, , é, ê mx + 1, if x £, If f ( x ) = ê, êsin x + n, if x >, êë, , p, p, 2, is continuous at x = ,, p, 2, 2, , then, , 8., , np, +1, 2, p, (d) m = n =, 2, , (b) m =, , mp, 2, Let f(x) = |sin x|. Then, (a) f is everywhere differentiable, , (c) n =, , 9., , æ 1 - x2, If y = log ç, ç 1 + x2, è, , (a), , 4x 3, 1- x4, , (b), , ö, dy, , is equal to, ÷ , then, ÷, dx, ø, , -4x, 1- x, , 4, , 10. If y = sin x + y , then, , (a), 11., , (c), , 1, 4-x, , 4, , (d), , -4x 3, 1- x4, , dy, is equal to, dx, , cos x, cos x, (b) 1 - 2 y, 2 y -1, , sin x, sin x, (c) 1 - 2 y (d) 2 y - 1, , The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is, (a) 2, , {x = 2np, n Î Z}, , p, np, ì, ü, ì, ü, ; n Î Zý, (c) íx = ( 2n +1) ;n Î Zý (d) íx =, 2, 2, î, þ, î, þ, The function f (x) = e| x | is, (a) continuous everywhere but not differentiable at x = 0, (b) continuous and differentiable everywhere, (c) not continuous at x = 0, (d) None of the above, , (a) m = 1, n = 0, , p, , n Î Z., 2, (d) None of these, x = ( 2n + 1), , 4 - x2, , (a) R, 4., , (b) f is everywhere continuous but not differentiable at, x = np, n Î Z, (c) f is everywhere continuous but not differentiable at, , (c), , (b), , 2, x, , -1, 2 1- x2, , (d) 1 – x2, , 12. If x = t2 and y = t3, then, , d2y, dx 2, , is equal to, , 3, 3, 3, 3, (b), (c), (d), 4t, 2t, 2t, 2, 13. The value of c in Rolle's theorem for the function f (x) = x3, , (a), , – 3x in the interval éë0, 3 ùû is, (a) 1, , (b) – 1, , (c), , 3, 2, , (d), , 1, 3, , 1, 14. For the function f ( x ) = x + , x Î [1, 3] , the value of c for, x, mean value of theorem is, , (a) 1, (c) 2, , (b) 3, (d) None of these, , Past Year MCQs, 15. If f and g are differentiable functions in [0, 1] satisfying f, (0) = 2 = g(1), g(0) = 0 and f (1) = 6, then for some c Î]0,1[, [JEE MAIN 2014, C], (a) f ¢(c) = g ¢(c), (b) f ¢(c) = 2g ¢(c), (c) 2f ¢(c) = g¢(c), (d) 2f ¢(c) = 3g¢(c)

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EBD_7762, 276, , 16., , MATHEMATICS, , 1, ì, if x ¹ 1, ï( x –1) sin, Let f ( x) = í, x –1, ïî, 0, if x = 1, , Then which one of the following is true?, [BITSAT 2014, A], (a) f is differentiable at x = 0 and x =1, (b) f is differentiable at x = 0 but not at x = 1, (c) f is differentiable at x = 1 but not at x = 0, (d) f is neither differentiable at x = 0 nor at x =1, 17., , 18., , d, (cos–1 x + sin–1 x) is, dx, (a) p/2, , (b) 0, , (c) 2 / 1 - x 2, , (d) None of these, , [BITSAT 2014, C], , If the function., ïì k x + 1, 0 £ x £ 3, g(x) = í, is differentiable, then the value, ïî m x + 2, 3 < x £ 5, of k + m is :, [JEE MAIN 2015, A], , 10, (a), 3, , 19., , 16, (b) 4, (c) 2, (d), 5, 2, d y, dy, n, 2, 1 + x 2 ) , then (1 + x ) 2 + x dx is, dx, , If y = (x +, , 23., , 24., , x, x, +, 1 + x ( x + 1)( 2x + 1), , +, , x, , ( 2x + 1)( 3x + 1), , + ....¥ ,, , then at x = 0, f(x), [BITSAT 2015, S], (a) has no limit, (b) is not continuous, (c) is continuous but not differentiable, (d) is differentiable, For x Î R, f(x) = |log2 – sinx| and g(x) = f(f(x)), then :, [JEE MAIN 2016, A], (a) g'(0) = – cos(log2), (b) g is differentiable at x = 0 and g'(0) = – sin(log2), (c) g is not differentiable at x = 0, (d) g'(0) = cos(log2), For any differentiable function y of x,, d2x, , æ dy ö, , 3, , d2y, , ç ÷ + 2 =, dy 2 è dx ø, dx, , (a) 0, , (b) y, , [BITSAT 2016, A], (c) – y, , (d) x, , é1 - (log x)2 ù, ú, 2 then the value of f ' (e) is equal, ëê1 + (log x) ûú, , 28. If f (x) = cos–1 ê, to, , [BITSAT 2017, A], , (a) 1, , [BITSAT 2015, A], (a) n2y, (b) – n2y, (c) –y, (d) 2x2y, 20. The number of real roots of the equation ex–1 + x – 2 = 0 is, [BITSAT 2015, A], (a) 1, (b) 2, (c) 3, (d) 4, 21. If g is the inverse of function f and f ¢(x) = sin x, then, g¢(x) is equal to, [BITSAT 2015, A], (a) cosec {g(x)}, (b) sin {g(x)}, 1, (c) sin {g ( x )}, (d) None of these, 22. If a function f(x) is given by, f (x) =, , 25. Let f ( x) = x - 1 + x + 24 - 10 x - 1;, 1 < x < 26 be real valued function. Then f ¢(x) for 1 < x < 26, is, [BITSAT 2016, A], 1, (a) 0, (b), x -1, (c) 2 x - 1 - 5, (d) None of these, 1, 26. The number of points at which the function f ( x ) =, log | x |, is discontinuous is :, [BITSAT 2016, C], (a) 1, (b) 2, (c) 3, (d) 4, dy, 27. If (2 + sin x), + (y + 1) cos x = 0 and y(0) = 1, then, dx, æ pö, y ç ÷ is equal to :, [JEE MAIN 2017, A], è 2ø, 4, 1, 2, 1, (a), (b), (c) (d) 3, 3, 3, 3, , (b) 1, e, , (c), , 2, e, , (d) 22, e, , ì x log cos x, ,x ¹ 0, ï, 29. If f (x) = í log(1 + x 2 ), then f(x) is, ï, 0, ,, x, =, 0, î, , (a), (b), (c), (d), , [BITSAT 2017, S], continuous as well as differentiable at x = 0, continuous but not differentiable at x = 0, differentiable but not continuous at x = 0, neither continuous nor differentiable at x = 0, , 30. The function f ( x ) = x - | x - x 2 |, - 1 £ x £ 1 is continuous, on the interval, [BITSAT 2017, A], (a) [–1, 1], (b) (–1, 1), (c) {–1, 1] – { 0 }, (d) (–1, 1) – {0}, 31. Let S = { t Î R : f (x) = | x - p | (e|x| - 1)sin | x | is not, differentiable at t}. Then the set S is equal to :, [JEE MAIN 2018, S], (a) {0}, (b) {p}, (c) {0, p}, (d) f (an empty set), 32., , If f ( x ) = sin x , when x is rational ü, ý, = cos x , when x is irrational þ, Then the function is, [BITSAT 2018, A], (a) discontinuous at x = np + p/4, (b) continuous at x = np + p/4, (c) discontinuous at all x, (d) none of these, , 3p, ì, ï1, when 0 < x £ 4, 2, 3p, 33. If f (x) = íï, [BITSAT 2018, A], 2sin x, when, <x<p, 9, 4, î, , (a) f (x) is continuous at x = 0

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CONTINUITY AND DIFFERENTIABILITY, , 277, , (b) f (x) is continuous at x = p, 3p, (c) f (x) is continuous at x =, 4, (d) f (x) is discontinuous at x =, , x, x +1, d2y, +, , then, at x = 1 is equal to, x +1, x, dx 2, [BITSAT 2018, A], 7, 7, 1, -7, (b), (c), (d), 4, 8, 4, 8, , 35. If y =, 3p, 4, , (a), , 34. The value of c in (0, 2) satisfying the mean value theorem, for the function f(x) = x(x – 1)2, x Î [0, 2] is equal to, [BITSAT 2018, C], 3, 4, 1, 2, (b), (c), (d), (a), 4, 3, 3, 3, , 36. Let y =, , æ d 2 y ö æ d 2x ö, [BITSAT 2018, A], ç 2 ÷ ç 2 ÷ is, è dx ø è dy ø, (b) e–2x, (c) 2e–2x (d) –2e–2x, , e2x. Then, , (a) 1, , Exercise 3 : Try If You Can, 1., , Let f ¢¢(x) be continuous at x = 0 and f ¢¢(0) = 4., Then value of lim, , 2f (x) - 3f (2x) + f (4x), x2, (c) 6, , x ®0, , 2., , is, , ( f (7))2 + ( f (2)) 2 + f (2) f (7), is,, 3, , (a) Only (i), 7., , 4., , (b) 0, (d) None of these, , æ 5p ö, represents greatest integer function, then f ¢ ç, is, ç 3 ÷÷, è, ø, equal to, , (c), , 5p, 3, -2, , 5p, 3, , 1, , 5., , Let f ( x) = 3 - x, , 1, 5 - 3x, , (b) 2, , 5p, 3, , (d) -, , 5p, 3, , 1, 2, , 3, , 3 x - 1 . Then which of the, , 2 x 2 - 1 3 x5 - 1 7 x8 - 1, , following is/are correct?, (a) f (x) = 0 has at least two real roots, (b) f¢ (x) = 0 has at least one real root., (c) f (x) is many-one function, (d) All of the above, , (b) Only (ii), , (c) Only (iii), (d) All of these, 2 h ( x ) + | h( x ) |, If g ( x) =, where h( x ) = sin x - sin n x ,, 2h ( x) - | h ( x ) |, æ p ö æp, ö, x Î ç 0,, ÷È ç , p ÷, 2ø è2, è, ø, p, x=, 2, , ì, ïï[ g ( x)],, f ( x) = í, ï3,, ïî, , æp, 2ö, If f ( x) = sin ç [ x] - x ÷ , where 2 < x < 3 and [ . ], è3, ø, , (a), , ìï 0, x ³ 0, (ii) f ( x) = í 2, ïî x , x < 0, , n Î R + , where R+ is the set of positive real numbers, and, , æ pö, , n > 1, then f ' çè ÷ø is, 2, (a) 1, (c) – 1, , f(x) = min {x, sin x}, , (iii) f(x) = x2 sgn(x), , where, , c Î [2, 7] ., (a) 5f 2(c) f '(c) (b), 5f '(c), (c) f (c) f '(c), (d), None of these, If f (x) = cos x cos 2x cos2 2 x cos 23 x ..... cos 2 n–1 x and, , Which of the following functions is not differentiable at, x = 0?, (i), , (a) 12, (b) 10, (d) 4, Let f :[2, 7] ® [0, ¥) be a continuous and differentiable, function. Then,, ( f (7) - f (2)), , 3., , 6., , 8., , Where [x] denotes the greatest integer function , then, p, (a) f (x) is continuous and differentiable at x = , when, 2, 0 < n <1, p, (b) f (x) is continuous and differentiable at x = , when, 2, n >1, p, (c) f (x) is continuous but not differentiable at x = ,, 2, when 0 < n < 1, p, (d) f (x) is continuous but not differentiable at x = ,, 2, when n > 1, Let 0 < x < p and y(x) be given by, (1+sin x) y3 – (cos x) y2 + 2 (1+sin x) y –2 cos x = 0., The derivative of y with respect to tan, (a), , 1, 2, , (b) -, , 1, 2, , (c) 2, , x, p, at x = is :, 2, 2, , (d) –2

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EBD_7762, 278, , 9., , MATHEMATICS, , 13. If f (x) be a differentiable function satisfying, , Number of points where function f (x) defined as, ì, ï3 - cos x ï, f :[0, 2p] ® R, f ( x) = í, ï 2 + cos x +, ïî, , non-differentiable is, (a) 2, (b) 3, The function f defined by, , 10., , (c) 4, , 1, 1, , | sin x |<, 2, 2, 1, 1, , | sin x |³, 2, 2, , is, , (d) 5, , ìï (1 + sin p x ) - 1 üï, f ( x) = lim í, ý is, t, t ®¥ î, ï (1 + sin p x) + 1þï, (a) every where continuous, (b) discontinuous at all integer values of x, (c) continuous at x = 0, (d) None of these, Let F(x) = f(x) g(x) h (x) for all x, where g(x) and h(x) are, differentiable functions. At some point x0, F¢(x0) = 21F(x0),, f ¢(x0) = 4f(x0), g¢(x0) = –7g(x0) and h¢(x0) = kh(x0). Then, k is:, (a) 1, (b) 7, (c) 24, (d) 5, If f is a real valued differentiable function satisfying, t, , 11., , 12., , | f (x) – f (y) | £ ( x - y )2 , x, y Î R and f (0) = 0, then f (1), equals, (a) – 1, , (b) 0, , (c) 2, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, , (d), (c), (d), (d), (b), (b), (c), (a), (a), (d), (c), (d), (d), (a), (b), (a), , 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, , (a), (c), (b), (b), (a), (a), (d), (d), (c), (b), (d), (c), (b), (d), (d), (d), , 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, , (c), (b), (c), (a), (c), (b), (c), (b), (a), (a), (b), (c), (c), (c), (b), (c), , 1, 2, 3, 4, , (d), (c), (b), (a), , 5, 6, 7, 8, , (a), (a), (c), (b), , 9, 10, 11, 12, , (b), (a), (a), (b), , 1, 2, , (a), (a), , 3, 4, , (a), (b), , 5, 6, , (d), (d), , (d) 1, , æxö, f ( y ) f ç ÷ = f ( x )"x, y Î R, y ¹ 0 and f (1) ¹ 0 ,, è yø, f ¢(1) = 3 , then which of the following is/are correct?, , (a) sgn( f ( x)) is non-differentiable at exactly one point, (b), , x 2 (cot x - 1), =0, x®0, f ( x), lim, , f ( f ( x)) - f 3 ( x) = 0 has infinitely many solutions, (d) All of the above, (c), , 14. If f n ( x ) = e fn –1 ( x ) for all n Î N and f 0 (x) = x, then, d, { f n ( x)} is equal to, dx, , (a), , f n ( x )., , d, { f n –1 ( x )}, dx, , n, , (b), , Õ f i ( x), i =1, , (c) fn (x). fn – 1 (x)... f2 (x). f1 (x), (d) All of the above, 15. If f (x) =, (a) – 1, , æ xx – x – x ö, ç, ÷ , then f ¢ (1) is, 2, è, ø, (b) 1, (c) log 2 (d) – log 2, , cot–1, , ANSW ER KEYS, Exercise 1 : Topic-w ise MCQs, (c), (a), (c) 97, (c), 49, 65, 81, (d), (b), (c), (a), 50, 66, 82, 98, (d), (b), (b) 99, (d), 51, 67, 83, (d), (d), (a) 100 (c), 52, 68, 84, (d), (c), (c) 101 (d), 53, 69, 85, (b), (a), (d) 102 (d), 54, 70, 86, (c), (c), (b) 103 (c), 55, 71, 87, (a), (d), (c) 104 (a), 56, 72, 88, (c), (c), (d) 105 (c), 57, 73, 89, (c), (b), (b) 106 (a), 58, 74, 90, (d), (c), (d) 107 (a), 59, 75, 91, (a), (d), (c) 108 (b), 60, 76, 92, (b), (a), (c) 109 (c), 61, 77, 93, (a), (c), (b) 110 (c), 62, 78, 94, (b), (d), (d) 111 (a), 63, 79, 95, (a), (a), (b) 112 (b), 64, 80, 96, Exercise 2 : Exempl ar & Past Year MCQs, (a), (b), (c) 25, (a), 13, 17, 21, (b), (c), (b) 26, (c), 14, 18, 22, (b), (a), (d) 27, (b), 15, 19, 23, (b), (a), (a), (b), 16, 20, 24, 28, Exerci se 3 : Try If You Can, (b), (c), (c) 13 (d), 7, 9, 11, (b), (b), (b) 14 (d), 8, 10, 12, , 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, , (b), (a), (b), (a), (a), (b), (c), (c), (a), (d), (d), (b), (d), (c), (d), (d), , 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, , (a), (a), (a), (c), (b), (c), (b), (b), (a), (a), (d), (b), (b), (c), (c), (a), , 29, 30, 31, 32, , (a), (a), (d), (b), , 33, 34, 35, 36, , (c), (b), (a), (d), , 15, , (a), , 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, , (d), (b), (a), (d), (c), (d), (a), (d), (c), (c), (a), (c), (a), (c), (d), (d)

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22, , APPLICATION OF DERIVATIVES, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 7, 6, 5, JEE MAIN, , 4, , BITSAT, 3, 2, 1, 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 6, 9, , Critical Concepts, , Rate of Change of Quantities,, Increasing & Decreasing Functions,, Tangents &Normals, Maxima &, Minima of Functions, , Rating of, Difficulty Level, , CUS, (chapter utility score), Out of 10, , 4.5/5, , 7.5

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APPLICATION OF DERIVATIVES, , 281

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EBD_7762, 282, , MATHEMATICS, , Topic 1 : Rate of Change of Quantities., 1., , 9., , A point on the parabola y2 = 18x at which the ordinate, increases at twice the rate of the abscissa is, æ9 9ö, , (a) ç 8 , 2 ÷, è, ø, , (b), , (2, - 4), , æ -9 9ö, çè , ÷ø, (d) (2, 4), 8 2, A stone is dropped into a quiet lake and waves moves in, circles at the speed of 5 cm/s. If at a instant, the radius of, the circular wave is 8 cm, then the rate at which enclosed, area is increasing, is, (a) 20 p cm2/s, (b) 40 p cm2/s, 2, (c) 60 p cm /s, (d) 80 p cm2/s, A particle moves along the curve 6y = x3 + 2. The point, ‘P’ on the curve at which the y-coordinate is changing 8, times, (c), , 2., , 3., , 4., , 5., , 31 ö, æ, as fast as the x-coordinate, are (4, 11) and ç -4, - ÷ ., 3ø, è, (a) x-coordinates at the point P are ± 4, -31, (b) y-coordinates at the point P are 11 and, 3, (c) Both (a) and (b), (d) None of the above, A ball is dropped from a platform 19.6m high. Its position, function is –, (a) x = – 4.9t2 + 19.6 (0 £ t £ 1), (b) x = – 4.9t2 + 19.6 (0 £ t £ 2), (c) x = – 9.8t2 + 19.6 (0 £ t £ 2), (d) x = – 4.9t2 – 19.6 (0 £ t £ 2), A lizard, at an initial distance of 21 cm behind an insect,, , 10., , 11., , 12., , 13., , 2, , 6., , 7., , 8., , moves from rest with an acceleration of 2 cm / s and pursues, the insect which is crawling uniformly along a straight line at, a speed of 20 cm/s. Then the lizard will catch the insect after, (a) 20 s (b) 1 s, (c) 21 s, (d) 24 s, The radius of a cylinder is increasing at the rate of 3 m/s, and its altitude is decreasing at the rate of 4 m/s. The rate, of change of volume when radius is 4 m and altitude is, 6m, is, (a) 20 p m3/s, (b) 40 p m3/s, (c) 60 p m3/s, (d) None of these, The radius of a sphere initially at zero increases at the rate, of 5 cm/sec. Then its volume after 1 sec is increasing at, the rate of :, (a) 50 p, (b) 5 p, (c) 500 p, (d) None of these, If a circular plate is heated uniformly, its area expands, 3c times as fast as its radius, then the value of c when, the radius is 6 units, is, (a) 4 p, (b) 2 p, (c) 6 p, (d) 3 p, , 14., , A man is moving away from a tower 41.6 m high at a rate, of 2 m/s. If the eye level of the man is 1.6 m above the, ground, then the rate at which the angle of elevation of the, top of the tower changes, when he is at a distance of 30 m, from the foot of the tower, is, 4, 2, (a) rad/s, (b) rad/s, 125, 25, 1, (c) rad/s, (d) None of these, 625, A ladder is resting with the wall at an angle of 30°. A man, is ascending the ladder at the rate of 3 ft/sec. His rate of, approaching the wall is, 3, ft / sec, (a) 3 ft / sec, (b), 2, 3, 3, ft / sec, ft / sec, (c), (d), 2, 4, The length x of a rectangle is decreasing at the rate of, 5 cm/min and the width y is increasing at the rate of, 4 cm/min. If x = 8 cm and y = 6 cm, then which of the, following is correct?, I. The rate of change of the perimeter is – 2 cm/min., II. The rate of change of the area of the rectangle is, 12 cm2/min., (a) Only I is correct, (b) Only II is correct, (c) Both I and II are correct, (d) Both I and II are incorrect, The rate of increase of bacteria in a certain culture is, proportional to the number present. If it doubles in 5 hours, then in 25 hours, its number would be, (a) 8 times the original, (b) 16 times the original, (c) 32 times the original (d) 64 times the original, A football is inflated by pumping air in it. When it, acquires spherical shape its radius increases at the rate, of 0.02 cm/s. The rate of increase of its volume when the, radius is 10 cm is ___________ p cm/s, (a) 0, (b) 2, (c) 8, (d) 9, Assertion: The ordinate of a point describing the circle, x2 + y2 = 25 decreases at the rate of 1.5 cm/s. The rate of, change of the abscissa of the point when ordinate equals 4, cm is 2 cm/s., Reason: xdx + ydy = 0., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., , bö, æ, 15. The cost of running a bus from A to B, is ` ç av + ÷ ,, vø, è, where v km/h is the average speed of the bus. When the, bus travels at 30 km/h, the cost comes out to be ` 75 while

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APPLICATION OF DERIVATIVES, , 283, , at 40 km/h, it is ` 65. Then the most economical speed (in, km/ h) of the bus is :, (a) 45, (b) 50, (c) 60, (d) 40, 16. Two men A and B start with velocities v at the same, time from the junction of two roads inclined at 45° to, each other., Statement I: If they travel by different roads, then the, rate at which th ey are being separated, is, , (, , ), , p, at the uniform rate of 2 cm2/s is the surface area, 4, through a tiny hole at the vertex of the bottom. When the, slant height of cone is 4 cm, then rate of decrease of the, slant height of water is, , angle, , 2, cm / s, 3p, , (b), , 2, cm / s, p, , 2, cm / s, (d) None of these, 4p, 19. The altitude of a cone is 20cm and its semi-vertical angle is, 30°. If the semi-vertical angle is increasing at the rate of 2°, per second, then the radius of the base is increasing at the, rate of–, , (c), , (a) 30 cm/sec, (c) 10 cm/sec, , 160, cm/sec, 3, (d) 160 cm/sec., , (b), , Topic 2 : Increasing and Decreasing Functions., 20. The function f(x) = tan x – 4x is strictly decreasing on, , æ p pö, (a) ç - , ÷, è 3 3ø, , æ p pö, (b) ç , ÷, è3 2ø, , æ p pö, ç- , ÷, è 3 2ø, , æp ö, (d) ç , p ÷, è2 ø, , (c), , f ( x) =, , 2 - 2 v unit / s ., , Statement II: If they travel by different roads, then the rate, at which they are being separated, is 2v sin p/8 unit/s., (a) Only statement I is true, (b) Only statement II is true, (c) Both the statements are true, (d) Both the statements are false, 17. A kite is moving horizontally at a height of 151.5. If the, speed of kite is 10 m/s, then the rate at which the string is, being let out; when the kite is 250 m away from the boy, who is flying the kite and the height of the boy is 1.5 m, is, (a) 4 m/s (b) 6 m/s, (c) 7 m/s, (d) 8 m/s, 18. Water is dripping out from a conical funnel of semi-vertical, , (a), , 22. On which of the following intervals is the function, x 100 + sin x - 1 decreasing?, (a) (0, p/2), (b) (0, 1), (c) (p/2, p), (d) None of these, 23. The intervals in which the function f given by, , x, x, , where 0 < x £ 1, then, and g ( x) =, sin x, tan x, in this interval,, (a) both f(x) and g(x) are increasing functions, (b) both f(x) and g(x) are decreasing functions, (c) f(x) is an increasing function, (d) g(x) is an increasing function, , 21. If f ( x ) =, , 4 sin x - 2 x - x cos x, is increasing in x Î (0, 2p) is, 2 + cos x, , (a), , (0, p) È (2p,4p), , æ p ö æ 3p ö, (b) ç 0, ÷ È ç ,2p ÷, è 2ø è 2, ø, , (c), , æ pö æp ö, ç 0, 4 ÷ È ç 2 , p ÷, è, ø è, ø, , (d) None of these, , 24. The interval in which the function f(x) =, decreasing is :, , æ 1 1ö, (a) ç - , ÷, è 2 2ø, (c) (–1, 1), , 4x 2 + 1, is, x, , é 1 1ù, ê- 2 , 2 ú, ë, û, (d) [–1, 1], 2x, 25. The function f ( x ) = log(1 + x) is increasing on, 2+ x, , 26., , (b), , (a), , (0, ¥), , (b), , (c), , (-¥, ¥), , (d) None of these, , f (x) =, , (-¥, 0), , log(p + x ), is, log(e + x ), , (a) increasing in [0, ¥), , (b) decreasing in [0, ¥), , ép ö, é pù, (c) decreasing in ê0, ú & increasing in ê , ¥ ÷, ëe ø, ë eû, ép ö, é pù, (d) increasing in ê0, e ú & decreasing in ê , ¥ ÷, û, ë, ëe ø, 27. Assertion : Let f : R ® R be a function such that, f(x) = x3 + x2 + 3x + sin x. Then f is one-one., Reason : f(x) neither increasing nor decreasing function., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 28. If f (x) = 3x4 + 4x3 – 12x2 + 12, then f (x) is, (a) increasing in (– ¥ , – 2) and in (0, 1), (b) increasing in ( – 2, 0) and in (1, ¥ ), (c) decreasing in ( – 2, 0) and in (0,1), (d) decreasing in ( – ¥ , – 2) and in (1, ¥ ), 29. If f ( x) = x + sin x, g ( x) = e- x , u = c + 1 - c ,, , v = c - c - 1, (c > 1), then, (a), , fog (u) < fog (v), , (b), , gof (u) < gof (v ), , (c), , gof (u ) > gof (v), , (d), , fog (u ) < fog (v)

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EBD_7762, 284, , 30., , MATHEMATICS, , The maximum area of a right angled triangle with, hypotenuse h is :, (a), , h2, 2 2, , (b), , h2, 2, , (c), , h2, 2, , (d), , h2, 4, , æ e 2x - 1 ö, 31. f ( x ) = ç 2x, ÷ is, ç e +1 ÷, è, ø, (a) an increasing function (b) a decreasing function, (c) an even function, (d) None of these, 32. The function f(x) = tan–1(sin x + cos x) is an increasing, function in, , æ p pö, (a) ç , ÷, è4 2ø, (c), , æ pö, ç 0, ÷, è 2ø, , Topic 3 : Tangents and Normals, Angle of Intersection of, Two Curves, Orthogonal Curves., 38. The equation of all lines having slope 2 which are tangent, 1, , x ¹ 3 , is, x -3, (a) y = 2, (b) y = 2x, (c) y = 2x + 3, (d) None of these, 39. The slope of the normal to the curve, , to the curve y =, , (a) x = a cos3q, y = a sin 3q at q =, , æ p pö, (b) ç - , ÷, è 2 4ø, , (b) x = 1 – a sinq, y = b cos2q at q =, , æ p pö, (d) ç - , ÷, è 2 2ø, , 40., 3, , 33., , The number of solutions of the equation 3 tan x + x = 2 in, , 34., , æ pö, ç 0, 4 ÷ is, è, ø, (a) 1, (b) 2, (c) 3, (d) infinite, The difference between greatest and least value of, é 3p ù, f (x) = 2 sin x + sin 2x, x Î ê0, ú is –, ë 2û, (a), , 3 3, 2, , (b), , æ p 2p ö, (c) Decreasing in çè , ÷ø, 2 3, (d) All of the above, , 41., , 42., , 3 3, -2, 2, , 3 3, +2, (d) None of these, 2, 35. Statement I: The logarithm function is strictly increasing, on (0, ¥)., Statement II: The function f given by f(x) = x2 – x + 1 is, neither increasing nor decreasing strictly on (– 1, 1), (a) Only statement I is true, (b) Only statement II is true, (c) Both the statements are true, (d) Both the statements are false, 36. If f(x) = cosx, g(x) = cos 2x, h(x) = cos3x and I(x) = tanx,, then which of the following option is correct?, (a) f(x) and g(x) are strictly decreasing in (0, p/2), (b) h(x) is neither increasing nor decreasing in (0, p/2), (c) I(x) is strictly increasing in (0, p/2), (d) All are correct, 37. The function f (x) = 3 cos4x + 10 cos3x + 6 cos2x – 3,, (0 £ x £ p) is –, æ p 2p ö, (a) Increasing in çè , ÷ø, 2 3, æ p ö æ 2p ö, (b) Increasing in çè 0, ÷ø È çè , p÷ø, 2, 3, (c), , p, is 0, 4, , 43., , a, p, is, 2b, 2, , (c) Both (a) and (b) are true, (d) Both (a) and (b) are not true, The point of intersection of the tangents drawn to the curve, x2y = 1 – y at the points where it is met by the curve, xy = 1 – y, is given by, (a) (0, –1), (b) (1, 1), (c) (0, 1), (d) None of these, The curve given by x + y = exy has a tangent parallel to the, Y-axis at the point, (a) (0, 1), (b) (1, 0), (c) (1, 1), (d) None of these, If y = (4x – 5) is a tangent to the curve y2 = px3 + q at (2, 3),, then, (a) p = – 2, q = – 7, (b) p = – 2, q = 7, (c) p = 2, q = – 7, (d) p = 2, q = 7, The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0, intersect at an angle of, p, p, p, p, (a), (b), (c), (d), 3, 6, 4, 2, , 44. The total number of parallel tangents of f1 ( x) = x2 - x + 1, and f2 ( x) = x3 - x 2 - 2 x + 1 is, (a) 2, (b) 0, (c) 1, (d) infinite, 45. The angle of intersection to the curve y = x2,, 6y = 7 – x3 at (1, 1) is :, p, p, p, (b), (c), (d) p, 2, 4, 3, 46. If tangent to the curve x = at2, y = 2at is perpendicular to, x-axis, then its point of contact is, (a) (a, a) (b) (0, a), (c) (0, 0), (d) (a, 0), 47. The slope of the tangent to the curve x = 3t2 + 1, y= t3 –1 at, x = 1 is:, , (a), , 1, (b) 0, (c) –2, (d) ¥, 2, 48. The chord joining the points (5, 5) and (11, 227), on the curve y = 3x2 – 11x – 15 is parallel to tangent at, a point on the curve. Then the abscissa of the point is, (a) – 4, (b) 4, (c) – 8, (d) 8, , (a)

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APPLICATION OF DERIVATIVES, , 285, , 49. The condition that the curves ax2 + by2 = 1, and a1x2 + b1y2 = 1 may cut each other orthogonally is, (a), , a - a 1 b - b1, =, aa1, bb1, , (b), , (c), , a - a1 b - b1, =, a + a 1 b + b1, , (d) None of these, , a + a1 b + b1, =, aa1, bb1, , (a), , x (7x – 6), where the tangent is parallel to x-, , 1, 2, 6, 1, (b), (c), (d), 3, 7, 7, 2, 51. The distance between the point (1, 1) and the tangent to, the curve y = e2x + x2 drawn at the point x = 0 is, , (a), , (a), , –, , 1, , -1, , (b), (c), (d), 5, 5, 5, 5, 52. Assertion: The curves x = y2 and xy = k cut at right angle,, if 8k2 = 1., Reason: Two curves intersect at right angle, if the tangents, to the curves at the point of intersection are perpendicular, to each other i.e., product of their slope is – 1., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 53. The equation of one of the tangents to the curve, y = cos(x +y), -2p £ x £ 2p that is parallel to the line, x + 2y = 0, is, (a) x + 2y = 1, (b) x + 2y = p/2, (c) x + 2y = p/4, (d) None of these, 54. The locus of all th e points on the curve, xö, æ, y 2 = 4a ç x + a sin ÷ at which the tangent is parallel to, aø, è, x-axis is :, (a) y = 4a, (b) y = 4ax, , (c) y2 = 4ax, , (d) y2 = 4a2 sin, , x, a, , 55. If the tangent at any point (4m2 , 8m3 ) of x3 - y 2 = 0 is a, normal to the curve x3 - y 2 = 0, then find the values of m, are, , 56., , 1, 2, 2, 9, (a) ±, (b) ±, (c) ±, (d) ±, 3, 3, 9, 2, If the parabola y = f (x), having axis parallel to the y-axis,, touches the line y = x at (1, 1), then, , (a), , 2 f '(0) + f (0) = 1, , (b), , (b), , p, - tan -1 9, 2, , p, p, + tan -1 9, (d), 2, 2, 58. The points at which the tangent passes through the origin, for the curve y = 4x3 – 2x5 are, (a) (0, 0), (2, 1) and (– 1, – 2), (b) (0, 0), (2, 1) and (– 2, – 1), (c) (2, 0), (2, 1) and (– 3, 1), (d) (0, 0), (1, 2) and (– 1, – 2), 59. The angle of intersection of the curve y2 = x and x2 = y is, , -2, , 2, , tan -1 9, , (c), , 50. What is the x-coordinate of the point on the curve, f (x) =, axis?, , æ 5 -3 ö, y = x 2 + 4 x - 17 at çè 2 , 4 ÷ø is, , 2 f (0) + f '(0) = 1, , (c) 2 f (0) - f '(0) = 1, (d) 2 f '(0) - f (0) = 1, 57. Angle formed by the positive Y-axis and the tangent to, , (a), , æ3ö, tan -1 ç ÷, è2ø, , (b), , æ3ö, tan -1 ç ÷, è4ø, , æ1ö, -1 æ 1 ö, tan -1 ç ÷, (d) tan ç ÷, è 5ø, è2ø, 60. The shortest distance between the line y – x = 1 and the, curve x = y2 is, (c), , (a), , 3 2, 8, , (b), , 2 3, 8, , (c), , 3 2, 5, , (d), , 3, 4, , kx, , 61. The angle at which the curve y = ke intersects the, y-axis is :, (a) tan–1(k2), (b) cot–1(k2), æ 1 ö, -1, ÷, 1+ k4, sin -1 ç, (d) sec, ç, 4 ÷, è 1+ k ø, The number of tangents to the curve x3/2 + y3/2 = 2a3/2, a> 0,, which are equally inclined to the axes, is, (a) 2, (b) 1, (c) 0, (d) 4, The curve y – exy + x = 0 has a vertical tangent at the point:, (a) (1, 1) (b) at no point (c) (0, 1), (d) (1, 0), If the curves x2 = 9A (9 – y) and x2 = A(y + 1) intersect, orthogonally, then the value of A is, (a) 3, (b) 4, (c) 5, (d) 7, The equation of the tangent to 4x2 – 9y2 = 36 which is, perpendicular to the straight line 5x + 2y – 10 = 0 is, , (c), , 62., 63., 64., 65., , æ, 11 ö, ÷, (a) 5 ( y - 3) = 4 çç x 2 ÷ø, è, , (b), , 2x - 5y + 10 - 12 3 = 0, , 2x - 5y + 10 + 12 3 = 0, (d) None of these, 66. If the tangent at P(1, 1) on y2 = x(2 – x)2 meets the curve, again at Q, then Q is, (a) (2, 2), (b) (– 1, – 2), (c), , (c), , æ 9 3ö, ç , ÷, è 4 8ø, , (d) None of these

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EBD_7762, 286, , MATHEMATICS, , ax - b, has a turning point P(2, – 1), then, ( x - 1)( x - 4 ), the value of a and b respectively, are, (a) 1, 2, (b) 2, 1, (c) 0, 1, (d) 1, 0, , 67. If y =, , Topic 4 : Approximations and Errors, 68. If f(x) = x3 – 7x2 + 15, then the approximate value of, f(5.001) is, (a) 34.995, (b) – 30.995, (c) 24.875, (d) None of these, 69. If the error committed in measuring the radius of sphere,, then ... will be the percentage error in the surface area., (a) 1%, (b) 2%, (c) 3%, (d) 4%, 70. The approximate change in the volume V of a cube of, side x meters caused by increasing the side by 2%, is, (a) 1.06x3m3, (b) 1.26x3m3, 3, 3, (c) 2.50x m, (d) 0.06x3m3, 71. The approximate value of {(3.92)2 + 3(2.1)4}1/6 is, (a) 2.466 (b) 3.567, (c) 1.562, (d) 2.577, 72. Assertion: If the radius of a sphere is measure as 9 m, with an error of 0.03 m, then the approximate error in, calculating its surface area is 2.16 pm2., , æ ds ö, Reason: We have, DS = ç ÷ Dr where,, è dr ø, DS = Approximate error in calculating the surface area,, Dr = Error in measuring radius r., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 73. There is an error of 0.04 cm in the measurement of the, diameter of a sphere. When the radius is 10 cm, the, percentage error in the volume of the sphere is, (a) ± 1.2 (b) ± 1.0, (c) ± 0.6, (d) ± 0.8, 74. If the radius of a spherical balloon increases by 0.2%. Find, the percentage increase in its volume, (a) 0.8% (b) 0.12%, (c) 0.6%, (d) 0.3%, 75. The approximate value of (0.007)1\3, 23, 27, 19, 17, (a), (b), (c), (d), 120, 120, 120, 120, 76. If the error k% is made in measuring the radius of a sphere,, then percentage error in its volume is, k, %, (a) k%, (b) 3k%, (c) 2k%, (d), 3, 77. If the radius of a sphere is measured as 9 cm with an error, of 0.03 cm, then find the approximating error in, calculating its volume., (a) 2.46p cm3, (b) 8.62p cm3, 3, (c) 9.72p cm, (d) 7.6p cm3, Topic 5 : Maxima and Minima, 78. If the function f be given by f(x) = |x|, x Î R, then, (a) point of minimum value of f is x = 1, , (b) f has no point of maximum value in R, (c) Both (a) and (b) are true, (d) Both (a) and (b) are not true, 79. If at x = 1, the function x4 – 62x2 + ax + 9 attains its, maximum value on the interval [0, 2], then the value of a is, (a) 110, (b) 10, (c) 55, (d) None of these, ln x, in ( 2, ¥ ) is, x, (a) 1, (b) e, (c) 2/e, (d) 1/e, 81. The difference between the greatest and least values of, , 80. The maximum value of, , é -p p ù, the function f(x) = sin2x – x, on ê , ú is, ë 2 2û, p, 2, , p, 4, p, The function f (x) = 1 + x (sin x) [cos x], 0 < x £, 2, (where [ . ] is G.I.F.), , (a), 82., , (b) p, , (c), , 3p, 2, , (d), , p, (a) is continuous on æç 0, ö÷, è 2ø, (b) is strictly increasing in æ 0, p ö, çè, ÷, 2ø, , æ pö, (c) is strictly decreasing in çè 0, ÷ø, 2, (d) has global maximum value 2, 83. The range of the function f ( x) = 2 x - 2 + 4 - x is, (a), , (, , 2, 10, , (, , ), , (b) éë 2, 10, , ), , (d) éë 2, 10 ùû, 84. If for a function f (x), f '(a) = 0, f "(a) = 0, f '''(a) > 0, then, at x = a, f (x) is, (a) Minimum, (b) Maximum, (c) Not an extreme point (d) Extreme point, (c), , 2, 10 ùû, , 85. On the interval [0, 1] the function x 25 (1 - x)75 takes its, maximum value at the point, 1, 1, 1, (c), (d), 4, 3, 2, 86. The maximum area of rectangle inscribed in a circle of, diameter R is, , (a) 0, , (b), , 2, , R, R2, R2, (c), (d), 8, 2, 4, 87. If sum of two numbers is 3, the maximum value of the, product of first and the square of second is, (a) 4, (b) 3, (c) 2, (d) 1, 88. If A > 0, B > 0 and A + B = p/3, then the maximum value, of tan A tan B is, 1, 1, (a), (b), (c) 3, (d), 3, 3, 3, , (a) R2, , (b)

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APPLICATION OF DERIVATIVES, , 287, , 89. Let f(x) be a function defined as follows :, , 90., , 91., , 92., 93., , 94., , f ( x) = sin( x 2 - 3 x), x £ 0; and 6 x + 5 x 2 , x > 0, Then at x = 0, f(x), (a) has a local maximum (b) has a local minimum, (c) is discontinuous, (d) None of these, A right circular cylinder which is open at the top and has a, given surface area, will have the greatest volume if its, height h and radius r are related by, (a) 2h = r (b) h = 4r, (c) h = 2r (d) h = r, A wire 34 cm long is to be bent in the form of a quadrilateral, of which each angle is 90°. What is the maximum area, which can be enclosed inside the quadrilateral?, (a) 68 cm2, (b) 70 cm2, 2, (c) 71.25 cm, (d) 72. 25 cm2, f(x) = sin(sinx) for all x Î R, (a) – sin 1 (b) sin 6, (c) sin 1, (d) – sin 3, Let AP and BQ be two vertical poles at points A and B, respectively. If AP = 16 m, BQ = 22 m and AB = 20 m,, then the distance of a point R on AB from the point A, such that RP2 + RQ2 is minimum, is, (a) 5 m, (b) 6 m, (c) 10 m, (d) 14 m, Find the greatest value of the function, f ( x) =, , sin 2 x, on the interval, pö, æ, sin ç x + ÷, è, 4ø, , é pù, ê0, 2 ú, ë, û, , (a) 1, (b) 2, (c) 3, (d) None of these, 95. If the function f be given by, f(x) = x3 – 3x + 3, then, I. x = ± 2 are the only critical points for local maxima, or local minima., II. x = 1 is a point of local minima., III. local minimum value is 2., IV. local maximum value is 5., (a) Only I and II are true, (b) Only II and III are true, (c) Only I, II and III are true, (d) Only II and IV are true, 96. The local minimum value of the function f given by, f(x) = 3 + |x|, x Î R is, (a) 1, (b) 2, (c) 3, (d) 0, 97. The minimum value of the function y = x4 – 2x2 + 1 in, , é1 ù, the interval ê , 2ú is, ë2 û, (a) 0, (b) 2, (c) 8, (d) 9, 98. The maximum value of the function, y = – x2 in the interval [–1, 1] is, (a) 0, (b) 2, (c) 8, (d) 9, 99. The fuel charges for running a train are proportional to the, square of the speed generated in miles per hour and costs, ` 48 per hour at 16 miles per hour. The most economical, speed if the fixed charges i.e. salaries etc. amount to ` 300 per, hour is, (a) 10, (b) 20, (c) 30, (d) 40, , 100. Tangent is drawn to ellipse, x2, + y 2 = 1 at 3 3 cos q, sin q [where q Î (0, p/2)]. Then, 27, the value of q such that sum of intercepts on axes made by, this tangent is minimum, is, , (, , ), , (a) p 3, (b) p 6, (c) p 8, (d) p 4, 101. Assertion: If the length of three sides of a trapezium other, than base are equal to 10 cm, then the area of trapezium, when it is maximum, is 75 3 cm2., Reason: Area of trapezium is maximum at x = 5., (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 102. An isosceles triangle of vertical angle 2q is inscribed in a, circle of radius a., Statement I: The area of triangle is maximum when, p, q=, 6, Statement II: The area of triangle is minimum when, p, q=, 6, (a) Only statement I is true, (b) Only statement II is true, (c) Both the statements are true, (d) Both the statements are false, 103. The largest distance of the point (a, 0) from the curve, 2x2 + y2 – 2x = 0, is given by, (a), , (1 - 2a + a 2 ), , (b), , (1 + 2a + 2a 2 ), , (c), , (1 + 2a - a 2 ), , (d), , (1 - 2a + 2a 2 ), , 104. The maximum value of the function sin x + cos x is, 2, (a) 1, (b), (c) 2, (d) None of these, 105. The minimum value of the function, , 1ö, æ, 3/2, -3/2, - 4 ç x + ÷ for all permissible real x, is, f (x) = x + x, è, xø, (a) – 10 (b) –6, (c) –7, (d) – 8, 106. The maximum value of [x(x – 1) + 1]1/3, 0 £ x £ 1 is, 1/ 3, , æ1ö, (a) ç ÷, è 3ø, , (b), , 1, 2, , (c) 1, , ( 2x2 -2x +1) sin 2 x, , (d) zero, , 107. The minimum value of e, is, (a) 0, (b) 1, (c) 2, (d) 3, 108. If a point on the hypotenuse of a triangle is at distance a, and b from the sides of triangle, then the minimum length, of the hypotenuse is

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EBD_7762, 288, , MATHEMATICS, , 2ö, æ 2, ç, 3, (a) a + b 3 ÷, ç, ÷, è, ø, , (b), , 3, 2 ö2, æ 2, ça 3 + b3 ÷, , ç, è, , ÷, ø, , 3, , 1 ö2, æ 1, (c) ç a 3 + b 3 ÷, (d) None of these, ç, ÷, è, ø, 109. The function f(x) = x2 log x in the interval [1, e] has, (a) a point of maximum and minimum, (b) a point of maximum only, (c) no point of maximum and minimum in [1, e], (d) no point of maximum and minimum, , 110. If the function f ( x) = 2 x3 - 9ax 2 + 12a 2 x + 1 , where, a > 0 , attains its maximum and minimum at p and q, respectively such that p 2 = q , then a equals, 1, (a), (b) 3, (c) 1, (d) 2, 2, 111. If sum of two numbers is 6, the minimum value of the sum, of their reciprocals is, 6, 3, (b), 5, 4, 112. Consider the function, , (a), , (c), , 2, 3, , (d), , 1, 2, , p, ì, ïï sin x for 0 < x £ 2, f (x) = í, ï 1, for x = 0, ïî 2, Assertion: f has a local maximum value at x = 0., , Reason: f ¢(0) = 0 and f ¢¢ ( 0 ) < 0, (a) Assertion is correct, reason is correct; reason is a, correct explanation for assertion., (b) Assertion is correct, reason is correct; reason is not, a correct explanation for assertion, (c) Assertion is correct, reason is incorrect, (d) Assertion is incorrect, reason is correct., 113. Find the maximum profit that a company can make, if the, , profit function is given by P(x) = 41 + 24x – 18x2., (a) 25, (b) 43, (c) 62, (d) 49, 114. The coordinates of the point on the parabola y2 = 8x which, is at minimum distance from the circle x2 + (y + 6)2 = 1, are, (a) (2, – 4), (b) (18, – 12), (c) (2, 4), (d) None of these, 115. Find the height of the cylinder of maximum volume that, can be inscribed in a sphere of radius a., (a) 2a/3, , (b), , 2a, , 2, -2sin, 116. f ( x ) = (sin x )e, , (a), , 1, , e, (c) 1, , 2, , (c) a/3, , 3, 2, , x, , (d) a/5, , ; max. f(x) – min. f(x) =, 1, 1, 2e e 2, (d) None of these, , (b), , 117. LL' is the latus rectum of the parabola y 2 = 36x and PP', is double ordinate drawn between the vertex and the latus, rectum. The area of the trapezium PP'L'L is maximum, when the distnace of PP' from the vertex is, (a) 1, (b) 4, (c) 9, (d) 36, p, 118. Find the minimum value of 64 secx +27 cosecx , 0 < x <, 2, (a) 130, (b) 120, (c) 125, (d) None of these, 119. If f be a function defined on an interval I and there exists, a point c in I such that f(c) > f(x), for all x Î I, then, (a) function ‘f’ is said to have a maximum value in I, (b) the number f(c) is called the maximum value of f in I, (c) the point c is called a point of maximum value of f in I, (d) All the above are true, 120. Let (h, k) be a fixed point, where h > 0, k > 0. A straight, line passing through this point cuts the positive direction, of the coordinate axes at the points P and Q. Which of the, following is the minimum area of the triangle OPQ, O, being the origin?, (a) hk, (b) 2hk, 1, hk, (c), (d) None of these, 2, , Exercise 2 : Exemplar & Past Year MCQs, 1., , NCERT Exemplar MCQs, Each side of an equilateral triangle expands at the rate, of 2 cm/s. What is the rate of increase of area of the triangle, when each side is 10 cm?, (a) 10 2 cm 2 / s, , (b) 10 3 cm 2 /s, , (c) 10 cm2/s, , (d) 5 3 cm 2 /s, , 2., , A ladder, 5 m long, standing on a horizontal floor, leans, against a vertical wall. If the top of the ladder slides, downwards at the rate of 10 cm/s, then the rate at which, the angle between the floor and the ladder is decreasing, when lower end of ladder is 2 m from the wall is, 1, rad/s, 10, (c) 20 rad/s, , (a), , 1, rad/s, 20, (d) 10 rad/s, , (b)

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APPLICATION OF DERIVATIVES, , 3., , 4., , 5., 6., 7., , 8., , 9., , 1, x5, , The curve y =, at (0, 0) has, (a) a vertical tangent (parallel to y-axis), (b) a horizontal tangent (parallel to x-axis), (c) no oblique tangent, (d) no tangent, The equation of normal to the curve 3x2 – y2 = 8 which is, parallel to the line x + 3y = 8 is, (a) 3x – y = 8, (b) 3x + y + 8 = 0, (c) x + 3y ± 8 = 0, (d) x + 3y = 0, If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1,, 1), then the value of a is, (a) 1, (b) 0, (c) – 6, (d) 6, If y = x4 – 10 and x changes from 2 to 1.99, then what is, the change in y?, (a) 0.32 (b) 0.032, (c) 5.68, (d) 5.968, The equation of tangent to the curve y(1 + x2) = 2 – x,, where it crosses X-axis is, (a) x + 5y = 2, (b) x – 5y = 2, (c) 5x – y = 2, (d) 5x + y = 2, The points at which the tangent to the curve y = x3 – 12x + 18, are parallel to X-axis are, (a) (2, – 2), (– 2, – 34), (b) (2, 34), (– 2, 0), (c) (0, 34), (– 2, 0), (d) (2, 2), (– 2, 34), The tangent to the curve y = e2x at the point (0, 1) meets, X-axis at, , æ 1 ö, (a) (0, 1) (b) ç - , 0 ÷ (c) (2, 0), (d) (0, 2), è 2 ø, 10. The slope of tangent to the curve x = t2 + 3t – 8,, y = 2t2 – 2t – 5 at the point (2, – 1) is, 22, 6, -6, (b), (c), (d) – 6, 7, 7, 7, The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0, intersect at an angle of, , (a), , 11., , p, p, p, p, (b), (c), (d), 3, 6, 4, 2, The interval on which the function f(x) = 2x3 + 9x2 + 12x –1, is decreasing, is, , (a), 12., , (a), , [ -1, ¥ ), ( -¥, -2], , (b) [– 2, – 1], , (c), (d) [– 1, 1], 13. If f : R ® R be definend by f (x) = 2x + cos x, then f, (a) has a minimum at x = p, (b) has a maximum at x = 0, (c) is a decreasing function, (d) is an increasing function, 14. If y = x (x – 3)2 decreases for the values of x given by, (a) 1 < x < 3, (b) x < 0, 3, (c) x > 0, (d) 0 < x <, 2, 15. The function f (x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is, strictly, , æ 3p ö, (a) increasing in ç p, ÷, è 2 ø, æp ö, (b) decreasing in ç , p ÷, è2 ø, , 289, , é -p p ù, (c) decreasing in ê , ú, ë 2 2û, é pù, (d) decreasing in ê 0, ú, ë 2û, , æ pö, 16. Which of the following function is decreasing on ç 0, ÷ ?, è 2ø, (a) sin 2x (b) tan x, (c) cos x, (d) cos 3x, 17. The function f (x) = tan x – x, (a) always increases, (b) always decreases, (c) never increases, (d) sometimes increases and sometimes decreases, 18. If x is real, then the minimum value of x2 – 8x + 17 is, (a) – 1, (b) 0, (c) 1, (d) 2, 19. The smallest value of the polynomial x3 – 18x2 + 96x in, [0, 9] is, (a) 126, (b) 0, (c) 135, (d) 160, 20. The function f(x) = 2x3 – 3x2 – 12x + 4, has, (a) two points of local maximum, (b) two points of local minimum, (c) one maxima and one minima, (d) no maxima or minima, 21. The maximum value of sin x . cos x is, 1, 1, (a), (b), (c), (d) 2 2, 2, 4, 2, 5p, 22. At x =, , f ( x ) = 2 sin 3 x + 3cos 3 x is, 6, (a) maximum 1, (b) minimum, (c) zero, (d) neither maximum nor minimum, 23. The maximmum slope of curve y = – x3 + 3x2 + 9x – 27 is, (a) 0, (b) 12, (c) 16, (d) 32, 24. The function f (x) = xx has a stationary point at, (a) x = e, , x=, , (b), , 1, e, , (c) x = 1, , (d), , x= e, , x, , æ1ö, 25. The maximum value of ç ÷ is, èxø, (a) e, , (b) e, , e, , (c), , 1, ee, , 1, , æ 1 öe, (d) ç ÷, èeø, , Past Year MCQs, 26. If x = –1 and x = 2 are extreme points of, f ( x ) = a log x + b x 2 + x then, , (a), , a = 2, b = -, , 1, 2, , (b), , [JEE MAIN 2014, A], a = 2, b =, , 1, 2, , 1, 1, (d) a = -6, b = 2, 2, 27. The fuel charges for running a train are proportional to, the square of the speed generated in miles per hour and, costs ` 48 per hour at 16 miles per hour. The most, economical speed if the fixed charges i.e. salaries etc., amount to ` 300 per hour is, [BITSAT 2014, A], (a) 10, (b) 20, (c) 30, (d) 40, , (c), , a = -6, b =

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EBD_7762, 290, , 28., , 29., , 30., , 31., , 32., , 33., , 34., , 35., , 36., , MATHEMATICS, , The interval in which the function 2x3 + 15 increases less, rapidly than the function 9x2 – 12x, is –, [BITSAT 2014, A], (a) (–¥, 1), (b) (1, 2), (c) (2, ¥), (d) None of these, The normal to the curve, x2 + 2xy – 3y2 = 0, at (1, 1), [JEE MAIN 2015, A], (a) meets the curve again in the third quadrant., (b) meets the curve again in the fourth quadrant., (c) does not meet the curve again., (d) meets the curve again in the second quadrant., p, If 0 < x < , then, [BITSAT 2015, A], 2, (a) tan x < x < sin x, (b) x < sin x < tan x, (c) sin x < tan x < x, (d) None of these, If a and b are non-zero roots of x2 + ax + b = 0 then the, least value of x2 + ax + b is, [BITSAT 2015, A], 9, 2, (a), (b) –, 3, 4, 9, (c), (d) 1, 4, x 2, The function f ( x ) = + has a local minimum at, 2 x, [BITSAT 2015, A], (a) x = 2, (b) x = –2, (c) x = 0, (d) x = 1, The slope of the tangent to the curve y = ex cos x is, minimum at x = a, 0 £ a £ 2p, then the value of a is, [BITSAT 2015, A], (a) 0, (b) p, (c) 2p, (d) 3p/2, Tangents are drawn from the origin to the curve y = cos x., Their points of contact lie on, [BITSAT 2015, A], (a) x2y2 = y2 – x2, (b) x2y2 = x2 + y2, (c) x2y2 = x2 – y2, (d) None of these, The line which is parallel to X-axis and crosses the curve, [BITSAT 2015, A], y = x at an angle of 45°, is, 1, 1, (a) x =, (b) y =, 4, 4, 1, (c) y =, (d) y = 1, 2, æ 1 + sin x ö, æ, ö, ÷, x Î ç0,p÷. A normal, Consider f (x) = tan -1 ç, ç, ÷, ç 1 - sin x ÷, è 2ø, è, ø, p, to y = f(x) at x = a so passes through the point :, 6, [JEE MAIN 2016, A], æp ö, æp ö, ç , 0÷, ÷, ç , 0÷, ÷, (a) ç, (b) ç, è6 ø, è4 ø, , æ 2pö, ç0, ÷, ÷, (d) ç, è 3 ø, A wire of length 2 units is cut into two parts which are, bent respectively to form a square of side = x units and a, circle of radius = r units. If the sum of the areas of the, square and the circle so formed is minimum, then:, [JEE MAIN 2016, A], (a) x = 2r, (b) 2x = r, (c) 2x = (p + 4)r, (d) (4 – p) x =pr, The curve y = xex has minimum value equal to, [BITSAT 2016, A], , 1, 1, (a) (b), e, e, (c) – e, (d) e, 39. At an extreme point of a function f (x), the tangent to the, curve is, [BITSAT 2016, C], (a) parallel to the x-axis, (b) perpendicular to the x-axis, (c) inclined at an angle 45° to the x-axis, (d) inclined at an angle 60° to the x-axis, 40. Consider the following statements in respect of the function, [BITSAT 2016, A], , 41., , 42., , 43., , 44., , (c) (0, 0), , 37., , 38., , 45., , f (x) = x3 – 1, x Î [ -1, 1], I. f (x) is increasing in [– 1, 1], II. f (x) has no root in (– 1, 1)., Which of the statements given above is/are correct?, (a) Only I, (b) Only II, (c) Both I and II, (d) Neither I nor II, A wire 34 cm long is to be bent in the form of a quadrilateral, of which each angle is 90°. What is the maximum area, which can be enclosed inside the quadrilateral?, [BITSAT 2016, A], (a) 68 cm2, (b) 70 cm2, (c) 71.25 cm2, (d) 72. 25 cm2, What is the x-coordinate of the point on the curve, f (x) = x (7x – 6), where the tangent is parallel to x-axis?, [BITSAT 2016, A], 1, 2, 6, 1, (a) –, (b), (c), (d), 3, 7, 7, 2, Match List I with List II and select the correct answer using, the code given below the lists:, [BITSAT 2016, C], List I, List II, (A) f (x) = cos x, 1. The graph cuts y-axis in, infinitenumber of points, (B) f (x) = ln x, 2. The graph cuts x - axis, in two points, (C) f (x) =, 3. The graph cuts y-axis, x2 – 5x + 4, in only one point, (D) f (x) = ex, 4. The graph cuts x-axis, in only one point, 5. The graph cuts x-axis in, infinite number of points, Codes:, (A), (B) (C), (D), (a) 1, 4 5, 3, (b) 1, 3 5, 4, (c) 5, 4 2, 3, (d) 5, 3 2, 4, The set of all values of a for which the function, f(x) = (a2 – 3a + 2) (cos2x/4 –sin 2x/4) + (a –1) x + sin 1, does not possess critical points is, [BITSAT 2016, A], (a) [1, ¥), (b) (0, 1) È (1, 4), (c) (–2, 4), (d) (1, 3) È (3, 5), The normal to the curve y(x – 2)(x – 3) = x + 6 at the, point where the curve intersects the y-axis passes through, the point:, [JEE MAIN 2017, A], æ 1 1ö, æ 1 1ö, (a) çè , ÷ø, (b) ç - , - ÷, 2 3, è 2 2ø, , æ 1 1ö, (c) ç , ÷, è 2 2ø, , æ 1 1ö, (d) ç , - ÷, è 2 3ø

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APPLICATION OF DERIVATIVES, , 291, , 46. Twenty metres of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum, area (in sq. m) of the flower-bed, is :, [JEE MAIN 2017, A], (a) 30, (b) 12.5, (c) 10, (d) 25, 47. The eccentricity of an ellipse whose centre is at the, 1, origin is . If one of its directices is x = – 4, then the, 2, æ 3ö, equation of the normal to it at ç1, ÷ is :, è 2ø, [JEE MAIN 2017, A], (a) x + 2y = 4, (b) 2y – x = 2, (c) 4x – 2y = 1, (d) 4x + 2y = 7, 48. A cylindircal gas container is closed at the top and open at, 5, the bottom. if the iron plate of the top is, time as thick, 4, as the plate forming the cylindrical sides. The ratio of the, radius to the height of the cylinder using minimum material, for the same capacity is, [BITSAT 2017, A], 2, 4, 1, 1, (a), (b), (c), (d), 3, 5, 3, 2, 49. If f (x) = xx, then f (x) is increasing in interval :, [BITSAT 2017, A], , é 1ù, ê0, e ú, ë, û, (c) [0, 1], (d) None of these, 1, 1, 50. Let f (x) = x 2 +, and g(x) = x - , x Î R - {-1,0,1} ., 2, x, x, f (x), , then the local minimum value of h(x) is :, If h(x) =, g(x), [JEE MAIN 2018, A], (a) – 3, (b) -2 2, (d) 3, (c) 2 2, (a) [0, e], , (b), , 51. If the curves y2 = 6x,9x 2 + by 2 = 16 intersect each other, at right angles, then the value of b is :, [JEE MAIN 2018, A], 7, 9, (b) 4, (c), (d) 6, 2, 2, 52. A ball is dropped from a platform 19.6m high. Its position, function is –, [BITSAT 2018, A], (a) x = – 4.9t2 + 19.6 (0 £ t £ 1), (b) x = – 4.9t2 + 19.6 (0 £ t £ 2), (c) x = – 9.8t2 + 19.6 (0 £ t £ 2), (d) x = – 4.9t2 – 19.6 (0 £ t £ 2), , (a), , Exercise 3 : Try If You Can, 1., , 2., , æ p pö, Let the function g : ( -¥, ¥) ® ç - , ÷ be given by, è 2 2ø, p, g (u ) = 2 tan -1 (eu ) - . Then, g is, 2, (a) even and is strictly increasing in (0, ¥ ), (b) odd and is strictly decreasing in ( -¥, ¥), (c) odd and is strictly increasing in ( -¥, ¥), (d) neither even nor odd, but is strictly increasing in, ( -¥, ¥), If A denotes the arithmetic mean of the real numbers a1,, a2, ...........,an, then, , 3., , n, , å ( x – ai ), , 2, , 4., , is the point of local minima is 2 5, (b) f (x) is increasing for x Î [1, 2 5 ], (c) f(x) has local minima at x = 2, (d) the value of f(0) = 15, 5., , has a minimum at., , " x1 > x2 . Then the solution set of, , 6., , f ( g (a 2 - 2a)) > f ( g (3a - 4)), is, (a), , a Î [1, 4), , (b), , (c), , a Î (1, 4], , (d) None of these, , Tangent at P1 (2,3) on the curve 3 y = x3 + 1 meets the, curve again at P2 . The tangent at P2 meets the curve at, , i =1, , (a) x = 0 (b) x = 1, (c) x = A, (d) x = A+1, Let f(x) and g(x) be two continuous functions defined from, R ® R, such that f ( x1 ) > f ( x2 ) and g ( x1 ) > g ( x2 ), , f (x) is cubic polynomial with f(2) = 18 and f (1) = –1. Also, f (x) has local maxima at x = –1 and f '(x) has local minima, atx = 0, then, (a) the distance between (–1, 2) and (a f(a)), where x = a, , a Î (1, 4), 7., , P3 and so on. If the sum of the ordinates for, æ 2183 - 8 ö, P1, P2 , P3 ,.....P60 be S then S + ç, ÷ is equal to 5k,, è 27 ø, when k is equal to, (a) 3, (b) 4, (c) 5, (d) 6, For the curve y = 3 sinq cosq, x = eq sin q, 0 £ q £ p, the, tangent is parallel to x-axis when q is:, p, 3p, p, p, (a), (b), (c), (d), 6, 4, 2, 4, p, n, 2, 2, If f ( x ) = (sin x - 1) (2 + cos x), then x = is a point, 2, of (where, n Î N )

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EBD_7762, 292, , MATHEMATICS, , (a), (b), (c), (d), 8., , local maximum, if n is odd, Both (a) and (b), Neither (a) nor (b), local maximum, if n is even, , then the range of a so that f(x) has maxima at x = – 2 is, (a) | a |³ 1 (b) | a |< 1, (c) a > 1, (d) a < 1, n2, , The largest term in the sequence, an = 3, is, n + 200, (a) a6, (b) a7, (c) a8, (d) None of these, If f (x) is differentiable and strictly increasing function,, , 9., , f (x 2 ) - f (x), is, x ®0 f ( x )-f (0), (b) 0, (c) –1, , then the value of lim, (a) 1, , x, , Domain of the function f (x) if 3 + 3, , 10., , (d) 2, f ( x), , = minimum of, , f(t ) where f(t ), = minimum of {12t 3 - 15t 2 + 36t - 25, 2| sin t |; 2 £ t £ 4} is, (-¥, log 3 e), , (a), , (-¥,1), , (b), , (c), , (0, log 3 2), , (d) (-¥, log 3 2), , The greatest of the numbers 1, 21 / 2 , 31 / 3 , 41 / 4 , 51 / 5 , 61 / 6, , 11., , and 71 / 7 is, (a), , 21 / 2, , (b) 31 / 3, , (c), , 71 / 4, , (d) All but 1 are equal, , ìï 2 - | x 2 + 5 x + 6 |;, f ( x) = í, 2, ;, ïî a + 1, , 12., , x ¹ -2, x = -2, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, , (a), (d), (c), (b), (c), (d), (c), (a), (a), (b), (a), (c), , 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, , (c), (b), (c), (c), (d), (c), (b), (a), (c), (d), (b), (a), , 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, , (a), (b), (c), (b), (c), (d), (a), (b), (a), (c), (c), (d), , 1, 2, 3, 4, 5, 6, , (b), (b), (b), (c), (d), (a), , 7, 8, 9, 10, 11, 12, , (a), (d), (b), (b), (c), (b), , 13, 14, 15, 16, 17, 18, , (d), (a), (b), (c), (a), (c), , 1, 2, , (c), (c), , 4, 5, , (b), (c), , 6, 7, , (c), (c), , 13. Let f and g be functions from the interval [0, ¥ ) to the, interval [0, ¥), f being an increasing and g being a, decreasing function. If f {g(0)} = 0 then, (a), , f {g( x )} ³ f {g(0)}, , (b) f {g(2)} = 0 , g{f ( x )} £ g{f (0)}, (c) f {g(2)} = – 1, (d) None of these, 14. If composite function f1 ( f 2 ( f3 (.....( f n ( x))......) is an, increasing function and if r of f 'i s are decreasing, functions while rest are increasing, then maximum value, of r (n - r ) is :, (a), , n2 -1, when n is an even number, 4, , n2, when n is an odd number, 4, (c) Both (a) and (b), (d) Neither (a) nor (b), 15. The set of all values of a for which the function, f(x) = (a2 – 3a + 2) (cos2x/4 –sin 2x/4) + (a –1) x + sin 1, does not possess critical points is, (a) [1, ¥), (b) (0, 1) È (1, 4), (c) (–2, 4), (d) (1, 3) È (3, 5), (b), , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (b) 73, (c), 37 (a) 49 (a) 61, (b) 74, (c), 38 (d) 50 (b) 62, (d) 75, (a), 39 (d) 51 (c) 63, (b) 76, (b), 40 (c) 52 (a) 64, (d) 77, (c), 41 (b) 53 (b) 65, (c) 78, (b), 42 (c) 54 (c) 66, (d) 79, (d), 43 (c) 55 (a) 67, (b) 80, (d), 44 (d) 56 (b) 68, (d) 81, (b), 45 (a) 57 (b) 69, (d) 82, (a), 46 (c) 58 (d) 70, (a) 83 (d), 47 (b) 59 (b) 71, (a) 84, (c), 48 (d) 60 (a) 72, Exercise 2 : Exemplar & Past Year MCQs, (b) 37, (a), 19 (b) 25 (c) 31, (d) 38, (a), 20 (c) 26 (a) 32, (b) 39, (a), 21 (b) 27 (d) 33, (c) 40, (a), 22 (d) 28 (b) 34, (c) 41, (d), 23 (b) 29 (b) 35, (d) 42, (b), 24 (b) 30 (d) 36, Exercise 3 : Try If You Can, (b) 10 (d) 12, (a) 14, (d), 8, (c) 11 (b) 13, (b) 15, (b), 9, , 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, , (b), (b), (a), (b), (b), (d), (d), (c), (c), (a), (d), (c), , 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, , (a), (a), (d), (b), (a), (a), (d), (b), (a), (c), (b), (b), , 43, 44, 45, 46, 47, 48, , (c), (b), (c), (d), (c), (c), , 49, 50, 51, 52, , (b), (c), (c), (b), , 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, , (c), (d), (c), (c), (d), (c), (b), (d), (a), (c), (d), (b)

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23, , INTEGRALS, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 4, , 3, JEE MAIN, BITSAT, , 2, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 6, 2, , Critical Concepts, , Rating of Difficulty, , CUS, (chapter utility score), out of 10, , Methods of integration,, Integration by, Partial Fractions, Integration, by Parts, Fundamental, Theorem of Calculus,, Evaluation of Definite, Integrals by Substitution,, Properties of Definite, Integral, , 4/5, , 8.5

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INTEGRALS, , 295

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EBD_7762, 296, , MATHEMATICS, , Topic 1 : Standard Integrals, Integration, by Substitution, Integration by Parts, 1., , òx, , x, , (1 + log x)dx is equal to, , (a) x x, (c) xx log x, 2., , òx, , (a), , 51, , (b) x2x, (d) 1/2 (1 + log x)2, 8., , 52, , (a), , x, (tan –1 x + cot –1 x) + c, 52, , (b), , x52, (tan –1 x - cot –1 x) + c, 52, , 9., , -, , 4., , Let, , ò, , 1, 16, , (b), , x1 / 2, 1 - x3, , 10., , 1, 16, , (c), , 1, 8, , (d), , -, , 1, 8, , 2, gof ( x) + C , then, 3, , dx =, , 11., , f ( x) = x, , (a), , (b) f (x) = x3/2 and g(x) = sin–1x, (c) f (x) = x2/3, (d) None of these, 5., , ò sec, , 2/3, , (a), +c, (c) 3(tanx)–1/3 + c, , dx, , ò cos x +, , 3 sin x, , (c), 7., , –3(tanx)–1/3, , (b), +c, (d) (tan x) –1/3 + c, , equals, , æx p ö, (a) log tan ç + ÷ + C, è 2 12 ø, , (b) log tan æç x - p ö÷ + C, è 2 12 ø, , 1, 1, æx p ö, log tan æç x + p ö÷ + C (d), log tan ç - ÷ + C, 2, 2, è 2 12 ø, è 2 12 ø, , Evaluate:, , 2x, , x, , 2, 2 x, ò 2 2 2 dx, , (d), , 1, , x, , ( log 2 )3, 1, , ( log 2 ), , 4, , 22 + C, , 22, , 2x, , +C, , 1ö, , (a) f ( x ) = x 2, , (b) g(x) = log x, , (c) L = 1, , (d) None of these, , ò, , ìï, 1 - x üï, cos í2 tan -1, ý dx is equal to, 1 + x ïþ, ïî, , (a), , 1 2, ( x - 1) + k, 8, , (b), , (c), , 1, x+k, 2, , None of these, , òe, , 3 log x, , (d), , (c), 12., , ò, , 1 2, x +k, 2, , ( x 4 + 1) -1 dx is equal to, , (a) log ( x 4 + 1) + C, , xcosec 4 / 3 x dx =, , –3(tanx)1/3, , 6., , 22 + C, , (b), , dx is equal to, 10x + x10, (a) 10x – x10 + C, (b) 10x + x10 + C, x, 10, –1, (c) (10 – x ) + C, (d) loge(logx + x10) + C, If, , cos8 x + 1, , ò cot 2 x - tan 2 x dx = A cos8 x + k ,, , where k is an arbitrary constant, then A is equal to :, (a), , 2, , 1, 2, , x, p, + +c, 52 2, , If the integral, , +C, , 10x9 + 10x loge 10, , 52, , 3., , 2x, , x, , ( log 2 ), , æ, , px, p, + +c, 104 2, , (d), , 22, , ò x log çè 1+ x ÷ø dx = f (x) log(x +1) + g(x)x 2 + Lx + C, then, , 52, , (c), , ò, , ( log 2 )3, 1, , (c), , (tan –1 x + cot –1x) dx, , 1, , - log (x 4 + 1) + C, , (b), , (d) None of these, , 3, , x + x2 + 6 x, x (1 + 3 x ), , dx is equal to, , (a), , 3 2/3, + 6 tan -1 x1 / 6 + C, x, 2, , (b), , 3 2/3, - 6 tan -1 x1 / 6 + C, x, 2, , (c), , -, , 3 2/3, + 6 tan -1 x1 / 6 + C, x, 2, , (d) None of these, , 1, log ( x 4 + 1) + C, 4

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INTEGRALS, , 13. If, , ò sin, , 297, , 3, , x cos5 x dx = A sin 4 x + Bsin 6 x + Csin8 x + D ., , Then, (a), , 1, 1, 1, A = , B =- , C = , DÎR, 4, 3, 8, , (b), , 1, 1, 1, A = , B = , C = , DÎR, 8, 4, 3, , (a), (b), (c), (d), , Statement I is true, Statement II is true, Both statements are true, Both statements are false, 1, dx, 19. Evaluate: ò, sin 3 x cos5 x, , 2, , (a), , 1, 1, A = 0, B = - , C = , D Î R, 6, 8, (d) None of these., , tan x, , (c), , 14., , æ, , 1ö, , ò çè x + x ÷ø, , n +5 æ, , 2, ö, ç x - 1 ÷ dx is equal to :, ç x2 ÷, è, ø, , (a), , é x ù, êë x 2 +1úû, , (c), 15., , +c, , é x 2 + 1ù, ê 2 ú, ëê x ûú, , (b), , n +6, , sin 8 x - cos 8 x, , ò 1 - 2 sin 2 x cos 2 x dx, , (c), , -, , 2, tan x, 2, , ( n + 6) + c, , 20. Evaluate:, , n +6, , (n + 6) + c, , -, , 2, ( tan x )3 / 2 + C, 3, +, , 2, ( tan x )3/ 2 + C, 3, , -, , 2, ( tan x )3 / 2 + C, 3, , tan x, (d) None of these, , n +6, , 1ö, æ, çx + ÷, xø, è, n+6, , (b), , -, , (a), , 1, tan -1 ( 2 tan x ) + C, 6, , (b), , (c), , 1, æ 2 tan x ö, tan -1 ç, ÷+C, 6, è 3 ø, , (d) None of these, , (d) None of these, , is equal to, 21., , ò tan, , -1, , 1, sin 2 x + C, 2, , (b) - sin 2 x + C, , (a), , ( x + 1) tan -1, , (c), , 1, - sin x + C, 2, , (d) - sin 2 x + C, , (b), , x tan -1 x - x + C, , (c), , x - x tan -1 x + C, , (d), , x - ( x + 1) tan -1 x + C, , ò cos, , n, , (a) 0, 17. Evaluate:, , x sin x dx = -, , (b) 1, , 6, , cos x, + C , then n =, 6, (c) 2, (d) 5, , 3, 3, ò sin x cos x dx, , 22., , tan -1 ( 2 tan x ) + C, , x dx is equal to, , (a), , 16. If, , 1, 2, , 1, , ò 1 + 3sin 2 x + 8 cos 2 x dx, , x - x +C, , sin 8 x - cos 8 x, , ò 1 - 2 sin 2 x cos 2 x dx, , is equal to, , (a), , 1 ì3, 1, ü, í cos 2x - cos 6x ý + C, 32 î 2, 6, þ, , (a), , 1, sin 2x + c, 2, , (b) - sin 2x + c, , (c), , 1, - sin x + c, 2, , (d) - sin 2 x + c, , (b), , 1 ì 3, 1, ü, í- cos 2x + cos 6x ý + C, 32 î 2, 6, þ, , (c), , 1 ì 3, 1, ü, í- cos 2x - cos 6x ý + C, 32 î 2, 6, þ, , 23., , (d) None of these, 18. Consider the following statements, Statement-I: The value of, , ò, , Statement-II: The value of, , 1, 2, , æ 3 - 4t ö, sin -1 ç, ÷+C., è 3 ø, , dx, , ò, , 16 - 9x, dt, , 2, , 3t - 2t 2, , is, , 1 -1 3x, sin, +C, 3, 4, , is, , 1, 2, , 1, , ò x log x log(log x) dx =, (a), , log log(x ) + c, , (b) log [ log(log x )] + c, , (c), , - log log ( log x ) + c, , (d) none of these, , 24. If, , sin x, , ò sin(x - a) dx = Ax + Blog sin(x - a) + C,, , then value, , of (A, B) is, (a), , (- cos a, sin a ), , (b) (cos a, sin a), , (c), , (- sin a, cos a), , (d) (sin a, cos a )

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EBD_7762, 300, , 46., , MATHEMATICS, , then the value of B is, (a) 3, (b) 4, 47., , dx, , ò, , If, , 2, , a -x, , (a) 3, 48., , 2, , = sin -1, , (c) 6, , x3 -1, , ò x 3 + x dx, , 53. If, , x, + c , then a =, 3, , (b) 4, , (c) 6, , (d) 8, , (b), , x - log x +, , (c), , 1, x + log x + log( x 2 + 1) + tan -1 x + C, 2, , 1, log(x 2 + 1) - tan -1 x + C, 2, , 50. Evaluate:, , 51., , ò, , 1, 9 + 8x - x 2, , (c) 2, , (d) 5, , æ x-4ö, - sin -1 ç, ÷+C, è 5 ø, , (b), , æx+4ö, - sin -1 ç, ÷+C, è 5 ø, , (c), , æ x-4ö, sin -1 ç, ÷+C, è 5 ø, , (d) None of these, , log sec x + tan x - 2 tan ( x / 2 ) + C, , (b), , log sec x - tan x - 2 tan ( x / 2 ) + C, , (c), , log sec x + tan x + 2 tan ( x / 2 ) + C, , (d) None of these, 56. If, , = A, , (a), , (b), , Reason : The value of the integral e x {f ( x ) + f ¢ ( x )} dx, is ex f (x) + C., (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., , ò x 4 - 9 dx, , (a), , 1, 1, x2 + 3, log x 4 - 9 + log, +C, 4, 12, x2 - 3, , (b), , 1, 1, x2 - 3, log x 4 - 9 - log 2, +C, 4, 12, x +3, , 1, , ò ( sin x + 4)( sin x - 1) dx, , Assertion : The value of the integral ò e x [tan x + sec2 x]dx, is ex tan x + C, , 1 - cos x, , ò cos x (1 + cos x ) dx, , (a), , dx, , x3 + x, , + e12x )1/3dx is equal to, , 55. Evaluate:, , (a), , 52. Evaluate:, , 9x, , (d) All of these, , (c) (1/3) (27 + e3x)4/3 + C (d) (1/4) (27 + e3x)4/3 + C, , ò ex - e- x dx = log (e2x – 1) – Ax + C,, (b) 1, , ò (27e, , 1, 2, , (b) f(x) = x2 + 2x + 2, , (a) (1/4) (27 + e3x)1/3 + C (b) (1/4) (27 + e3x)2/3 + C, , ex + e- x, , then A =, (a) 0, , k=-, , (c), 54., , x - log x + log( x 2 + 1) - tan - 1 x + C, , 3x + 4, , ò x3 - 2x - 4dx = log x - 2 + k log f ( x ) + c , then, , (a) f(x) = |x2 + 2x + 2|, , is equal to, , (a), , If, , (c), , (d) 8, , (d) None of these, 49., , 1, 1, x2 - 3, log x 4 - 9 + log, +C, 4, 12, x2 + 3, (d) None of these, , 3x + 1, -5, B, dx = ò, dx + ò, dx ,, If ò, (x - 3)(x - 5), (x - 3), (x - 5), , (c), , (d), , 1, + B tan -1 ëéf ( x ) ûù + C1 , then, x, æ, ö, ç tan - 1 ÷, 2 ø, è, A=-, , 1, -2, 4 tan x + 3, , f (x) =, , B=, 5, 5 15, 15, , æxö, 4 tan ç ÷ + 1, 1, 1, è 2ø, A=- , B=, , f (x) =, 5, 15, 15, 2, 2, 4 tan x + 1, , B = - , f (x) =, 5, 5, 5, æxö, 4 tan ç ÷ + f, 2, -2, è2ø, A= , B=, , f (x) =, 5, 5 15, 15, A=, , x2 + 1, 57. Value of ò, dx is, (x - 1)(x - 2), (a), , (c), , é ( x - 2 )5 ù, é ( x - 1)2 ù, ú + C (b) x + log ê, ú+C, x + log ê, ê ( x - 1)2 ú, ê ( x - 2 )5 ú, ë, û, ë, û, é ( x - 2 )5 ù, ú + C (d) None of these, x - log ê, ê ( x - 1) 2 ú, ë, û

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INTEGRALS, , 58. If, , 301, , òx, , 13 / 2, , = A (1 +, , x5/2)7/2, , + B (1 +, , x5/2)5/2, , + C (1 +, , x5/2)3/2 +, , D, then, (a) 1, , 4, 8, 4, ,B=–, ,C=–, 35, 25, 15, , 64. The value of, , (a), , (a), , (c), 60., , 2sin, , -1, , b-a, +C, x-a, , x+a, +C, b-a, , if (b > a) is, , (b), , 2sin -1, , 0, , x-a, +C, b-a, , (a), , (c), , (e x + e - x ), 1, , +C, , (b), , (d), , -1, (e x + e - x ), 1, (e x + e - x ), , 3, , 4 æ x -1 ö, ç, ÷, 3èx+2ø, , (b), , 1/ 4, , (c), , 1 æ x -1 ö, ç, ÷, 3è x + 2ø, , (d), , e, , 1æ x+2ö, ç, ÷, 3 è x -1 ø, , 1, , 68. The value of, , ò tan, , (a) 1, , (b) 0, , +C, , 2p, , 69., , ìï e x - 1 üï, sin í x, ý dx equals, îï e + 1 þï, log1/2, , (a), , cos, , 1, 1, (b) sin, 3, 2, , dx is, , (d) 1, , (b), , e1000 - 1, e -1, , (d), , e -1, 1000, , 2x - 1 ö, ç 1 + x + x 2 ÷ dx is, è, ø, (c) – 1, , (d), , p, 4, , (b) p/2, , (c), 70., , p(a + b), a -b, , log p, , òlog, , p2, , (d), , (b), , 3, , 71. Value of, , ò, , 2, , (a) 2, , p 2, (a - b 2 ), 2, , ö, æ1, e 2 x sec 2 ç e 2 x ÷ dx is equal to :, ø, è3, , 8, , (d) 0, , x, 9- x + x, , (d) 3 - 3, , æ a + b sec x ö, , (a), (c) 2 cos 2, , 3+ 3, , ò logçè a - b sec x ÷ødx =, , log 2, , ò, , 1 3, ln, 6 2, , -1 æ, , 0, , (a) 0, , Topic 3 : Evaluation of Definite Integral by Substitution,, , 62., , (d), , 0, , +C, , Properties of Definite Integrals,, Definite Integral as the Limit of a Sum, , (c), , (c) 2, , e1000 - 1, , (c) 1000(e – 1), , 4æx+2ö, ç, ÷, 3 è x -1 ø, , 3, 2, , dx is, , +C, , 1/ 4, , +C, , 1000 x -[x], , ò0, (a), , 1/ 4, , +C, , (b) 3/2, , +C, , ò [(x - 1)3 ( x + 2)5 ]1 / 4 dx is equal to, 1/ 4, , ò, , dx is, , dx, , 3, , x, xö, æ, ç cos + sin ÷, 2, 2ø, è, 2+ 2, , ln, , (c), , cos x, , (d) None of these, , 1, , (a), , 1 3, ln, 2 2, , 6, , 67., , +C, , ( e x + 1) 2, , sin x 2 + sin(ln6 - x 2 ), , 66. The value of integral,, I=, , ò (ex + e- x )2 =, -e - x, , ln 2, , 2 - 2 (b), , (a) 1/2, , (d) 0, , x sin x 2, , ò, , ò, , 1, 3, , (c), , ln 3, , 1 3, ln (b), 4 2, , 65. Evaluate:, , 2dx, , (a), , 61., , (x - a )(b - x), , 2sin -1, , 7, 3, , p/2, , dx, , ò, , dx as limit of sums., , (b), , (d) None of these, 59. Value of, , 2, , 1, , 4, 8, 4, ,B=–, ,C=, 35, 25, 15, , (c) A =, , òx, , 63. Evaluate, , 4, 8, 4, ,B=–, ,C=, 35, 25, 15, , (a) A = –, (b) A =, , 2, , . (1 + x 5 / 2 )½ dx, , 1, 3, , 10 - x, x + 10 - x, , (b) 3, , (c), , 3 3, 2, , (d), , 1, 2 3, , dx is, , (c) 4, , (d) 5

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EBD_7762, 302, , MATHEMATICS, 1, , ò, , 72. The value of, , 1, , ( x - [x]) dx (where [ . ] denotes greatest, , Statement-II : The value of integral, , -1, , -1, , integer function) is, (a) 0, (c) 2, 2, , 73., , (a), (b), (c), (d), , (b) 1, (d) None of these, , æ1+ x ö, , ò log çè 1 - x ÷ødx = 0 ., , Assertion :, , 4, , 80., , -2, , -a, , (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., , (c), , 75., , If m is an integer, then, , 76., , (a) 1, (b) 2, The value of, sin 2 x, , ò0, , (a), , p, , 77. The value of, , sin x, , ò0, , p, 4, , (b), 1, , 78. The value of, , æ 4 + 3sin x ö, , 3, 4, , ò tan, , b-a, f ( x) dx, 2 ò, , b, , (a) p, 84., , is, , a, , (d) – 2, , (c) – 1, , ò, 0, , x 4 (1 - x ), 1+ x2, , satisfies, , the, , conditions, , ò, , cos 2 x, , -p 1 + a, , x, , An =, , ò, , (d) P = 5, Q = – 6, R = 3, , dx, a > 0, is, (c) p/2, , sin(2n - 1)x, dx; B n =, sin x, , p/2, , ò, 0, , (d) 2 p, 2, , æ sin nx ö, ÷ dx;, ç, è sin x ø, , For n Î N, then, , (d), , p, 4, , 79. Consider the following statements, Statement-I : The value of, , a+b, f ( x ) dx, 2 ò, , b, , (b) a p, , 0, , 2x - 1 ö, ç, ÷ dx is, è 1 + x - x2 ø, , 1, , (d), , [f (x) - Rx]dx =, , p, , (d) 1, , (c) 0, , (b) 0, , (b), , a+b, f (b + x) dx, 2 aò, , 39, are given by, 2, (a) P = 2, Q = – 3, R = 4 (b) P = – 5, Q = 2, R = 3, , ò0, , 83. The value of, , -1 æ, , 0, , (a) 1, , (c), , p/2, , ò log çè 4 + 3cos x ÷ø dx, , is equal to, , f (0) = -1, f ' (log 2) = 31 and, , p, , 0, , (a) 2, , (a), , log 4, , cos -1 t dt is, (c), , ò x f ( x) dx, , a, , dx is equal to :, , (d), , (d) 8, , b, , b, , a+b, f (b – x ) dx, 2 òa, , (c) P = 5, Q = – 2, R = 3, cos 2 x, , p, 2, , p, 2, , (c) 6, , a, , (d) p, , (c) 0, , sin -1 t dt +, (b), , (b) 4, , f (x) = Pe 2x + Qe x + Rx, , p, 2, , p sin( 2 mx ), , ò0, , 0, , 82. The numbers P, Q and R for which the function, , 0, , p, (b), 4, , 4, , b, , ò log(tan x) dx is, , dx is 2., , 8, 4, 2, (x 8 - x 4 + x 2 + 1) dx = 2ò (x - x + x + 1)dx ,, , 81. If f (a + b – x) = f ( x) , then, , p, 2, , 74. The value of definite integral, , x+2, , Statement I is true, Statement II is true, Both statements are true, Both statements are false, , then a =, (a) 3, , ò f (x )dx = 0 ., , Reason : If f is an odd function, then, , ò, , -a, , a, , (a) 0, , x+2, , ò, , 4, , (a), , A n +1 = A n , Bn+1 -Bn = An+1, , (b), , Bn +1 = B n, , (c), , A n +1 - A n = B n +1, , (d) None of these, p, , 85. Evaluate:, , 1, , ò 5 + 4 cos x dx, 0, , 22, -p., dx is, 7, , (a), , p, 3, , (b), , p, 2, , (c), , p, 4, , (d), , p, 6

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INTEGRALS, , 303, , p, , p/4, , 0, , 0, , ò ln sin x dx = k , then value of, , 86. If, , ò, , ln (1 + tan x ) dx is, , k, k, k, k, (a) (b), (c) (d), 8, 8, 4, 4, 87. Let f(x) be a function satisfying f '(x) = f(x) with, f(0)=1 and g(x) be a function that satisfies, , (a), , 1, , (a) [0, 4], (c), , e+, , e2 3, 2 2, , (d) e -, , e2 3, 2 2, , (c), , ìïx 2 , 0 £ x £ 1, 88. If f ( x ) = í, then, ïî x , 1 £ x £ 2, (a), (c), , 2, , ò f (x)dx =, 0, , (b), , 4 2 -1, , (d) None of these, , x, , 89. If g(x) =, , ò, , 94., , ò, , (b), , é 1 ù, ê- 4 , 2ú, ë, û, , p 2 x (1 + sin x ), , 2, - p 1 + cos x, , 0, , p2, (b) p2, (c) zero, (d), 4, 95. If f and g are defined as f (x) = f (a – x) and, , (b) g (x) – g (p), , (c) f (x) g (p), , (d), , g( x ), g ( p), , 90. Assertion : I =, , ò, , tan x dx =, , a, , (a), , a, , ò f ( x ) dx, , (b), , 0, a, , (c), , 2, , a, , ò g ( x ) dx, , (d) 2ò g ( x ) dx, 0, , 96. Consider the following statements, l, , Statement-I:, , ò, , 3p 4, , òp 4, , 2 +1, , fdf, is equal to, 1 + sin f, , (b), , y dy, y+l, , is equal to, , (, , ), , 2, 2+ 2 l l ., 3, , 2, , 0, , | sin x - cos x | dx is equal to :, , 2 2, , (c), 92., , p 2, , 2ò f ( x ) dx, 0, , 1, , Reason: tan x = t2 makes the integrand in I as a rational, function., (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., , ò0, , is equal to, , æ ax -1ö, Statement-II: 3a ò ç, dx is equal to (a – 1) + (a – 1)2., è a -1 ÷ø, , p, , 0, , (a), , ò f ( x ) g ( x ) dx, 0, , 0, , p, 2, , p, 2, , a, , 0, , (a) g(x) + g(p), , 91. The integral, , dx is, , (a), , 4 2, , cos 4 t dt , then g(x + p) equals, , é 1 ù, ê- 4 , 4ú, ë, û, , (d) None of these, , g (x) + g (a – x) = 4, then, , 1, 3, , 2 -1, , then range of f (x) is, , ò f ( x ) g ( x ) dx, is, e2 5, 2 2, , 2 +1, , ò, , 0, , (b) e -, , 2 -1, , p, , (d), , x, , 1, , e2 5, +, 2 2, , p, , (c), , 93. Let f(x) be defined by f ( x ) = t ( t 2 - 3t + 2)dt , 1 £ x £ 3,, , f(x) + g(x) = x 2 . Then the value of the integral, , (a) e +, , 1, , 2 - 1 (b), , 2( 2 - 1), , (d) None of these, , (a), (b), (c), (d), , Statement I is true, Statement II is true, Both statements are true, Both statements are false, p, , 97. Value of ò | cos x | dx is, (a) 2, (c) 1, , 0, , p /2, , 98. Value of, , ò, , 0, , (b) –2, (d) None of these, sin x, sin x + cos x, , dx is, , (a), , p, 2, , (b), , -p, 2, , (c), , p, 4, , (d) None of these

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EBD_7762, 304, , MATHEMATICS, p/4, , 99. If, , ò sec, , k, , 2, , q sin qdq =, , 0, , (a), , ò(, 0, , 9, 16, , (b), , 106. The value of the integral, , dx, x + x + k), , 9, 25, , (c), , , then k =, , 16, 25, , (d), , 3, , æ, , 1, , è, , ò çç tan, 25, 16, , 1, , x, , -1, , x 2 +1, , (a) p, , + tan -1, , (b) 2p, , (b) –2I, (d) None of these, p /3, , ò, , p /6, , 1, 1 + cot x, , (a), , (b), , p, 12, , (c), , 12, p, , (d), , None of these, x, , log t, 1, 102. Let F(x) = f (x) + f æç ö÷ ,where f ( x) = ò, dt , Then, 1+ t, è xø, l, , -1, , (c) 1/2, , 109., , (d) 0, , |x|, dx is, x, , (a) 0, (c) –1, , 1 p, +, 2 6, , (c), , -, , ò, , (a) A.P., , (b) G.P., , (c) H.P., , (d) None of these, dx, , (d), , 3, p, -12, 6, , p sin(p log e x), dx is equal to, x, , (b) 20, , (c), , 2, p, , (c) 2p, , ò, , p, , 0, , x (sin4 x cos4 x) dx is, , 3p 2, 64, , (b), , (c), , 3p 2, 256, , (d) None of these, , ò, , 3p 2, 128, , sin 2 x cos3 x dx =, , 0, , 0, , (c), , 3p, 4, , 3 p, +, 2 6, , (a), , p/2, , 111., , ò (x 2 + a 2 )(x 2 + b 2 ) is, , (a), , (d), , Topic 4 : Reduction Formulae for Definite Integration,, Gamma and Beta Function, Walli’s Formula,, Summation of Series by Integration, , a 2 - a 1 , a 3 - a 2 , a 4 - a 3 ,............ are in, , 105., , 1 p, 2 6, , (a) 2, , 0, , ¥, , (b) 1 -, , 110. The value of, , 2, , sin n x, dx then, 104. If a n =, sin x, , ò, , p, 2, , BEYOND NCERT, (b) 1, (d) None of these, , p, 2, , (c), , x sin ( p [x] - x) dx is equal to :, , (a), , 1, , (b) 2, , ò, , ò, , e 37, , F(e) equals, , 2, , (b) p, , 5- x, is, x-2, , p/3, , p, 6, , 103. Value of, , 3p, 2, , (d) None, , p/2, , 108., , dx is, , (a), , (a) 1, , ò, 2, , 0, , 101. Value of, , (c) 4p, 5, , 107. The value of the integral, , æ1 ö, 100. Value of ò log ç - 1÷ dx is, èx ø, , (a) 2I, (c) 0, , x 2 + 1 ö÷, dx is equal to, x ÷ø, , (a) 0, pab, a+b, , (b), , p, 2ab(a + b), , (d), , p, 2(a + b), p(a + b), 2ab, , (c), , 4, 15, , (b), , 2, 15, , (d) None of these, , 1, 1 ü, ì 1, +, + ..... +, 112. lim í, ý is equal to, n ®¥ n + 1, n+2, n + nþ, î, (a) 3 log 2, (b) log 2, (c) 2 log 2, (d) None of these

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EBD_7762, 306, , MATHEMATICS, , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, 1., , 2., , cos 2 x - cos 2q, dx is equal to, cos x - cos q, (a) 2(sin x + x cos q) + C, (b) 2(sin x – x cos q) + C, (c) 2(sin x + 2x cos q) + C, (d) 2(sin x – 2x cos q) + C, , ò, , (b), , (c), , sin ( x - b ), , sin ( x - a ), , cosec ( b - a ) log, , cosec ( b - a ) log, , sin ( x - b ), , sin ( x - a ), , (a), , ( x + 1) tan -1, , (b), , x tan -1 x - x + C, , +C, , +C, , 7., , 8., , x - ( x + 1) tan -1 x + C, , ( 4x + 1), , (c), , If, , dx is equal to, , 6, , 1 æ, 1 ö, ç4+ 2 ÷, 5x è, x ø, , 1 æ1, ö, ç + 4÷, 10x è x, ø, , (, , -5, , +C, , (b), , 1æ, 1 ö, ç4+ 2 ÷, 5è, x ø, , 9., , -5, , ö, 1æ 1, + 4÷, (d), ç, 10 è x 2, ø, , +C, , 2, , ), , a=, , -1, 2, ,b=, 10, 5, , (d) a =, , ò, , +C, , a=, , 1, 2, ,b =10, 5, 1, 2, ,b =, 10, 5, , x3, is equal to, x +1, , (a), , x+, , x 2 x3, +, - log |1 - x | + C, 2, 3, , (b), , x+, , x 2 x3, - log |1 - x | + C, 2, 3, , (c), , x-, , x 2 x3, - - log |1 + x | + C, 2, 3, , (d), , x-, , x 2 x3, + - log |1 + x | + C, 2, 3, , x + sin x, , ò 1 + cos x, , x3 dx, , If, , 2, , is equal to, , (, , = a 1 + x2, , 1, a = , b =1, 3, , (c), , a=, , dx, , p/4, , p/ 2, , (a), , (d) x . tan, , ), , 3/ 2, , x, +C, 2, , + b 1 + x 2 + C , then, , (b), , -1, , b = -1, 3, , ò-p / 4 1 + cos 2 x, ò0, , (b) log | x + sin x | + C, , x, +C, 2, , (a), , (a) 1, 10., , -5, , 1, = a log | 1 + x 2 | + b tan -1 x + log, 5, x +1, , dx, , ò ( x + 2), , -5, , +C, , (c), , 1+ x, , (d), , 2, , (b), , (c) x – tan, , x - x +C, , x - x tan -1 x + C, , ò, , -1, -2, ,b=, 10, 5, , (a) log |1 + cos x | + C, , (c), , x9, , a=, , +C, , -1, ò tan x dx is equal to, , (a), , 5., , sin ( x - b ), , sin ( x - b ), , (a), , +C, , sin ( x - a ), , sin ( x - a ), , (d) sin ( b - a ) log, , 4., , 6., , dx, is equal to, sin ( x - a ) sin ( x - b ), , (a) sin ( b - a ) log, , 3., , | x + 2 | + C , then, , a=, , -1, , b =1, 3, , 1, (d) a = , b = -1, 3, , is equal to, , (b) 2, , (c) 3, , (d) 4, , 1 - sin 2 x is equal to, , 2 2, , (c) 2, , (b), (d), , (, 2(, , 2, , ), 2 - 1), 2 +1

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EBD_7762, 308, , MATHEMATICS, , 24. If ò x log æç1+ ö÷ dx = f (x) log(x +1) + g(x)x 2 + Lx + c, then, è xø, 1, , p, 2, , (a), , [BITSAT 2017, A], (a), , 1, f (x) = x 2, 2, , (b) g(x) = log x, , (c) L = 1, cos x - 1, , ò sin x + 1 e, , x, , dx is equal to :, , If, , 26., , e x cos x, +C, (a), 1 + sin x, ex, (c) C 1 + sin x, The integral, , (a) p, , [BITSAT 2017, A], , e x sin x, (b) C 1 + sin x, e x cos x, (d) C 1 + sin x, , (c) p / 2, , ò (sin5 x + cos3x sin 2 x + sin3x cos2 x + cos5x)2 dx, 3(1 + tan 3 x), , -1, , +C, , (b), , 1, 1 + cot 3 x, , 27., , The value of, , ò, , p 1+ 2, 2, , x, , dx is :, , xdx, is:, x, +, a+b-x, a, [BITSAT 2018, A], 1, (b - a), (b), 2, (d) b – a, , ex, , dx is equal to :, , [BITSAT 2018, A], , 2, , 2, , (3 + x 2 ), , +k, , (b), , 1 ex, +k, 2 (3 + x 2 ) 2, , (d), , 1 ex, +k, 2 (3 + x 2 ), , 2, , 1 ex, +k, 4 (3 + x 2 ) 2, a, , ò, , 30. If, , a, , f (2a - x)dx = m and, , 0, , ò, , 2a, , f (x)dx = n, then, , 0, , 0, , [BITSAT 2018, C], (c) m – n, (d) m + n, , [JEE MAIN 2018, A], , Exercise 3 : Try If You Can, 1., , For 0 < x < 1, let, n, , f ( x ) = lim (1 + x)(1 + x 2 )(1 + x 4 ).....(1 + x 2 ), n ®¥, , f ( x), , then, , ò 1 - x log e x dx, , (a), , æ x ö, log e ç, ÷+c, è 1- x ø, , equals, , (b), , æ x ö log e x, - loge ç, +c, ÷+, è 1- x ø 1- x, , (c), , log e x, + log e (1 - x) + c, 1- x, , (d), , x loge x + loge (1 - x) + c, , ò f (x)dx, , is equal to, (a) 2m + n (b) m + 2n, , sin 2 x, , p, 8, , (d), , ò, , 2, , +C, , 1, , +C, , p, 2, , (3 + x 2 ) 2, , (c), , +C, (c), 1 + cot 3 x, 3(1 + tan3 x), (where C is a constant of integration), , (c), , ò, , e x (2x + x 3 ), , is equal to :, , [JEE MAIN 2018, S], -1, , p, 4, , 2, , 29., , (a), , sin 2 x cos 2 x, , (c), b, , 28. The value of the integral, , (d) None of these, , 25., , (a), , (b) 4p, , 2., , ò, , x2 - 1, x ( x 2 + ax + 1)( x 2 + bx + 1), , dx =, , (a), , ì x 2 + ax + 1 + x 2 + b x + 1 ü, ï, ï, 2log í, ý+c, x, îï, þï, , (b), , ì x 2 + ax + 1 - x 2 + b x + 1 ü, ï, ï, 2log í, ý+c, x, ïî, ïþ, , (c), , log{ x 2 + ax + 1 - x 2 + b x - 1} + c, , (d) None of these

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INTEGRALS, , 3., , If, , ò, , 309, , x5 – x, , 1, æ x 4 –1ö 1, dx = k tan– 1 ç, –, 8, 2÷, 1, x +1, è k2 x ø 2, x, , ò x8 + 1, , 2, , ò f ( x)dx =, , + C,, , then value of k1 k2 is, (where C is an integration constant), (a) 8, (b) 2, (c), 4., , If, , dx, , ò x 2 ( x n + 1)( n-1) / n = -[ f ( x)], , 1/ n, , n, , x +x, , (c), , -n, , The value of, , dx, , ò, , a 1+ x, , 2n +1, , + 1 + x4n + 2, , ò (t + 2016), , 7, , ±1, , (c), , ±2, , 1, 2, (d) None of these, , (b), , Let g ( x ) = ò f (t )dt , where f is such that, 0, , 1, 1, £ f (t ) £ 1, for t Î[ 0,1] and 0 £ f (t ) £ , for t Î[1, 2] ., 2, 2, Then g(2) satisfies the inequality, , where n Î N and a, , and b are local maximum and minimum point of g(x) =, x, , (a), , x, , 9., , (d) none of these, b, , 5., , + C , then f(x) is, , -n, (b) 1 + x, , (1 + x n ), , (a), , progession is, , (d) 4, , 4 2, , -4 , then the common difference of the, , 0, , 5, (t + 2014)110 ( t – 2016) dt respectively, is, , (a), , -, , 3, 1, £ g (2) <, 2, 2, , (b) 0 £ g (2) < 2, , 3, 5, < g (2) £, (d) 2 < g (2) < 4, 2, 2, 10. Let f : R ® R be a differentiable function an d, , (c), , 0, , 6., , f (x), , b, , I=, , e, , ò, , a, , e, b, (a), a, , æ f ( x) ö, +e, fç, è x – a ÷ø, , æ f ( x) ö, fç, è x – b ÷ø, , , then |I| is equal to, , (b), , 11., , b, 2a, , derivative and f ¢ ( x ) Î ( 0,1] "x Î [0,1]., , f ( x), , x ®0, , x, , 2, , (c) 4, , 1/ x, , f ( x) ö, æ, exists finitely and lim ç1 + x +, x ÷ø, x ®0 è, , then, , ò f ( x) log e x dx, , (a), , 2 3æ, 1ö, x ç loge x - ÷ + c, 3 è, 3ø, , (b), , x3 æ, 1ö, ç log e x - ÷ + c, 3 è, 3ø, , (c), , 2 3, x (log e x + 1) + c, 3, , (d), , 2 3, x (log e x - 1) + c, 3, , is equal to, , (i), , f ( - x) = - f ( x), , Let ò f ( t ) dt = g ( x ) ,f ( 0 ) = g ( 0 ) and, , (ii), , f ( x + 1) = f ( x) + 1, , x, , æ 1 ö f ( x), (iii) f ç ÷ = 2 " x ¹ 0, è xø, x, , 0, , ò (f ( t )), , 3, , dt = h(x), then, , 0, , (g(x))2, , (a) h (x) £ g(x), (b), ³ h(x), (c) 2g(x) ³ (f(x))2, (d) f(1) = 1, If p, q, r, s are in arithmetic progression and, f ( x) =, , p + sin x q + sin x, q + sin x r + sin x, , p - r + sin x, -1 + sin x, , r + sin x, , s - q + sin x, , s + sin x, , such that, , x, , then, , òe, , (a), , e x ( x - 1) + c, , (b) e x log x + c, , (c), , ex, +c, x, , (d), , f ( x ) dx is equal to, , ex, +c, x +1, , , if, , (d) 2, , 12. If f : R ® R is a function satisfying the following:, , x, , 8., , If lim, , (d) None of these, , f : [0, 1] ® R is a function with continuous second, , x -1, , x ®1, , f ' (1) = 2 is, (a) 16, (b) 8, , 2t dt, , 4, , f (1) = 4. Then th e value of lim, , æ f ( x)ö, fç, è x – a ÷ø, , b2 – 4ac, | 2a |, , (c), 7., , ò, , (a) 0, (b) 2016, (c) 2014, (d) 2n + 1, If a, b(b > a) are the roots of f (x) º ax2 + bx + c = 0, a ¹, 0 and f (x) is an even function and, , = e3,

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EBD_7762, 310, , 13., , MATHEMATICS, , cosec 2 x - 2010, , (b), , 1, log{log e x} - log{log e x} + log{loge x} + C, 2, , æpö, where f ç ÷ = 1; then the number of solutions of the, è4ø, , (c), , 1, log{log e ex} - log{log e e 2 x 2 } + log{loge e3 x3} + C, 2, , f ( x), = {x} in [0, 2p] is/are : (where {.}, g ( x), , (d), , 1, 1, log{log e ex} - log{loge e 2 x} + log{log e e3 x} + C, 2, 2, , ò, , If, , cos 2010 x, , equation, , dx = -, , f ( x), ( g ( x ))2010, , represents fractional part function), (a) 0, (b) 1, (c) 2, 1, , ò x{logex e × log e2 x e × loge3 x e} dx, , 14., , (a), , + C;, , (d) 3, , is equal to, , 15., , ù, ù é 10 2n +1, é 10 -2n, ê, sin 27 x dx ú + ê, sin 27 x dx ú =, ú, ú ê, ê, û, û ë n =1 2n, ë n =1 - 2n -1, , å ò, , (a), , 272, , åò, , (b) –54, , (c), , 54, , (d), , 0, , 1, log{log e e 2 x} - log{loge e3 x} + log{log e e4 x} + C, 2, , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, , (a), (a), (a), (b), (b), (c), (a), (d), (d), (b), (b), (a), (a), , 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, , (a), (b), (d), (b), (a), (b), (c), (a), (b), (b), (b), (b), (a), , 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, , (b), (b), (a), (b), (a), (b), (b), (c), (b), (a), (c), (b), (b), , 1, 2, 3, , (a), (c), (a), , 4, 5, 6, , (d), (c), (d), , 7, 8, 9, , (d), (d), (a), , 1, 2, , (b), (a), , 3, 4, , (a), (b), , 5, 6, , (b), (c), , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (b) 79, (c), 40 (b) 53 (d) 66, (c) 80, (b), 41 (b) 54 (d) 67, (b) 81, (d), 42 (a) 55 (a) 68, (a) 82, (d), 43 (c) 56 (d) 69, (a) 83, (c), 44 (c) 57 (a) 70, (b) 84 (a), 45 (b) 58 (c) 71, (b) 85, (a), 46 (d) 59 (b) 72, (a) 86, (c), 47 (a) 60 (a) 73, (a) 87, (d), 48 (b) 61 (a) 74, (c) 88, (d), 49 (b) 62 (d) 75, (c) 89, (a), 50 (c) 63 (b) 76, (c) 90, (a), 51 (a) 64 (a) 77, (b) 91, (b), 52 (c) 65 (a) 78, Exercise 2 : Exemplar & Past Year MCQs, (c) 19, (b), 10 (d) 13 (d) 16, (b) 20, (d), 11 (b) 14 (b) 17, (a) 21, (d), 12 (a) 15 (b) 18, Exercise 3 : Try If You Can, (b) 9, (b) 11, (a) 13, (a), 7, (a) 10 (a) 12, (a) 14, (d), 8, , 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, , (c), (c), (b), (b), (d), (a), (c), (a), (c), (b), (c), (b), (c), , 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, , (c), (a), (a), (b), (a), (c), (b), (b), (c), (d), (c), (c), (d), , 118, 119, 120, 121, 122, 123, 124, 125, , (d), (c), (b), (b), (d), (a), (c), (b), , 22, 23, 24, , (c), (c), (d), , 25, 26, 27, , (a), (a), (c), , 28, 29, 30, , (b), (d), (d), , 15, , (d)

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24, , APPLICATION OF INTEGRALS, , Chapter, , Number of Questions, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , JEE MAIN, , 2, , BITSAT, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 3, 2, , Critical Concepts, , Area of the Region bounded by a, Curve & y-axis or x-axis, Area between, Two curves, , Rating of, Difficulty Level, , CUS, (chapter utility score), Out of 10, , 2/5, , 6.7

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APPLICATION OF INTEGRALS, , 313

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EBD_7762, 314, , MATHEMATICS, , Exercise 1 : Topic-wise MCQs, Topic 1: Area of the Region Bounded by a Curve and, X-axis Between Two Ordinates, Area of the Region Bounded by, a Curve and Y-axis Between Two Abscissae., 1., , 2., , 3., 4., , 5., , The area of the region bounded by the ellipse, x2, y2, +, = 1 is, 16, 9, (a) 12 p, (b) 3 p, (c) 24 p, (d) p, The area bounded by the line y = x, x-axis and lines, x = – 1 to x = 2, is, (a) 0 sq. unit, (b) 1/2 sq. units, (c) 3/2 sq. units, (d) 5/2 sq. units, , The area enclosed between the curve y = log e ( x + e) and, the coordinate axes is, (a) 1, (b) 2, (c) 3, (d) 4, The slope of the tangent to a curve y = f(x) at (x, f(x)) is 2x, + 1. If the curve passes through the point (1, 2), then the, area of the region bounded by the curve, the x-axis and, the line x = 1 is, (a), , 5, sq unit, 6, , (b), , (c), , 1, sq unit, 6, , (d) 6 sq unit, , The area enclosed between the graph of y = x3 and the, lines x = 0, y = 1, y = 8 is, 45, 4, (c) 7, , (a), , 6., , 6, sq unit, 5, , (b) 14, , (a), , 3, 4, , (b), , 1, will be :, 2, , 3, 2, , 5, (d) None of these, 4, Area bounded by the curve y = log x and the coordinate, axes is, (a) 2, (b) 1, (c) 5, (d) 2 2, , (c), 7., , 8., 9., , (b), , 1, 2, , 1, (d) None of these, 3, 10. The area bounded by the curve y = sin–1x and the line x = 0,, , (c), , | y |=, , p, is, 2, , (a) 1, (b) 2, (c) p, (d) 2 p, 11. Area between the curve y = cos2 x, x-axis and ordinates, x = 0 and x = p in the interval (0, p) is, 2p, p, (b) 2 p, (c) p, (d), 3, 2, 12. The area (sq. units) bounded by the parabola y2 = 4ax and, the line x = a and x = 4a is :, , (a), , (a), , 35a 2, 3, , (b), , 4a 2, 3, , 7a 2, 56a 2, (d), 3, 3, 13. Which of the following is true, (a) The area bounded by the curve y =sin x between x = 0, and x = 2p is 2 sq. units., (b) The area bounded by the curve y = 2 cos x and the xaxis from x = 0 to x = 2p is 8 sq. units., (c) Both (a) & (b) are true., (d) Both (b) & (b) are false., 14. Area of the region bounded by y = |x – 1| and y = 1 is, (a) 2 sq. units, (b) 1 sq. unit, , (c), , 1, sq. units, (d) None of these, 2, 15. The area bounded by the curves y = f(x), the x-axis, and the, ordinates x = 1 and x = b is (b – 1) sin (3b + 4). Then f(x), is, (a) (x – 1) cos (3x + 4), (b) sin (3x + 4), (c) sin (3x + 4) + 3(x – 1) cos (3x + 4), (d) None of these, 16. If the ordinate x = a divides the area bounded by x-axis,, , (c), , (d) None of these, , The area bounded by y –1 = |x|, y = 0 and |x| =, , (a) 1, , The area of the region bounded by y = | x – 1 | and y = 1 is, (a) 2, (b) 1, (c) 1/2, (d) 1/4, Let f(x) be a continuous function such that the area bounded, by the curve y = f(x), x-axis and the lines, p, æpö, a2 a, x = 0 and x = a is, + sin a + cos a, then f ç ÷ =, 2, 2 2, è2ø, , 8, part of the curve y = 1 + 2 and the ordinates x = 2, x = 4, x, into two equal parts, then a is equal to, (a), (b) 2 2, 2, (c) 3 2, , (d) None of these, , 17. Assertion : The area bounded by the curves, 16a, sq. units., 3, Reason : The area enclosed between the parabola, , y2 = 4a2(x – 1) and lines x = 1 and y = 4a is

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APPLICATION OF INTEGRALS, , 315, , 8, sq. units., 3, (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is, not a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., The area of the region bounded by the curve x = 2y + 3 and, lines y = 1 and y = –1 is, , y2 = x2 – x + 2 and the line y = x + 2 is, , 18., , 3, sq. units, 2, (c) 6 sq. units, (d) 8 sq. units, 19. The area of the region bounded by the curves, y = | x - 2 |, x = 1, x = 3 and the x-axis is, (a) 4, (b) 2, (c) 3, (d) 1, 20. The area bounded by the parabola y2 = 36x,, the line x = 1 and x-axis is ______ sq. units., (a) 2, (b) 4, (c) 6, (d) 8, , (a) 4 sq. units, , (b), , p, 21. The area under the curve y = | cos x – sin x |, 0 £ x £ ,, 2, and above x-axis is :, , (a), , 2 2, , (b) 2 2 - 2, , (c) 2 2 + 2 (d) 0, , 22. The value of a (a > 0) for which the area bounded by the, curves y =, , x 1, + , y = 0, x = a and x = 2a has the least, 6 x2, , value is, (a) 2, (b), 2, 23. The area of the ellipse, , (c) 21/3, , (d) 1, , 27. The area bounded by the x-axis, the curve y = f(x) and the, b 2 + 1 - 2 for all b > 1,, , lines x =1, x =b, is equal to, then f(x) is, , x, , (a), , x - 1 (b), , x +1, , (c), , x + 1 (d), 2, , 1+ x 2, , 28. The area of the region (in sq. units), in the first quadrant, bounded by the parabola y = 9x2 and the lines x = 0, y = l, and y = 4, is :, (a) 7/9, , (b) 14/3, , (c) 7/3, , (d) 14/9, , 29. Area of triangle whose two vertices formed from the, x-axis and line y = 3 – |x| is, (a) 9 sq. units, , (b) 9/4 sq. units, , (c) 3 sq. units, , (d) None of these, , 30. Area of the region bounded by the curve y = |x + 1| + 1,, x = –3, x = 3 and y = 0 is, (a) 8 sq units, , (b) 16 sq units, , (c) 32 sq units, , (d) None of these, , Topic 2: Different Cases of Area Bounded Between the, Curves., 31. The area of the region bounded by the parabola y = x 2 and y, = |x| is, (a) 3, , (b), , 1, 2, , 1, (d) 2, 3, 32. The area of the region bounded by the curves, y = x2 + 2, y = x, x = 0 and x = 3 is, , (c), , x2 y2, +, = 1 in first quadrant is 6p sq. units., 9, 4, The ellipse is rotated about its centre in anti-clockwise, direction till its major axis coincides with y-axis. Now the, area of the ellipse in first quadrant is______ p sq. units., (a) 2, (b) 4, (c) 6, (d) 8, 24. The area under the curve y = x2 and the line x = 3 and x, axis is ______ sq. units., (a) 0, (b) 1, (c) 3, (d) 9, 25. For the area bounded by the curve y = ax, the line x = 2, and x-axis to be 2 sq. units, the value of a must be equal, to, (a) 2, (b) 4, (c) 5, (d) None of these, , 26. The area bounded by the curve y =, , 3, x , the line, 2, , x = 1 and x-axis is ______ sq. units., (a) 2, , (b) 1, , (c) 6, , (d) None of these, , (a), , 2, 21, , (b) 21, , 21, 9, (d), 2, 2, 33. The area of the region enclosed by the parabola x2 = y, the, line y = x + 2 and the x-axis, is, , (c), , 2, 9, (c) 9, , 9, 2, (d) 2, , (a), , (b), , x2, , (c), , ab ( p + 2 ), 2, , sq. units, , (d), , ab ( p - 2 ), 4, , +, , y2, , = 1,, a 2 b2, where OA = a, OB = b. The area between the arc AB and, chord AB of the ellipse is, (a) p ab sq. units, (b) (p – 2)ab sq. units, , 34. AOB is a positive quadrant of the ellipse, , sq. units

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EBD_7762, 316, , MATHEMATICS, , 35. The area bounded by the line y = 2x – 2, y = – x and x-axis, is given by, (a), , Y, , 43, (b), sq. units, 6, , 9, sq. units, 2, , A(– a, a2), , 35, (c), sq. units, (d) None of these, 6, 36. The area of the region formed by, , x 2 + y 2 - 6 x - 4 y + 12 £ 0, y £ x and x £, , æp, 3 + 1ö, (a) çç 6 - 8 ÷÷ sq unit, è, ø, , 5, is, 2, , (a) 1 sq. unit, , (b), , 1, sq. units, 3, , 2, 4, sq. units, (d), sq. units, 3, 3, 38. The area bounded by the curves y = sin x, y = cos x and x = 0 is, , (a), , (, , ), , 2 - 1 sq. units, , (c), , 2 sq. units, , 44., , 46, , (b) 1 sq. unit, (d), , (1 + 2 ) sq. units, , 39. The area bounded by f (x) = x2, 0 £ x £ 1,, , 47., , g(x) = - x + 2,1£ x £ 2 and x - axis is, , (a), , 3, 2, , (b), , 4, 3, , 8, (d) None of these, 3, 40. Area between the parabolas y2 = 4ax and x2 = 4ay is, , (c), , (a), , 2 2, a -5, 3, , (b), , 15 2, a +5, 4, , 48., , 16 2, 16 2, a +2, a, (d), 3, 3, 41. Area between the curves y = x and y = x3 is, , (c), , (a), , 3, 2, , (b), , 1, 2, , (c) 2 2, , (d), , 1, 4, , 1, , x = 2, y = log x, 2, x, and y = 2 , then the area of this region, is, , 42. The region bounded by the curves x =, , (a), , 4, sq. unit, 3, , (b), , 5, sq. unit, 3, , 3, sq. unit, (d) None of these, 2, 43. The figure shows as triangle AOB and the parabola y = x 2., The ratio of the area of the triangle AOB to the area of the, region AOB of the parabola y = x2 is equal to, , (c), , 3, 3, 7, 5, (b), (c), (d), 5, 4, 8, 6, Using the method of integration the area of the triangle, ABC, coordinates of whose vertices are A (2, 0), B (4, 5), and C (6, 3) is, (a) 2, (b) 4, (c) 7, (d) 8, Area bounded by the lines y = |x| – 2 and y = 1 – |x – 1| is, equal to, (a) 4 sq. units, (b) 6 sq. units, (c) 2 sq. units, (d) 8 sq. units, The area bounded by the curve y2 = 16x and line y = mx is, 2, , then m is equal to, 3, (a) 3, (b) 4, (c) 1, (d) 2, Assertion : The area bounded by the circle y = sin x and, y = –sin x from 0 to p is 3 sq. unit., Reason :The area bounded by the curves is symmetric, about x-axis., (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is, not a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., 2, The area bounded by the curves x = y2 and x =, is, 1 + y2, 2, 2, (a) p (b) p +, 3, 3, 2, (c) -p (d) None of these, 3, The area bounded by the curves x2 + y2 = 25, 4y = |4 – x2|, and x = 0, above x-axis is, 25 -1 4, 25 -1 4, (a) 2 + sin, (b) 2 + sin, 2, 5, 4, 5, , (a), , 45., , (c), , X, , O(0, 0), , æp, 3 -1ö, (b) çç 6 + 8 ÷÷ sq unit, è, ø, , æp, 3 -1 ö, (c) çç 6 - 8 ÷÷ sq unit (d) None of these, è, ø, 37. The area bounded by the curves x + 2y2 = 0 and x + 3y2 = 1 is, , B(a, a2), , 49., , 25 -1 1, sin, (d) None of these, 2, 5, 50. The area bounded by curves (x – 1)2 + y2 = 1 and, x2 + y2 = 1 is, , (c) 2 +, , æ 2p, 3ö, (a) çç 3 - 2 ÷÷, è, ø, , (b), , 2p, 3, , 3, 2, , (d), , 2p, 3, +, 3, 2, , (c)

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APPLICATION OF INTEGRALS, , 317, , 51. The area of the smaller region bounded by the ellipse, x y, x2, y2, +, = 1 and the line + = 1 is, 3 2, 9, 4, , (a) 3 (p – 2), , 3, p, 2, , (b), , 2, ( p - 2), 3, , 3, ( p - 2), (c), (d), 2, 52. The area of the region, {(x, y) : y2 £ 4x, 4x2 + 4y2 £ 9} is, , (a), , 2 9 p 9 -1 æ 1 ö, +, - sin ç ÷, 6, 8 4, è3ø, , (b), , 2 9p, 6, 8, , 2, , (, , 2 -1, , and curve y =, , (b), , 2–1, , 5, sq. units, 2, , (d), , p3 - 8, sq. units, 4, , (b), , p2 - 4, p2 - 8, sq. units, (d), sq. units, 8, 4, 59. Area enclosed between the curves y = sin 2x, y = cos2x and, y = 0 in the interval [0, p/2] is, 1, ( 2p - 1) sq. units, 3, , (a), , (b), , (a), , 4 2 2, a, b, sq. units, 3, b-a, , (b), , (d), , 1, sq. units, 2, , 8 8 2, b, a, sq. units, 3, b -a, , (c), , 4 2 2, b, a, sq. units, 3, b-a, , (d), , 8 8 2, a, b, sq. units, 3, b-a, , x2, c, =, y, =, and, be two curves lying in, 2, 1 + x2, 2, XY-plane, then, 1, , 1+ x2, , (b) area bounded by c1 and c2 is, , and y = 0 is, , p, 2, , p, -1, 2, , (c) area bounded by c1 and c2 is 1 -, , 1, , p, 2, , p, and x-axis is, 2, 1+ x2, 56. The area of the region whose boundaries are defined by, the curves y = 2 cos x, y = 3 tan x, and the y-axis is, , (d) area bounded by curve y =, , æ 2 ö, (a) 1 + 3ln ç, ÷ sq. units, è 3ø, , 3, (b) 1 + ln 3 - 3ln 2 sq. units, 2, , 1, ( p - 3) sq. units, 2, , 1, ( p - 2 ) sq. units (d) (2p + 3) sq. units, 4, 60. The area included between the parabolas y2 = 4a (x + a), and y2 = 4b(x – a), b > a > 0, is, , 3, sq. units, 2, , 1, , p3 - 4, sq. units, 8, , (c), , (b), , (a) area bounded by curve y =, , 2, ( 4 + 3p) sq. units, 3, , (c), , 1, and the positive x-axis is, x, , (a) 1 sq. unit, , 55. If c1 = y =, , (c) 6 + 3p sq. units, , (a), , ), , (b), , 4, (8 + 3p ) sq. units, 3, 58. Area of the region between the curves x2 + y2 = p2,, y = sin x and y-axis in first quadrant is, , (d), (c), 2+1, 2, 54. The area of the region enclosed by the lines y = x, x = e, , (c), , 1, ( 2 + 3p ) sq. units, 3, , (a), , 9p 9 -1 æ 1 ö, - sin ç ÷, (c), 8 4, è 3ø, (d) None of these, 53. The area bounded by the y-axis, y = cos x and y = sin x, p, when 0 £ x £ is, 2, (a), , 3, (c) 1 + ln 3 - ln 2 sq. units, 2, (d) ln 3 – ln 2 sq. units, 57. The area lying above x-axis and included between the circle, x2 + y2 = 8x and inside of parabola y2 = 4x is, , 61. The area common to the ellipse, x2, b2, , (a), , +, , y2, a2, , x2, a2, , +, , y2, b2, , = 1 and, , = 1 , 0 < b < a is, , ( a + b )2 tan -1, , b, a, , (b), , ( a + b )2 tan -1, , a, b, , b, -1 a, (d) 4ab tan, a, b, 62. Which of the following is not the area of the region, bounded by y = ex and x = 0 and y = e?, -1, (c) 4ab tan, , e, , (a) e – 1, , (b), , 1, , x, (c) e - ò e dx, 0, , ò ln(e + 1 - y)dy, , 1, e, , (d), , ò ln, , 1, , y dy

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EBD_7762, 318, , MATHEMATICS, , 63. The area of the triangle formed by the tangent and normal, , (, , ), , (a), , at the point 1, 3 on the circle x + y = 4 and the x-axis, 2, , is, (a) 3 sq. units, , 2, , (b), , 2 3 sq. units, (c) 3 2 sq. units, (d) 4 sq. units, 64. The ratio in which the area bounded by the curves, y2 = 12x and x2 = 12y is divided by the line x = 3 is, (a) 15 : 49, (b) 13 : 37, (c) 15 : 23, (d) 17 : 50, 65. The line y = mx bisects the area enclosed by lines, x =0, y = 0 and x = 3/2 and the curve y = 1 + 4x – x2. Then, the value of m is, (a), , 13, 6, , 13, 13, (d), (c), 5, 7, 2, 66. Area bounded by the circle x + y2 = 1 and the curve, | x | + | y | = 1 is, (a) 2 p, (b) p – 2, (c) p, (d) p + 3, 67. The area of the plane region bounded by the curves, x + 2y2 = 0 and x + 3y2 = 1is equal to, , (b), , 2, (d), 3, 68. The area bounded by the curve, , (c), , 1, 3, 4, 3, , y 2 (a 2 + x 2 ) = x 2 (a 2 - x 2 ) is, , (a), , a 2 (p - 2) sq unit, , (b) a 2 (p + 2) sq unit, , (d) a 2 (p + 1) sq unit, (c) a 2 (p - 1) sq unit, 69. Area bounded by the parabola y = x2 – 2x + 3 and tangents, drawn to it from the point P(1, 0) is equal to, , 13, 2, , (b), , 5, 3, , (a), , 4 2 sq. units, , (b), , 4 2, sq. units, 3, , (c), , 8 2, sq. units, 3, , (d), , 16, 2 sq. units, 3, , 70. The area (in sq. units) bounded by the curves y = x,, 2y – x + 3 = 0 and x-axis lying in the first quadrant is, (a) 9, (b) 36, (c) 18, , (d), , 27, 4, , Exercise 2 : Exemplar & Past Year MCQs, 6., , NCERT Exemplar MCQs, 1., , The area of the region boundned by the Y-axis y = cos x, , (a), , p, and y = sin x, where 0 £ x £ is, 2, , (a), (c), 2., , 4., , 5., , (b), , ), , (d), , 2 - 1 sq. units, , ( 2 + 1) sq. units, ( 2 2 - 1) sq. units, , 8, 9, , (b), , 9, 7, , (c), , 7, 9, , (d), , 9, 8, , The area of the region bounded by the curve y = 16 - x 2, and X-axis is, (a) 8p sq. units, (b) 20p sq. units, (c) 16p sq. units, (d) 256p sq. units, Area of the region in the first quadrant enclosed by the, X-axis, the line y = x and the circle x2 + y2 = 32 is, (a) 16p sq. units, (b) 4p sq. units, (c) 32p sq. units, (d) 24p sq. units, Area of the region bounded by the curve y = cos x between, x = 0 and x = p is, (a) 2 sq. unit, (b) 4 sq. unit, (c) 3 sq. unit, (d) 1 sq. unit, , 4, sq. units, 3, , (b) 1 sq. units, , 2, 1, sq. units, (d), sq. units, 3, 3, The area of the region bounded by the curve y = sin x, , (c), , 7., , p, and the X-axis is, 2, (b) 4 sq. units, (d) 1 sq. unit, , between the ordinates x = 0, x =, , Area between the parabola x2 = 4y and line x = 4y –2 is, (a), , 3., , (, , 2 sq. units, , The area of the region bounded by parabola y2 = x and the, straight line 2y = x is, , (a) 2 sq. units, (c) 3 sq. units, , x2 y 2, +, = 1 is, 25 16, (a) 20p sq. units, (b) 20p 2sq. units, (c) 16p2 sq. units, (d) 25p sq. units, 9. The area of the region bounded by the circle x2 + y2 = 1 is, (a) 2p sq. units, (b) p sq. units, (c) 3p sq. units, (d) 4p sq. units, 10. The area of the regionn bounded by the curve y = x + 1 and, the lines x = 2, x = 3 is, , 8., , The area of the region bounded by the ellipse, , (a), , 7, sq. units, 2, , (b), , 9, sq. units, 2, , (c), , 11, sq. units, 2, , (d), , 13, sq. units, 2

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APPLICATION OF INTEGRALS, , 319, , 11. The area of the region bounded by the curve x = 2y + 3 and, the lines y = 1, y = –1 is, , y2, , 16. The area (in sq. units) of the region {(x, y) :, ³ 2x, and x2 + y2 £ 4x, x ³ 0, y ³ 0} is : [JEE MAIN 2016, A], , 3, sq. units, 2, (d) 8 sq. units, , (a) 4 sq. units, , (b), , (c) 6 sq. units, , Past Year MCQs, 12. The area of the region described by, , A=, , {( x, y ) : x, , 2, , }, , + y 2 £ 1 and y 2 £ 1 - x is:, , (a) p-, , 4 2, 3, , (b), , (c) p-, , 4, 3, , (d) p-, , {(x, y) : x ³ 0, x + y £ 3, x2 £ 4y and y £ 1 +, , (a), , x -1, , (b), , x +1, , (c), , x2 +1, , (d), , (a), , 5, 2, , (b), , 59, 12, , (c), , 3, 2, , (d), , 7, 3, , p, ,, 2, [BITSAT 2017, A], , 18. The area under the curve y = | cos x – sin x |, 0 £ x £, and above x-axis is :, , x, , (a), 2, , x }is :, , [JEE MAIN 2017, A], , p 2, p 2, p 4, p 4, +, +, (b), (c), (d), 2 3, 2 3, 2 3, 2 3, 13. The area bounded by the x-axis, the curve y = f(x) and the, , (a), , b 2 + 1 - 2 for all b > 1,, [BITSAT 2014, C], , 8, 3, , 17. The area (in sq. units) of the region, , [JEE MAIN 2014, A], , lines x =1, x =b, is equal to, then f(x) is, , p 2 2, 2, 3, , 2 2 (b) 2 2 - 2, , (c) 2 2 + 2 (d) 0, , 1+ x, 14. The area (in sq. units) of the region described by, {(x, y) : y2 £ 2x and y ³ 4x – 1} is [JEE MAIN 2015, A], , 19. Let g(x) = cos x 2 ,f (x) = x , and a, b (a < b) be the, , 15, 9, 7, 5, (b), (c), (d), 64, 32, 32, 64, 15. The area of the region R = {(x, y):|x| £ |y| and x2 + y2 £ 1}, is, [BITSAT 2015, S], , the area (in sq. units) bounded by the curve y = (gof)(x), , roots of the quadratic equation 18x 2 - 9 px + p 2 = 0 . Then, , (a), , and the lines x = a, x = b and y = 0 , is :, [JEE MAIN 2018, S], , (a), , 3p, sq. units, 8, , (b), , 5p, sq. units, 8, , (a), , 1, ( 3 + 1), 2, , (b), , 1, ( 3 - 2), 2, , (c), , p, sq. units, 2, , (d), , p, sq. unit, 8, , (c), , 1, ( 2 - 1), 2, , (d), , 1, ( 3 - 1), 2, , Exercise 3 : Try If You Can, 1., , The area bounded by the curve y = x(3 – x)2, the x-axis and, the ordinates of the maximum and minimum points of the, curve, is given by, (a) 1 sq. unit, (b) 2 sq. unit, (c) 4 sq. unit, (d) None of these, , 4., , of the area bounded by the curve | y | +, , p, , 2., , f ( x) = min{2sin x, 1 - cos x, 1} then, , ò f ( x) dx is equal to, 0, , (a), , (b), , 2p, -1+ 3, 3, , 5p, 2p, +1- 3, (d), -1- 3, 6, 3, The area enclosed between the curve y = log e ( x + e) and, the coordinate axes is, (a) 1, (b)2, (c) 3, (d) 4, , (c), , 3., , p, +1 - 3, 3, , Let for all n Î N , n - 2 ³ 2 log e n then the minimum value, , 1, (b) 1, (c) 2, 2, The area bounded by the curve, , (a), , 5., , 1, £ e -| x| is, n, , (d) 4, , f ( x) = max.{4 - x 2 ,| x - 2|, ( x - 2)1/ 3 } for x Î[-2, 4] and, x-axis is, , (a), , 45, sq units, 4, , (b), , 23, sq units, 4, , (c), , 59, sq units, 4, , (d), , 53, sq units, 4

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EBD_7762, 320, , MATHEMATICS, , The area bounded by y = f (x) and the curve y =, , 6., , 2, 2, , 1+ x, where f is a continuous function satisfying the, conditions, f (x) · f (y) = f (xy). " x, y, Î R and f ¢ (1) = 2, f (1) = 1is ,, then, aö, aö, æ, æ, (a) ç p + ÷, (b) ç p - ÷, 3ø, 3ø, è, è, æ 2a ö, (c) ç p+ ÷, (d) None of these, 3ø, è, The area bounded by the curve y = f(x), y = x and the lines x, , 7., , 2, , = 1, x = t is (t + 1 + t ) - 2 - 1 sq unit, for all t > 1. If, f(x) satisfying f(x) > x for all x > 1, then f(x) is equal to, x, , (a) x + 1 +, , 1 + x2, , x, , (c) 1 +, , x, , (b) x +, , 1 + x2, , x, , (d), 1 + x2, 1 + x2, If the line y = mx + 2 cuts the parabola 2y = x2 at points, ( x1, y1 ) and (x2, y2) where (x1 < x2), then the value of m, , 8., , for which, , x2, , ò, , x1, , æ, x2 ö, ç mx + 2 - ÷ dx is minimum, is, 2ø, è, , 8, (c) 1, (d) 0, (a), (b), 2, 3, 3, Let f and g be continuous function on a £ x £ b and p (x), = max { f (x), g (x)} and q (x) = min { f (x), g (x)}. The area, bounded by the curves y = p (x), y = q (x) and the ordinates, x = a and x = b is given by, , 9., , b, , (a), , òa ( f ( x) - g ( x) )dx, , (c), , òa, , b, , (b), , p ( x ) - q( x ) dx, , b, , òa ( p ( x) - q ( x))dx, , (d) None of these, , 10. Area between the curve y = 2x 4 - x 2 , x axis and the, ordinates of two minima of curve is, 7, 9, 11, 13, (a), (b), (c), (d), 120, 120, 120, 120, , 1, 2, 3, 4, 5, 6, 7, , (a), (d), (a), (a), (a), (c), (b), , 8, 9, 10, 11, 12, 13, 14, , (b), (b), (b), (d), (d), (b), (b), , 15, 16, 17, 18, 19, 20, 21, , (c), (b), (c), (c), (d), (b), (b), , 1, 2, , (c), (d), , 3, 4, , (a), (b), , 5, 6, , (a), (a), , 1, 2, , (c), (d), , 3, 4, , (a), (c), , 5, 6, , (c), (b), , 11. The area of the region between the curves y =, and y =, , p, 1 - sin x, bounded by the lines x = 0 and x = is, 4, cos x, , 2 -1, , (a), , ò, , 0, , 1 + sin x, cos x, , 2 -1, , t, 2, , (1 + t ) 1 - t, , 2 +1, , ò, , 2, , ò, , dt (b), , 4t, 2, , (1 + t ) 1 - t 2, , 0, , 2 +1, , 4t, , t, , ò, , dt, 2, (1 + t ) 1 - t, 0 (1 + t ) 1 - t, 12. Maximum area of rectangle whose two sides are, x = x0 , x = p - x0 and which is inscribed in a region, , (c), , 0, , 2, , 2, , dt (d), , dt, , 2, , bounded by y = sin x and x-axis is obtained when x0 Î, æ p pö, (b) ç , ÷, è6 4ø, , p p, (a) æç , ö÷, è 4 3ø, , æ pö, (d) None of these, ç 0, ÷, è 6ø, 13. Find the area of the region R which is enclosed by the curve, , (c), , y ³ 1 - x 2 and max {| x |, | y |} £ 4 is, p, 2, 14. If [x] is the greatest integrer £ x , then, , (a) 4 + p, , (b) 6 – p, , (c) 8 -, , (d) 4 +, , p, 2, , 2, , ò min{x - [ x], - x - [ - x ]} dx =, , -2, , (a) 1, , (b) 2, , (c), , 3, 2, , (d) 0, , 2, , 15. The value of integral, , ò max{x+ | x |,, , x - [ x ]}dx where [.], , -2, , represents the greatest integer function is, 7, (a) 4, (b) 5, (c), (d), 2, , ANSW ER KEYS, Exercise 1 : Topic-w ise MCQs, (d), (d), (c) 43, (b), 22, 29, 36, (c), (b), (d) 44, (c), 23, 30, 37, (d), (c), (a), (a), 24, 31, 38, 45, (d), (c), (d) 46, (b), 25, 32, 39, (b), (b), (d) 47, (d), 26, 33, 40, (d), (d), (b) 48, (a), 27, 34, 41, (d), (d), (d) 49, (a), 28, 35, 42, Exercise 2 : Exempl ar & Past Year MCQs, (d), (b), (c) 13, (d), 7, 9, 11, (a), (a), (c) 14, (b), 8, 10, 12, Exerci se 3 : Try If You Can, (a), (c), (b) 13, (c), 7, 9, 11, (d), (a), (b) 14, (a), 8, 10, 12, , 9, 4, , 50, 51, 52, 53, 54, 55, 56, , (a), (c), (a), (b), (b), (b), (b), , 57, 58, 59, 60, 61, 62, 63, , (d), (a), (c), (b), (c), (c), (b), , 64, 65, 66, 67, 68, 69, 70, , (a), (a), (b), (d), (a), (c), (a), , 15, 16, , (c), (d), , 17, 18, , (a), (b), , 19, , (d), , 15, , (b)

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25, , DIFFERENTIAL EQUATIONS, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 4, , 3, JEE MAIN, BITSAT, , 2, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 3, 4, , Critical Concepts, , General & Particular solution of a, Differential Equation, Formation of a, Differential Equation whose General, solution is given, Homogenous, Differential Equation, Linear, Differential Equation., , Rating of, Difficulty Level, , CUS, (chapter utility score), Out of 10, , 3.5/5, , 6.7

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DIFFERENTIAL EQUATIONS, , 323

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EBD_7762, 324324, , MATHEMATICS, , Topic 1 : Ordinary Differential Equations,, Order and Degree of Differential Equations,, Formation of Differential Equations, 1., , 2., , 3., , 4., , 5., , 6., , 7., , The differential equation representing the family of, parabolas having vertex at origin and axis along positive, direction of x-axis is, (a) y2y¢¢ – 2xy¢ = 0, (b) y2 – 2xyy¢¢ = 0, 2, (c) y – 2xyy¢ = 0, (d) None of these, The differential equation of all non-horizontal lines in a, plane is, d2x, dx, d2 y, = 0 (c) dy = 0 (d), =0, (a), (b), 2, 2, dy, dx, dy, dx, The family of curves y = ea sin x, where a is an arbitrary, constant, is represented by the differential equation, dy, dy, (a) log y = tan x, (b) y log y = tan x, dx, dx, dy, dy, (c) y log y = sin x, (d) log y = cos x, dx, dx, The differential equation which represent the family of, curves y = aebx, where a and b are arbitrary constants., (a) y¢ = y2, (b) y¢¢ = y y¢, (c) y y¢¢ = y¢, (d) y y¢¢ = (y¢)2, 1 - y2, dy, =, dx, y, determines a family of circle with, (a) variable radii and fixed centre (0, 1), (b) variable radii and fixed centre (0, –1), (c) fixed radius 1 and variable centre on x-axis, (d) fixed radius 1 and variable centre on y-axis, Differential equation of y = sec (tan–1x) is :, dy, dy, (a) (1 + x 2 ), (b) (1 + x 2 ) = y - x, = y+x, dx, dx, dy, dy x, 2, = xy, (c) (1 + x ), (d) (1 + x 2 ), =, dx, dx y, Which of the following equation has, y = c1 ex + c2 e–x as the general solution?, , The differential equation, , d2 y, d2 y, +, y, =, 0, (b), –y=0, dx 2, dx 2, d2 y, d2 y, (c), +, 1, =, 0, (d), –1=0, dx 2, dx 2, The order of the differential equation of a family of curves, represented by an equation containing four arbitrary, constants, will be, (a) 2, (b) 4, (c) 6, (d) None of these, The differential equation which represents the three parameter, , (a), , 8., , 9., , 2, 2, family of circles x + y + 2 gx + 2 fy + c = 0 is, , (a), , y ''' =, , 3 y ' y''2, 1 + y ¢2, , (b), , y ''' =, , 3 y''2, 1 + y ¢2, , (c), , y ''' =, , 3y¢, 1+ y¢2, , (d), , y ''' =, , 3y¢, 1- y¢2, , 10. The order and degree of the differential equation, 1/3, , 11., , d 2 y æ dy ö, +, + x1/4 = 0 is, 2 ç dx ÷, è ø, dx, (a) order = 3, degree = 2 (b) order = 2, degree = 3, (c) order = 2, degree = 2 (d) order = 3, degree = 3, The differential equation representing the family of curves, y = A cos (x + B), where A, B are parameters, is, , (a), , d2 y, dx 2, d2 y, , +y=0, , (b), , d2 y, dx 2, , -y=0, , dy, dy, +y, +y=0, (d), dx, dx, dx, 12. The order and degree of the differential equation, (c), , 2, , =, , 2, , dy, æ dy ö, + a 2 ç ÷ + b2 is, è dx ø, dx, (a) order = 1, degree = 2 (b) order = 2, degree = 1, (c) order = 2, degree = 2 (d) None of these, 13. The order and degree of the differential equation whose, y= x, , solution is y = cx + c 2 - 3c3 2 + 2 , where c is a parameter, is, (a) order = 4, degree = 4 (b) order = 4, degree = 1, (c) order = 1, degree = 4 (d) None of these, 14. An equation which involves variables as well as derivative, of the dependent variable with respect to independent, variable, is known as, (a) differential equation (b) integral equation, (c) linear equation, (d) quadratic equation, d2 y, , + y = 0, if there is a, dx 2, function y = f (x) that will satisfy it, then the function, y = f (x) is called, (a) solution curve only, (b) integral curve only, (c) solution curve or integral curve, (d) None of the above, 16. The degree an d order of the differential equation, , 15. For the differential equation, , 2 2, 2, y = px + 3 a p + b , p =, , (a) 3, 1, , (b) 1, 3, , dy, , are respectively, dx, (c) 1, 1, (d) 3, 3

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DIFFERENTIAL EQUATIONS, , 325, , 17. The differential equation obtained by eliminating the, arbitrary constants a and b from xy = aex + be–x is, (a), , x, , (c), , x, , d2 y, dx 2, , d2 y, dx 2, , +2, , dy, - xy = 0 (b), dx, , +2, , dy, + xy = 0 (d), dx, , d2 y, dx 2, , d2 y, dx 2, , + 2y, , +, , dy, - xy = 0, dx, , dy, - 2y = 0, dx, , 18. A first order-first degree differential equation is of the form, (a), , d2 y, dx 2, , = F ( x, y), , (b), , 2, , (c), , æ dy ö, çè ÷ø = F ( x, y ), dx, , (d), , d3 y, dx 2, , = F ( x, y), , dy, = F ( x, y ), dx, , 19. If p and q are the order and degree of the differential, equation y, , dy, d2 y, + x3, + xy = cos x, then, dx, dx 2, , (a) p < q, (b) p = q, (c) p > q, (d) None of these, 20. The differential equation obtained by eliminating arbitrary, constants from y = aebx is, (a), , d2 y, , dy, y, +, =0, 2, dx, dx, , (b), , d2 y, , dy, =0, 2, dx, dx, , 2, , 21., , 22., , d2y æ dyö, -ç ÷ = 0, dx2 è dxø, , 2, , d2 y, , æ dy ö, +ç ÷ =0, 2, è, dx ø, dx, The elimination of constants A, B and C from, y = A + Bx – Ce–x leads the differential equation:, (a) y" + y"' = 0, (b) y" – y"' = 0, (c) y' + ex = 0, (d) y" + ex = 0, Consider the following statements, , (c), , I., II., , y, , (d), , y, , The order of the differential equation, The order of the differential equation, , dy, = e x is 1., dx, 2, , d y, , dx 2, III. The order of the differential equation, , + y = 0 is 2., , 3, , æ d3 y ö, æ 2 ö, 2 d y, +, x, ç 3÷, ç 2 ÷ = 0 is 3., è dx ø, è dx ø, , Choose correct option., (a) I and II are true, (b) II and III are true, (c) I and III are true, (d) All are true, 23. The degree of the differential equation, y32 / 3 + 2 + 3y2 + y1 = 0 is :, , (a) 1, , (b) 2, , (c) 3, , (d) None of these, , 24. The differential equation representing the family of curves, y 2 = 2c ( x + c ) , where c > 0, is a parameter, is of order, and degree as follows :, (a) order 1, degree 2, (b) order 1, degree 1, (c) order 1, degree 3, (d) order 2, degree 2, 25. Differential equation of all straight lines which are at a, constant distance from the origin is, , (a), , (y + xy1)2 = p2 (1+ y12 ) (b), 2, , 2, , (y - xy1 )2 = p 2 (1 - y12 ), , 2, , (c) (y - xy1) = p (1+ y1 ) (d) None of these, 26. The order of the differential equation of all tangent lines to, the parabola y = x2, is, (a) 1, (b) 2, (c) 3, (d) 4, 27. The order of the differential equation whose general, solution is given by, y = (C1 + C2) cos (x + C3) – C 4 e x + C5, where C1, C2, C3, C4, C5 are arbitrary constant, is, (a) 5, (b) 4, (c) 3, (d) 2, 28. Family y = Ax + A3 of curves will correspond to a differential, equation of order, (a) 3, (b) 2, (c) 1, (d) not infinite, 29. The degree of the differential equation satisfied by the curve, 1 + x - a 1 + y = 1, is, (a) 0, (b) 1, , 30., , (c) 2, , (d) 3, 2y, d, dy, + 2y , is, If y = ex (sin x + cos x), then the value of 2 - 2, dx, dx, (a) 0, (b) 1, (c) 2, (d) 3, , 31. The order of the differential equation whose solution is :, y = a cos x + b sin x + ce–x is :, (a) 3, (b) 2, (c) 1, (d) None of these, 32. The degree of the differential equation, 4, , 5, , æ d3 y ö æ d 2 y ö dy, + y = 0 is, ç 3 ÷ +ç 2 ÷ +, ç dx ÷ ç dx ÷ dx, è, ø è, ø, (a) 2, (b) 4, (c) 6, , (d) 8, , 33. A family of curves is given by the equation, , x2, , +, , y2, , = 1., a 2 b2, The differential equation representing this family of curves, , d2 y, , 2, , dy, æ dy ö, + Ax ç ÷ – y, = 0. The value of A is, 2, è, ø, dx, dx, dx, (a) 0, (b) 1, (c) 3, (d) 5, 34. Assertion: Order of the differential equation whose solution, is y = c1e x + c 2 + c3e x + c4 is 4., , is given by xy, , Reason: Order of the differential equation is equal to the, number of independent arbitrary constant mentioned in the, solution of differential equation., (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect Reason is correct

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DIFFERENTIAL EQUATIONS, , 327, , 50. For y = cos kx to be a solution of differential equation, , dy 1 - x, =, dx, y, is a family of curves which looks most like which of the, following?, , 57. The general solution of the differential equation, , 2, , d y, , + 4y = 0, the value of k is, dx 2, (a) 2, (b) 4, (c) 6, (d) 8, 51. The function f (x) satisfying the equation., f 2(x) + 4f '(x). f (x) + [f '(x)]2 = 0, 3) x, , (2 +, (b) f (x) = C.e, , 3) x, , 3 – 2) x, , (d) f (x) = C.e(2 +, , 3) x, , (2 –, (a) f (x) = C.e, (, (c) f (x) = C.e, , (a), , (b), , (c), , (d), , 52. For the function y = Bx2 to be the solution of differential, 3, , dy, æ dy ö, – 2xy = 0, the value of B is, equation ç ÷ – 15x2, è dx ø, dx, __________, given that B ¹ 0., (a) 2, (b) 4, (c) 6, (d) 8, 53. In the particular solution of differential equation, 1, dy, =, , the value of constant term is ________,, dx, x(3y 2 - 1), , given that y = 2 when x = 1., (a) 2, (b) 4, (c) 6, , (d) 8, , 54. Which of the following differential equation has y = x as, one of its particular solution ?, (a), , (c), , d2y, , dy, –x, + xy = x (b), 2, dx, dx, , d2y, , d2y, , d2y, , dx, , 2, , 2, , – x2, , dy, + xy = 0 (d), dx, , dy, + x + xy = x, 2, dx, dx, , dx, , 2, , +x, , dy, + xy = 0, dx, , 55. In a culture the bacteria count is 1,00,000. The number is, increased by 10% in 2 hours. In how many hours will the, count reach 2,00,000 if the rate of growth of bacteria is, proportional to the number present., (a), , (b), , log 2, log11, , 2 log 2, æ 11 ö, log ç ÷, è 10 ø, , log 2, æ 11 ö, log ç ÷, è 10 ø, The equation of the curve passing through the point, , (c), , 56., , 2, 11, log, 10, , (d), , æ pö, ç 0, 4 ÷ whose differential equation is, è, ø, sin x cos y dx + cos x sin y dy = 0, is, , 60. If f (x) =, , ò {f ( x )}, , (b) cos x cos y =, , (c) sec x =, , (d) cos y =, , 2, 2 sec y, , -2, , dx and f (1) = 0, then f (x) is equal to, , (a) {2 (x – 1)}1/4, , (b) {5 (x – 2)}1/5, , (c) {3 (x – 1)}1/3, , (d) None of these, , æ xö, 61. If (1 + e x / y ) dx + ç1 - ÷ e x / y dy = 0, then, è yø, (a) x – yex/y = c, (b) y – xex/y = c, x/y, (c) x + ye = c, (d) y + xex/y = c, 62. The female-male ratio of a village decreases continuously, at the rate proportional to their ratio at any time. If the ratio, , of female : male of the villages was 980 : 1000 in 2001 and, 920 : 1000 in 2011. What will be the ratio in 2021 ?, (a) 864 : 1000, (b) 864 : 100, (c) 1000 : 864, (d) 100 : 864, 63. A particular solution of, , æ dy ö, log ç ÷ = 3x + 4 y , y ( 0 ) = 0 is, è dx ø, (a), , e3 x + 3e -4 y = 4, , (b) 4e3 x - 3e -4 y = 3, , (d) 4e3 x + 3e -4 y = 7, 3e3 x + 4e 4 y = 7, 64. Solution of the differential equation, (c), , xdy – ydx =, , (a) sec x sec y =, , 2, 2 cos y, , 58. The population of a village increases continuously at the, rate proportional to the number of its inhabitants present, at any time. If the population of the village was 20, 000 in, 1999 and 25000 in the year 2004, what will be the population, of the village in 2009?, (a) 3125, (b) 31250, (c) 21350, (d) 12350, dy ax + 3, =, 59. If the solution of the differential equation, dx 2y + f, represents a circle, then the value of ‘a’ is, (a) 2, (b) – 2, (c) 3, (d) – 4, , x 2 + y 2 dx is, , (a) y = cx2, (c), , y + x 2 + y2 = cx 2, , (b) y = cx2 +, (d), , x 2 + y2, , y - x 2 - y2 = c

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EBD_7762, 328328, , MATHEMATICS, , 65. At any point (x, y) of a curve, the slope of tangent is twice, the slope of the line segment joining the point of contact to, the point (–4, –3). The equation of the curve given that it, passes through (–2, 1) is, (a) x + 4 = (y + 3)2, (b) (x + 4)2 = (y – 3), 2, (c) x – 4 = (y – 3), (d) (x + 4)2 = |y + 3|, 66., , 67., 68., , 69., , 70., , The equation of curve through the point (1, 0), if the slope, y -1, , is, of the tangent to the curve at any point (x, y) is, x2 + x, (a) (y + 1) (x – 1) + 2x = 0, (b) (y + 1) (x – 1) – 2x = 0, (c) (y – 1) (x – 1) + 2x = 0, (d) (y – 1) (x + 1) + 2x = 0, If dx +dy = (x + y) (dx – dy), then log (x + y) is equal to, (a) x + y + C, (b) x + 2y + C, (c) x – y + C, (d) 2x + y + C, If the slope of the tangent to the curve at any point P(x, y), y, y, is, – cos2, , then the equation of a curve passing, x, x, æ pö, through çè1, ÷ø is, 4, y, æ ö, æ yö, (a) tan ç ÷ + log x = 1, (b) tan çè ÷ø + log y = 1, x, èxø, x, æ ö, æ xö, (c) tan çè y ÷ø + log x = 1, (d) tan çè y ÷ø + log y = 1, y f (y / x), The solution of the differential equation y¢= +, x f¢ (y / x), is, (a) x f (y/x) = k, (b) f (y/x) = kx, (c) y f (y/x) = k, (d) f (y/x) = ky, The differential equation, (1 + y 2 ) x dx - (1 + x 2 ) y dy = 0, represents a family of :, (a) ellipses of constant eccentricity, (b) ellipses of variable eccentricity, (c) hyperbolas of constant eccentricity, (d) hyperbolas of variable eccentricity, , Topic 3 : Linear Differential Equation of, , 73. In order to solve the differential equation, dy, + y(x sin x + cos x) = 1, dx, the integrating factor is:, (a) x cos x, (b) x sec x, (c) x sin x, (d) x cosec x, x cos x, , 74. Assertion: x sin x, , 2, 1, æ pö, = sin x, y çè ÷ø = 1 - Þ lim y ( x ) =, p, 2, 3, x ®0, Reason: The differential equation is linear with integrating, factor x(1 – cos x)., (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is, not a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., 75. Consider the differential equation, æ, 1ö, y 2 dx + ç x - ÷ dy = 0. If y (1) = 1, then x is given by :, è, yø, , (a), , (c), , æ pö, The equation of a curve passing through çè1, ÷ø and having, 4, , 72., , sin 2 y, at (x, y), is, x + tan y, (a) x = tan y, (b) y = tan x, (c) x = 2 tan y, (d) y = 2 tan x, The integrating factor of the differential equation, dy, x, – y = 2x2 is, dx, 1, (a) e–x, (b) e–y, (c), (d) x, x, , slope, , 1, , 1, , 2 ey, 4- y e, , 1 ey, 3- +, y e, , 1+, , 1, y, , 1 e, y e, , (b), , (d) 1 -, , òe, , Adx, , 1, y, , 1 e, +, y e, , 76. If the I.F. of the differential equation, , dy, + 5y = cos x is, dx, , , then A =, , (a) 0, (b) 1, (c) 3, (d) 5, 77. The solution of the differential equation, , dy y, y, 2, + log y = 2 ( log y ) is, dx x, x, , 1ö, æ, log y = x ç cx 2 + ÷, è, 2ø, , (a), , y = log ( x 2 + cx ), , (b), , (c), , 1ö, æ, x = log y ç cx 2 + ÷, è, 2ø, , (d) None of these., , First order and its Solutions., 71., , dy, + (x + x cos x + sin x) y, dx, , 78. Solution of the differential equation, , dx x log x, ey, =, , if y(1) = 0, is, dy 1 + log x 1 + log x, y, , (a), , x x = e ye, , (c), , x x = ye y, , (b), , ey = xe, , y, , (d) None of these

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DIFFERENTIAL EQUATIONS, , 329, , 79. If y(t) is a solution of (1 + t ), the value of y (1) is, , dy, - ty = 1 and y(0) = –1, then, dt, , 1, 1, (b) (c) 2, (d) 1, 2, 2, 80. The general solution of the differential equation, (tan–1 y – x) dy = (1 + y2) dx is, , (a), , (a) x = (tan–1 y + 1) + Ce - tan, , -1, , (b) x = (tan–1 y – 1) + Ce - tan, , -1, , (c) x = (tan–1 x – 1) + Ce, , y, , (a), , 3, x, + ex = c, y, , (b), , 3, x, - ex = c, y, , (c), , 3, y, + ex = c, x, , (d), , 3, y, - ex = c, x, , 86. The expression which is the general solution of the differential, dy, x, y = x y is, +, equation, dx 1 - x 2, , y, , 1, , - tan -1 x, , 1, y + (1 - x2 ) = c (1 - x2 ) 4, 3, , (a), , -1, , (d) x = (tan –1 x + 1) + Ce- tan x, 81. The expression satisfying the differential equation, , (x, , 2, , ) dy, + 2 xy = 1 is, dx, , -1, , (b), , ( x 2 - 1) y = x + c, , ( y 2 - 1) x = y + c, , (d) None of these, , BEYOND NCERT, Topic 4 : Equation Reducible to Linear differential Equation of, First order, Differential Equation of the, form, , d2y, dx 2, , = f ( x ) , Solution by Inspection Method., , 82. Solution of differential equation (x2 – 2x + 2y2) dx + 2xy dy = 0 is, (a), , 1, c, y = 2x - x 2 +, (b), 4, x2, 2, , 2, c, y = x - x2 +, 3, x2, 2, , 2, x2, c, x+, (d) None of these, 3, 4 x2, 83. The solution of the differential equation, , (c), , y2 =, , dy, dx = x 2 + 2y2 +, dy, y-x, dx, x+y, , y4, x, , 2, , y, 1, =c, x x 2 + y2, , (a), , (b), , (c), , x, 1, =c, 2, y x + y2, , (d) None of these, , 84. The equation of the curve satisfying the differential, equation y2 (x2 + 1) = 2 xy, passing through the point (0, 1), and having slope of tangent at x = 0 as 3, is, (a) y = x3 + 3x + 1, (b) y = x2 + 3x + 1, (c) y = x3 + 3x, (d) y = x3 + 1, 85. The solution of the differential equation, ydx - x dy + 3x y e, , dx = 0 is, , = c (1 - x 2 ), , 1, 2 4, y (1 - x ), , 1, = (1 - x 2 ) + c, 3, , (d) None of these, dy, + 4 x 2 tan y = e x sec y, dx, satisfying y (1) = 0 is :, , 87. The solution of x 3, , (a), , x, -4, tan y = (x - 2) e x log x (b) sin y = e ( x - 1) x, , (c), , tan y = ( x - 1)e x x -3, , x, -3, (d) sin y = e (x - 1) x, , dy 1, + sin 2 y = x 3 cos 2 y, dx x, represents a family of curves given by the equation, , 88. The diffrerential equation, , 6x 2 tan y = x 6 + C, , (a), , x 6 + 6x 2 = C tan y, , (c), , sin 2 y = x 3 cos 2 y + C (d) none of these, , (b), , 89. Solution of the differential equation, dx x log x, ey, =, , if y(1) = 0, is, dy 1 + log x 1 + log x, , is, , y, 1, +, =c, 2, 4 x + y2, , 2 2 x3, , y (1 -, , (c), , (a) x2y – xy2 = c, (c), , (b), , 1, 2 4, x ), , y, , (a), , x x = e ye, , (c), , x x = ye y, , (b), , ey = xe, , y, , (d) None of these, , 90. The particular solution of the differential equation, æ d2y ö, dy, = 0 when x = 0, is, - 1÷ = x , where y =, sin -1ç, ç dx 2, ÷, dx, è, ø, (a), , x2, + x - sin x, 2, , y = x 2 + x - sin x, , (b), , y=, , x2 x, + - sin x, 2 2, , (d), , 2y = x 2 + x - sin x, , (c) y =

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DIFFERENTIAL EQUATIONS, , 331, , dy, + x = C represents, dx, (a) family of hyperbolas (b) family of parabolas, (c) family of ellipses, (d) family of circles, The general solution of ex cos ydx – ex sin ydy = 0 is, (a) ex cos y = k, (b) ex sin y = k, x, (c) e = k cos y, (d) ex = k sin y, , 16. The differential equation y, , 17., , 3, , d 2 y æ dy ö, 5, 18. The degree of differential equation 2 + ç ÷ + 6 y = 0 is, è dx ø, dx, (a) 1, (b) 2, (c) 3, (d) 5, , 26, , The differential equation of the family of curves, x2 + y2 – 2ay = 0, where a is arbitrary constant is, (a), , ( x2 - y 2 ) dydx = 2xy, , (c), , 2 x2 - y 2, , (, , ) dydx = xy, , 21., , 22., , ) dydx = xy, , 2 x2 + y 2, , (d), , ( x2 + y 2 ) dydx = 2 xy, , 27. The family Y = Ax + A3 of curves will correspond to a, differential equation of order, (a) 3, (b) 2, (c) 1, (d) not defined, 2, dy, = 2 x e x - y is, dx, , dy, + y = e - x , y ( 0 ) = 0 is, dx, (a) y = ex(x – 1), (b) y = xe–x, (c) y = xe–x + 1, (d) y = (x + 1)e–x, The integrating factor of differential equation, , 28. The general solution of, , dy, + y tan x - sec x = 0 is, dx, (a) cosx, (b) sec x, (c) ecos x, (d) esec x, , 29., , dy 1 + y 2, =, The solution of differential equation, is, dx 1 + x 2, (a) y = tan–1 x, (b) y – x = k(1 + xy), (c) x = tan–1 y, (d) tan(xy) = k, The integrating factor of differential equation, , dy, 30. The general solution of differential equation, =e, dx, is, , 19. The solution of, , 20., , (, , (b), , dy, 1+ y, +y =, is, dx, x, , (a), , (a), , (a), (c), , dy, + my = 0, dx, d2y, 2, , 2, , -m y =0, , 31., , (b), (d), , dy, - my = 0, dx, d2y, 2, , dx, dx, 24. The solution of differential equation, cos x sin y dx + sin x cos y dy = 0 is, , (d) cos x cos y = C, , (a), , dy, + y = e x is, dx, , ex k, y=, +, x x, , (c) y = xex + k, , 2, , /2, , (d), , ey k, +, y y, , 2, , + xy, , /2, 2, , 2, , 2x -1, =k, 2y + 3, , (b), , 2y +1, =k, 2x - 3, , (c), , 2x + 3, =k, 2 y -1, , (d), , 2x -1, =k, 2 y -1, , d2y, dx 2, d2y, dx 2, , + y =0, , (b), , + (a + b) y = 0, , (d), , d2y, dx 2, d2y, dx 2, , -y=0, + (a - b) y = 0, , dy, + y = e - x , y ( 0 ) = 0 is, dx, (a) y = e–x (x – 1), (b) y = xex, –x, (c) y = xe + 1, (d) y = xe–x, The order and degree of differential equation, 2, , x=, , y = Ce x, , 33. The solution of, , 34., , (b) y = xex + Cx, , (b), , (a), , (c), , (c) sin x + sin y = C, 25. The solution x, , y = Ce - x, , x2, 2, , x /2, (d) y = ( C - x ) e, y = ( x + C ) ex / 2, The solution of equation (2y – 1)dx – (2x + 3) dy = 0, , (a), , +m y =0, , (b) sin x sin y = C, , 2, , e- y + e x = C, , 32. The differential equation for which y = a cos x + b sin x is a, solution, is, , 2, , sin x, =C, sin y, , (a), , (b), , =C, , (c), , (b), , x, , -y, , 2, (d) e x 2 + y = C, e y = ex + y + C, The curve for which the slope of the tangent at any point is, equal to the ratio of the abcissa to the ordinate of the point is, (a) an ellipse, (b) parabola, (c) circle, (d) rectangular hyperbola, , x, , x, , 2, , (c), , e, x, e, (c) xex, (d) ex, 23. y = aemx + be–mx satisfies which of the following differential, equation?, , (a), , ex, , 4, æ d3y ö, d2y, æ dy ö, 4, ç 3 ÷ - 3 2 + 2ç ÷ = y, ç dx ÷, dx, è, ø, dx, è, ø, (a) 1, 4, (b) 3, 4, (c) 2, 4, (d) 3, 2

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DIFFERENTIAL EQUATIONS, , 333, , 50. An integrating factor of the differential equation, dy, sin x, + 2 y cos x = 1 is, [BITSAT 2018, A], dx, 2, (a) sin 2 x, (b), sin x, 1, (c) log |sin x|, (d), sin 2 x, , 51. The expression satisfying the differential equation, dy, x2 -1, + 2 xy = 1 is, [BITSAT 2018, A], dx, , (, , ), , (a), , x 2 y - xy 2 = c, , (b), , ( y 2 - 1) x = y + c, , (c), , ( x 2 - 1) y = x + c, , (d) none of these, , Exercise 3 : Try If You Can, 1., , General solution of differential equation of, dy, f ( x), = f 2 ( x) + f ( x) y + f '( x) y is : (c being arbitrary, dx, constant)., (b) y = - f ( x) + ce x, (a) y = f ( x) + ce x, , 2., , 7., , dy, xy, x4 + 2x, + 2, =, in (– 1, 1) satisfying f (0) = 0. Then, dx x - 1, 1 - x2, , (c) y = - f ( x) + ce x f ( x) (d) y = cf ( x) + e x, The function y = f (x) is the solution of the differential equation, , 3, 2, , ò f ( x) d ( x), , dy, xy, 7 x6 + 2 x, in (– 2, 2) satisfying f (0) = 1. If, + 2, =, dx x – 4, 4 – x2, , 3, 2, , 3, , ò, , p, (d) None of these, 24, The function y = f (x) is the solution of the differential equation, , (c) b – a £, , (a), , f ( x )dx = ap + b 3 , a, b are rational numbers, then, , – 3, , value of 3a + 2b is, (a) 4, (b) 5, 3., , 4., , 5., , (c) 6, , (d), , y, – y = 0,, x, y(1) = 0 is f (x), then [9 f (9)] equals (where [.] greatest integer, function), (a) 70, (b) 0, (c) 75, (d) 61, , If solution of the differential equation y' +, , If the solution of the differential equation, y (1 + 2xy sec2 (x2 – y)) dx – (x + y2 sec2 (x2 – y)) dy = 0 is, f(x, y) = c (where c is integration constant), then f(2, 1) is, equal to, (a) 2 – tan 3, (b) 2 + tan 3, (c) –2 + tan 1, (d) None of these, Let y = f (x) be a curve passing through (e, ee), which satisfy, the differential equation (2ny + xy logex) dx – x logex dy = 0,, x > 0, y > 0. If g ( x) = lim f ( x), then, n®¥, , 6., , lim f ( x) = 31/ 4 . Also f ¢ (x) ³ f 3 (x) +, -, , (a), , b–a³, , p, 4, , 1, then, f ( x), , (b) b – a £, , p, 4, , (b), , p, 3, 3 4, , p, 3, p, 3, (d), 6 4, 6 2, Let f(x) be continuously differentiable on the interval (0, ¥ ), t 2 f ( x ) - x 2 f (t ), = 1 for each x > 0,, t®x, t-x, , such that f(1) = 1, and lim, then f(x) is, 1 2 x2, +, 3x, 3, , (a), , 9., , 10., , (b) -, , 1 4 x2, +, 3x, 3, , 1 2, 1, (d), (c) - + 2, x x, x, Let f be a non-negative function defined on the interval [0, 1]., , If, , 1/ e, , x®a +, , x ®b, , 8., , g ( x ) dx equals to, , (a) e, (b) 1, (c) 0, (d) None of these, Lef f (x) be a positive, continuous and differentiable function, lim f ( x ) = 1 and, on the interval (a,b). If, , p, 3, 3 2, , (c), , e, , ò, , is, , x, , ò0, , 1 - ( f '(t )) 2 dt =, , x, , ò0, , f (t ) dt , 0 £ x £ 1 and f(0) = 0, then, , (a), , f (1/ 2) < 1/ 2 and f (1/ 3) > 1/ 3, , (b), , f (1/ 2) > 1/ 2 and f (1/ 3) > 1/ 3, , (c), , f (1/ 2) < 1/ 2 and f (1/ 3) < 1/ 3, , (d) f (1/ 2) > 1/ 2 and f (1/ 3) < 1/ 3, Through any point (x, y) of a curve which passes through, the origin, lines are drawn parallel to the co-ordinate axes., The curve, given that it divides the rectangle formed by, the two lines and the axes into two areas, one of which is, twice the other, represents a family of, (a) circles, (b) pair of straight lines, (c) parabolas, (d) rectangular hyperbolas

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EBD_7762, 334334, , MATHEMATICS, , 11. The equation of the curve passing through the points (3a, a), (a > 0) in the form x = f(y) which satisfy the differential, equation;, , a 2 dx x y, × = + - 2, is, xy dy y x, , æ 1 + e y -k ö, (a) x = y + a çç, y-k ÷, ÷, è 1 - 2e, ø, y, k, æ1+ e, ö, (c) y = x + a çç, y-k ÷, ÷, è 1- e, ø, , ö, ÷÷, ø, , (d) None of these, , æ d, ö, t ç ( g ( x )) ÷ - t 2, ø, is, 12. Solution of differential equation dt = è dx, dx, g ( x), g ( x), g ( x), +c, (a) t =, (b) t = 2 + c, x, x, g ( x), (c) t =, (d) t = g ( x) + x + c, x+c, 13. Let f (x) be differentiable function on the interval (0, ¥ ), æ t 3 f ( x) - x3 f (t ) ö 1, lim, ÷÷ = " x > 0,, such that f (1) = 1 and t ® x çç, t 2 - x2, è, ø 2, then f (x) is :, , (c), (b), (b), (d), (c), (c), (b), (b), (a), , 10, 11, 12, 13, 14, 15, 16, 17, 18, , (b), (a), (a), (c), (a), (c), (a), (a), (d), , 19, 20, 21, 22, 23, 24, 25, 26, 27, , (c), (c), (a), (d), (b), (c), (c), (a), (c), , 1, 2, 3, 4, 5, 6, , (a), (d), (a), (c), (b), (d), , 7, 8, 9, 10, 11, 12, , (c), (d), (a), (c), (b), (b), , 13, 14, 15, 16, 17, 18, , (b), (c), (c), (d), (a), (a), , 1, 2, , (c), (c), , 3, 4, , (c), (b), , 5, 6, , (c), (c), , 1 3 x2, +, 4x, 4, , (b), , 1 3 x3, +, 4x, 4, 14. The solution of, , (c), , æ 1 + e y -k, (b) x = y + a çç, y-k, è 1- e, , 1, 2, 3, 4, 5, 6, 7, 8, 9, , (a), , e, , x, , ( y 2 -1), y, , (d), , 3 x3, +, 4x 4, 1, 4 x3, , +, , 3x, 4, , {xy 2 dy + y 3dx} + { y dx - x dy} = 0, is, , (a) e xy + e x / y + C = 0, , (b) e xy - e x / y + C = 0, , (c) e xy + e y / x + C = 0, , (d) e xy - e y / x + C = 0, , 15. The solution of differential equation, æ -2 x, ö, e, y dx, ç, ÷, = 1; x ¹ 0 is, è, x, x ø dy, , (a), , ye 2 x = 2 x + C, , (b), , ye x = x + C, , (c), , ye 2 x = y + C, , (d), , ye 2 x = 2 x + C, , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (a) 55, (b), 28 (b) 37 (a) 46, (d) 56, (a), 29 (b) 38 (b) 47, (c) 57, (a), 30 (a) 39 (c) 48, (c) 58, (b), 31 (a) 40 (d) 49, (a) 59, (b), 32 (b) 41 (a) 50, (c) 60, (b), 33 (b) 42 (d) 51, (a) 61, (c), 34 (d) 43 (b) 52, (c) 62, (a), 35 (d) 44 (d) 53, (d) 63, (d), 36 (c) 45 (a) 54, Exercise 2 : Exemplar & Past Year MCQs, (c) 37, (a), 19 (b) 25 (a) 31, (a) 38, (a), 20 (b) 26 (a) 32, (d) 39, (a), 21 (b) 27 (c) 33, (d) 40, (c), 22 (b) 28 (c) 34, (c) 41, (c), 23 (c) 29 (d) 35, (d) 42, (a), 24 (b) 30 (c) 36, Exercise 3 : Try If You Can, (b) 9, (c) 11, (b) 13 (a ), 7, (a) 10 (c) 12, (c) 14, (a), 8, , 64, 65, 66, 67, 68, 69, 70, 71, 72, , (c), (d), (d), (c), (a), (b), (d), (a), (c), , 73, 74, 75, 76, 77, 78, 79, 80, 81, , (b), (a), (c), (d), (c), (a), (b), (b), (c), , 43, 44, 45, 46, 47, 48, , (c), (a), (a), (a), (a), (b), , 49, 50, 51, , (b), (a), (c), , 15, , (a), , 82, 83, 84, 85, 86, 87, 88, 89, 90, , (c), (b), (a), (a), (a), (b), (b), (a), (b)

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26, , VECTOR ALGEBRA, , Chapter, , Trend, Analysis, , off JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 4, , 3, JEE MAIN, BITSAT, , 2, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 3, 5, , Critical Concepts, , Properties of Vector Addition, Section, Formula, Scalar Product of two vectors,, Projection of a Vector on a line, Vector, product of two vectors, Scalar Triple, product and Vector Triple product, , Rating of, Difficulty Level, , 3.5/5, , CUS, (chapter utility score), Out of 10, 7

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VECTOR ALGEBRA, , 337

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EBD_7762, 338, , MATHEMATICS, , Topic 1 : Vectors, Types of Vectors, Addition and Subtraction, of Vectors, Multiplication of Vectorrs by Scalar, Position, Vector, Modulus of a Vector, Section Formula, uur uur, uur uur, r, r, 1., If | a + b | = | a - b | then the vectors a and b are, adjacent sides of, (a) a rectangle, (b) a square, (c) a rhombus, (d) None of these, 2., , ABC is a triangle and P is any point on BC such that PQ is, , 3., , the resultant of the vectors AP, PB and PC , then, (a) the position of Q depends on position of P, (b) Q is a fixed point, (c) Q lies on AB or AC, (d) None of these, ABCDEF is a regular hexagon where centre O is the origin., If the position vectors of A and B are î - ĵ + 2k̂ and, 2î + ˆj - k̂ respectively, then BC is equal to, , 4., , (a), , ˆi + ˆj - 2kˆ, , (b) - î + ˆj - 2k̂, , (c), , 3î + 3ˆj - 4k̂, , (d) None of these, , If a = î + 2 ĵ + 3k̂ , b = 2î + 3 ĵ + k̂ , c = 3î + ĵ + 2k̂, and a a + b b + g c = - 3( î - k̂ ) , then the ordered triplet, , 5., , 6., , (a, b, g ) is, (a) (2, –1, –1), (b) (–2, 1, 1), (c) (–2, –1, 1), (d) (2, 1, –1), Consider points A, B, C and D with position, vect or s 7iˆ - 4 ˆj + 7kˆ, iˆ - 6 ˆj + 10kˆ , - iˆ - 3 ˆj + 4kˆ an d, 5iˆ - ˆj + 5kˆ respectively. Then ABCD is a, (a) parallelogram but not a rhombus, (b) square, (c) rhombus, (d) None of these, If A,B and C are the vertices of a triangle whose position, r r, r, vectors are a, b and c respectively and G is the centroid of, uuur uuur uuur, the DABC, then GA + GB + GC is, r r r, r, (a) 0, (b) a + b + c, r r r, r r r, a+b+c, a-b-c, (c), (d), 3, 3, , 7., , If the position vectors of the vertices A, B, C of a triangle, ABC are 7 ˆj + 10kˆ , -iˆ + 6 ˆj + 6kˆ an d -4iˆ + 9 ˆj + 6kˆ, respectively, the triangle is :, (a) equilateral, (b) isosceles, (c) scalene, (d) right angled and isosceles also, 8., In triangle ABC, which of the following is not true?, uuur uuur uuur r, uuur uuur uuur r, (a) AB + BC + CA = 0 (b) AB + BC - AC = 0, uuur uuur uuur r, uuur uuur uuur r, (c) AB + BC - CA = 0 (d) AB – CB + CA = 0, r, 9., If a is a non-zero vector of magnitude a and l a non-zero, r, scalar, then l a is a unit vector if., 1, (a) l = 1, (b) l = – 1, (c) a = |l| (d) a = | l |, 10. A zero vector has, (a) any direction, (b) no direction, (c) many directions, (d) None of these, 11. If two vertices of a triangle are i – j and j + k, then the third, vertex can be, (a) i + k, (b) i –2j – k and –2i – j, (c) i – k, (d) All the above, 12. Which among the following is correct statement?, (a) A quantity that has only magnitude is called a vector, (b) A directed line segment is a vector, denoted as |AB| or |a|, (c) The distance between initial and terminal points of a, vector is called the magnitude of the vector, (d) None of the above, 13. If l(3iˆ + 2 ˆj - 6kˆ) is a unit vector, then the values of l are, 1, 1, (b) ± 7, (c) ± 43 (d) ±, ±, 43, 7, The figure formed by the four points, , (a), , 14., , iˆ + ˆj - kˆ , 2iˆ + 3 ˆj , 5 ˆj - 2kˆ and kˆ - ˆj is, (a) trapezium, (b) rectangle, (c) parallelogram, (d) None of these, 15. If two vectors a and b are such that a = b, then, (a) they have same magnitude and direction regardless of, the positions of their initial points, (b) they have same magnitude and different directions, (c) Both (a) and (b) are true, (d) Both (a) and (b) are false, 16. A vector whose magnitude is the same as that of a given, vector, but direction is opposite to that of it, is called, (a) negative of the given vector, (b) equal vector, (c) null vector, (d) collinear vector

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VECTOR ALGEBRA, , 339, , 17. Which of the following represents graphically the, displacement of 40 km, 30° East of North?, , N, , (a), , N, , P, , 40° km, 40, E, , W, , km, 4030°, , (b) W, , S, , P, E, , S, , P, (c), , W, , E, , then the vector sum | BC | IA + | CA | IB+ | AB | IC is, (a) zero, , (b), , IA + IB + IC, 3, , (c) 3 ( IA + IB + IC), (d) None of these, 23. If two vectors a and b represented by two adjacent sides of, a parallelogram in magnitude and direction then a + b is, represented as, , N, 30° km, 40, , r r, r rr r, r r, 21. If p, q and rr are perpendicular to q + r, r + p and p + q, r r, r r, respectively and if | p + q |= 6, | q + r |= 4 3 and, r r, r r r, | r + p | = 4 then | p + q + r | is, (a) 5 2, (b) 10, (c) 15, (d) 5, 22. A point I is the centre of a circle inscribed in a triangle ABC,, , (d) None of these, , a, 2, 1, , S, , C, , A, , B, (a) AB – BC = AC, (b) AC = AB – BC, (c) AC = AB + BC, (d) None of these, 19. Consider the figure given below, , a, , (a) diagonal 1 (as shown), (b) diagonal 2 (as shown), (c) sides opposite to either of the side, (d) None of the above, 24. Which of the following is an example of two different vectors, with same magnitude?, , (b), , b, A, , a, , (a), , C, , B, , C¢, , Here, it is shown that a vector BC¢ is having same magnitude, as the vector BC, but its direction is opposite to that of it., Based on above information which of the following is true?, (a) AC¢ = a + b, (b) AC¢ = a – b, (c) Difference of a and b is AC, (d) None of these, uuur uuur, 20. In a triangle ABC three forces of magnitudes 3AB , 2AC, uuur, and 6CB are acting along the sides AB, AC and CB, respectively. If the resultant meets AC at D, then the ratio, DC : AD will be equal to :, (a) 1 : 1, (b) 1 : 2, (c) 1 : 3, (d) 1 : 4, , b, , b, , 18. If a girl moves from A to B and then from B to C (as shown)., Then, net displacement made by the girl from A to C, is, , (c), , ( 2iˆ + 3jˆ + kˆ ) and ( 2iˆ + 3jˆ - kˆ ), ( 3iˆ + 5ˆj + kˆ ) and ( 3iˆ + 4ˆj + kˆ ), ( ˆj + kˆ ) and ( 2ˆj + 3kˆ ), , (d) None of the above, 25. ABCD be a parallelogram an d M be the point of, intersection of the diagonals, if O is any point, then, OA + OB + OC + OD is equal to, (a) 3 OM (b) 2 OM, (c) 4 OM (d) OM, 26. If a is a vector of magnitude 50, collinear with the vector, 15, b = 6iˆ - 8jˆ - kˆ and makes an acute angle with the positive, 2, direction of Z-axis, then a is equal to, (a) -24iˆ + 32ˆj + 30kˆ, (b) 24iˆ - 32jˆ - 30kˆ, , -12iˆ + 16jˆ - 15kˆ, , (c), ®, , (d) 12iˆ - 16jˆ - 15kˆ, ®, , ®, 27. If a = 2iˆ – 2 ˆj + kˆ and c = –iˆ + 2kˆ then | c | . a is equal, to :, (a) 2 5 iˆ + 2 5 ˆj + 5 kˆ (b) 2 5 iˆ – 2 5 ˆj + 5 kˆ, , (c), , 5 iˆ + 5 ˆj + 5 kˆ, , (d), , 5 iˆ + 2 5 ˆj + 5 kˆ

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EBD_7762, 340, , 28., , 29., , 30., , MATHEMATICS, , If the middle points of sides BC, CA & AB of triangle ABC, are respectively D, E, F then position vector of centre of, triangle DEF, when position vector of A, B, C are respectively, iˆ + ˆj, ˆj + kˆ, kˆ + iˆ is, 1 ˆ ˆ ˆ, i + j +k, (a), (b) iˆ + ˆj + kˆ, 3, 2 ˆ ˆ ˆ, i + j +k, (c) 2 iˆ + ˆj + kˆ, (d), 3, If ABCDE is a pentagon, then resultant of AB, AE, BC, DC,, ED and AC is, (a) 2AC, (b) 3AC, (c) AB, (d) None of these, , (, (, , 32., , 33., , 34., , 35., , 36., , (, , ), , (, , ), , Magnitude of the vector joining the points P(x1, y1, z1) and, Q(x2, y2, z2) is, (a) (x2 – x1) + (y2 – y1) + (z2 – z1), (b) (x2 – y2 + z2) + (x1 + y1 + z1), (c), , 31., , ), ), , ( x 2 - x1 ), , 2, , + ( y 2 - y1 ), , AB + AC + AD + AE + AF = n AD . Then n is, (b) 2, , 1, æ, ö, and ç - ˆi - ˆj + 4kˆ ÷ is, 2, è, ø, 1, (a), sq. units, (b) 1 sq. units, 2, (c) 2 sq units, (d) 4 sq. units., 38. ABCDEF is a regular hexagon with centre at origin such, that AD + EB + FC = lED, then l is equal to, (a) 2, (b) 4, (c) 6, (d) 3, 39. If G is the centroid of triangle ABC, then the value of, uuur uuur uuur, GA + GB + GC is, , (a), , 2, , (d) None of the above, If C is the middle point of AB and P is any point outside AB,, then, uuur uuur uuur, uuur uuur, uuur, (a) PA + PB = PC, (b) PA + PB = 2 PC, uuur uuur uuur r, uuur uuur, uuur r, (c) PA + PB + PC = 0, (d) PA + PB + 2 PC = 0, ABCD is a parallelogram whose diagonals meet at P. If O is, uuur uuur uuur uuur, a fixed point, then OA + OB + OC + OD equals, uuur, uuur, uuur, uuur, (a) OP, (b) 2 OP, (c) 3 OP, (d) 4 OP, Statement I : The position vector of point R which divides, the line joining two points P(2a + b) and Q(a – 3b) externally, in the ratio 1 : 2, is 3a + 5b., Statement II : P is the mid-point of the line segment RQ., (a) Only statement I is true, (b) Only statement II is true, (c) Both statements are true, (d) Both statements are false, uuur, uuur, If the vectors AB = -3iˆ + 4kˆ and AC = 5iˆ - 2 ˆj + 4kˆ are the, sides of a triangle ABC, then the length of the median, through A is, (a), (b) 18, (c), (d) 4, 14, 29, r r, Let the position vectors of the points A, B and C be a, b, and cr respectively. Let Q be the point of intersection of, uuur uuur uuur, the medians of the triangle D ABC. Then QA + QB + QC =, r r r, r, a +b +c, (a), (b) 2ar + b + cr, 2, r, r, (c) ar + b + cr, (d) 0, If ABCDEF is a regular hexagon and, , (a) 1, , 37. Area of rectangle having vertices A, B, C and D with, position vector, æ ˆ 1 ˆ, ˆ ö æˆ 1 ˆ, ˆö, ˆö æˆ 1 ˆ, ç - i + 2 j + 4k ÷ , ç i + 2 j + 4k ÷ , ç i - 2 j + 4k ÷, è, ø è, ø è, ø, , (c) 3, , (d), , 5, 2, , (c), , 1 uuur uuur, GB + GC, 2, 1 uuur uuur, GB - GC, 2, , (, , ), , (b) 0, , (, , ), , (d) None of these, , 40. The vectors AB = 3iˆ + 4 kˆ & AC = 5iˆ - 2 ˆj + 4kˆ, are the sides of a triangle ABC. The length of the median, through A is, (a), , 41., , (d) 33, 72, r, r, r, If position vector of a point A is a + 2b and any point P a, uuur, divides AB in the ratio of 2 : 3, then position vector of B is, r, r, r, r, (a) 2ar - b (b) b - 2ar, (c) ar - 3b (d) b, , 288, , (b), , (c), , 18, , (), , Topic 2: Scalar or Dot Product of Two Vectors, Projection of a, vector along any other vector collinearity of three points., r r r, , 42. Let a , b , c, , be three vectors of magnitudes 3, 4 and 5, , respectively. If each one is perpendicular to the sum of the, r, , r, , r, , other two vectors, then | a + b + c |=, (a) 5, 43. If, , (b) 3 2, , (c) 5 2, , r, r, r, | a | = 5, | b | = 4, | c | =3 ,, , then, , (d) 12, th e, , value, , of, , ® ® ®, , a.b + b.c + c.a , is equal to ( given that a + b + c = 0), , (a) 25, (b) 50, (c) –25, (d) –50, r r r, r r r, 44. a , b , c are 3 vectors, such that a + b + c = 0 ,, r, rr rr rr, r, r, a = 1 , b = 2, c = 3, , then a.b + b .c + c .a is equal to, (a) 1, (b) 0, (c) – 7, (d) 7, 45. If vector a = 2i – 3j + 6k and vector b = – 2i + 2j – k, then, Projection of vector a on vector b, =, Projection of vector b on vector a, 3, 7, (a), (b), (c) 3, (d) 7, 7, 3

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VECTOR ALGEBRA, , 46., , 341, , a , b , c are three vectors of which every pair is, , noncollinear. If the vector a + b and b + c are collinear, with c and a respectively then a + b + c is, (a) a unit vector, (b) the null vector, , 47., 48., , 49., , 50., , 51., , (c) equally inclined to a , b , c, (d) None of these, Three points (2, –1, 3), (3, –5, 1) and (– 1, 11, 9) are, (a) Non-collinear, (b) Non-coplanar, (c) Collinear, (d) None of these, If three points A, B and C have position vectors (1, x, 3),, (3, 4, 7) and (y, – 2, – 5) respectively and, if they are collinear,, then (x, – y) is equal to, (a) (2, – 3) (b) (– 2, 3) (c) (2, 3) (d) (– 2, – 3), r r, r r, The angle between the vectors a + b and a - b , where, r, r, a = (1, 1, 4) and b = (1, –1, 4) is, (a) 90°, (b) 45°, (c) 30°, (d) 15°, r r r, r, r, The two variable vectors 3x i + yj - 3k and x i - 4yj + 4k, are orthogonal to each other, then the locus of (x, y) is, (a) hyperbola, (b) circle, (c) straight line, (d) ellipse, Assertion : The projection of the vector a = 2iˆ + 3jˆ + 2kˆ, , r, 55. If the vectors a = (2, log 3 x, a ) and, r, b = (-3, a log 3 x, log 3 x ) are inclined at an acute angle then, (a) a = 0, (b) a < 0, (c) a > 0, (d) None of these, 56. Which among the following figure correctly represents, projection of AB on a line l ?, B, a, , C, , (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., 52. The non-zero vectors a, b and c are related by a = 8b and c, = – 7b, then the angle between a and c is, (b) 0, , (c), , p, 4, , P, , (d), , p, 2, , 53. Which of the following is true?, ˆ ˆ =0, (a) ˆi.iˆ = ˆj.jˆ = k.k, , ˆ ˆ=0, (b) ˆi.jˆ = ˆj.kˆ = k.i, (c) Both (a) and (b) are true, (d) Both (a) and (b) are not true, r, 54. If a = 4iˆ + 6 ˆj and bˆ = 3 ˆj + 4kˆ, then the component of a, along b is, (a), , 18, (3 ˆj + 4kˆ), 10 3, , (b), , (c), , 18 ˆ, (3 j + 4kˆ), 3, , (d) (3 ˆj + 4kˆ), , 18 ˆ, (3 j + 4kˆ), 25, , l, , q, A, , a, , (b), B, , 180° < q< 270°, , P, , q, A, , 1, Reason The projection of vector a on vector b is a ( a.b ) ., , l, , A, P, 90° < q < 180°, , C, , r, 5, 6., on the vector b = ˆi + 2jˆ + kˆ is, 3, , (a) p, , q, , (a), , C, A, , l, , a, , (c), , B, 270° < q < 360°, (d) All of these, 57. A unit vector in xy- plane makes an angle of 45° with the, vector ˆi + ˆj and an angle of 60° with the vector 3iˆ - 4ˆj is, (a), , 13 ˆ 1 ˆ, i+ j, 7, 7, , (b), , 7 ˆ 7 ˆ, i+ j, 13 15, , (c), , 13 ˆ 1 ˆ, i+ j, 14 14, , (d) None of these, , 58. If the vectors aiˆ ∗ 2jˆ ∗ 3kˆ and ,ˆi ∗ 5jˆ ∗ akˆ are, perpendicular to each other then a is equal to:, (a) 5, (b) – 6, (c) – 5, (d) 6, 59. The vectors (2iˆ - mjˆ + 3mkˆ) & {(1 + m ) iˆ - 2mjˆ + kˆ}, include an acute angle for, (a) all values of m, , (b) m < – 2 or m > – 1/2, 1ù, é, (c) m = – 1/2, (d) m Î ê -2, - ú, 2û, ë, 2, 2, 60. The two vectors ( x - 1)î + ( x + 2)ˆj + x k̂ and 2î - xĵ + 3k̂, are orthogonal, (a) for no real value of x (b) for x = –1, 1, 1, (c) for x =, (d) for x = – and x = 1, 2, 2

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EBD_7762, 342, , MATHEMATICS, , r r r, 61. If a, b, c are mutually perpendicular unit-vector, then, r r r, a + b - c equals :, , 62., , 2, (a) 1, (b), (c) 3, (d) 2, uur, uur, uur, uur, If | a | = 3,| b | = 4, then a value of l for which a + l b, uur uur, is perpendicular to a - l b is :, 9, 4, 3, 3, (b), (c), (d), 16, 3, 4, 2, If p, q, r be three non-zero vectors, then equation, p .q = p . r implies:, (a) q = r, (b) p is orthogonal to both q and r., (c) p is orthogonal to q – r., (d) either q = r or p is perpendicular to q – r., r, r, If a and b are unit vectors inclined at an angle of 30° to, each other, then which one of the following is correct ?, r r, r r, (a) | a + b |> 1, (b) 1 <| a + b |< 2, r r, r r, (c) | a + b |= 2, (d) | a + b |> 2, r r r r, If | a + b |=| a - b |, then which one of the following is, correct?, r, r, (a) a = lb for some scalar l, r, r, (b) a is parallel to b, r, r, (c) a is perpendicular to b, r r r, (d) a = b = 0, r r r, r, If a, b , c are the 3 vectors such that | a | = 3 ,, r, r, r r r, | b | = 4, | c | = 5 , | a + b + c | = 0 then the value of, r r r r r r, a . b + b . c + c . a is :, (a) – 20, (b) – 25, (c) 25, (d) 50, r, r r r, Four vectors a, b , c and x satisfy the relation, r r, r r r r r, (a × x )b = c + x where b × a ¹ 1. The value of xr in terms of, r r, r, a, b and c is equal to, r r r r r r, r, ( a × c )b - c (a × b - 1), c, r, r r, (a), (b) r, ( a × b - 1), a × b -1, r, r r r r, r r, r, 2(a × c )c + c, 2(a × c )b + c, (c), (d), r r, r r, a × b -1, a × b -1, Statement I : The point A(1, –2, – 8), B(5, 0, – 2) and C(11, 3, 7), are collinear., Statement II : The ratio in which B divides AC, is 2 : 3, (a) Only statement I is true, (b) Only statement II is true, (c) Both statements are true, (d) Both statements are false, , (a), , 63., , 64., , 65., , 66., , 67., , 68., , 69. If the scalar product of the vector ˆi + ˆj + kˆ with a unit vector, along the sum of vectors 2iˆ + 4jˆ - 5kˆ and lˆi + 2jˆ + 3kˆ is, equal to one then the value of l is, (a) 0, , (b) – 1, , 70. Angle between the vectors, , (c), , 1, 2, , (d) 1, , uur uur, uur uur uur uur, 3(a ´ b ) and b - (a . b ) a is, , p, p, p, (b) 0, (c), (a), (d), 4, 3, 2, 71. Choose the correct option, r, r, r, r r, r, (a) | a | = | b | = | a - b | = 1 then angle between a and b is, p/2. r, r, a +b, (b), is a vector in the direction of angle bisector of, 2, r, r, vectors a and b ., , (c), , r, r, r, (a.iˆ)2 + (a. ˆj)2 + (a.kˆ)2 = a2., , (d) Vector perpendicular to -iˆ + ˆj + kˆ and coplanar with, , iˆ + ˆj + kˆ and -iˆ + ˆj + kˆ is 3iˆ + 2 ˆj + kˆ ., 72. Let there be two points A, B on the curve y = x 2 in the plane, uuur, uuur, OXY satisfying OA . ˆi = 1 and OB . iˆ = -2 , then the length, of the vector 2OA - 3 OB is, (c) 3 41 (d) 2 41, (b) 2 51, 14, r r r r, 73. If a, b, c, d are the position vectors of points A, B, C and D, r r r r, r r r r, respectively such that (a - d).(b - c) = (b - d).(c - a) = 0 ,, then D is the, (a) centroid of D ABC, (b) circumcentre of D ABC, (c) orthocentre of D ABC (d) None of these, 74. The angle between any two diagonal of a cube is, (a) 45°, (b) 60°, (a), , -1, (d) tan (2 2 ), , (c) 30°, , r, 75. If the projection of b on ar is twice the projection of ar on, r, r, r, b , then | b | - | a | is equal to, r, r, r, r r, r, (a) | a - b | (b) | a | + | b | (c) | b |, (d) | a |, r r, r, r r r r, r r, 76. a , b and c are perpendicular to b + c , c + a and a + b, r r, r r, r r, respectively and if a + b = 6 , b + c = 8 and c + a = 10 ,, r r r, then a + b + c is equal to, (a), , 5 2, , (b) 50, , (c) 10 2, , (d) 10, , 77. The dot product of a vector with the vectors ˆi + ˆj - 3kˆ ,, , ˆi + 3jˆ - 2kˆ and 2iˆ + ˆj - 4kˆ are 0, 5 and 8 respectively. Find, the vector., (a) ˆi + 2jˆ + kˆ, (b) - ˆi + 3jˆ - 2kˆ, (c), , ˆi + 2ˆj + 3kˆ, , (d) ˆi - 3jˆ - 3kˆ

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VECTOR ALGEBRA, , 343, , 78. If a, b and c are unit vectors, then, |a – b|2 + |b – c|2 + |c – a|2 does not exceed, (a) 4, (b) 9, (c) 8, (d) 6, uuur, uuur, uuur, 79. In a parallelogram ABCD, | AB | = a,| AD | = b and | AC | = c ,, , r, r, r, 86. lf a = iˆ + 2 ˆj + 2kˆ,| b | = 5 and the angle between ar and b, p, , then the area of the triangle formed by these two, 6, vectors as two sides is, , is, , uuuur uuur, , the value of DB . AB is, (a), , 3a 2 + b2 - c 2, 2, , (b), , a 2 + 3b 2 - c 2, 2, , (c), , a 2 - b 2 + 3c 2, 2, , (d), , a 2 + 3b 2 + c 2, 2, , 87., , Topic 3: Vector or cross product of two vectors Area of, Triangle, Area of Parallelogram, Vector Inequality., 80. A unit vector perpendicular to the plane ABC, where, A, B and C are respectively the points (3, –1, 2), (1, –1, –3) and, (4, –3, 1), is, (a), , 1, , -, , 29, 1, , (c), , 26, , (2î + 5k̂), , (b), , ( 4î - 3 ĵ + k̂ ), , 1, 6, , (î - 2 ĵ - k̂ ), , 1, , (d) -, , 165, , 3, 7, , (10î + 7 ĵ - 4k̂ ), , 26, 3 26, (c), (d) 26, 7, 7, 82. A unit vector perpendicular to the plane formed by the points, (1, 0, 1), (0, 2, 2) and (3, 3, 0) is, , (a), , (c), 83., , (b), , 1, 5 3, 1, 5 3, , 1, , (5î - ˆj + 7 k̂ ), , (5î - ĵ - 7 k̂), , (b), , (5î + ĵ + 7k̂ ), , (d) None of these, , 5 3, , ®, , ®, , ®, , a = 3iˆ - 5jˆ and b = 6iˆ + 3jˆ are two vectors and c is a, ®, , ® ®, , ®, , ®, , ®, , vector such that c = a ´ b then | a | : | b | : | c |, (a), , (b), 34 : 45 : 39, 34 : 45 : 39, (c) 34 : 39 : 45, (d) 39 : 35 : 34, r, r, r r 2 rr 2, 84. If a ´ b + a.b = 676 and b = 2 , then a is equal to, , (, , 88., , (c) ˆi ´ kˆ = -ˆj, (d) All of these, 89. If q is the angle between any two vectors a and b, then, , a.b = a ´ b , where q is equal to, , 81. The perpendicular distance of A(1, 4, –2) from BC, where, coordinates of B and C are respectively (2, 1, –2) and, (0, –5, 1) is, (a), , 15, 15, 15 3, (b), (c) 15, (d), 4, 2, 2, r r r r, r r r, a, +, b, +, c, =, 0, Let a , b , c be unit vectors such that, . Which, one of the following is correct ?, r r, r r r r, (a) a ´ b = b ´ c = c ´ a = 0, r r, r r r r, (b) a ´ b = b ´ c = c ´ a ¹ 0, r r, r r r r, (c) a ´ b = b ´ c = a ´ c ¹ 0, r r, r r r, (d) a ´ b , b ´ c , c ´ a are muturally perpendicular, Which of the following is true?, (a) ˆj´ ˆi = kˆ, (b) kˆ ´ ˆj = ˆi, , (a), , ) ( ), , (a) 13, (b) 26, (c) 39, (d) None of these, uuur uuur, 85. If AB ´ AC = 2iˆ - 4jˆ + 4kˆ , then the area of DABC is, (a) 3 sq. units, (b) 4 sq. units, (c) 16 sq. units, (d) 9 sq. units, , (a) zero, , p, 4, , (b), , (c), , p, 2, , (d) p, , r, r, 90. Vectors, at an angle q = 120°. If, a and b are inclined, r, r, r r, r r 2, é, | a |= | b |= 2, then ë (a + 3b) ´ (3a + b) ùû is equal to, (a) 190, (b) 275, (c) 300, (d) 192, 91. The area of the parallelogram whose diagonals are, 3ˆ 1 ˆ ˆ, i + j - k and 2 iˆ - 6 ˆj + 8 kˆ is :, 2 2, , (c) 25 3 (d) 25 2, (b) 5 2, 5 3, r, r, 92. If a an d b are the two vectors such that, r r, r r, a × b = 0 and a ´ b = 0, then, r, r, (a) a is parallel to b ., r, r, (b) a is perpendicular to b ., r, r, (c) either a or b is a null vector ., (d) None of these., 93. For any two vectors a and b, (a × b)2 equals, (a) a2b2 – (a.b)2, (b) a2 + b2, 2, 2, (c) a – b, (d) None of these, r, r r 2 rr 2, 94. Assertion : If a ´ b + a.b = 144 and a = 4 ,then, (a), , (, , r, b =9., , ) ( ), , r, r, r r, Reason : If a and b are any two vectors, then a ´ b, , (, , r, equal to a, , r, , rr, , ( ) ( b ) - ( a.b ), 2, , 2, , 2, , ), , 2, , is

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EBD_7762, 344, , 95., , MATHEMATICS, , (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., r, r, r, Let u = iˆ + ˆj, v = iˆ - ˆj and w = iˆ + 2jˆ + 3kˆ ., r, r, If n̂ is a unit vector such that u .nˆ = 0 and v .nˆ = 0 , then, r, w. nˆ is equal to, , (a) 3, , 96., , 97., , 98., , (b) 0, (c) 1, (d) 2, r, r, If angle between a = iˆ – 2 ˆj + 3kˆ and b = 2iˆ + ˆj + kˆ is q, then the value of sin q is, 3, -2, 4, 5, (a), (b), (c), (d), 2 7, 7, 3 7, 2 7, If liˆ + mjˆ + nkˆ is a unit vector which is perpendicular to, vectors 2iˆ – ˆj + kˆ and 3iˆ + 4 ˆj – kˆ then find the value of | l |., 3, -5, -3, 5, (a), (b), (c), (d), 155, 155, 155, 155, For what value of m, are the points with position vector, ˆ 12iˆ - 5jˆ and miˆ + 11jˆ collinear ?, 10iˆ + 3j,, (a) – 8, , 99., , (b) 8, , (c) 4, , (d) – 4, , The acute angle that the vector 2î - 2ˆj + k̂ makes with the, plane contained by the two vectors 2î + 3ˆj - k̂ and, î - ĵ + 2k̂ is given by, , (a), , æ 1 ö, ÷÷, cos -1 çç, è 3ø, , ( ), , -1 æ 1 ö÷, (b) sin çç, ÷, è 5ø, , (d) cot -1 ( 2 ), r, r r r r, 100. If a = iˆ + ˆj, bˆ = 2jˆ – kˆ and r ´ a = b ´ a ,, r, r r r r, r, r ´ b = a ´ b , then what is the value of r ?, |r|, (c), , (a), (c), , tan -1 2, , ˆ, (iˆ + 3jˆ – k), 11, ˆ, (iˆ + 3jˆ + k), , (b), (d), , ˆ, (iˆ – 3jˆ + k), 11, , ˆ, (iˆ – 3jˆ – k), , 11, 11, r, r r r r, r ˆ ˆ ˆ rr, 101. If a = i + j + k , a.b = 1 and a ´ b = j - k , then b is, , (a), , iˆ - ˆj + kˆ, , (b) 2 ˆj - kˆ, , (c) 2iˆ, (d) iˆ, 102. A vector perpendicular to the plane containing the vectors, î - 2ˆj - k̂ and 3î - 2 ĵ - k̂ is inclined to the vector î + ˆj + k̂, at an angle, (a) tan -1 14, (c), , tan -1 15, , (b) sec -1 14, (d) None of these, , 103. The altitude through vertex C of a triangle ABC, with position, r r r, vectors of vertices a , b, c respectively is :, r r r r r r, r r r, b´c + c´a + a´b, a+b+c, r r, r r, (a), (b), b-a, b-a, r r r r r r, b´c + c´a + a´b, r r, (c), (d) None of these, b´ a, 104. Let A = 2 î + k̂ , B = î + ĵ + k̂ and C = 4î - 3ˆj + 7 k̂ . The, vector R which satisfies the equations, R ´ B = C ´ B and R . A = 0 is given by, , (a), , (b) - î - 8 ĵ + 2k̂, , - 2î + k̂, , 1, , (c), , 6, , (î - ĵ + 2k̂ ), , (d) None of these, , BEYOND NCERT, Topic 4 : Linear Dependence and Independence of vectors,, colinearity of Three vectors, Coplanarity of three & four, vectors, Coplanarity of Four points., r, r, r, ˆ, ˆ, 105. If a = i + j + kˆ , b = 4iˆ + 3jˆ + 4kˆ and c = ˆi + aˆj + bkˆ are, r, linearly dependent vectors and c = 3 , then, (a) a = 1, b = – 1, (b) a = 1, b = ± 1, (c) a = – 1, b = ± 1, (d) a = ± 1, b = 1, 106. Let a, b, g be distinct real numbers. The points with position, vectors a$i + b $j + g k$ , b$i + g $j + a k$ and g$i + a $j + bk$, (a), (b), (c), (d), , are collinear, form an equilateral triangle, form a scalene triangle, form a right-angled triangle, , r r r, , 107. a, b, c are three non-zero vectors; no two of them are parallel., r r, r r, r, r, If a + b is collinear to c and b + c is collinear to a,, r, r, r, a + b + c is equal to, r, (a) ar, (b) b, (c) cr, (d) 0, r, r, 108. The vectors a = xiˆ + 2jˆ + 5kˆ and b = iˆ + yjˆ - zkˆ are, collinear, if, (a) x = 1, y = – 2, z = – 5 (b) x = 1/2, y = – 4, z = – 10, (c) x = – 1/2, y= 4, z = 10, (d) All of these, r, r, ˆ, ˆ, ˆ b = 4iˆ - 2ˆj + 3kˆ and, 109. If the vectors a = 2i + j + 4k,, , r, c = 2iˆ - 3jˆ - lkˆ are coplanar, then the value of l is equal to, (a) 2, (b) 1, (c) 3, (d) –1, ur r r r ur, r r, 110. Assertion : If the point P = a + b - c , Q = 2a + b, ur, r r, r r r, and R = b + tc are collinear, where a , b , c are three non-, , (, , (, , ), , coplanar vectors, then the value of t is –2., Reason : If P, Q, R are collinear, then, PQ P PR or PQ = l PR , l Î R, , ), , (, , )

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VECTOR ALGEBRA, , 345, , (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., r r, r, 111. Let a, b and c be three non-zero vectors such that no two, r, r, of these are collinear. If the vector a + 2b is collinear with, r r, r, r, c and b + 3c is collinear with a (l being some non-zero, r r, r, scalar) then a + 2b + 6c equals, r, r, r, (a) 0, (b) lb, (c) lc, (d) la, r r r, 112. If a, b , c are non-coplanar vectors and l is a real number,,, then the vectors a + 2b + 3c , lb + 4c and (2l - 1)c are, non coplanar for, (a) no value of l, (b) all except one value of l, (c) all except two values of l (d) all values of l, 113. If the vector, pi$ + $j + kˆ, i$ + q $j + kˆ and iˆ + ˆj + rkˆ ( p ¹ q ¹ r ¹ 1), are coplanar, then the value of pqr - ( p + q + r ) is, (a) 2, (b) 0, (c) – 1, (d) – 2, r, r, 114. Two vectors a and b are non-zero and non-collinear. What, is the value of x for which the vectors, r, r, r r, r r, p = ( x – 2) a + b and q = (x + 1) a – b are collinear?, (a) 1, , (b), , 1, 2, (c), 2, 3, BEYOND NCERT, , (d) 2, , Topic 5: Scalar Tripple product, Vector Tripple Products, Scalar product of Four vectors, reciprocal system of, vectors, Application of vectors in Mechanics., r, r, r, r, r r, 115. If a is perpendicular to b and c , a = 2 , b = 3 , c = 4, r r r, 2p, r, r, and the angle between b and c is, , then éëa b c ùû is, 3, equal to, (a) 4 3, (b) 6 3, (c) 12 3 (d) 18 3, r, r, 116. Let a = ˆi - kˆ , b = xiˆ - ˆj + (1 - x ) kˆ and, r, c = yiˆ + xjˆ + (1 + x - y ) kˆ ., r r r, Then, éëa b c ùû depends on, (a) neither x nor y, (b) both x and y, (c) only x, (d) only y, 117. Which of the following statement is correct?, (a) [a b c] is a scalar quantity, (b) The magnitude of the scalar triple product is the volume, of a parallelopiped formed by adjacent sides given by, three vectors a, b and c, (c) The volume of a parallelopiped form by three vectors, a, b and c is equal to |a. (b × c)|, (d) All are correct, , 118. If a = 2iˆ + ˆj + 3kˆ , b = -ˆi + 2jˆ + kˆ and c = 3iˆ + ˆj + 2kˆ then, a.(b × c) is equal to, (a) – 15, (b) 15, (c) – 10, (d) – 5, r, r, 119. If a and b are two non-zero non-collinear vectors then, r r r, r r, r r, r r, 2 [a , b , iˆ] iˆ + 2 [a , b , ˆj ] ˆj - 2 [a , b , kˆ] kˆ +[a, b , a ] is equal to, r, r r, (a) 2(a ´ b ), (b) ar ´ b, r, (c) ar + b, (d) None of these, r r, r r r r, 120. If ((a ´ b ) ´ (c ´ d )).( a ´ d ) = 0 , then which of the following, is always true ?, r r, (a) ar, b , cr, d are necessarily coplanar, r, r r, r, (b) either a or d must lie in the plane of b and c, r r, r, r, (c) either b or c must lie in the plane of a and d, r, r r, r, (d) either a or b must lie in the plane of c and d, 121. Which of the following statement is correct?, (a), , a. ( b ´ c ) = [ b c a ], , (b), , [a, [c, , b c ] = [ c a b], , a b ] = c. ( a ´ b ) = ( a ´ b ) .c, (c), (d) All are correct, ® ® ®, , 122. The vector a ´(b´c) is:, ®, , (a) parallel to a ., , ®, , (b) perpendicular to a ., , ®, , ®, , (c) parallel to b ., (d) perpendicular to b ., 123. If a, b, c are the p th , q th . r th terms of an HP and, r r r, r, r, r r i j k, r, u = (q - r ) i + (r - p) j + (p - q)k, v = + + then, a b c, r r, (a) u, v are parallel vectors, r r, (b) u, v are orthogonal vectors, r r =1, (c) u.v, r r r r r, (d) u ´ v = i + j + k, 124. For non zero, non collinear vectors p and q , the value of, [ î p q ]î + [ ĵ p q ]ˆj + [ k̂ p q ]k̂ is, , (b) 2( p ´ q ), , (a), , 0, , (c), , (q ´ p), , (a), , iˆ - ˆj + kˆ, , (b) 2 ˆj - kˆ, , (c), , iˆ, , (d) 2iˆ, , (d) ( p ´ q ), r, r, r, r, r r, 125. If a = (iˆ + ˆj + kˆ), a.b = 1 and a ´ b = ˆj - kˆ, then b is, , ®, , ®®®, , ® ®, , 126. If a , b , c are vectors such that [ a b c ] = 4, then, ® ®® ® ® ®, , [a´ b b´ c c´ a]=, (a) 16, , (b) 64, , (c) 4, , (d) 8

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EBD_7762, 346, , MATHEMATICS, , ur r ur ur r ur, 127. If the vertices of any tetrahedron be A = J + 2K, B = 3I + K,, ur, r r ur, ur, r r, C = 4I + 3J + 6K and D = 2I + 3J + 2K, then its volume is, , 1, (a), units, (b) 6 units, 6, (c) 36 units, (d) None of these, r ˆ ˆ ˆ r ˆ ˆ, r, 128. Let a = i + j + k , b = i - j + 2kˆ and c = xiˆ + ( x - 2) ˆj - kˆ ., r, r, r, If the vector c lies in the plane of a and b , then x equals, (a) – 4, (b) – 2, (c) 0, (d) 1., r r ˆ r ˆ r ˆ r rˆ r ˆ r ˆ, 129. Let, a = a1i + a2 j + a3k , b = b1i + b2 j + b3k &, r r, r, r, r, c = c1iˆ + c2 ˆj + c3kˆ be three non-zero vectors such that c, r r, is a unit vector perpendicular to both a & b . If the angle, , a1, p, r r, between a & b is , then b1, 6, c1, , a2, b2, c2, , a3, b3, c3, , 2, , is equal to, , (a) 0, , (b) 1, 1 r 2 r 2, 3 r 2 r 2, |a | |b |, |a | |b |, (c), (d), 4, 4, 130. Let R1 and R2 respectively be the maximum ranges up and, down an inclined plane and R be the maximum range on the, horizontal plane. Then R1, R , R2 are in, (a) H.P, (b) A.G..P, (c) A.P, (d) G..P., 131. Two forces whose magnitudes are 2 gm wt, and 3 gm wt act, on a particle in the directions of the vectors 2î + 4 ĵ + 4k̂, and 4î + 4 ĵ + 2k̂ resepectively. If the particle is displaced, from the origin to the point (1, 2, 2), the work done is, (the unit of length being 1 cm) :, (a) 6 gm-cm, (b) 4 gm-cm, (c) 5 gm-cm, (d) None of these, ®, , ®, , ®, , 132. If a , b and c are unit coplanar vectors, then the scalar, é ® ® ® ® ® ®ù, triple product ê2 a - b , 2 b - c , 2 c - a ú =, ë, û, , (a) 0, (b) 1, (c) - 3 (d), r, r, r, 133. If a = ˆi + ˆj + kˆ , b = ˆi + ˆj , c = iˆ and, r r r, r, r, a ´ b ´ c = la + mb , then l + m is equal to, , (, , 3, , (a) 0, (b) 1, (c) 2, (d) 3, r r r r r r, r r, r, 134. If ( a ´ b ) ´ c = a ´ (b ´ c ) where a, b and c are any three, r r, r, r, r, vectors such that ar.b ¹ 0 , b . c ¹ 0 then a and c are, (a), (b), (c), (d), , (a), , 2p, 3, , (b), , p, 3, , (c) p, , (d), , p, 2, , 136. The resultant moment of three forces î + 2ˆj - 3k̂ ,, 2î + 3ˆj + 4k̂ and -î - ˆj + k̂ acting on a particle at a point P, (0, 1, 2) about the point A (1, –2, 0) is, , (a), , (c), (b) 140, (d) None, 6 2, 21, 137. The acute angle between the medians drawn through the, acute angle of an isosceles right angled triangle is, (a), , æ2ö, cos -1 ç ÷, è3ø, , -1 æ 3 ö, (b) cos ç ÷, è4ø, , (c), , æ4ö, cos -1 ç ÷, è5ø, , -1 æ 5 ö, (d) cos ç ÷, è6ø, , 138. If b and c are any two non-collinear mutually perpendicular, unit vectors and a is any vector, then, ( a × b) b + ( a . c ) c +, (a), , a, , (b) 2 a, , a . ( b ´ c), | b ´ c |2, , ( b ´ c) is equal to :, , (c) 3 a, , (d) None, , 139. The three vectors î + ˆj , ˆj + k̂ , k̂ + î taken two at a time, form three planes. The three unit vectors drawn perpendicular, to these three planes form a parallelopiped of volume :, 4, 1, 3 3, (d), (b) 4, (c), 3 3, 3, 4, 140. | (a × b).c| = |a| |b||c| , if, (a) a.b = b. c = 0, (b) b.c = c. a = 0, (c) c.a = a.b = 0, (d) a.b = b.c = c.a = 0, 141. The sum of two forces is 18 N and resultant whose direction, is at right angles to the smaller force is 12 N. The magnitude, of the two forces are, (a) 13, 5, (b) 12, 6, (c) 14, 4, (d) 11, 7, , (a), , 142. If a = x ( a ´ b ) + y( b ´ c) + z ( c ´ a ) and, , ), , p, inclined at an angle of, between them, 3, p, between them, inclined at an angle of, 6, perpendicular, parallel, , 135. The angle between two forces each equal to P when their, resultant is also equal to P is :, , 1, [ a b c] = , then x + y + z =, 8, , (a), (c), , 8 a . ( a + b + c), ur ur ur, 8( a + b + c), , (b) a . ( a + b + c), , (d) None of these., r, r, r, r, 143. If a is any vector, then ˆi ´ a ´ ˆi + ˆj ´ a ´ ˆj + kˆ ´ a ´ kˆ is, , ( ), , equal to, r, (a) a, , ( ), , (, , ), , r, r, (b) 2a, (c) 3a, (d) 0, r, p, 144. If unit vector c makes an angle with ˆi + ˆj , then minimum, 3, r, and maximum values of ˆi ´ ˆj .c respectively are, , ( )

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VECTOR ALGEBRA, , (a), , 0,, , (c), , - 1,, , 347, , 3, 2, , (b) -, , 3, 2, , 3 3, ,, 2 2, , (d) None of these, , (c) -iˆ + ˆj (d) 3iˆ - 3jˆ, (a) 3iˆ + 3jˆ (b) 3iˆ + ˆj, 148. A girl walks 4 km towards West. Then, she walks 3 km in a, direction 30° East to North and stops. The girls displacement, from her initial point of departure is, , 145. If OA = a ; OB = b ; OC = 2 a + 3 b ;, OD = a - 2 b , the length of OA is three times the length, , of OB and OA is perpendicular to DB then, ( BD ´ AC ).(OD ´ OC) is, (a) 7 | a ´ b | 2, (c) 0, , (b) 42 | a ´ b |2, (d) None of these, , 146. A particle is acted upon by constant forces 4iˆ + ˆj - 3kˆ, and 3iˆ + ˆj - kˆ which displace it from a point iˆ + 2 ˆj + 3kˆ to, the point 5iˆ + 4 ˆj + kˆ . The work done in standard units by, the forces is given by, (a) 15, (b) 30, (c) 25, (d) 40, 147. The moment about the point ˆi + 2ˆj + 3kˆ of a force, , (a), , 3 3 3ˆ, - iˆ +, j, 2, 2, , (c), , 5 3 3ˆ, - iˆ +, j, 2, 2, , 5, 3, (b) - ˆi + ˆj, 2, 2, , (d) None of these, r, r, 149. Resolved part of vector ar along vector b is a1 and that, r, r, r r, perpendicular to b is a2 , then a1 ´ a2 is equal to, r, r, r, r r r, (a ´ b ) × b, (a × b ) a, r, (a), (b), r, | b |2, | a |2, r r r r, r r r r, (a × b ) (b ´ a ), ( a × b ) (b ´ a ), r r, r, (c), (d), |b ´a |, | b |2, uur uur uur, uur, 150. If a , b , c are three unit vectors such that b is not parallel, uur uur uur, uur, uur, 1 uur, to c and a ´ ( b ´ c ) = b , then the angle between a, 2, uur, and c is, , represented by ˆi + ˆj + kˆ acting through the point 2i + 3j +, k is, , (a), , p, 3, , (b), , p, 2, , (c), , p, 6, , (d), , p, 4, , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, 1., , 4., , The vector in the direction of the vector ˆi - 2jˆ + 2kˆ that, has magnitude 9 is, , 2., , 3., , (a), , ˆi - 2jˆ + 2kˆ, , (c), , 3 ˆi - 2ˆj + 2kˆ, , (, , (b), , ), , (a), , ˆi - 2ˆj + 2kˆ, 3, , (, , (d) 9 iˆ - 2ˆj + 2kˆ, , 5., , ), , The position vector of the point which divides the join of, r r, r r, points 2a - 3b and a + b in the ratio 3 : 1, is, r r, r r, 3a - 2b, 7a - 8b, (a), (b), 2, 4, r, r, 3a, 5a, (c), (d), 4, 4, The vector having initial and terminal points as (2, 5, 0), and (–3, 7, 4), respectively is, (a), , - $i + 12 $j + 4k$, , (b) 5$i + 2 $j - 4k$, , (c), , - 5$i + 2 $j + 4k$, , (d) $i + $j + k$, , r, r, The angle between two vectors a and b with magnitudes, r r, 3 and 4 respectively and a × b = 2 3 is, p, 6, , p, 3, , (c), , p, 2, , (d), , 3, 2, , (d), , 5p, 2, , r, Find the value of l such that the vectors a = 2i$ + l $j + k$, r, and b = i$ + 2 $j + 3k$ are orthogonal., , (a) 0, 6., , (b), , (b) 1, , (c), , -5, 2, , The value of l for which the vectors 3$i - 6 $j + k$ and, 2$i - 4 $j + l k$ are parallel, is, 2, 2, 3, 5, (b), (c), (d), 3, 5, 2, 2, The vectors from origin to the points A and B are, r, r, a = 2$i + -3 $j + 2k$ and b = 2$i + -3 $j + k$ respectively , then, , (a), 7., , the area of DOAB is equal to, (a) 340, , (b), , 25, , (c), , 229, , (d), , 1, 229, 2

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EBD_7762, 348, , 8., , MATHEMATICS, , r, , For any vector a , the value of, r, r, r, 2, 2, 2, a ´ ˆi + a ´ ˆj + a ´ kˆ is equal to, , ( ) ( ) (, , (a), , 9., , r2, a, , 11., , 12., , 13., , 14., 15., , r2, (c) 4a, , r2, (d) 2a, , (c) 14, (d) 16, $, $, $, The vector are coplanar, if li + j + 2k , $i + l $j - k$ and, , (b) 10, , 2i$ - $j + l k$ are coplanar, if, (a) l = – 2 (b) l = 0, (c) l = 1, (d) l = – 1, r, r r r r, r r, If a , b and c are unit vectors such that a + b + c = 0 , then, r r r r r r, the value of a × b + b × c + c × a is, (a) 1, (b) 3, , (c), , 3, 2, , (d) None of these, r, r, The projection vector of a on b is, r r, r r, æ a ×b ö, æ a ×b ö r, (a) ç uur ÷ b, (b) ç uur ÷, ç |b| ÷, ç |b| ÷, è, ø, è, ø, r r, r r, æ a ×b ö, æ a ×b ö, (c) ç r ÷, (d) çç r 2 ÷÷ b$, ç |a| ÷, è|a | ø, è, ø, r r, r r r r, r, If a , b and c are three vectors such that a + b + c = 0 and, r, r, r, r r r r, | a | = 2,| b | = 3 and | c | = 5, then the value of a × b + b × c, r r, + c × a is, (a) 0, (b) 1, (c) – 19, (d) 38, ur, r, If | a | = 4 and – 3 £ l £ 2 , then the range of | l a | is, (a) [0, 8], (b) [– 12, 8] (c) [0, 12] (d) [8, 12], The number of vectors of unit length perpendicular to the, r, r, vectors a = 2$i + $j + 2k$ and b = $j + k$, (a) one, , (b) two, , (c) three, , (d) infinite, , Past Year MCQs, 16., , 17., , r r r r r r, rrr 2, If éë a ´ b b ´ c c ´ a ùû = l éë a b c ùû then l is equal to, [JEE MAIN 2014, C, BN], (a) 0, (b) 1, (c) 2, (d) 3, If the middle points of sides BC, CA & AB of triangle ABC, are respectively D, E, F then position vector of centre of, triangle DEF, when position vector of A, B, C are respectively, [BITSAT 2014, C], iˆ + ˆj, ˆj + kˆ, kˆ + iˆ is, , (a), , ), , (b), , (, , ), , (d), , (, , iˆ + ˆj + kˆ, , (, , ), , ), , 2 ˆ ˆ ˆ, i + j +k, 3, The angle between any two diagonal of a cube is, [BITSAT 2014, C], , (c), 18., , (, , 1 ˆ ˆ ˆ, i + j +k, 3, , 2 iˆ + ˆj + kˆ, , (a) 45° (b) 60°, , ®, , 1® ®®, b c a . If q is the, 3, ®, ®, angle between vectors b and c , then a value of sin q is :, [JEE MAIN 2015, A, BN], 2, - 2, 2 2, -2 3, (a), (c), (d), (b), 3, 3, 3, 3, 20. Let a, b and c be three vectors satisfying a × b = (a ×c),, |a| = |c| = 1, |b| = 4 and |b × c| = 15 ., If b – 2c = la, then l equals, [BITSAT 2015, A], (a) 1, (b) – 1, (c) 2, (d) – 4, ®, , r, r r, rr, r, If a = 10 , b = 2 and a.b = 12 , then the value of a ´ b is, , (a) 5, 10., , r2, (b) 3a, , ), , ® ®, , 19. Let a , b and c be three non-zero vectors such that no two, , (c) 30° (d) tan -1 (2 2 ), , ®, , ®, , of them are collinear and (a ´ b) ´ c =, , ®, , ® ®, , 21. Let a , b and c be three unit vectors such that, ®, ®, æ® ® ö, 3 æç® ® ö, a ´ çç b´ c ÷÷ =, b+ c ÷, . If b is not parallel to c , then, ç, ÷, è, ø 2 è, ø, , ®, , ®, , ®, , the angle between a and b is:, [JEE MAIN 2016, A, BN], (a), , 2p, 3, , 5p, 6, , (b), , (c), , r, 22. Let a = 2iˆ + ˆj - 2kˆ and, r r, r r, that | c - a | = 3, a ´ b, r r, and a ´ b be 30°. Then, , (, , ), , 3p, 4, , (d), , p, 2, , r, r, b = ˆi + ˆj . Let c be a vector such, r, ´ c = 3 and the angle between rc, , rr, a.c is equal to :, , [JEE MAIN 2017, S, BN], (a), , 1, 8, , 25, (b), 8, , (c) 2, , (d) 5, , 23. The dot product of a vector with the vectors ˆi + ˆj - 3kˆ ,, , ˆi + 3jˆ - 2kˆ and 2iˆ + ˆj + 4kˆ are 0, 5 and 8 respectively. The, vector is, [BITSAT 2017, A], (a) ˆi + 2jˆ + kˆ, (b) - ˆi + 3jˆ - 2kˆ, ˆi + 2ˆj + 3kˆ, (d) ˆi - 3jˆ - 3kˆ, r r r, 24. Let a, b & c be non-coplanar unit vectors equally inclined, rrr, to one another at an acute angle q. Then | [a b c] | in terms of, q is equal to, [BITSAT 2017, A, BN], (c), , (a), , (1 + cos q) cos 2q, , (b), , (1 + cos q) 1 - 2cos 2q, , (c) (1 - cos q) 1 + 2cos 2q, (d) None of these, r, 25. Let u be a vector coplanar with the vectors, r, r, r, r, a = 2iˆ + 3jˆ - kˆ and b = ˆj + kˆ . If u is perpendicular to a, r r, r, and u × b - 24 , then | u |2 is equal to : [JEE MAIN 2018, A], (a) 315, (b) 256, (c) 84, (d) 336

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EBD_7762, 350, , 10., , MATHEMATICS, , Let p, q, r be three mutually perpendicular vectors of the, same magnitude. If a vector x satisfies the equation, p ´ {( x - q ) ´ p} + q ´ {( x - r )) ´ q}, + r ´ {( x - p) ´ r} = 0 then x is given by, 1, 1, ( p + q - 2 r), ( p + q + r), (a), (b), 2, 2, 1, 1, ( p + q + r), (2 p + q - r ), (c), (d), 3, 3, r r, r, r r, r r, r r r, Let r = (a ´ b )sin x + (b ´ c ) cos y + 2(c ´ a ) where a, b , c, are three non-coplanar vectors. If rr is perpendicular to, r r r, a + b + c , then minimum value of x2 + y2 is, p2, (a) p2, (b), 4, 5p2, (c), (d) None of these, 4, uur, If b is vector whose initial point divides the join of 5iˆ and, , 11., , 12., , 5 ˆj in the ratio k : 1 and terminal point is origin and, uur, | b | £ 37 , then k does not belong to, 1ù, é, ê- 6, - 6 ú, ë, û, , (a), , 1, (b) (-¥, - 6) È éê - , ¥ ÷ö, ë 6, ø, , é 1 ö, (d) ê - , 6 ÷, ë 6 ø, , (c) [ 0, 6], , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, , (a), (b), (b), (a), (d), (a), (d), (a), (d), (b), (d), (c), (a), (d), (a), , 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, , (a), (c), (c), (b), (b), (a), (a), (a), (a), (c), (a), (b), (d), (b), (d), , 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, , (b), (d), (c), (b), (d), (c), (c), (b), (b), (d), (c), (c), (a), (c), (b), , 1, 2, 3, , (c), (d), (c), , 4, 5, 6, , (b), (d), (a), , 7, 8, 9, , (d), (d), (d), , 1, 2, , (b), (b), , 3, 4, , (d), (d), , 5, 6, , (b), (a), , 13. If p and q are two unit vectors inclined at an angle a to, each other then | p + q |< 1 if, (a), , 2p, 4p, <a<, 3, 3, , (c), , 0<a<, , (b), , p, 3, , 4p, < a < 2p, 3, , (d) a =, , p, 2, , 14. If b and c are any two non-collinear mutually perpendicular, unit vectors and a is any vector, then, a . ( b ´ c), , ( b ´ c) is equal to :, | b ´ c |2, (a) a, (b) 2 a, (c) 3 a, (d) None, uur, uuur, uuur, uur, 15. Two vectors m1 & m2 with m1 = 2, m2 = 1 and angle, , ( a × b) b + ( a . c ) c +, , uuur 2p, uur, r, ur, ur r ur, ur, between m1 & m2 is, . If a = 2tm1 + 7 m2 , b = m1 + tm2, 3, r ur, ur, & c = tm1 - 2m2 , (t ¹ 0) are three vectors such that, r r, a ×b, r r < 0, then number of integral values of t will be, b ×c, , (a) 5, , (b) 3, , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (a), 46 (b) 61 (c) 76 (d) 91, (c), 47 (c) 62 (b) 77 (a) 92, (a), 48 (c) 63 (d) 78 (b) 93, (d), 49 (a) 64 (b) 79 (a) 94, (a), 50 (a) 65 (c) 80 (d) 95, (d), 51 (a) 66 (b) 81 (c) 96, (a), 52 (a) 67 (a) 82 (b) 97, (b), 53 (d) 68 (c) 83 (b) 98, (d), 54 (b) 69 (d) 84 (a) 99, 55 (d) 70 (a) 85 (a) 100 (a), 56 (d) 71 (c) 86 (a) 101 (d), 57 (c) 72 (d) 87 (b) 102 (a), 58 (c) 73 (c) 88 (c) 103 (a), 59 (b) 74 (d) 89 (b) 104 (b), 60 (d) 75 (d) 90 (d) 105 (d), Exercise 2 : Exemplar & Past Year MCQs, 10 (a), 13 (c), 16 (b), 19 (c), 11 (c), 14 (c), 17 (d), 20 (d), 12 (a), 15 (b), 18 (d), 21 (b), Exercise 3 : Try If You Can, 7 (c), 9 (c), 11 (c), 13 (a), 8 (b), 10 (b), 12 (a), 14 (a), , 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, , (c) 4, , (a), (d), (d), (b), (a), (c), (c), (d), (b), (c), (a), (d), (c), (a), (c), , 22 (c), 23 (a), 24 (c), 15 (c), , 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 25, 26, 27, , (d) 2, , (d), (a), (b), (d), (c), (a), (b), (b), (c), (a), (a), (a), (a), (d), (a), (d), (d), (d), , 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, , (b), (c), (a), (d), (d), (a), (a), (b), (b), (b), (d), (d), (c), (c), (a)

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27, , THREE DIMENSIONAL, GEOMETRY, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , 5, , Number of Questions, , 4, JEE MAIN, , 3, , BITSAT, 2, 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , Weightage, , Critical Concepts, , JEE Main, , 6, , Direction cosines and Direction ratios, , BITSAT, , 2, , Equation of a line, Angle between, Two lines, Shortest Distance Between, Two lines, Coplanarity of Two lines,, Equation of a plane, Angle, between Two planes, Distance of a, point From a plane, Angle between a, Line & a plane., , Rating of, Difficulty Level, , 4.5/5, , CUS, (chapter utility score), Out of 10, , 8.1

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THREE DIMENSIONAL GEOMETRY, , 353

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EBD_7762, 354, , MATHEMATICS, , Topic 1: Direction Cosines and Direction Ratios of a Line,, Angle Between Two Lines in Terms of d.c. and d.r. Conditions, of Parallelism and Perpendicularity of Two Lines., 1., , Under what condition do, , 1 1, , ,k, 2 2, , represent direction, , cosines of a line?, (a), , 1, 2, , (b), , k=–, , 1, 2, , 1, (d) k can take any value, 2, A line makes the same angle q, with each of the x and z axis., If the angle b, which it makes with y-axis, is such that, , (c), , 2., , k=, , k=±, , sin 2 b = 3 sin 2 q, then cos2q equals, , (a), 3., , 2, 5, , 1, 5, , (b), , 3, 5, , 2, 3, , (d), , æ1 1 1ö, If the direction cosines of a line are ç , , ÷ then, èc c cø, (a) 0 < c < 1, (b) c > 2, (c) c > 0, , 4., , (c), , (d), , c=± 3, , The direction cosines l, m, n, of one of the two lines, connected by the relations, , l - 5m + 3n = 0, 7l 2 + 5m2 - 3n2 = 0 are, (a), (c), , 5., , 6., , 1, 14, 1, , ,, ,, , 2, 14, -2, , ,, ,, , 3, 14, 3, , (b), (d), , -1, 14, 1, , ,, ,, , 2, 14, 2, , ,, ,, , 3, 14, -3, , 14 14 14, 14 14 14, The direction cosines l, m, n of two lines are connected by, the relations l + m + n = 0, lm = 0, then the angle between, them is :, (a) p / 3 (b) p / 4, (c) p / 2, (d) 0, The angle between a line whose direction ratios are in the, ratio 2 : 2 : 1 and a line joining (3, 1, 4) to (7, 2, 12) is, (a) cos–1(2/3), (b) cos–1(–2/3), –1, (c) tan (2/3), (d) None of these, , 1 1 1, , ,, , The, bc ca ab, , 7., , Direction ratios of two lines are a, b, c and, , 8., , lines are, (a) Mutually perpendicular (b) Parallel, (c) Coincident, (d) None of these, Three lines with direction ratios, <1, 1, 2> < 3 - 1, - 3 - 1, 4 > and < - 3 - 1, 3 - 1, 4 >, form, , (a) a right angled triangle (b) a scalene triangle, (c) an equilateral triangle (d) None of these, é l1, A = êêl2, êë l3, , n1 ù, é p1 q1 r1 ù, ú, ê, ú, n, 9., 2 ú and B = ê p2 q2 r2 ú , where pi, qi,, m3 n3 úû, êë p3 q3 r3 úû, ri are the co-factors of the elements l1, mi, ni for i = 1, 2, 3. If, (l1, m1, n1), (l2, m2, n2) and (l3, m3, n3) are the direction, cosines of three mutually perpendicular lines then, (p1, q1, r1), (p2, q2, r2) and (p3, q3, r3) are, (a) the direction cosines of three mutually perpendicular, lines, (b) the direction ratios of three mutually perpendicular, lines which are not direction cosines, (c) the direction cosines of three lines which need not be, perpendicular, (d) the direction ratios but not the direction cosines, 10. Which of the following are true?, I. If a, b and c are the direction ratios of a line, then, ka, kb and kc is also a set of direction ratios., II. The two sets of direction ratios of a line are in, proportion., III. There exists two sets of direction ratios of a line., (a) I and II are true, (b) II and III are true, (c) I and III are true, (d) All are true, m1, , m2, , Topic 2 : Equation of a Straight Line in Symmetrical Form,, Angle Between Two Lines, Condition for, Coplanarity of Two Lines., 11. The points A(1, 2, 3), B (–1, –2, –3) and C(2, 3, 2) are, three vertices of a parallelogram ABCD. The equation of, CD is, (a), , x y z, = =, 1 2 2, , (b), , x+2 y+3 z-2, =, =, 1, 2, 2, , x y z, x-2 y-3 z-2, = =, =, =, (d), 2 3 2, 1, 2, 2, 12. The points A(1, 2, 3), B (–1, –2, –3) and C(2, 3, 2) are, three vertices of a parallelogram ABCD. The equation of, CD is, , (c), , (a), , x y z, = =, 1 2 2, , (b), , x+2 y+3 z-2, =, =, 1, 2, 2, , (c), , x y z, = =, 2 3 2, , (d), , x-2 y-3 z-2, =, =, 1, 2, 2

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THREE DIMENSIONAL GEOMETRY, , 355, , r r, r, r r, r, 13. Two lines r = a1 + l b1 and r = a 2 + mb 2 are said to be, coplanar, if, (a) (a2 – a1) . (b1 × b2) = 0, , (b), , x1, , y1, , z1, , a1, , b1, , c1 = 0,, , a2, , b2, , c2, , 1, , 2, , r r, a 2 × b1, (d) cos q = r r, a 2 b1, , 15. A line makes angles of 45° and 60° with the positive axes of X, and Y respectively. The angle made by the same line with the, positive axis of Z, is., (a) 30° or 60°, (b) 60° or 90°, (c) 90° or 120°, (d) 60° or 120°, 16. The ordered pair (l, µ) such that the points, (l, µ, –6),(3, 2, –4) and (9, 8, –10) become collinear is, (a) (3, 4) (b) (5, 4), (c) (4, 5), (d) (4, 3), 17. The vector equation of the symmetrical form of equation of, straight line, , x -1 y -1 z -1, =, =, , then its, 3, 0, 4, perpendicular distance from the origin is, , perpendicular to the line, , (a), , where (x1, y1, z1) are the coordinates of a point on any of, the line and a1, b1, c1 and a2, b2, c2 are the direction ratio of, b1 and b2, (c) Both (a) and (b), (d) None of the above, r, r, 14. If the equations of two lines l1 and l2 are given by r = a1 + lb1, r, r, and r = a 2 + lb 2 , where l, m are parameter then angle q, between them is given by, r r, r r, a1 × a 2, b2 × b1, (a) cos q = r r, (b) cos q = r r, a1 a 2, b b, r r, a1 × b2, (c) cos q = r r, a1 b2, , 19. If a plane passes through the point (1, 1, 1) and is, , x -5 y + 4 z -6, =, =, is, 3, 7, 2, , r, r = ( 3iˆ + 7ˆj + 2kˆ ) + m ( 5iˆ + 4j - 6kˆ ), r, (b) r = ( 5iˆ + 4jˆ - 6kˆ ) + m ( 3iˆ + 7 j + 2kˆ ), r, (c) r = ( 5iˆ - 4jˆ - 6kˆ ) + m ( 3iˆ - 7 j - 2kˆ ), r, (d) r = ( 5iˆ - 4jˆ + 6kˆ ) + m ( 3iˆ + 7 j + 2kˆ ), 18. Consider the following statements, Statement I : The vector equation of a line passing through, two points whose position vectors are a and b, is r = a + l, (b – a) " l Î R., Statement II : The cartesian equation of a line passing, through two points (x1, y1, z1) and (x2, y2, z2) is, , (a), , x - x1, y - y1, z - z1, =, =, x 2 - x1 y2 - y1 z 2 - z1, Choose the correct option., (a) Statement I is true, (b) Statement II is true, (c) Both statements are true, (d) Both statements are false, , 3, 4, , (b), , 4, 3, , (c), , 20. If vector equation of the line, , 21., , 7, 5, , (d) 1, , x - 2 2y - 5, =, = z + 1, is, 2, -3, , r æ, 5, 3, ö, æ, ö, r = ç 2iˆ + ˆj - kˆ ÷ + l ç 2iˆ - ˆj + pkˆ ÷ then p is equal to, è, 2, ø, è, 2, ø, (a) 0, (b) 1, (c) 2, (d) 3, The lines whose vector equations are, ˆ, r = 2iˆ - 3jˆ + 7kˆ + l (2iˆ + pjˆ + 5k), , ˆ, and r = iˆ - 2jˆ + 3kˆ + m (3iˆ + pjˆ + pk), are perpendicular for all values of l and m if p =, (a) 1, (b) –1, (c) – 6, (d) 6, 22. The equation of two lines through the origin, which intersect, the line, , x -3 y-3 z, p, =, = at angles of, each, are, 2, 1, 1, 3, , (a), , x y z x y z, = = ; = =, 1 2 1 1 1 2, , (b), , x y, z x, y, z, = = ;, = =, 1 2 -1 -1 1 -2, , x y z x, y, z, = = ; =, =, 1 2 -1 1 -1 -2, (d) None of the above, 23. The angle between two lines, , (c), , x ∗1 y ∗ 3 z , 4, x , 4 y ∗ 4 z ∗1, <, <, <, <, is:, and, 2, 2, ,1, 1, 2, 2, , (a), , æ1ö, cos,1 çç ÷÷÷, çè 9 ø, , (b), , æ4ö, cos,1 çç ÷÷÷, çè 9 ø, , (c), , æ2ö, cos,1 çç ÷÷÷, çè 9 ø, , (d), , æ 3ö, cos,1 çç ÷÷÷, çè 9 ø, , Topic 3 : Perpendicular Distance of point From a Line, Skew, Lines, Shortest Distance Between Two Skew Lines., Condition for Two Lines to Intersect Each Other,, Distance between Two Parallel Lines., , 24. If the straight lines, , x -1 y - 2 z - 3, =, =, and, k, 2, 3, , x - 2 y - 3 z -1, =, =, intersect at a point, then the integer, 3, k, 2, k is equal to, (a) –5, (b) 5, (c) 2, (d) –2

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EBD_7762, 356, , MATHEMATICS, , 25. The line which passes through the origin and intersect the, two lines, x - 1 y + 3 z - 5 x - 4 y + 3 z - 14, , is, =, =, ,, =, =, 2, 4, 3, 2, 3, 4, , (a), , x, y z, =, =, 1 -3 5, , (b), , x, y z, = =, -1 3 5, , x y, z, x y, z, = =, = =, (d), 1 3 -5, 1 4 -5, The length of the perpendicular drawn from the point, , (c), 26., , x, 2, , (3, -1, 11) to the line =, (a), 27., , 29, , (b), , 33, , (c), , 53, , The coordinates of a point on the line, at a distance of, , 6, 2, , (d), , 66, , x + 2 y +1 z - 3, =, =, 3, 2, 2, , from the point (1, 2, 3) is, , (a) (56,,43, 111), 28., , y - 2 z -3, =, is :, 3, 4, , (b), , æ 56 43 111ö, çè , ,, ÷, 17 17 17 ø, , (c) (2, 1, 3), (d) (–2, –1, –3), The angle between two lines, x ∗1 y ∗ 3 z , 4, x , 4 y ∗ 4 z ∗1, <, <, <, <, and, is:, 2, 2, ,1, 1, 2, 2, , (a), , 29., , æ1ö, cos,1 çç ÷÷÷, çè 9 ø, , (b), , æ4ö, cos,1 çç ÷÷÷, çè 9 ø, , æ2ö, æ 3ö, (c) cos,1 çç ÷÷÷, (d) cos,1 çç ÷÷÷, çè 9 ø, çè 9 ø, The foot of the perpendicular from (2, 4, –1) to the line, 1, 1, ( y + 3) = – ( z – 6), 4, 9, (a) (– 4, 1, – 3), (b) (4, –1, –3), (c) (–4, –1, 3), (d) (– 4, – 1, –3), The shortest distance between the lines x = y + 2 = 6z – 6, and x + 1 = 2y = – 12z is, x +5 =, , 30., , 31., , 1, 3, (b) 2, (c) 1, (d), 2, 2, The distance between the lines given by, r, r, r = ˆi + ˆj + l ( ˆi - 2ˆj + 3kˆ ) and r = ( 2iˆ - 3kˆ ) + m ( iˆ - 2jˆ + 3kˆ ), is, , 32., , 118, 59, (d), 7, 7, r, Assertion: The pair of lines given by r = ˆi - ˆj + l ( 2i + k ), r, and r = 2iˆ - kˆ + m ( i + ˆj - k ) intersect., , (a), , (a), , 59, 14, , (b), , 59, 7, , (c), , Reason: Two lines intersect each other, if they are not, parallel and shortest distance = 0., , (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., 33. The length intercepted by a line with direction ratios, 2, 7, –5 between the lines, x -5 y -7 z + 2, x +3 y-3 z -6, =, =, =, =, and, is, -1, -3, 3, 1, 2, 4, , (a), , 75, , (b), , (c), , 83, , (d) None of these, , 78, , 34. The equation of the line which passes through the point, x –1, y–2, z –3, =, =, and, (1, 1, 1) and intersect the lines, 2, 3, 4, x+2, y –3, z +1, =, =, is, 1, 2, 4, (a), , x –1, y –1 z –1, =, =, 3, 10, 17, , (b), , x –1, y –1, z –1, =, =, 2, 3, –5, , (c), , x –1, z –1, y –1, =, =, –2, –4, 1, , (d), , y –1, x –1, z –1, = –2 =, 8, 3, , Topic 4 : Equation of A Plane, Equation of Plane Passing, Through The Intersection of Two Given Planesd,, Angle Between Two Planes., 35. The equation of the plane passing through three nonr r r, collinear points with position vectors a, b, c is, r r r r r r r, (a) r. ( b ´ c + c ´ a + a ´ b ) = 0, r r r r r r r, rrr, (b) r. ( b ´ c + c ´ a + a ´ b ) = ëéa b c ûù, , r r r r, rrr, r. a ´ ( b + c ) = éëa b c ùû, r r r r, (d) r. ( a + b + c ) = 0, 36. The plane x + 3y + 13 = 0 passes through the line of, intersection of the planes 2x – 8y + 4z = p and 3x – 5y + 4z, + 10 = 0. If the plane is perpendicular to the plane 3x – y, – 2z – 4 = 0, then the value of p is equal to, (a) 2, (b) 5, (c) 9, (d) 3, 37. What is the condition for the plane ax + by + cz + d = 0 to be, perpendicular to xy-plane ?, (a) a = 0, (b) b = 0, (c) c = 0, (d) a + b +c = 0, 38. The d.r. of normal to the plane through (1, 0, 0), (0, 1, 0), (c), , (, , ), , which makes an angle, (a) 1, 2 ,1(b) 1, 1,, , p, with plane x + y = 3 are, 4, 2, , (c) 1, 1, 2, , (d), , 2 , 1, 1

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THREE DIMENSIONAL GEOMETRY, , 357, , r, 39. A vector n is inclined to x-axis at 45°, to y-axis at 60° and at, r, an acute angle to z-axis. If n is a normal to a plane passing, , through the point, , (, , ), , 2, -1,1 then the equation of the plane is, , :, (a), , 4 2x + 7 y + z - 2, , (b), , 2x + y + 2z = 2 2 + 1, , (c) 3 2 x - 4 y - 3z = 7, (d), 2x - y - z = 2, 40. Which one of the following planes contains the z-axis?, (a) x – z = 0, (b) z + y = 0, (c) 3x + 2y = 0, (d) 3x + 2z = 0, 41. If the plane x – 3y + 5z = d passes through the point, (1, 2, 4), then the length of intercepts cut by it on the axes of, X, Y, Z are respectively, is, (a) 15, –5, 3, (b) 1, –5, 3, (c) –15, 5, –3, (d) 1, –6, 20, x – 2 y –1 z + 2, lie in the plane, =, =, 3, –5, 2, x + 3y – az + b = 0. Then (a, b) equals, (a) (–6, 7), (b) (5, –15), (c) (–5, 5), (d) (6, –17), 43. If two lines L1 and L2 in space, are defined by, , 42. Let the line, , {, L2 = { x =, , ( l - 1) , z = ( l - 1) y + l} and, m y + (1 - m ) , z = (1 - m ) y + m}, , L1 = x = l y +, , then L1 is perpendicular to L2, for all non-negative reals l, and m, such that :, (a), (c), , l + m =1, , l+m = 0, , 135, r, (b) r . -iˆ + 2 ˆj - kˆ =, 2, r ˆ, 135, =0, (c) r . 5i + 3 ˆj - 11kˆ +, 2, 135, r, (d) r . 5iˆ + 3 ˆj - 11kˆ =, 2, 47. The volume of the tetrahedron included between the plane, 3x + 4y – 5z – 60 = 0 and the coordinate planes is, (a) 60, (b) 600, (c) 720, (d) None of these, 48. The plane ax + by = 0 is rotated through an angle a about, its line of intersection with the plane z = 0. Then the, equation to the plane in new position., , (, (, (, , ), , ), ), , (a), , ax - by ± z a 2 + b2 tan a = 0, , (b), , ax + by ± z a 2 + b2 tan a = 0, , ax - by ± z a 2 - b2 tan a = 0, (d) None of these, (c), , Topic 5 : Angle between A Line and a Plane, Distance, between Two Parallel Planes, Intersection Point of A, Line and A Plane, Division of A Line Segment by Plane,, Distance of A Point from A Plane., 49. The vector equation of the plane which is at a distance of, , 6, 29, , from the origin and its normal vector from the origin, , (b), , l¹m, , ˆ is, is 2iˆ - 3jˆ + 4k,, , (d), , l =m, , (a), , 44. Consider three planes, P1 : x – y + z = 1, P2 : x + y – z = 1, P3 : x – 3y + 3z = 2, Let L1, L2, L3 be the lines of intersection of the planes P2, and P3, P3 and P1, P1 and P2, respectively., Assertion : At least two of the lines L1, L2 and L3 are nonparallel, Reason : The three planes doe not have a common point., (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., 45. Four points (0, –1, –1) (–4, 4, 4) (4, 5, 1) and (3, 9, 4) are, coplanar. Find the equation of the plane containing them., (a) 5x + 7y + 11z – 4 = 0, (b) 5x – 7y + 11z + 4 = 0, (c) 5x – 7y –11z – 4 = 0, (d) 5x + 7y – 11z + 4 = 0, 46. Equation of the plane through the mid–point of the line, segment joining the points P(4,5,–10) and Q(–l,2,l) and, perpendicular to PQ is, r æ3, 7, 9 ö, (a) r . ç iˆ + ˆj - kˆ ÷ = 45, 2, 2 ø, è2, , r, 6, r . ( 2iˆ - 3jˆ + 4kˆ ) =, 29, , r æ 2, ˆi - 3 ˆj + 4 kˆ ö = 6, r .ç, ÷, è 29, 29, 29 ø, 29, (c) Both (a) and (b), (d) None of the above, , (b), , x – 2 y –1 z + 2, =, =, lie in the plane, 3, –5, 2, x + 3y – az + b = 0. Then (a, b) equals, (a) (–6, 7) (b) (5, –15), (c) (–5, 5), (d) (6, –17), y -1 z - 3, =, 51. If the angle between the line x =, and the plane, 2, l, æ 5 ö, x + 2y + 3z = 4 is cos–1 ç, ÷ , then l equals, è 14 ø, 3, 2, 5, 2, (b), (c), (d), (a), 2, 5, 3, 3, 52. If a plane passes through the point (1, 1, 1) and is, , 50. Let the line, , x -1 y -1 z -1, =, =, , then its, 3, 0, 4, perpendicular distance from the origin is, , perpendicular to the line, , (a), , 3, 4, , (b), , 4, 3, , (c), , 7, 5, , (d) 1

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EBD_7762, 358, , MATHEMATICS, , 53. Distance between the parallel planes, 2x – y + 3z + 4 = 0 and 6x – 3y + 9z – 3 = 0 is:, (a), , 54., , 55., , 5, , 4, , 61. The distance of the point (–5, –5, –10) from the point of, , 5, , 3, , (d), 14, 2 3, x +1, y -1, z -2, If the angle q between the line, =, =, and, 1, 2, 2, the plane 2x – y + l z + 4 = 0 is such that, 1, sin q = then the value of l is, 3, 5, 3, -3, -4, (a), (b), (d), (c), 3, 5, 4, 3, The angle between the line, (b), , 3, , (c), , 6, , x-2 y -2 z -2, =, =, and the plane ax + by + cz + 6 = 0 is, a, b, c, , æ, ö, 1, sin -1 çç, ÷ (b) 45°, 2, 2, 2 ÷, è a +b +c ø, (c) 60°, (d) 90°, The projections of the segment PQ on the, co-ordinate axes are –9, 12, –8 respectively. The direction, cosines of the line PQ are, , (a), 56., , 57., , 58., , <, , (c), , <-, , 17, , ,, , 12, 17, , ,, , -8, , >, , (b), , 9, 12 -8, ,, ,, > (d), 289 289 289, , < -9,12, - 8 >, <-, , 9 12 -8, ,, ,, >, 17 17 17, , x – x0, y – y0 z – z0, =, =, is parallel to the, l, m, n, plane ax + by + cz = d then :, a b c, = =, (a), (b) al = bm = cn, l m n, a b c, + +, (c), (d) al + bm + cn = 0, l m n, The coordinates of the point where the line through the, points A (3, 4, 1) and B (5, 1, 6) crosses the XY - plane are, , æ 13 23 ö, ç , ,0÷, è5 5 ø, , (b), , æ 13 23 ö, ç - , ,0÷, è 5 5 ø, , 2ab z = 1?, (b) 1/(a + b), (d) ab, , The distance of a point (2, 5, –3) from the plane, r × ( 6iˆ - 3jˆ + 2kˆ ) = 4 is, , (a) 13, , (b), , 13, 7, , (c), , 13, 5, , (d), , and the plane r · ( ˆi - ˆj + kˆ ) = 5 is, (d) 10 2, (a) 13, (b) 12, (c) 4 15, 62. Let A(1, 1, 1) , B(2, 3, 5) and C(– 1, 0, 2) be three points, then, equation of a plane parallel to the plane ABC which is at, distance 2 is, (a) 2x – 3y + z + 2 14 = 0 (b) 2x – 3y + z – 14 = 0, (c) 2x – 3y + z + 2 = 0, (d) 2x – 3y + z – 2 = 0, 63. If O be the origin and the coordinates of P be (1, 2, –3), then, the equation of the plan e passing through P and, perpendicular to OP is, (a) x + 2y + 3z = –5, (b) x + 2y + 3z = – 14, (c) x + 2y – 3z =14, (d) x + 2y – 3z =5, 64. Consider the following statements, Statement I: The angle between two planes x + 2y + 2z = 3, æ 19 2 ö, ., and –5 x + 3y + 4z = 9 is cos -1 ç, è 30 ÷ø, , Statement II: The angle between the line, , 37, 7, , x -1 y - 2 z + 3, =, =, 2, 1, -2, , and the plane x + y + 4 = 0 is 45°., Choose the correct option., (a) Statement I is true, (b) Statement II is true, (c) Both statements are true, (d) Both statements are false, 65. What is the angle between the line 6x = 4y = 3z and the plane, 3x + 2y – 3z = 4 ?, (a) 0, (b) p/6, (c) p/3, (d) p/2, 66. A rectangular parallelopiped is formed by drawing planes, through the points (–1, 2, 5) and (1, –1, –1) and parallel to the, coordinate planes. The length of the diagonal of the, parallelopiped is, (a) 2, (b) 3, (c) 6, (d) 7, 67. The line, , æ 13 -23 ö, æ 13 23 ö, ,0÷, (c) ç , - , 0 ÷, (d) ç - ,, 5, 5, 5, è 5, ø, è, ø, What is the length of the perpendicular from the origin to, the plane ax + by +, (a) 1/(ab), (c) a + b, , 60., , 17, , The straight line, , (a), , 59., , -9, , (a), , intersection of the line r . = 2iˆ - ˆj + 2kˆ + l ( 3iˆ + 4ˆj + 2kˆ ), , x -1 y - 2 z -1, =, =, and the plane x + 2y + z = 6, -2, 1, 3, , meet at, (a) no point., (b) only one point., (c) infinitely many points., (d) none of these., 68. If the distance between planes, 4x – 2y – 4z + 1 = 0 and, 4x – 2y – 4z + d = 0 is 7, then d is:, (a) 41 or – 42, (b) 42 or – 43, (c) – 41 or 43, , (d) – 42 or 44, , 69. The plane passing through the point (–2, –2, 2) and, containing the line joining the points (1, 1, 1) and (1, –1, 2), makes intercepts on the coordiantes axes, the sum of whose, lengths is, (a) 3, (b) – 4, (c) 6, (d) 12

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THREE DIMENSIONAL GEOMETRY, , 359, , 70. The three planes x + y = 0, y + z = 0 and x + z = 0, (a) meet in a unique point, (b) meet in a line, (c) meet taken two at a time in parallel lines, (d) None of these, 71. The value of l, k for which the line, , x -1 y z + 2, =, =, lies, -1, 1, 2, , in the plane 2x + 3y + lz + k = 0, (a) l = 1/2, k = –1, (b) l = 1/2, k = 1, (c) l = –1/2, any value of k (d) l = –1/2, k = –3, 72. If the points (1, 1, p) and (–3, 0, 1) be equidistant from the, , parallel to the line y = 0, z = 0 is, (a) (ab'-a ' b) x + (bc '-b' c) y + (ad'-a ' d) = 0, (b) (ab'-a ' b) x + (bc'- b' c) y + (ad'-a ' d )z = 0, (c) (ab'-a ' b) y + (ac'-a ' c)z + (ad'-a ' d) = 0, (d) None of these, 78. A variable plane remains at constant distance p from the, origin.If it meets coordinate axes at points A, B, C then the, locus of the centroid of D ABC is, , plane r. ( 3iˆ + 4j - 12kˆ ) + 13 = 0 , then the value of p is, , 3, 7, 4, 3, (a), (b), (c), (d), 7, 3, 3, 4, 73. The equation of the line passing through (1, 2, 3) and parallel, , (, , ), , (, r = ( ˆi + 2ˆj + 3kˆ ) + l ( 2iˆ + 3jˆ + 4kˆ ), r = ( -3iˆ + 5jˆ + 4kˆ ) + l ( iˆ + 2ˆj + 3kˆ ), r = ( ˆi + 2jˆ + 3kˆ ) + l ( -3iˆ + 5jˆ + 4kˆ ), r = l ( -3iˆ + 5jˆ + 4kˆ ), , ), , to the planes r. ˆi - ˆj + 2kˆ = 5 and r. 3iˆ + ˆj + kˆ = 6 is, (a), (b), (c), , (d), 74. The equation of two lines through the origin, which intersect, the line, , x -2 + y-2 + z -2 = 9p -2, , (b), , x -3 + y -3 + z -3 = 9p -3, , (c), , x 2 + y2 + z 2 = 9p 2, , (d), , x 3 + y3 + z3 = 9p 3, , BEYOND NCERT, Topic 6 : Projection of a Point and Line Segment, Equation of, Straight Line in Unsymmetrical Form, Bisectors of, The Angles between The Lines, Image of a Point in a, Line, Equation of a Plane Bisecting Angle between two, Planes, Image of a Point in a Plane,, Projection of a Line on a Plane., 79. The projection of the line segment joining the points (–1, 0, 3), and (2, 5, 1) on the line whose direction ratios are (6, 2, 3) is, , x -3 y-3 z, p, =, = at angles of, each, are, 2, 1, 1, 3, , 22, (d) 3, 7, The shortest distance between the Z-axis and the line, x + y + 2z - 3 = 0, 2x + 3y + 4z - 4 = 0 is, , (a) 6, , (b) 7, , (c), , (a), , x y z x y z, = = ; = =, 1 2 1 1 1 2, , 80., , (b), , x y, z x, y, z, = = ;, = =, 1 2 -1 -1 1 -2, , 1, (c) 0, (d) 1, 2, 81. The distance between two points P and Q is d and the, length of their projections of PQ on the co-ordinate planes, are d1, d2, d3. Then d12 + d 22 + d32 = kd 2 where ‘k’ is, , x y z x, y, z, = = ; =, =, 1 2 -1 1 -1 -2, (d) None of the above, 75. The locus of a point, such that the sum of the squares of its, distances from the planes x + y + z = 0, x – z =0 and, x – 2y + z = 0 is 9, is, , (c), , (a), , x 2 + y2 + z2 = 3, , (b) x 2 + y 2 + z 2 = 6, , (c), , x 2 + y2 + z2 = 9, , (d) x 2 + y 2 + z 2 = 12, , 76. If the angle q between the line, the plane 2x – y +, sin q =, , x +1, y -1, z-2, =, =, and, 1, 2, 2, , l z + 4 = 0 is such that, , 1, then the value of l is, 3, , 5, 3, -3, -4, (b), (d), (c), 3, 5, 4, 3, The equation of the plane through the line of intersection of, planes ax + by + cz + d = 0, a ' x + b' y + c' z + d' = 0 and, , (a), , 77., , (a), , (a) 2, , (b), , (a) 1, , (b) 5, , (c) 3, , (d) 2, , 82. If two points are P (7, –5, 11) and Q (–2, 8, 13), then the, projection of PQ on a straight line with direction cosines, 1 2 2, , , is, 3 3 3, , (a), , 1, 2, , (b), , 26, 3, , (c), , 4, 3, , (d) 7, , 83. What is the length of the projection of 3iˆ + 4jˆ + 5kˆ on the, xy-plane ?, (a) 3, (b) 5, (c) 7, (d) 9, 84. The projection of line joining (3, 4, 5) and (4, 6, 3) on the line, joining (–1, 2, 4) and (1, 0, 5) is –, (a), , 4, 3, , (b), , 2, 3, , (c), , 8, 3, , (d), , 1, 3

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EBD_7762, 360, , 85., , 86., , MATHEMATICS, , Consider the following statements:, I. Equations ax + by + cz + d = 0,, a'x + b'y + c'z + d' = 0 represent a straight line., II. Equation of the form, x - a y -b z - g, =, =, l, m, n, represent a straight line passing through the point (a, b, g), and having direction ratio proportional to l, m, n., Which of the statements given above is/are correct ?, (a) Only I, (b) Only II, (c) Both I and II, (d) Neither I nor II, Let L be the line of intersection of the planes 2x + 3y + z = 1, and x + 3y + 2z = 2. If L makes an angle a with the positive, x-axis, then cos a equals, , 87., , 88., , 1, , 1, , 1, 2, 2, 3, The projections of the segment PQ on the co-ordinate axes, are –9, 12, –8 respectively. The direction cosines of the line, PQ are, -9 12 -8, ,, ,, > (b) < -9,12, - 8 >, (a) <, 17 17 17, , (a) 1, , (b), , (c), , (d), , 9, 12 -8, 9 12 -8, ,, ,, > (d) < - ,, ,, >, 289 289 289, 17 17 17, The projections of a line segment on the coordinate axes are, 12, 4, 3. The direction cosine of the line are:, , (c), , <-, , (a), , ,, , 12, 4 3, ,, ,, 13 13 13, , (b), , 12, 4 3, ,, ,, 13 13 13, , 12 4 3, , ,, (d) None of these, 13 13 13, The lines x = ay + b, z = cy +d and x = a¢y + b¢, z = c¢y +d¢ are, perpendicular if, (a) aa¢ +bb¢ + cc¢ + 1 = 0 (b) aa¢+bb¢+1 = 0, (c) bb¢+cc¢+1 = 0, (d) aa¢ + cc¢ +1 = 0, The shortest distance between the z-axis and the line,, x + y + 2z - 3 = 0 , 2x + 3y + 4z - 4 = 0 is, , (c), , 89., , 90., , (a) 1, 91., , 92., , 93., , (b) 2, , (c) 3, , (d) None, , The equation of the right bisector plane of the segment, joining (2, 3, 4) and (6, 7, 8) is, (a) x + y + z + 15 = 0, (b) x + y + z – 15 = 0, (c) x – y + z – 15 = 0, (d) None of these, The equation of the plane which bisects the angle between, the planes 3x – 6y + 2z + 5 = 0 and 4x – 12y + 3z – 3 = 0 which, contains the origin is, (a) 33x – 13y + 32z + 45 = 0 (b) x – 3y + z – 5 = 0, (c) 33x + 13y + 32z + 45 = 0 (d) None of these, The lines which intersect the skew lines y = mx, z = c;, y = – mx, z = – c and the x – axis lie on the surface, (a) cz = mxy, (b) xy = cmz, (c) cy = mxz, (d) None of these, , BEYOND NCERT, Topic 7 : Equation of a Sphere, Equation of Concentric, Spheres, Condition of Tangency of a Plane to a Sphere,, Plane Section of a Sphere, Intersection of two Spheres,, Condition for Orthogonality of two Spheres,, Intersection of Straight Line and a Sphere, 94. The shortest distance from the plane 12 x + 4 y + 3z = 327, to the sphere x 2 + y 2 + z 2 + 4 x - 2 y - 6 z = 155 is, (a) 39, , (b) 26, , (c), , 11, , 4, 13, , (d) 13, , 95. The equation of the sphere whose centre has the position, vector (3$i + 6 $j – 4k$ ) and which touches the plan e, r $ $ $, r × (2i – 2 j – k ) = 10 is, r, r, (a) | r – (3$i + 6 $j – 4k$ ) | = 2 (b) | r – (3$i + 6 $j + 4k$ ) | = 2, r, (c) | r – (3$i + 6 $j – 4k$ ) | = 4 (d) None of these, r r r r, 96. Chord AB is a diameter of the sphere | r – 2i – j + 6k | = 18 ., If the coordinates of A are (3, 2, – 2), then the coordinates of, B are, (a) (1, 0, 10), (b) (1, 0, – 10), (c) (– 1, 0, 10), (d) None of these, uur r, 97. Radius of the circle r 2 + r × (2$i – 2 $j – 4k$ ) – 19 = 0,, r $ $ $, r × (i – 2 j + 2k ) + 8 = 0, is, (a) 5, (b) 4, (c) 3, (d) 2, 98. The vector form of the sphere, 2(x2+ y2 + z2) – 4x + 6y + 8z – 5 = 0 is, r r r, r r, 2, (a) r × [r - (2i + j + k)] =, 5, r r, r, r r, 1, (b) r × [r - (2i - 3 j - 4k)] =, 2, r r, r, r r, 5, (c) r × [r - (2i + 3 j + 4k)] =, 2, r, r, r, r r, 5, (d) r × [r - (2i - 3 j - 4k)] =, 2, 99. If the plane 2ax – 3ay + 4az + 6 = 0 passes through the, midpoint of the line joining the centres of the spheres, x 2 + y 2 + z 2 + 6 x - 8 y - 2 z = 13 and, x 2 + y 2 + z 2 - 10 x + 4 y - 2 z = 8 then a equals, (a) – 1, (b) 1, (c) – 2, (d) 2, 100. The locus of a point, such that the sum of the squares of its, distances from the planes x + y + z = 0, x – z =0 and x – 2y +, z = 0 is 9, is, (a) x 2 + y 2 + z 2 = 3, (b) x 2 + y 2 + z 2 = 6, , (c), , x 2 + y2 + z2 = 9, , (d) x 2 + y 2 + z 2 = 12, , 101. If the radius of the sphere, x2 + y2 + z2 – 6x – 8y + 10z + l = 0 is unity, what is the value, of l?, (a) 49, (b) 7, (c) – 49, (d) – 7

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THREE DIMENSIONAL GEOMETRY, , 361, , 102. If (2, 3, 5) is one end of a diameter of the sphere, x2 + y2 + z2 – 6x – 12y – 2z + 20 = 0, then the cooordinates of, the other end of the diameter are, (a) (4, 3, 5), (b) (4, 3, – 3), (c) (4, 9, – 3), (d) (4, –3, 3), 103. Under what condition does the equation, x2 + y2 + z2 + 2ux + 2uy + 2wz + d = 0 represent a real sphere?, (a) u2 + v2 + w2 = d2, (b) u2 + v2 + w2 > d, (c) u2 + v2 + w2 < d, (d) u2 + v2 + w2 < d2, 104. The radius of the sphere, x 2 + y 2 + z 2 = 49 , 2x + 3y - z - 5 14 = 0 is, , (a), (b) 2 6, (c) 4 6, (d) 6 6, 6, 105. Two spheres of radii 3 and 4 cut orthogonally The radius of, common circle is, (a) 12, , (b), , 12, 5, , 12, 5, , (c), , (d), , 12, , 106. The equation of the sphere circumscribing the tetrahedron, x y z, whose faces are x = 0, y= 0 z = 0 and + + = 1 is, a b c, 2, 2, 2, 2, 2, 2, (a) x + y + z = a + b + c, (b), , 107. The plane 2x – 2y + z + 12 = 0 touches the sphere, x2 + y2 + z2 – 2x – 4y + 2z – 3 = 0 at the point, (a) (1, – 4, – 2), (b) (–1, 4, – 2), (c) (–1, – 4, 2), (d) (1, 4, – 2), 108. What is the centre of the sphere ax2 + by2 + cz2 – 6x = 0 if the, radius is 1 unit?, (a) (0, 0, 0), (b) (1, 0, 0), (c) (3, 0, 0), (d) cannot be determined as values of a,b, c are unknown, 109. The intersection of the spheres, x 2 + y 2 + z 2 + 7 x - 2 y - z = 13 and, x 2 + y 2 + z 2 - 3x + 3 y + 4 z = 8 is the sa me as the, intersection of one of the sphere and the plane, (a) 2 x - y - z = 1, (b) x - 2 y - z = 1, (c) x - y - 2 z = 1, (d) x - y - z = 1, 110. The radius of the circle in which the sphere, x 2 + y 2 + z 2 + 2 x - 2 y - 4 z - 19 = 0 is cut by the plane, x + 2 y + 2 z + 7 = 0 is, (a) 4, (b) 1, (c) 2, (d) 3, , x 2 + y 2 + z 2 - ax - by - cz = 0, , (c) x 2 + y 2 + z 2 - 2ax - 2by - 2cz = 0, (d) None of these, , Exercise 2 : Exemplar & Past Year MCQs, 5., , NCERT Exemplar MCQs, 1., , Distance of the point (a, b, g) from y-axis is, (a) b, (b) | b |, (c) | b | + | g |, , 2., , (d), , If the directions cosines of a line are k, k, k, then, (a) k > 0, (b) 0 < k < 1, (c) k = 1, , 3., , 4., , 6., , a2 + g 2, , (d) k =, , 1, 3, , or -, , 1, , 7., , 3, , r æ2, 3, 6 ö, The distance of the plane r . ç iˆ + ˆj - kˆ ÷ = 1 from the, 7, 7 ø, è7, origin is, (a) 1, (b) 7, , 8., , 1, (c), (d) None of these, 7, The sine of the angle between th e straight line, , x-2 y-3 z-4, =, =, and the plane 2x – 2y + z = 5 is, 3, 4, 5, , (a), , 10, 6 5, , (b), , 4, 5 2, , (c), , 2 3, 5, , (d), , 2, 10, , The reflection of the point (a, b, g) in the XY-plane is, (a) (a, b, 0), (b) (0, 0, g), (c) (–a, –b, g), (d) (a, b, –g), The area of the quadrilateral ABCD where A(0, 4, 1),, B(2, 3, –1), C(4, 5, 0) and D (2, 6, 2) is equal to, (a) 9 sq. units, (b) 18 sq. units, (c) 27 sq. units, (d) 81 sq. units, The locus represented by xy + yz = 0 is, (a) a pair of perpendicular lines, (b) a pair of parallel lines, (c) a pair of parallel planes, (d) a pair of perpendicular planes, If the plane 2x – 3y + 6z – 11 = 0 makes an angle sin–1 a, with X-axis, then the value of a is, (a), , 3, 2, , (b), , 2, 3, , (c), , 2, 7, , (d), , 3, 7, , Past Year MCQs, 9., , The image of the line, , x -1 y - 3 z - 4, in the plane, =, =, 3, 1, -5, , 2 x - y + z + 3 = 0 is the line:, , [JEE MAIN 2014, A, BN]

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EBD_7762, 362, , MATHEMATICS, , (a), , x-3 y +5 z - 2, =, =, 3, 1, -5, , (b), , x-3 y +5 z - 2, =, =, -3, -1, 5, , x+3 y -5 z - 2, x+3 y -5 z + 2, =, =, =, =, (d), 3, 1, -5, -3, -1, 5, The angle between the lines whose direction cosines satisfy, , (c), 10., , the equations l + m + n = 0 and l 2 + m 2 + n 2 is, [JEE MAIN 2014, A], p, p, p, p, (b), (c), (d), 6, 3, 2, 4, The equation of the right bisector plane of the segment, joining (2, 3, 4) and (6, 7, 8) is, [BITSAT 2014, A], (a) x + y + z + 15 = 0, (b) x + y + z – 15 = 0, (c) x – y + z – 15 = 0, (d) None of these, , (a), 11., , 12. Find the angle between the line, plane 10x + 2y – 11z = 3., , x +1 y z - 3, = =, and the, 2, 3, 6, [BITSAT 2014, C], , æ 8ö, (a) sin -1 ç ÷, è 21ø, , æ 5ö, (b) sin -1 ç ÷, è 21ø, , æ 7ö, (c) sin -1 ç ÷, è 21ø, , æ 1ö, (d) sin -1 ç ÷, è 21ø, , 13. The equation of the plane containing the line 2x – 5y + z =, 3; x + y + 4z = 5, and parallel to the plane, x + 3y + 6z = 1,, is :, [JEE MAIN 2015, A], (a) x + 3y + 6z = 7, (b) 2x + 6y + 12z = – 13, (c) 2x + 6y + 12z = 13, (d) x + 3y + 6z = –7, 14. The distance of the point (1, 0, 2) from the point of, intersection of the line, x – y + z = 16, is, , x - 2 y +1 z - 2, =, =, and the plane, 3, 4, 12, [JEE MAIN 2015, A], , (a) 3 21 (b) 13, (c) 2 14, (d) 8, 15. A line makes the same angle q with each of the X and Zaxes. If the angle b, which it makes with Y-axis, is such, that sin 2 b = 3sin2 q, then cos2 q equals, [BITSAT 2015, A], (a) 2/5, (b) 1/5, (c) 3/5, (d) 2/3, y, z, 16. Two lines L1 : x = 5,, =, ,, 3 - a -2, y, z, =, are coplanar. Then, a can take, -1 2 - a, value (s), [BITSAT 2015, C], (a) 1, 4, 5 (b) 1, 2, 5, (c) 3, 4, 5 (d) 2, 4, 5, 17. The distance of the point (1, –5, 9) from the plane x – y +, z = 5 measured along the line x = y = z is :, [JEE MAIN 2016, A], L2 : x = a,, , (a), , 10, 3, , (b), , 20, 3, , (c) 3 10, , (d) 10 3, , x -3 y+2 z+4, =, =, lies in the plane, lx + my – z =, -1, 2, 3, 9, then l2 + m2 is equal to :, [JEE MAIN 2016, A], (a) 5, (b) 2, (c) 26, (d) 18, 19. If the image of the point P(1, –2, 3) in the plane,, 2x + 3y – 4z + 22 = 0, measured par allel to, , 18. If the line,, , line,, , x y z, =, =, is Q, then PQ is equal to :, 1 4 5, [JEE MAIN 2016, S, BN], , (a), , (c) 2 42, (d), 6 5 (b) 3 5, 42, 20. The distance of the point (1, 3, –7) from the plane passing, through the point (1, –1, –1), having normal, perpendicular to both the lines, , x -1 y + 2 z - 4, and, =, =, 1, -2, 3, , x - 2 y +1 z + 7, =, =, , is :, 2, -1, -1, , (a), , 10, , (b), , 20, , [JEE MAIN 2017, A], (c), , 10, , 5, , (d), , 74, 83, 83, 74, 21. If the line through the points A (k, 1, –1) and B (2k, 0, 2) is, perpendicular to the line through the points B and, C (2 + 2k , k, 1), then what is the value of k?, [BITSAT 2017, A], (a) –1, (b) 1, (c) –3, (d) 33, 22. The ratio in which the join of ( 2, 1, 5) and (3, 4, 3) is, 1, divided by the plane x + y – z = is: [BITSAT 2017, A], 2, (a) 3 : 5 (b) 5 : 7, (c) 1 : 3, (d) 4 : 5, 23. The length of the projection of the line segment joining, the points (5, –1, 4) and (4, –1, 3) on the plane,, x + y + z = 7 is:, [JEE MAIN 2018A, BN], (a), , 2, 3, , (b), , 1, 3, , 2, 3, , (c), , (d), , 2, 3, , 24. If L 1 is the line of intersection of the planes, 2x - 2y + 3z - 2 = 0, x - y + z + 1 = 0 and L2 is the line, of intersection of the planes, , x + 2y - z - 3 = 0,, , 3x - y + 2z - 1 = 0 , then the distance of the origin from the, plane, containing the lines L1 and L2, is:, [JEE MAIN 2018, S], (a), , 1, 3 2, , (b), , 1, 2 2, , (c), , 1, 2, , (d), , 1, 4 2, , 25. The projection of line joining (3, 4, 5) and (4, 6, 3) on the, line joining (–1, 2, 4) and (1, 0, 5) is, [BITSAT 2018 A, BN], (a), , 4, 3, , (b), , 2, 3, , (c), , 8, 3, , (d), , 1, 3

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THREE DIMENSIONAL GEOMETRY, , 363, , Exercise 3 : Try If You Can, 1., , P is a point and PM, PN are perpendiculars from P to the, ZX and XY planes respectively. If OP makes angles q, a, b,, g with the plane OMN and the XY, YZ, ZX plane respectively, , 7., , (a) 0, , (b) 1, , x - 2 y- 2 z -3, =, =, and which passes through the point, 1, -1, -2, of intersection of l and p is –, , (c) –1, , (d) None of these, , (a), , then sin 2 q (cosec2 a + cosec 2b + cosec 2 g ) is equal to, , 2., , 3., , In R3, let L be a straight line passing through the origin., Suppose that all the points on L are at a constant distance, from the two planes P 1 : x + 2y – z + 1 = 0 and, P2 : 2x – y + z – 1 = 0. Let M be the locus of the feet of the, perpendiculars drawn from the points on L to the plane P1., Which of the following points lie (s) on M?, (a), , 2ö, æ 5, ç 0, 6 , – 3 ÷, è, ø, , (b), , (c), , 1ö, æ 5, ç – ,0, ÷, 6ø, è 6, , 1, 2, (d) æç – ,0, ö÷, 3ø, è 3, , 8., , 1 1ö, æ 1, ç– , – , ÷, 3 6ø, è 6, , If the lattice point P ( x, y , z ); x, y , z > 0 and x, y, z Î I, with least value of z such that the 'P' lies on the planes, 7 x + 6 y + 2 z = 272 and x - y + z = 16, then the value of, , (b) 3, , (c) 2, , (d) None of these, , The line which contains all points (x, y, z) which are of the, form (x, y, z) = (2, –2, 5) + l (1, –3, 2) intersects the plane, 2x – 3y + 4z = 163 at P and intersects the YZ-plane at Q. If the, , 6., , (b) 95, , (c) 27, (d) None of these, From a point P(l, l, l), perpendiculars PQ and PR are drawn,, respectively, on the lines y = x, z = 1 and y = – x, z = – 1. If, ÐQPR is a right angle, then the possible value(s) of l is/are, (a) 2, , (b) 1, , (c) – 1, , (d) –, , 9., , x + 2 y +1 z +1, x - 2 y -1 z - 1, =, =, =, =, (d), -1, -1, 2, 1, 2, 1, The point of intersection of the line passing through (0, 0, 1), and intersecting the lines x + 2y + z = 1, – x + y – 2z = 2 and, x + y = 2, x + z = 2 with xy plane is, , (a), , æ5 1 ö, çè , – , 0÷ø, 3 3, , (b) (1, 1, 0), , (c), , æ2 1 ö, çè , – , 0÷ø, 3 3, , (d), , æ 5 1 ö, çè – , , 0÷ø, 3 3, , r r r, r, r, Let A( a ) and B( b ) be points on two skew line r = a + l, r, r, r, and r = b + u q and the shortest distance between the, ur, r, skew line is 1, where p and q are unit vectors forming, , 1, (b) 2, 2, (c) 1, (d) l Î R – {0}, 10. The point P is the intersection of the straight line joining, the points Q(2, 3, 5) and R(1, – 1, 4) with the plane, 5x – 4y – z = 1. If S is the foot, of the perpendicular drawn, from the point T(2, 1, 4) to QR, then the length of the line, segment PS is, , (a), , (a), , 2, , y, z, If lines x = y = z and x =, =, and third line passing, 2, 3, , through (1, 1, 1) form a triangle of area 6 units, then the, point of intersection of third line with the second line will be, (a) (1, 2, 3), (b) (2, 4, 6), , æ 4 8 12 ö, (c) çè , , ÷ø, 3 3 3, , x -1 y - 3 z - 5, =, =, 3, 5, -1, , 1, 2, units. If an angle between AB and the line of shortest, distance is 60°, then AB =, , distance PQ is a b , where a, b Î N and a > 3, then, (a + b) is equal to, , 5., , (b), , adjacent sides of a parallelogram enclosing an area of, , (a) 0, , (a) 23, , x - 2 y -1 z - 1, =, =, 3, 5, -1, , (c), , ( x + y + z - 42) is equal to, , 4., , Equation of line in the plane p º 2x – y + z – 4 = 0 which, is perpendicular to the line l whose equation is, , 11., , 2, , (b), , 2, , (c) 2, , (d), , 2 2, , x+6, y + 10, z + 14, =, =, is the hypotenuse of an, 5, 3, 8, isosceles right-angled triangle whose opposite vertex is, (7, 2, 4). Then which of the folloiwng is not the side of the, triangle?, , The line, , (a), (d) None of these, , 1, , x–7, z–4, y–2, =, =, (b), 2, 6, –3, , x–7, y–2, z–4, =, =, 3, 6, 2

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EBD_7762, 364, , MATHEMATICS, , (c), , x–7, y–2, z–4, =, =, (d) None of these, 3, 5, –1, , é1 2 3 ù, 12. Consider matrices A = ê 4 1 2 ú ; B =, ê, ú, êë1 -1 1 úû, , é2 1 3 ù, ê 4 1 -1ú ;, ê, ú, êë 2 2 3 úû, , é14 ù, C = êê12 úú ; D =, êë 2 úû, , é13ù, é xù, ê11ú ; X = ê y ú such that solutions of, ê ú, ê ú, êë14 úû, êë z úû, equation AX = C and BX = D represents two points, P(x1, y1, z1) and Q(x2, y2, z2) respectively in three dimensional, space. If P'Q' is the reflection of the line PQ in the plane, x + y + z = 9, then the point which does not lie, on P'Q' is :, , (a) (3, 4, 2) (b) (5, 3, 4), , (c) (7, 2, 3), , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, , (c), (c), (d), (a), (a), (a), (b), (c), (a), (a), (d), , 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, , (b), (a), (b), (d), (b), (d), (c), (d), (a), (d), (b), , 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, , (b), (a), (a), (c), (b), (c), (a), (b), (b), (a), (b), , 1, 2, 3, , (d), (d), (a), , 4, 5, 6, , (d), (d), (a), , 7, 8, 9, , (d), (c), (c), , 1, 2, , (b), (b), , 3, 4, , (d), (a), , 5, 6, , (c), (b), , (d) (1, 5, 6), , 13. Projection of the line, , x –1, z –3, y +1, =, =, on the plane x, 2, 4, –1, , + 2y + z = 9 is the line of intersection of plane x + 2y + z, = 9 with the plane, (a) 9x + 2y – 5z + 8 = 0 (b) 3x + 4y – 5z + 16 = 0, (c) x + 3y – 7z + 4 = 0, (d) 9x – 2y – 5z + 4 = 0, 14. If the distance between the plane Ax – 2y + z = d and the, plane containing the lines, , x –1, y–2, z –3, =, =, and, 2, 3, 4, , x–2, y –3, z–4, =, =, is 6 , then the value of |d| is, 3, 4, 5, (a) 4, (b) 5, (c) 6, (d) 7, 2, 2, 2, 15. The intersection of the spheres: x + y + z + 7x – 2y – z = 13, and x2 + y2 + z2 – 3x + 3y + 4z = 8 is the same as the, intersection of one of the spheres and the plane, (a) x – y – z = 1, (b) x – 2y – z = 1, (c) x – y – 2z = 1, (d) 2x – y – z = 1, , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (b), (d) 67 (c), 34 (a) 45, 56, (d), (d) 68 (c), 35 (b) 46, 57, (b), (a) 69 (b), 36 (d) 47, 58, (b), (b) 70 (a), 37 (c) 48, 59, (b), (b) 71 (a), 38 (b) 49, 60, (a), (a) 72 (b), 39 (b) 50, 61, (d), (a) 73 (c), 40 (c) 51, 62, (c), (c) 74 (b), 41 (a) 52, 63, (c), (b) 75 (c), 42 (a) 53, 64, (a), (a) 76 (a), 43 (d) 54, 65, (d), (d) 77 (c), 44 (d) 55, 66, Exercise 2 : Exemplar & Past Year MCQs, (a), (a) 19 (c), 10 (c) 13, 16, (b), (d) 20 (c), 11 (b) 14, 17, (c), (b) 21 (d), 12 (a), 15, 18, Exercise 3 : Try If You Can, (b), (b), (b) 13 (d), 7, 9, 11, (a) 10, (a), (a) 14 (c), 8, 12, , 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, , (a), (c), (a), (d), (d), (b), (a), (c), (c), (d), (c), , 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, , (d), (b), (b), (d), (c), (d), (c), (b), (b), (d), (c), , 22, 23, 24, , (b), (c), (a), , 25, , (a), , 15, , (d), , 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, , (c), (a), (c), (b), (b), (b), (b), (b), (d), (a), (d)

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28, , LINEAR PROGRAMMING, , Chapter, , Trend, Analysis, , of JEE Main and BITSAT (Year 2010-2018), , Number of Questions, , 4, , 3, JEE MAIN, BITSAT, , 2, , 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 0, 2, , Critical Concepts, , Mathematical Formulation of LPP,, Graphical Method of Solving LPP., , Rating of, Difficulty Level, , 2/5, , CUS, (Chapter Utility Score), Out of 10, 4

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LINEAR PROGRAMMING, , 367

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EBD_7762, 368, , 1., , 2., , 3., , 4., , 5., , 6., , 7., , 8., , MATHEMATICS, , L.P.P is a process of finding, (a) Maximum value of objective function, (b) Minimum value of objective function, (c) Optimum value of objective function, (d) None of these, L.P.P. has constraints of, (a) one variables, (b) two variables, (c) one or two variables, (d) two or more variables, The solution set of constraints x + 2y ³ 11,, 3x + 4y £ 30, 2x + 5y £ 30 and x ³ 0 , y ³ 0 , includes, the point, (a) (2, 3) (b) (3, 2), (c) (3, 4), (d) (4, 3)., Corner points of feasible region of inequalities gives, (a) optional solution of L.P.P., (b) objective function, (c) constraints., (d) linear assumption, The optimal value of the objective function is attained at, the points, (a) Given by intersection of inequations with axes only, (b) Given by intersection of inequations with x- axis only, (c) Given by corner points of the feasible region, (d) None of these., x y, x y, Consider + ³ 1 and +, £ 1, x, y ³ 0. Then, 2 4, 3 2, number of possible solutions are :, (a) Zero, (b) Unique, (c) Infinite, (d) None of these, Which of the following statement is correct?, (a) Every L.P.P. admits an optimal solution, (b) A L.P.P. admits a unique optimal solution, (c) If a L.P.P. admits two optimal solutions, it has an, infinite number of optimal solutions, (d) The set of all feasible solutions of a L.P.P. is not a, convex set., If a point (h, k) satisfies an inequation ax + by ³ 4, then, the half plane represented by the inequation is, (a) The half plane containing the point (h, k), but excluding the points on ax + by = 4, (b) The half plane containing the point (h, k), and the points on ax + by = 4, (c) Whole xy-plane, (d) None of these, , 9., , The maximum value of z = 5x + 2y, subject to the, constraints x + y £ 7, x + 2y £ 10, x, y ³ 0 is, , (a) 10, (b) 26, (c) 35, 10. Shaded region is represented by, , (d) 70, , Y, , 4x – 2y = –3, A=(0,3/2), (3,0), B, O, (– 3 , 0), 4, , X, , (a) 4x – 2y £ 3, (b) 4x – 2y £ –3, (c) 4x – 2y ³ 3, (d) 4x – 2y ³ –3, 11. The maximum value of xy subject to x + y = 8 is:, (a) 8, (b) 16, (c) 20, (d) 24, 12. The maximum value of P = x + 3y such that, 2x + y £ 20, x + 2y £ 20, x ³ 0, y ³ 0 is, (a) 10, (b) 60, (c) 30, (d) None of these, 13. For the following feasible region, the linear constraints are, Y, , (0,6), 0, 11, 3, O, , (4,0), , (11,0), , X, , (a) x ³ 0, y ³ 0, 3x + 2y ³ 12, x + 3y ³ 11, (b) x ³ 0, y ³ 0, 3x + 2y £ 12, x + 3y ³ 11, (c) x ³ 0, y ³ 0, 3x + 2y £ 12, x + 3y £ 11, (d) None of these, 14. Feasible region for an LPP is shown shaded in the following, figure. Minimum of Z = 4 x + 3 y occurs at the point., Y, , D (0, 8), , Feasible, Region, C (2, 5), B (4, 3), , O, , (a) (0, 8) (b) (2, 5), , A (9, 0), , (c) (4, 3), , X, , (d) (9, 0)

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LINEAR PROGRAMMING, , 369, , 15. The feasible region for LPP is shown shaded in the figure., Let f = 3 x – 4 y be the objective function, then maximum, value of f is, (6,16), Y, (6,12), (0,4), (0,0), , O, , (6,0), , X, , (a) 12, (b) 8, (c) 0, (d) –18, 16. Which of these terms is not used in a linear programming, problem?, (a) Slack variables, (b) Objective function, (c) Concave region, (d) Feasible solution, 17. Z = 6x + 21y, subject to x + 2y ³ 3, x + 4y ³ 4, 3x + y ³ 3,, x ³ 0, y ³ 0. The minimum value of Z occurs at, æ 1ö, çè 2, ÷ø (d) (0, 3), 2, 18. If the constraints in a linear programming problem are, changed, (a) The problem is to be re-evaluated, (b) Solution is not defined, (c) The objective function has to be modified, (d) The change in constraints is ignored, 19. Maximize Z = 4x + 6y, subject to 3x + 2y £ 12, x + y ³ 4,, x, y ³ 0, is, (a) 16 at (4,0), (b) 24 at (0, 4), (c) 24 at (6, 0), (d) 36 at (0, 6), 20. Shamli wants to invest `50,000 in saving certificates and, PPE. She wants to invest atleast `15,000 in saving, certificates and at least `20,000 in PPF. The rate of interest, on saving certificates is 8% p.a. and that on PPF is 9%, p.a. Formulation of the above problem as LPP to determine, maximum yearly income, is, , (a) (4, 0) (b) (28, 8), , (c), , (a) Maximize Z = 0.08x + 0.09y, Subject to, x + y £ 50,000, x ³ 15000, y ³ 20,000, (b) Maximize Z = 0.08x + 0.09y, Subject to, x + y £ 50,000, x ³ 15000, y £ 20,000, (c) Maximize Z = 0.08x + 0.09y, Subject to, x + y £ 50,000, x £ 15000, y ³ 20,000, (d) Maximize Z = 0.08x + 0.09y, Subject to, x + y £ 50,000, x £ 15000, y £ 20,000, 21. Corner points of the feasible region for an LPP are (0, 2), (3, 0) (6, 0), (6, 8) and (0, 5). Let F = 4x + 6y be the, objective function., The minimum value of F occurs at, (a) (0, 2) only, (b) (3, 0) only, (c) the mid-point of the line segment joining the points, (0, 2) and (3, 0) only, (d) any point on the line segment joining the points, (0, 2) and (3, 0), , 22. If the number of available constraints is 3 and the number, of parameters to be optimized is 4, then, (a) The objective function can be optimized, (b) The constraint are short in number, (c) The solution is problem oriented, (d) None of these, 23. The point at which the maximum value of ( 3x + 2y) subject, to the constraints x + y £ 2, x ³ 0, y ³ 0 is obtained, is, (a) (0, 0), (b) (1.5, 1.5), (c) (2, 0), (d) (0, 2), 24. For the constraint of a linear optimizing function z = x1 + x2,, given by x1+ x2 £ 1, 3x1+ x2 ³ 3 and x1, x2 ³ 0,, (a) There are two feasible regions, (b) There are infinite feasible regions, (c) There is no feasible region, (d) None of these., 25. The maximum value of z = 3x + 2y subject to, x + 2y ³ 2, x + 2y £ 8, x, y ³ 0 is :, (a) 32, (b) 24, (c) 40, (d) None of these, 26. The area of the feasible region for the following constraints, 3y + x ³ 3, x ³ 0, y ³ 0 will be, (a) Bounded, (b) Unbounded, (c) Convex, (d) Concave, 27. For the constraints of a L.P. problem given by, x1 + 2x2 £ 2000, x1 + x2 £ 1500 and x 2 £ 600 and x1, x2, ³ 0, which one of the following points does not lie in the, positive bounded region?, (a) (1000, 0), (b) (0, 500), (c) (2, 0), (d) (2000, 0), 28. The maximum value of z = 4x + 2y subject to constraints, 2x + 3y £ 18, x + y ³10 and x, y ³ 0, is, (a) 36, (b) 40, (c) 20, (d) None, 29. The maximum value of P = x + 3y such that 2x + y £ 20,, x + 2y £ 20, x ³ 0, y ³ 0 is, (a) 10, (b) 60, (c) 30, (d) None, 30. Inequations 3x – y ³ 3 and 4x – y ³ 4, (a) Have solution for positive x and y, (b) Have no solution for positive x and y, (c) Have solution for all x, (d) Have solution for all y, 31. The maximum value of z = 6x + 8y subject to constraints, 2x + y £ 30, x + 2y £ 24 and x ³ 0, y ³ 0 is, (a) 90, (b) 120, (c) 96, (d) 240, 32. A wholesale merchant wants to start the business of cereal, with ` 24000. Wheat is ` 400 per quintal and rice is ` 600, per quintal. He has capacity to store 200 quintal cereal. He, earns the profit ` 25 per quintal on wheat and ` 40 per, quintal on rice. If he store x quintal rice and y quintal, wheat, then for maximum profit, the objective function is, (a) 25 x + 40y, (b) 40x + 25y, (c) 400x + 600y, , (d), , 400, 600, x+, y, 40, 25, , 33. The constraints, –x1 + x2 £ 1, –x1 +3x2 £ 9, x1,x2 ³ 0 define on, (a) Bounded feasible space, (b) Unbounded feasible space, (c) Both bounded and unbounded feasible space, (d) None of these

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EBD_7762, 370, , 34., , MATHEMATICS, , The true statement for the graph of inequations, 3x + 2y £ 6 and 6x + 4y ³ 20 , is, (a) Both graph are disjoint (b) Both contain (0,3), (c) Both contain point (1, 1)(d) None of these, , 35. Graph of the constraints, , x y, + £ 1, x ³ 0, y ³ 0 is, 3 4, , (a), , 4, , 3, , X, , (b), , 36. The lines 5x + 4y ³ 20, x £ 6, y £ 4 form, (a) A square, (b) A rhombus, (c) A triangle, (d) A quadrilateral, 37. Maximum value of 12x +3y subject to constraints x ³ 0,, y ³ 0, x + y £ 5 and 3x + y £ 9 is, (a) 15, (b) 36, (c) 60, (d) 40, 38. The graph of inequations x £ y and y £ x + 3 is located in, (a) II quadrant, (b) I, II quadrants, (c) I, II, III quadrants, (d) II, III, IV quadrants, 39. The number of corner points of the L.P.P., Max Z = 20x + 3y subject to the constraints, x + y £ 5, 2x + 3y £ 12, x ³ 0, y ³ 0 are, (a) 4, (b) 3, (c) 2, (d) 1, 40. Consider the objective function Z = 40x + 50y. The, minimum number of constraints that are required to, maximize Z are, (a) 4, (b) 2, (c) 3, (d) 1, 41. The no. of convex polygon formed bounding the feasible, region of the L.P.P. Max. Z = 30x + 60y subject to the, constraints 5x + 2y £ 10, x + y £ 4, x ³ 0, y ³ 0 are, (a) 2, (b) 3, (c) 4, (d) 1, 42. Assertion : The region represented by the set, {(x, y) : 4 £ x 2 + y 2 £ 9} is a convex set., , 4, , 3, , X, , 43., (c), , 44., , –3, , X, , 45., –4, , (d), , –3, , X, , 46., –4, , Reason : The set {(x, y) : 4 £ x 2 + y 2 £ 9} represents the, region between two concentric circles of radii 2 and 3., (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., The solution region satisfied by the inequalities, x + y £ 5, x £ 4, y £ 4,, x ³ 0, y ³ 0, 5x + y ³ 5, x + 6y ³ 6 ,, is bounded by, (a) 4 straight lines, (b) 5 straight lines, (c) 6 straight lines, (d) unbounded, The region represented by the inequalities x ³ 6, y ³ 2,, 2x + y £ 10, x ³ 0, y ³ 0 is, (a) unbounded, (b) a polygon, (c) exterior of a triangle (d) None of these, A company manufactures two types of products A and B., The storage capacity of its godown is 100 units. Total, investment amount is ` 30,000. The cost price of A and B, are ` 400 and ` 900 respectively. Suppose all the products, have sold and per unit profit is ` 100 and ` 120 through A, and B respectively. If x units of A and y units of B be, produced, then two linear constraints and iso-profit line, are respectively, (a) x + y = 100; 4x + 9y = 300, 100x + 120y = c, (b) x + y £ 100; 4x + 9y £ 300, x + 2y = c, (c) x + y £ 100; 4x + 9y £ 300,100 x + 120y = c, (d) x + y £ 100; 9x + 4y £ 300, x + 2y = c, Which of the following cannot be considered as the, objective function of a linear programming problem?, (a) Maximize z = 3x + 2y, (b) Minimize z = 6x + 7y + 9z, (c) Maximize z = 2x, (d) Minimize z = x2 + 2xy + y2

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LINEAR PROGRAMMING, , 371, , 47. Inequation y – x £ 0 represents, (a) The half plane that contains the positive X-axis, (b) Closed half plane above the line y = x, which contains, positive Y-axis, (c) Half plane that contains the negative X-axis, (d) None of these, 48. Graph of the inequalities x ³ 0, y ³ 0, 2x + 3y ³ 6,, 3x + 2y ³ 6 is, , 3, , 2, , (d), 3, , X¢, , X, , 2, , 3, , 2, , (a), X¢, , 3, , 2, , X, , 49. A printing company prints two types of magazines A and, B. The company earns `10 and `15 on each magazine A, and B respectively. These are processed on three machines, I, II & III and total time in hours available per week on, each machine is as follows:, Magzine ® A(x) B(y) Time available, ¯ Machine, I, 2, 3, 36, II, 5, 2, 50, III, , 3, , 2, , (b), 3, , X¢, , 2, , X, , 3, , 2, , 2, , 6, , 60, , The number of constraints is, (a) 3, (b) 4, (c) 5, (d) 6, 50. Children have been invited to a birthday party. It is, necessary to give them return gifts. For the purpose, it, was decided that they would be given pens and pencils in, a bag. It was also decided that the number of items in a, bag would be atleast 5. If the cost of a pen is `10 and cost, of a pencil is `5, minimize the cost of a bag containing, pens and pencils. Formulation of LPP for this problem is, (a) Minimize C = 5x + 10y subject to x + y £ 10, x ³ 0, y ³ 0, (b) Minimize C = 5x + 10y subject to x + y ³10, x ³ 0, y ³ 0, (c) Minimize C = 5x + 10y subject to x + y ³ 5, x ³ 0, y ³ 0, (d) Minimize C = 5x + 10y subject to x + y £ 5, x ³ 0, y ³ 0, , (c), X¢, , 3, 2, , X, , Exercise 2 : Exemplar & Past Year MCQs, (a), (b), (c), (d), , NCERT Exemplar MCQs, 1., , The corner points of the feasible region determined by the, system of linear constraints are (0, 0), (0, 40), (20, 40),, (60, 20), (60, 0). The objective function is Z = 4x + 3y., Compare the quantity in column A and column B, Column A, , Column B, , Maximum of Z, , 325, , 2., , The quantity in column A is greater, The quantity in column B is greater, The two quantities are equal, The relationship cannot be determined on the basis, of the information supplied., The feasible region for an LPP is shown shaded in the, figure. Let Z = 3 x – 4 y be the objective function. Minimum, of Z occurs at

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EBD_7762, 372, , MATHEMATICS, , 8., , Y, (4, 10), , (6, 5), , Feasible, Region, , (0, 0) O, 3., 4., , 5., , 9., , (6, 8), , (0, 8), , (5, 0), , X, , (a) (0, 0) (b) (0, 8), (c) (5, 0), (d) (4, 10), Refer to question 2. Maximum of Z occurs at, (a) (5, 0) (b) (6, 5), (c) (6, 8), (d) (4, 10), Refer to question 2, maximum value of Z + minimum value, of Z is equal to, (a) 13, (b) 1, (c) –13, (d) –17, The feasible region for LPP is shown in the following figure., Let F = 3x – 4y be the objective function. Maximum value, of F is, (12, 6), , Refer to question 7. Maximum of F – minimum of F is, equal to, (a) 60, (b) 48, (c) 42, (d) 18, Corner points of the feasible region determined by the, system of linear constraints are (0, 3), (1, 1) and (3, 0) ., Let Z = px + qy, where p, q > 0. Condition on p and q so, that the minimum of Z occurs at (3, 0) and (1, 1) is, q, (a) p = 2 q, (b) p =, 2, (c) p = 3 q, (d) p = q, Past Year MCQs, x y, x y, + ³ 1 and + £ 1, x, y ³ 0. Then number of, 2 4, 3 2, possible solutions are :, [BITSAT 2014, A], (a) Zero, (b) Unique, (c) Infinite, (d) None of these, , 10. Consider, , n m, , 11., , Minimise Z = åå cij xij, m, , Subject to, , j=1 i =1, , å x ij = b j , j = 1, 2,..., n, i =1, , n, , å x ij = b j , i = 1, 2,..., m, , is a LPP with number of, , j=1, , constraints, (a) m – n, , (12, 0), , 6., , 7., , (a) 0, (b) 8, (c) 12, (d) –18, Refer to question 5. Minimum value of F is, (a) 0, (b) –16, (c) 12, (d) Does not exist, Corner points of the feasible region for an LPP are, (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let F = 4 x + 6 y be, the objective function., The minimum value of F occurs at, (a) (0, 2) only, (b) (3, 0) only, (c) the mid point of the line segment joining the points, (0, 2) and (3, 0)., (d) any point on the line segment joining the points, (0, 2) and (3, 0)., , [BITSAT 2015, C], , m, n, 12. The maximum value of z = 3x + 2y subject to, x + 2y ³ 2, x + 2y £ 8, x, y ³ 0 is :, [BITSAT 2017, A], (a) 32, (b) 24, (c) 40, (d) None of these, 13. Which of the following statements is correct?, [BITSAT 2018, C], (a) Every L.P.P. admits an optimal solution., (b) A L.P.P. admits a unique optimal solution., (c) If a L.P.P. admits two optimal solutions, it has an infinite, number of optimal solutions., (d) The set of all feasible solutions of a L.P.P. is not a, convex set., 14. If the constraints in a linear programming problem are, changed then, [BITSAT 2018, C], (a) The problem is to be re-evaluated., (b) Solution is not defined., (c) The objective function has to be modified., (d) The change in constraints is ignored., , (c) m + n, , (0, 4), , (b) mn, (d)

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LINEAR PROGRAMMING, , 373, , Exercise 3 : Try If You Can, 1., , 2., , 3., , 4., , 5., , An oil company required 12000, 20000 and 15000 barrels, of high-grade, medium grade and low grade oil,, respectively. Refinery A produces 100, 300 and 200 barrels, per day of high-grade, medium-grade and low-grade oil,, respectively, while refinery B produces 200, 400 and 100, barrels per day of high-grade, medium-grade and lowgrade oil, respectively. If refinery A costs ` 400 per day, and refinery B costs ` 300 per day to operate, then the, days should each be run to minimize costs while satisfying, requirements are, (a) 30, 60 (b) 60, 30, (c) 40, 60 (d) 60, 40, Every gram of wheat provides 0.1 g of proteins and 0.25 g, of carbohydrates. The corresponding values of rice are 0.05, g and 0.5 g respectively. Wheat costs ` 4 per kg and rice, ` 6. The minimum daily requirements of proteins and, carbohydrates for an average child are 50 g and 200 g, respectively. Then in what quantities should wheat and, rice be mixed in the daily diet to provide minimum daily, requirement of proteins and carbohydrates at minimum, cost, (a) 400, 200, (b) 300, 400, (c) 200, 400, (d) 400, 300, A shop-keeper deals in the sale of TVs and VCPs. He has, ` 5.2 lacs to invest. He has only space for 50 pieces. A TV, costs `. 20, 000/-, and a VCP costs ` 8, 000/- From a TV and, VCP he earns a profit of ` 1500/- and ` 800/- respectively., Assuming that he sells all the items that he purchases, the, number of TVs and VCPs he should buy in order to maximize, his profit, is equal to, (a) 60,000 (b) 55,000, (c) 51,000, (d) 47,000, The corner points of the feasible region determined by the, system of linear constraints are (0, 10), (5, 5) (15, 15),, (0, 20). Let Z = px + qy, where p, q > 0. Condition on p, and q so that the maximum of Z occurs at both the points, (15, 15) and (0, 20) is, (a) p = q (b) p = 2 q, (c) q = 2 p (d) q = 3 p, A brick manufacture has two depots A and B, with stocks, of 30000 and 20000 bricks respectively. He receive orders, from three builders P, Q and R for 15000, 20,000 and, 15000 bricks respectively. The cost (in `) of transporting, 1000 bricks to the builders from the depots as given in the, table., To, From, , Transportation cost, per 1000 bricks (in, `), P, Q, R, , A, , 40, , 20, , 20, , B, , 20, , 60, , 40, , The manufacturer wishes to find how to fulfill the order, so that transportation cost is minimum. Formulation of, the L.P.P., is given as, , 6., , (a) Minimize Z = 40x – 20y, Subject to, x + y ³ 15, x + y £ 30, x ³ 15, y £ 20, x ³ 0, y ³ 0, (b) Minimize Z = 40x – 20y, Subject to, x + y ³ 15, x + y £ 30, x £ 15, y ³ 20, x ³ 0, y ³ 0, (c) Minimize Z = 40x – 20y, Subject to, x + y ³ 15, x + y £ 30, x £ 15, y £ 20, x ³ 0, y ³ 0, (d) Minimize Z = 40x – 20y, Subject to, x + y ³ 15, x + y £ 30, x ³ 15, y ³ 20, x ³ 0, y ³ 0, A furniture manufacturer produces tables and bookshelves, made up of wood and steel. The weekly requirement of, wood and steel is given as below., Material, Product ¯, , Wood, , Steel, , Table (x), , 8, , 2, , Book shelf (y), , 11, , 3, , The weekly variability of wood and steel is 450 and 100, units respectively. Profit on a table `1000 and that on a, bookshelf is `1200. To determine the number of tables, and bookshelves to be produced every week in order to, maximize the total profit, formulation of the problem as, L.P.P. is, (a) Maximize Z = 1000x + 1200 y, Subject to, 8x + 11y ³ 450, 2x + 3y £ 100, x ³ 0, y ³ 0, (b) Maximize Z = 1000x + 1200 y, Subject to, 8x + 11y £ 450, 2x + 3y £ 100, x ³ 0, y ³ 0, (c) Maximize Z = 1000x + 1200 y, Subject to, 8x + 11y £ 450, 2x + 3y ³ 100, x ³ 0, y ³ 0, (d) Maximize Z = 1000x + 1200 y, Subject to, 7., , 8., , 9., , 8x + 11y ³ 450, 2x + 3y ³ 100, x ³ 0, y ³ 0, Maximize Z = 3x + 5y, subject to x + 4y £ 24, 3x + y, £ 21, x + y, £ 9, x ³ 0, y ³ 0, is, (a) 20 at (1, 0), (b) 30 at (0, 6), (c) 37 at (4, 5), (d) 33 at (6, 3), Which of the following is not a vertex of the positive region, bounded by the inequalities 2x + 3y £ 6, 5x + 3y £ 15 and, x, y ³ 0 ?, (a) (0, 2) (b) (0, 0), (c) (3, 0), (d) None, Consider Max. z = – 2x – 3y subject to, x y, x y, + £ 1,, + £ 1 , x, y ³ 0, 3 2, 2 3

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EBD_7762, 374, , MATHEMATICS, , The max value of z is :, (a) 0, (b) 4, (c) 9, (d) 6, 10. Z = 7x + y, subject to 5x + y ³ 5, x + y ³ 3, x ³ 0, y ³ 0., The minimum value of Z occurs at, , æ1 5ö, (a) (3, 0) (b) ç , ÷, (c) (7, 0), (d) (0, 5), è2 2ø, The linear inequations for which the shaded area in the, following figure is the solution set, are, , 11., , 13. Shaded region in the following figure is represented by, Y, (0,20) x + y = 20, 20 , 40, B, C(10,16), 3 3, 2x+5y = 80, X, (40,0), , A(20,0), , x–y=1, , X, , X¢, , x + 2y = 8, , 2x + y = 2, , (a) x + y £ 1, 2x + y ³ 2, x – 2y ³ 8, x £ 0, y ³ 0, (b) x – y ³ 1, 2x + y ³ 2, x + 2y ³ 8, x ³ 0, y ³ 0, (c) x – y £ 1, 2x + y ³ 2, x + 2y £ 8, x ³ 0, y ³ 0, (d) x + y ³ 1, 2x + y £ 2, x + 2y ³ 8, x ³ 0, y ³ 0, The maximum value of z = 2x + 5y subject to the constraints, 2x + 5y £ 10, x + 2y ³1, x – y £ 4, x ³ y ³ 0, occurs at, (a) exactly one point, (b) exactly two points, (c) infinitely many points, (d) None of these, , 12., , 1, 2, 3, 4, 5, , (c), (d), (c), (a), (c), , 6, 7, 8, 9, 10, , (c), (c), (b), (c), (d), , 11, 12, 13, 14, 15, , (b), (c), (a), (b), (c), , 1, 2, , (b), (b), , 3, 4, , (a), (d), , 5, 6, , (c), (b), , 1, 2, , (b), (a), , 3, 4, , (d), (d), , 5, 6, , (c), (b), , (a) 2x + 5y ³ 80, x + y £ 20, x ³ 0, y £ 0, (b) 2x + 5y ³ 80, x + y ³ 20, x ³ 0, y ³ 0, (c) 2x + 5y £ 80, x + y £ 20, x ³ 0, y ³ 0, (d) 2x + 5y £ 80, x + y £ 20, x £ 0, y £ 0, 14. The solution set of the following system of inequations:, x + 2y £ 3, 3x + 4y ³ 12, x ³ 0, y ³1, is, (a) bounded region, (b) unbounded region, (c) only one point, (d) empty set, 15. Consider : z = 3x + 2y, Minimize subject to : x + y ³ 8, 3x + 5y £ 15, x, y ³ 0, It has :, (a) Infinite feasible solutions, (b) Unique feasible solution, (c) No feasible solution, (d) None of these, , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (b), 16 (c) 21 (d) 26 (b) 31, (b), 17 (c) 22 (b) 27 (d) 32, (b), 18 (a) 23 (c) 28 (d) 33, (a), 19 (d) 24 (c) 29 (c) 34, (b), 20 (a) 25 (b) 30 (a) 35, Exercise 2 : Exemplar & Past Year MCQs, (d) 9, (b) 11 (c) 13, (c), 7, (c) 10 (c) 12 (b) 14, (a), 8, Exercise 3 : Try If You Can, (c), (a) 11 (c) 13, (c), 7, 9, (d) 10 (d) 12 (c) 14, (d), 8, , 36, 37, 38, 39, 40, , (d), (b), (c), (a), (c), , 15, , (c), , 41, 42, 43, 44, 45, , (d), (d), (b), (d), (c), , 46, 47, 48, 49, 50, , (d), (a), (c), (c), (a)

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29, , PROBABILITY-2, , Chapter, , Trend, Analysis, , off JEE Main and BITSAT (Year 2010-2018), , 5, , Number of Questions, , 4, JEE MAIN, , 3, , BITSAT, 2, 1, , 0, 2010, , 2011, , 2012, , 2013, , 2014, , 2015, , 2016, , 2017, , 2018, , Year, , Chapter Utility Score (CUS), Exam, , JEE Main, BITSAT, , Weightage, , 3, 2, , Critical Concepts, , Conditional Probability, Independent, Events, Bayes Theorem, Mean &, Variance of a Random Variable,, Binomial Distribution, , Rating of, Difficulty Level, , CUS, (chapter utility score), Out of 10, , 4/5, , 7.5

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PROBABILITY-2, , 377

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EBD_7762, 378, , MATHEMATICS, , Topic 1 : Independent Events, Conditional Probability, 1., , 2., 3., , 4., , 1, , P (B) = 0, then P (A/B) is, 2, 1, (a) 0, (b), 2, (c) not defined, (d) 1, A and B are events such that P (A/B) = P (B/A) then, (a) A Ì B, (b) B = A, (c) A Ç B = f, (d) P(A) = P (B), The chances to fail in Physics are 20% and the chances to, fail in Mathematics are 10%. What are the chances to fail in, at least one subject?, (a) 28%, (b) 38%, (c) 72%, (d) 82%, If two events A and B are such that, P(A') = 0.3, P(B) = 0.4 and P ( A Ç B¢ ) = 0.5, then, , If P (A) =, , æ, P çè, 5., , 6., , B ö, ÷ =, A È B¢ ø, , P{(E1 È E2) Ç (E1 Ç E 2 )} is equal to, , 8., , 1, 4, , (b) >, , 1, 4, , 1, (d) None of these, 2, Let three fair coins be tossed. Let A = {all heads or all tails},, B = {atleast two heads}, and C = {atmost two tails}. Which, of the following events are independent?, (a) A and C, (b) B and C, (c) A and B, (d) None of these, A student appears for tests I, II and III. The student is, successful if he passes either in tests I and II or tests I and, IV. The probabilities of the student passing in tests I, II, III, , (c), , 7., , <, , ³, , are p, q and, , 1, respectively. The probability that the student, 2, , is successful is, , A coin is tossed three times is succession. If E is the event, that there are at least two heads and F is the event in which, first throw is a head, then P(E/F) equal to:, 1, 3, 3, 1, (a), (b), (c), (d), 8, 8, 4, 2, 10. The probability of the simultaneous occurrence of two, events A and B is p. If the probability that exactly one of the, events occurs is q, then which of the following is not correct?, (a), , P(A' ) + P (B' ) = 2 + 2q - p, , (b), , P(A' ) + P(B' ) = 2 - 2p - q, p, P(A Ç B | A È B) =, p+q, , (c), , (d) P(A 'Ç B') = 1 - p - q, 11. A person has a bunch of n keys, only one of which can, open a lock. The person tries the keys at random rejecting, those which do not open the lock. The probability that the, lock is opened at the kth (£ n) trial is, , (a) 1/4, (b) 1/5, (c) 3/5, (d) 2/5, It is given that the events A an d B are such that, 1, 1, 2, P ( A) = , P ( A | B ) = and P ( B | A) = . Then P(B) is, 4, 2, 3, 1, 1, 2, 1, (b), (c), (d), (a), 6, 3, 3, 2, For any two independent events E1 and E2, , (a), , 9., , 1, then the relation between p and q is, 2, , given by, (a) pq + p = 1, , (b) p2 + q = 1, , (c) pq – 1 = p, , (d) none of these., , (a), , 1, n, , (b), , k, n, k -1, , 1, k -1, æ 1ö, (d) ç1 - ÷ ., n, n, è nø, 12. One ticket is selected at random from 50 tickets numbered, 00,01,02,...,49. Then the probability that the sum of the digits, on the selected ticket is 8, given that the product of these, digits is zero, equals, , (c), , 1, 1, 5, 1, (b), (c), (d), 7, 50, 14, 14, 13. Two aeroplanes I and II bomb a target in succession. The, probabilities of I and II scoring a hit correctly are 0.3 and 0.2,, respectively. The second plane will bomb only if the first, misses the target. The probability that the target is hit by, the second plane is, (a) 0.2, (b) 0.7, (c) 0.06, (d) 0.14, 14. Let A and B be two events associated with an experiment, such that, , (a), , P ( A Ç B ) = P ( A ) P ( B), Assertion : P(A|B) = P(A) and P(B|A) = P(B), Reason : P ( A È B) = P ( A ) + P ( B), (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct.

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PROBABILITY-2, , 379, , 15. If A and B be two events such that P(A) = 0.6, P(B) = 0.2 and, P(A/B) = 0.5, then P ( A¢ / B¢ ) is equal to, 1, 3, 3, 6, (b), (c), (d), 10, 10, 8, 7, 16. If A and B are independent events, then which of the, following is not true ?, (a) P(A/B) = P(A), (b) P(B/A) = P(B), (c) P(A/B) = P(B/A), (d) None of these, 17. If A and B play a series of games in each of which probability, that A wins is p and that B wins is q = 1 – p. Therefore the, chance that A wins two games before B wins three is, , (a), , p2, 1 + 3q, , (b), , (c), , p 2 (1+2q+3 q 2 ), , (d) None of these, , (b), (c), (d), , Face, 1, P, 0.10, , (a), 25., 1, 5, , ( ), P (E | F) + P ( E | F ) = 1, P ( E | F) + P ( E | F) = 1, P ( E | F) + P ( E | F) = 0, , 16, 21, , 28, 143, , 36, 145, , (c), , (d), , 36, 143, , 2, 0.32, , 3, 0.21, , 4, 0.15, , 5, 0.05, , 6, 0.17, , 1, 4, 1, (b), (c), (d) 1, 5, 5, 2, 22. If E 1 and E 2 are two events such that P(E 1 ) = 1/4,, P(E2/E1) = 1/2 and P(E1/ E2) = 1/4, then choose the incorrect, statement, (a) E1 and E2 are independent, (b) E1 and E2 are exhaustive, (c) E2 is twice as likely to occur as E1, (d) Probabilities of the events E1 Ç E2 , E1 and E2 are in, G.P., , (b), , 1, 10, , 5, 16, , (c), , (d), , 5, 21, , Consider and event E = E1 Ç E2 Ç E3 find the value of P(E), if P(E1) = 2/5, P(E2/E1) =1/5 and P(E3/E1E2) = 1/10, (a) 2/125, , (b) 1/125, , (c) 3/125, , (d) None of these, , P (E | F) + P E | F = 1, , 20. A fair coin is tossed two times, I, The first and second tosses are independent of each, other., II, The sample space for the experiment is, S = {HH, HT, TH, TT}, III Getting head in both the tosses is a sure event., (a) Only I is correct, (b) Only I and II are correct, (c) All are correct, (d) Only III is correct, 3, 4, 1, 21. If P ( B ) = , P ( A | B ) =, an d P ( A È B ) = , then, 5, 5, 2, P ( A È B )¢ + P ( A ¢ È B ) =, (a), , (b), , 24. For a biased dice, the probability for the different faces to, turn up are, , 1 + 3q 2, , 1, 1, 1, (b), (c), (d), 3, 6, 4, 19. If E and F are events such that 0 < P(F) < 1, then, , 32, 145, , The dice is tossed and it is told that either the face 1 or face, 2 has shown up, then the probability that it is face 1, is, , 18. If P ( A Ç B ) = 0.15 , P ( B¢ ) = 0.10 , then P(A/B) =, , (a), , (a), , p2, , (a), , (a), , 23. A bag contains 12 white pearls and 18 black pearls. Two, pearls are drawn in succession without replacement. The, probability that the first pearl is white and the second is, black, is, , Topic 2 : Baye’s Theorem, 26. Girl students constitute 10% of I year and 5% of II year at, Roorkee University. During summer holidays 70% of the I, year and 30% of II year students are given a project. The, girls take turns on duty in canteen. The chance that I year, girl student is on duty in a randomly selected day is, 14, 3, 3, 7, (b), (c), (d), 17, 10, 17, 10, 27. Bag P contains 6 red and 4 blue balls and bag Q contains 5, red and 6 blue balls. A ball is transferred from bag P to bag Q, and then a ball is drawn from bag Q. What is the probability, that the ball drawn is blue?, , (a), , 7, 8, 4, 8, (b), (c), (d), 15, 15, 19, 19, A and B are two independent witnesses (i.e. there is no, collision between them) in a case. The probability that A will, speak the truth is x and the probability that B will speak the, truth is y. A and B agree in a certain statement. The probability, that the statement is true is, , (a), 28., , (a), , x–y, x+y, , (b), , xy, 1 + x + y + xy, , x–y, xy, (d), 1 – x – y + 2xy, 1 – x – y + 2 xy, 29. An urn contains five balls. Two balls are drawn and found to, be white. The probability that all the balls are white is, , (c), , (a), , 1, 10, , (b), , 3, 10, , (c), , 3, 5, , (d), , 1, 2

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EBD_7762, 380, , MATHEMATICS, , 30. A signal which can be green or red with probability 4/5 and, 1/5 respectively, is received by station A and then trasmitted, to station B. The probability of each station receiving the, signal correctly is 3/4. If the signal received at station B is, given, then the probability that the original signal is green,, is, 3, 6, 20, 9, (a), (b), (c), (d), 5, 7, 23, 20, 31. A box contains N coins, m of which are fair and the rest are, biased. The probability of getting a head when a fair coin is, 2, 1, , while it is when a biased coin is tossed. A, 3, 2, coin is drawn from the box at random and is tossed twice., Then the probability that the coin drawn is fair, is, , 35. A laboratory blood test is 99% effective in detecting a certain, disease, when it is in fact present. However, the test also, yields a false positive result for 0.5% of the healthy person, tested (that is, if a healthy person is tested, then, with, probability 0.005, the test will imply he has the disease). If, 0.1 percent of the population actually has the disease, then, probability that a person has the disease given that his test, result is positive., (a), , tossed is, , 9m, 8N + m, , (a), , (b), , 9m, 8N - m, , 9m, 9m, (d), 8m - N, 8m + N, 32. By examining the chest X-ray, the probability that TB is, detected when a person is actually suffering is 0.99. The, probability of an healthy person diagnosed to have TB is, 0.001. In a certain city, 1 in 1000 people suffers from TB, A, person is selected at random and is diagnosed to have TB., Then, the probability that the person actually has, TB is, 110, 2, 110, 1, (a), (b), (c), (d), 221, 223, 223, 221, 33. Coloured balls are distributed in four boxes as shown in, the following table, , (c), , Box, I, II, III, IV, , Colo ur, Black W hite Red, 3, 4, 5, 2, 2, 2, 1, 2, 3, 4, 3, 1, , (a), , 1, 29, , (b), , 28, 29, , (c), , 15, 29, , (d), , 14, 29, , 11, 133, , (c), , 33, 133, , (d), , 1, 133, , 36. Examples of some random variables are given below :, 1. Number of sons among the children of parents with, five children., 2. Number of sundays in some randomly selected months, with 30 days., 3. Number of apples in some 3 kg packets, purchased, from a retail shop., Which of the above is expected to follow binomial, distribution?, (a) Variable 1, (b) Variable 2, (c) Variable 3, (d) None of these, , æ 1ö, 37. The probability of safe arrival of one ship out of five is çè ÷ø ., 5, The probability of safe arrival of atleast 3 ship is:, 1, 184, 3, 181, (b), (c), (d), 3125, 52, 31, 3125, If x has binomial distribution with mean np and variance, , (a), , npq, then, , A box is selected at random and then a ball is randomly, drawn from the selected box. The colour of the ball is black., Probability that the ball drawn from Box III, is, (a) 0.161 (b) 0.162 (c) 0.165, (d) 0.104, 34. Assume that the chances of a patient having a heart attack, is 40%. It is also assumed that a meditation and yoga course, reduce the risk of heart attack by 30% and prescription of, certain drug reduces its chances by 25%. At a time a patient, can choose any one of the two options with equal, probabilities. It is given that after going through one of the, two options the patient selected at random suffers a heart, attack. The probability that the patient followed a course of, meditation and yoga is, , (b), , Topic 3 : Binomial Distribution, Expectation, , 38., Blue, 6, 2, 1, 5, , 22, 133, , P( x = k ), is equal to :, P( x = k - 1), , (a), , n-k p, ., k -1 q, , (b), , n - k +1 p, ., k, q, , (c), , n +1 q, ., k p, , (d), , n -1 q, ., k +1 p, , 39. The mean and variance of a random variable X having binomial, distribution are 4 and 2 respectively, then P (X = 1) is, 1, 1, 1, 1, (c), (d), (b), 32, 16, 8, 4, 40. Consider the following statement:, "The mean of a binomial distribution is 3 and variance is 4.", Which of the following is correct regarding this statement?, (a) It is always true, (b) It is sometimes true, (c) It is never true, (d) No conclusion can be drawn, , (a)

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PROBABILITY-2, , 381, , 41. Five fair coins are tossed. If p is the probability that not, more than two heads appear and q is the probability that, not less than three heads appear, then, (a) p > q, (b) p = q (c) p < q, (d) pq = 1, 42. The probability that a man hits a target is p = 0.1. He fires, n = 100 times. The expected number n of times he will hit the, target is :, (a) 33, (b) 30, (c) 20, (d) 10, 43. If the mean of a binomial distribution is 25, then its standard, deviation lies in the interval given below, (a) [0, 5), (b) (0, 5] (c) [0, 25), (d) (0,25], 44. In a meeting, 70% of the members favour and 30% oppose a, certain proposal. A member is selected at random and we, take X = 0, if he opposed and X = 1, if he is in favour. Then,, E(X) and Var (X) respectively are, (a), , 3 5, ,, 7 17, , (b), , 13 2, ,, 15 15, , 7 21, 7 23, ,, ,, (d), 10 100, 10 100, 45. Two dice are thrown, simultaneously. If X denotes the, number of sixes, then the expected value of X is, 1, 3, , E (X) =, , (c), , 1, E (X) =, 6, , 2, 3, , (b), , E (X) =, , (d), , 5, E (X) =, 6, , at least one success is greater than or equal to, , 9, , then n is, 10, , greater than:, 1, log10 4 + log10 3, , (c), , 4, log10 4 – log10 3, , 1, , 2, , P (X = x), , 1, 3, , 1, 2, , 0, , 3, 1, 6, , Then,, (a), , m = s2 = 2, , (b), , m = 1, s2 = 2, , (c), , m = s2 = 1, , (d), , m = 2, s2 = 1, , 50. In a binomial distribution, the mean is 4 and variance is 3., Then its mode is :, (a) 4, (b) 5, (c) 6, (d) 7, 51. In a box containing 100 bulbs, 10 are defective. The, probability that out of a sample of 5 bulbs, none is defective, is :, 5, , æ1ö, æ 9ö, 9, (c) ç ÷, (d), ç ÷, 10, è2ø, è 10 ø, 52. A die is thrown again and again until three sixes are obtained., The probability of obtaining third six in the sixth throw of, the die, is, 625, 621, 625, 620, (a), (b), (c), (d), 23329, 25329, 23328, 23328, , (a) 10–1, , (b), , 9, ; the, 10, chance that out of 5 expected ships, at least 4 will arrive, safely at the port 2, is, 91854, 32805, (a), (b), 100000, 100000, , 53. If the chance that a ship arrives safely at a port is, , 1ö, æ, 46. In a binomial distribution B ç n, p = ÷ , if the probability of, è, 4ø, , (a), , 0, , 5, , (c), , (a), , X=x, , (b), , 9, log10 4 – log10 3, , (d), , 1, log10 4 – log10 3, , 47. The random variable X has the following probability, distribution:, x:, -3, -1, 0, 1, 3, P ( X = x ) : 0.05 0.45 0.20 0.25 0.05, Then, its mean is, (a) – 0.2, (b) 0.2, (c) – 0.4, (d) 0.4, 48. The mean of the numbers obtained on throwing a die having, written 1 on three faces, 2 on two faces and 5 on one face is, 8, (a) 1, (b) 2, (c) 5, (d), 3, 49. If m and s2 are the mean and variance of the random variable, X, whose distribution in given by, , 59049, 26244, (d), 100000, 100000, 54. Consider the following statements, Statement I: An experiment succeeds twice as often as it, fails. Then, the probability that in the next six trials, there, , (c), , 4, , will be atleast 4 successes is, , 31 æ 2 ö, ç ÷ ., 9 è 3ø, , Statement II: The number of times must a man toss a fair, coin so that the probability of having atleast one head is, more than 90% is 4 or more than 4., (a) Statement I is true, (b) Statement II is true, (c) Both statements are true, (d) Both statements are false, 55. Five bad eggs are mixed with 10 good ones. If three eggs are, drawn one by one with replacement, then the probability, distribution of the number of good eggs drawn, is, , (a), , X, , 0, , 1, , 2, , 3, , P (X), , 4, 9, , 5, 9, , 7, 9, , 1, 9

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EBD_7762, 382, , MATHEMATICS, , (b), , (c), , X, , 0, , 1, , 2, , 3, , P (X), , 5, 54, , 7, 54, , 2, 27, , 7, 27, , X, , 0, , 1, , 2, , 3, , P (X), , 1, 13, , 2, 13, , 9, 26, , 3, 26, , 0, , 1, , 2, , 3, , X, , (c) Assertion is correct, Reason is incorrect, (d) Assertion is incorrect, Reason is correct., 63. There is 30% chance that it rains on any particular day. Given, that there is at least one rainy day, then the probability that, there are at least two rainy days is, 6, , (a), , 1, 2, 4, 8, 27, 9, 9, 27, 56. One hundred identical coins, each with probability p of, showing up heads, are tossed. If 0 < p < 1 and the probability, of heads showing on 50 coins is equal to that of heads, showing on 51 coins. The value of p is, 49, 50, 51, 1, (a), (b), (c), (d), 101, 101, 101, 2, 57. The probability of a man hitting a target is 1/4. The number, of times he must shoot so that the probability he hits the, target, at least once is more than 0.9, is, [use log 4 = 0.602 and log 3 = 0.477], (a) 7, (b) 8, (c) 6, (d) 5, 58. Two dice are tossed 6 times. Then the probability that 7 will, show an exactly four of the tosses is:, , (d), , P (X), , 116, 117, 125, 225 (b), (c), (d), 17442, 20003, 15552, 18442, A box contains 20 identical balls of which 10 are blue and 10, are green. The balls are drawn at random from the box one at, a time with replacement. The probability that a blue ball is, drawn 4th time on the 7th draw is, , (a), 59., , 60., , 5, 27, 5, 1, (a), (c), (b), (d), 32, 32, 64, 2, If X follows Binomial distribution with mean 3 and variance, 2, then P(X ³8) is equal to :, , (a), 61., , 9, , (b), , 18, , 1, 2, , 6, 1, , 9, , (c), , 19, , (d), , (b), , 9, , 57, 67, 1, , 1, , +, (d), 76, 67 67, Assertion : For a binomial distribution B(n, p),, Mean > Variance, Reason : Probability is less than or equal to 1, (a) Assertion is correct, Reason is correct; Reason is a, correct explanation for assertion., (b) Assertion is correct, Reason is correct; Reason is not, a correct explanation for Assertion, , (c), , 6, , (b), , 7, , æ7ö, 1+ ç ÷, è 10 ø, , (c), , 13 æ 7 ö, ´ç ÷, 5 è 10 ø, , (a), , æ3 1ö, ç + ÷, è 4 4ø, , æ 7 ö 14, ç ÷ è 10 ø 17, , 1-, , 6, , (d), , 6, , 14 æ 7 ö, ´ç ÷, 15 è 10 ø, 7, , æ 7ö, 1- ç ÷, è 10 ø, 64. In a binomial distribution, mean is 3 and standard deviation, 3, is , then the probability function is, 2, 12, , 12, , æ 1 3ö, ç + ÷, è 4 4ø, , (b), , 9, , 9, , æ1 3ö, æ3 1ö, (d) ç + ÷, ç + ÷, è4 4ø, è4 4ø, 65. In a hurdle race, a player has to cross 10 hurdles. The, (c), , 5, . Then, the, 6, probability that he will knock down fewer than 2 hurdles is, , probability that he will clear each hurdle is, , (a), , (c), , 59, , (b), , 2 ´ 69, 59, 2 ´ 610, , 20, , 3, 3, 3, 39, If the mean and variance of a binomial variate x are, 35, 35, respectively, and, , then the probability of x > 6 is :, 6, 36, (a), , 62., , 17, , 14 æ 7 ö, ´ç ÷, 5 è 10 ø, , 510, 2 ´ 610, 510, , (d), , 2 ´ 69, BEYOND NCERT, , Topic 4 : Probability Regarding letters and, their Envelopes, Poisson Distribution, 66. There are n letters and n addressed envelopes, the, probability that all the letters are not kept in the right, envelope, is, (a), , 1, n!, , (b) 1 –, , 1, n!, , 1, (d) None of these, n, 67. There are four letters and four envelopes, the letters are, placed into the envelopes at random, the probability that all, letters are placed in the wrong envelopes is, (c) 1 –, , (a), , 1, 8, , (b), , 3, 8, , (c), , 5, 8, , (d) 1

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PROBABILITY-2, , 383, , 68. If X is a Poisson variate such that P(X = 1) = P (X = 2), then, P (X = 4) is equal to, 1, 1, 2, (b), (c), (d), 2e 2, 3e2, 3e 2, Suppose that X has a Poisson distribution. If, , (a), , 69., , 73., , 1, e2, , P(X = 1) = 0.3 , and P(x = 2) = 0.2 then P(X = 0)=, (a) e1/3, (b) e–3/2 (c) e–4/3, (d) 0.5, 70. If mean of a poisson distribution of a random variable X is 2,, then the value of P(X > 1.5) is, (a), (c), , 3, e, , (b), , 2, , 1–, , 3, e, , 3, e, , (d) 1 –, , (a), , 6, 5, , (b), , e, , 5, 6, , (c), , (d), , e, 2, , (b), , e, 6, , (c), , 1, 6e, , (d), , 1, 3, , (c), , 1, 6, , 5, 6, , (d), , 1 1 1 1, 1, – + – + ....(–1) n, 1! 2! 3! 4!, n!, , (b), , 1 1 1 1, 1, + + – + ...., 2! 3! 4! 5!, n!, , (c), , 1 1 1 1, n 1, – + – + ....( -1), 2! 3! 4! 5!, n!, , (d) None of these, , 6, e5, , 72. If x is poisson variate and P(x = 0) = P(x = 1), then, P (x = 2) =, (a), , (b), , (a), 3, , 6, 55, , 1, 2, , 74. n letters to each of which corresponds on addressed envelope are placed in the envelope at random. Then the probability that n letter is placed in the right envelope, will be :, , e2, 71. At a telephone enquiry system the number of phone cells, regarding relevant enquiry follow Poisson distribution with, an average of 5 phone calls during 10 minute time intervals., The probability that there is at the most one phone call during, a 10-minute time period is, , (a), , Three letters are written to three different persons and, addresses on the three envelopes are also written. Without, looking in the addresses, the letters are kept in these, envelopes. The probability that all the letters are not placed, into their right envelopes is, , 1, 2e, , 75. A machine is producing 4% defective products. Find the, probability of getting exactly 4 defectives in a sample of 100, is [Given log 2 = 0.30102, log 3 = 0.4771, log e = 0.4343,, antilog (.2908) =1.954], (a) 0.192, (b) 0.156, (c) 0.182, (d) 0.1954, , Exercise 2 : Exemplar & Past Year MCQs, NCERT Exemplar MCQs, 1., , If P(A) =, (a), , 2., , 3., , 4, 7, and P(A Ç B) =, , then P(B/A) is equal to, 5, 10, , 1, 10, , If P(A Ç B) =, (a), , 4., , 14, 17, , (b), , 1, 8, , (c), , 7, 8, , (d), , (b), , 17, 20, , (c), , 7, 8, , (d), , (a), 5., , 1, 8, , 3, 2, 3, , P(B) =, and P(A È B) = , then, 10, 5, 5, P(B/A) + P(A/B) equals to, , (a), , 1, (b), 3, , 5, (c), 12, , (d), , 7, 12, , 5, 6, , (b), , (c), , 25, 42, , (d), , 1, , 1, 1, , P(B) =, 2, 3, , 1, , then P(A' È B') equals to, 4, , 1, 3, 1, 3, (b), (c), (d), 12, 4, 4, 16, If P(A) = 0.4, P(B) = 0.8 and P(B/A) = 0.6, then P(A È B) is, equal to, (a) 0.24, (b) 0.3, (c) 0.48, (d) 0.96, , (a), , 6., , 5, 7, , If A and B are two events such that P(A) =, and P(A/B) =, , If P(A) =, 1, 4, , 1, 2, 3, , P ( B) =, and P ( A Ç B ) = , then, 5, 5, 10, , P ( A¢ | B¢ ) . P ( B¢ | A¢ ) is equal to, , 17, 20, , 7, 17, and P(B) =, , then P(A/B) equals to, 10, 20, , If P ( A ) =

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EBD_7762, 384, , 7., , MATHEMATICS, , If A and B are two events and A ¹ f, then, (a) P(A/B) = P(A)× P(B), , 8., , 2, 3, , (b), , 1, 2, , (c), , 3, 10, , If A and B are two events such that P(B) =, and P(A È B) =, , 3, 1, 1, (b), (c), 10, 5, 2, In question 9 (above), P(B/A) is equal to, , (a), 11., , 13., , (b), , 3, 10, , (c), , 1, 2, , 3, 1, , P(A/B) =, 5, 2, , (b), , 4, 5, , (c), , 1, 2, , (d), , (d), , 3, 5, , 6, 4, 4, 5, (a), (b), (c), (d), 13, 13, 9, 9, If A and B are two events such that P(A) ¹ 0 and, , 1 - P(A È B), P(B), , 19., , P(B) =, , 21., , 22., , 23., , 3, and, 5, , 1, 1, 2, 4, (b), (c), (d), 2, 3, 3, 7, If a die is thrown and a card is selected at random from a, deck of 52 playing cards, then the probability of getting an, even number on the die and a spade card is, , 4, , then P(A' Ç B') equals to, 9, , 4, 8, 2, 1, (b), (c), (d), 15, 45, 9, 3, If two events are independent, then, (a) they must be mutually exclusive, (b) the sum of their probabilities must be equal to 1, (c) Both (a) and (b) are correct, (d) None of the above is correct, , 1, 1, 1, 3, (b), (c), (d), 2, 4, 8, 4, A box contains 3 orange balls, 3 green balls and 2 blue, balls. Three balls are drawn at random from the box without, replacement. The probability of drawing 2 green balls and, one blue ball is, , (a), 24., , (a), 15., , 1, 4, 15, 5, (b), (c), (d), 3, 7, 28, 28, Three persons, A, B and C, fire at a target in turn, starting, with A. Their probability of hitting the target are 0.4, 0.3, and 0.2 respectively. The probability of two hits is, (a) 0.024, (b) 0.188, (c) 0.336, (d) 0.452, Assume that in a family, each child is equally likely to be a, boy or a girl. A family with three children is chosen at, random. The probability that the eldest child is a girl given, that the family has atleast one girl is, , (a), , (a), , P(A), P(B), , If A and B are two independent events with P(A) =, , 45, 135, 15, 15, (b), (c), (d), 196, 392, 56, 29, Refer to question 19 above. If the probability that exactly, two of the three balls were red, then the first ball being red,, is, , (a), , æ ö, (b) 1 - P A, ç B÷, è ø, , (d), , 2, 3, 1, 1, (b), (c), (d), 7, 35, 70, 7, A bag contains 5 red and 3 blue balls. If 3 balls are drawn at, random without replacement, then the probability of getting, exactly one red ball is, , (a), , 20., , (d) 1, , 7, 9, 4, , P(B) =, and P(A Ç B) =, , then, 13, 13, 13, P(A'/B) is equal to, , (c), , 2, 3, 3, 6, (b), (c), (d), 5, 8, 20, 25, If the events A and B are independent, then P(A Ç B) is, equal to, (a) P(A) + P(B), (b) P(A) – P(B), (c) P(A) × P(B), (d) P(A) / P(B), , P ( E È F ) = 0.5, then P(E | F) – P(F | E) equals, , 3, 5, , If P(A) =, , (a) 1 - P æç A ö÷, è Bø, , 3, , then P(A/B) × P(A'/B) is equal to, 4, , 18. Two events E and F are independent. If P(E) = 0.3,, , æ Aö, P(B) ¹ 1, then P ç ÷ =, è Bø, , 14., , 17., , 3, 1, 4, , P(A/B) =, and P(A È B) = , then, 5, 2, 5, P(A È B)' + P(A' È B) is equal to, 1, 5, , 3, 5, , P(B) = and, 8, 8, , (a), , If P(B) =, , (a), 12., , 1, 5, , 1, 5, , 4, , then P(A) equals to, 5, , (a), 10., , (d), , If A and B be two events such that P(A) =, P(A È B) =, , (c) P(A/B) × P(B/A) = 1, (d) P(A/B) = P(A)/P(B), If A and B are events such that P(A) = 0.4, P(B) = 0.3 and, P(A È B) = 0.5, then P(B' Ç A) equals to, (a), , 9., , P ( A Ç B), P ( B), , (b) P(A/B) =, , 16., , 3, 2, 1, 167, (b), (c), (d), 28, 21, 28, 168, A flashlight has 8 batteries out of which 3 are dead. If two, batteries are selected without replacement and tested, then, probability that both are dead is, , (a), 25., , (a), , 33, 56, , (b), , 9, 64, , (c), , 1, 14, , (d), , 3, 28

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PROBABILITY-2, , 26., , 385, , If eight coins are tossed together, then the probability of, getting exactly 3 heads is, 1, 7, 5, 3, (b), (c), (d), 256, 32, 32, 32, Two dice are thrown. If it is known that the sum of the, numbers on the dice is less than 6, the probability of getting, a sum 3 is, , 35., , Suppose X follows a binomial distribution with parameters, n and p, where 0 < p < 1, if P(X = r)/P(X = n – r) is independent, of n and r, then, , (a), , 27., , 1, 2, 1, 5, (b), (c), (d), 8, 5, 5, 18, Which one is not a requirement of a binomial distribution?, (a) There are 2 outcomes for each trial, (b) There is a fixed number of trials, (c) The outcomes must be dependent on each other, (d) The probability of success must be the same for all, the trial, If two cards are drawn from a well shuffled deck of 52, playing cards with replacement, then the probability that, both cards are queens, is, , (a), 28., , 29., , (a), , 1 1, ´, 13 13, , (a) p =, , (a), , 31., , 32., , 33, , 7, 64, , (b), , 37., , correctly are, , 2, , 3, , 4, , 5, , P(X), , 7, k, , 9, k, , 11, k, , The value of k is, (a) 8, (b) 16, (c) 32, For the following probability distribution., , 1, and they obtain the, 20, same answer, then the probability of their answer to be, correct is, , 38., , 1, , 2, , 3, , 4, , P(X), , 1, 10, , 1 3, 5 10, , 2, 5, , E(X)2 is equal to, (a) 3, (b) 5, , (c) 7, , 5, , (a), , æ 9ö, çè ÷ø, 10, , (c), , 1æ 9 ö, ç ÷, 2 è 10 ø, , 39., , 4, , 1æ 9 ö, ç ÷, 2 è 10 ø, , 4, , 5, , 1æ 9 ö, æ 9ö, (d) ç ÷ + ç ÷, è 10 ø, 2 è 10 ø, , 4, , (, , (, , (d) 48, , ), , Let A and B be two events such that P A È B =, , ), , P AÇ B =, , ( ), , P A =, , 1, ,, 6, , 1, and, 4, , 1, , where A stands for the complement of the event, 4, , A. Then the events A and B are, , 40., , (d) 10, , (b), , Past Year MCQs, , E(X) is equal to, (a) 0, (b) – 1, (c) – 2, (d) – 1.8, For the following probability distribution, X, , 1, 1, 13, 10, (b), (c), (d), 12, 40, 120, 13, If a box has 100 pens of which 10 are defective, then what, is the probability that out of a sample of 5 pens drawn one, by one with replacement atmost one is defective?, , (a), , X, – 4 – 3 – 2 –1 0, P(X) 0.1 0.2 0.3 0.2 0.2, , 34., , 1, 1, and , respectively. If the probability of, 3, 4, , their making a common error is,, , 45, 7, (c), (d), 1024, 41, If the probability that a person is not a swimmer is 0.3, then, the probability that out of 5 persons 4 are swimmers is, (a) 5C4(0.7)4 (0.3), (b) 5C1(0.7)(0.3)4, (c) 5C4(0.7) (0.3)4, (d) (0.7)4 (0.3), The probability distribution of a discrete random variable, X is given below, , 5, k, , 1, 2, 9, 1, (b), (c), (d), 10, 5, 20, 3, A and B are two students. Their chances of sloving a problem, , (a), , 7, 128, , X, , 1, 3, , 1, (d) None of these, 4, 36. In a college, 30% students fail in physics, 25% fail in, Mathematics and 10% fail in both. One student is chosen at, random. The probability that she fails in Physics, if she has, failed in mathematics, is, , (c), 30., , (b) p =, , (c) p =, , 1 1, +, (b), 13 13, , 1 1, 1 4, ´, ´, (d), 13 17, 13 51, The probability of guessing correctly atleast 8 out of 10, answers on a true false type examination is, , 1, 2, , [JEE MAIN 2014, A], (a) independent but not equally likely., (b) independent and equally likely., (c) mutually exclusive and independent., (d) equally likely but not independent., A coin is tossed 7 times. Each time a man calls head. Find the, probability that he wins the toss on more occasions., [BITSAT 2014, A], 2, 1, 3, 1, (a), (b), (c), (d), 3, 2, 4, 3

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EBD_7762, 386, , 41., , 42., , MATHEMATICS, , A bag contains n + 1 coins. It is known that one of these, coins shows heads on both sides, whereas the other coins, are fair. One coin is selected at random and tossed. If the, 7, , then the value, probability that toss results in heads is, 12, of n is., [BITSAT 2014, A], (b) 4, , (c) 5, , (d) None of these, , If in a binomial distribution n = 4, P(X = 0) =, P(X = 4) equals, , 43., , 44., , 1, 16, , (b), , 16, , then, 81, , 1, 81, , (c), , 1, 27, , (c) 3/5, , (d) None of these, , A bag contains (2n + 1) coins. It is known that n of these, coins have a head on both sides, whereas the remaining (n +, 1) coins are fair. A coin is picked up at random from the bag, and tossed. If the probability that the toss results in a head, is 31/42, then n is equal to, [BITSAT 2015, A], (b) 11, , (c) 12, , (d) 13, , 45. Let two fair six-faced dice A and B be thrown simultaneously., If E1 is the event that die A shows up four, E2 is the event, that die B shows up two and E3 is the event that the sum of, numbers on both dice is odd, then which of the following, statements is NOT true ?, [JEE MAIN 2016, A], E1 and E3 are independent., , (a), , 231´, , æ 6 ö, (b) 462 ´ ç ÷, è 25 ø, , 35, 510, , 7, 55, , (b), , 6, 55, , (c), , 12, 55, , (d), , 14, 55, , For the events E = {X is a prime number} and F = {X < 4},, then P(E È F) is, (a) 0.50, , [BITSAT 2017, A], , (b) 0.77, , (c) 0.35, , (d) 0.87, , 50. One mapping is selected at random from all mappings of the set, S = {1, 2, 3, ......n} into itself. The probability that it is one-one is, 3, . Then the value of n is, [BITSAT 2017, C], 32, (a) 3, (b) 4, (c) 5, (d) 6, 51. A bag contains 4 red and 6 black balls. A ball is drawn at, random from the bag, its colour is observed and this ball, along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the, bag, then the probability that this drawn ball is red, is :, , 52., , [BITSAT 2016, S], , (c), , 4, , X, 1, 2, 3, 4, 5, 6, 7, 8, p(X) 0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05, , (a), , A man takes a step forward with probability 0.4 and backward, with probability 0.6. The probability that at the end of eleven, steps he is one step away from the starting point is, , 510, , (d), , 49. A random variable X has the probability distribution, , E1 and E2 are independent., , 2 5.35, , (c) 6, , [JEE MAIN 2018, S], , (d) E2 and E3 are independent., , (a), , 12, 5, , If two different numbers are taken from the set (0, 1, 2, 3,, ......., 10), then the probability that their sum as well as, absolute difference are both multiple of 4, is :, , (b) E1, E2 and E3 are independent., , 46., , (b), , [JEE MAIN 2017, C], , 1, 8, , (d), , (b) 2/5, , (c), , 6, 25, , [BITSAT 2015, A], , (a) 3/10, , (a), , [JEE MAIN 2017, A], , 48., , A bag contains 3 red and 3 white balls. Two balls are drawn, one by one. The probability that they are of different colours, is., [BITSAT 2015, A], , (a) 10, , A box contains 15 green and 10 yellow balls. If 10 balls, are randomly drawn, one-by-one, with replacement, then, the variance of the number of green balls drawn is :, , (a), , (a) 3, , (a), , 47., , 5, , (d) None of these, , 2, 5, , (b), , 1, 5, , (c), , 3, 4, , (d), , 3, 10, , In a binomial distribution, the mean is 4 and variance is 3., Then its mode is :, [BITSAT 2018, A], (a) 5, (b) 6, (c) 4, (d) None of these

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PROBABILITY-2, , 387, , Exercise 3 : Try If You Can, 1., , Let a, b, g Î R and point P(a, b, g) in R3, consider three, planes P1, P2, P3 in R3 where, P1 : 2x + 3y + 6z + 30 = 0, P2 : 2x + 3y + 6z + 1 = 0, P3 : 2x + 3y + 6z – 5 = 0, If, , 2., , 4., , a, is the probability that length of perpendicular from, b, , point P to plane P2 is less than equal to 1 given that point P, lies between plane P1 and P3, then the value of |3a – b| is, greater than, [Note : a, b are coprime numbers], (a) 2, (b) 3, (c) 4, (d) 5, Mr. A randomly picks 3 distinct numbers form the set {1, 2,, 3, ...., 9} and arranges them descending order to form a three, digit number. Mr. B randomly picks 3 distinct numbers from, the set {1, 2, 3, ...., 8} and also arranges them in descending, order to form 3 digit numbers, (a), , (c), , 5., , (c), , A set R has 6 elements. A subset P of R is selected at, random. Returning the element(s) of P, the set R is formed, again and then a subset Q is selected from it. If the, , 1, 84, , (a), , 17, 42, , (b), , (c), , 13, 72, , (d) dependent on m., , probability that Mr. A’s 3 digit number is greater then, 6., , 29, , (a), , 1, Mr. B’s 3 digit number is, 3, , Let the probability Pn that a family has exactly n children be, apn, when n ³ 1 and P0 = 1 – ap (1 + p + p2 + ...). Suppose, that all sex distributions of n children have the same, probability. If k ³ 1 , then the probability that a family, contains exactly k boys is, 2a, , (a), , ( 2 - p )k +1, , (b), , pk, , ( 2 - p )k +1, , 29, , C10, , (b), , 10, , (20), 29, , (c), 7., , 17, 36, , Let A and B be the sets {1, 2, …10} and {1, 2, …20} respectively., A function is selected randomly from A to B the probability, that the function is non-decreasing is, , (d) probability that Mr. A’s 3 digit number is greater then, , 3., , 1, 1, , while it is when a biased coin is tossed. A, 3, 2, , coin is drawn from the box at random and tossed which fell, headwise. The probability that the same coin when tossed, again will show head is, , 1, 3, , 37, Mr. B’s 3 digit number is, 56, , a, ,, b, , then the value of b – a is, [Note: a, b are coprime numbers], (a) 3667, (b) 421, (c) 2141, (d) 1793, A box contains 3m coins, m of which are fair and 2m are, biased. The probability of getting a head when a fair coin is, tossed is, , (b) probability that Mr. A and Mr. B has the same 3 digit, numbers is, , (d) None of these, , ( 2 - p )k +1, , probability that P and Q have no common element is, , probability that Mr. A’s 3 digit number is always greater, then Mr. B’s 3 digit number is, , 2a . p k, , C19, , C20, , (20)10, , (d) None of these, , (20)10, , A die is thrown 2n + 1 times, n Î N. The probability that, faces with even numbers show odd numbers of times, is, 2n + 1, 1, (a), (b) less than, 4n + 3, 2, (c) greater than, , 1, 2, , (d) None of these

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EBD_7762, 388, , MATHEMATICS, , If a Î[– 20, 0], then probability that the graph of the function, y =16x2 + 8(a + 5) x – 7a – 5 is strictly above the x-axis is, , 8., , (a), 9., , 7, 20, , (b), , 13, 20, , (c), , 17, 20, , (d), , (a), , 1, 2, , (b), , (c), , 2, 3, , (d) None of these, , 1, 4, , The probablilty that the length of a randomly chosen chord, of a circle lies between, , (a), 11., , 5, 16, , (b), , 2, 5, and, of its diameter is, 3, 6, , 1, 16, , (c), , 1, 4, , (d), , 5, 12, , A doctor is called to see a sick child. The doctor knows, (prior to the visit) that 90% of the sick children in that, neighbourhood are sick with the flu, denoted by F, while, 10% are sick with the measles, denoted by M. A well-known, symptom of measles is a rash, denoted by R. The probability, of having a rash for a child sick with the measles is 0.95., However, occasionally children with the flu also develop a, rash, with conditional probability 0.08. Upon examination, the child, the doctor finds a rash. Then what is the probability, that the child has the measles?, (a) 91/165 (b) 90/163, (c) 82/161 (d) 95/167, , 1, 2, 3, 4, 5, 6, 7, 8, , (c), (d), (a), (a), (b), (a), (c), (a), , 9, 10, 11, 12, 13, 14, 15, 16, , (a), (a), (a), (d), (d), (c), (c), (c), , 17, 18, 19, 20, 21, 22, 23, 24, , (c), (c), (a), (b), (d), (b), (c), (d), , 1, 2, 3, 4, 5, 6, , (c), (a), (d), (c), (c), (d), , 7, 8, 9, 10, 11, 12, , (b), (d), (c), (d), (d), (d), , 13, 14, 15, 16, 17, 18, , (c), (d), (d), (d), (c), (c), , 1, 2, , (c), (c), , 3, 4, , (c), (a), , 5, 6, , (a), (c), , f ( x) =, , 3, 20, , A wire of length l is cut into three pieces. Then the probability, that the three pieces form a triangle is, , 10., , 12. If the probability density function of a random variable X is, x, in 0 £ x £ 2, then P ( X > 1.5 | X > 1) is equal to, 2, , 7, 21, 3, 7, (b), (c), (d), 16, 64, 4, 12, X follows a binomial distribution with parameters n and p,, and Y follows a binomial distribution with parameters m and, p. Then, if X and Y are independent,, P(X = r | X + Y = r + s ) = _______, with r, s ³ 0, , (a), , 13., , (a), , ( m C r )( n C s ) / m + n C r + s, , (b), , m, , C r n Cs /, , (c), , m, , C r / n Cs, , m+ n, , Cr, , (d) None of these, 14. A bag contains p white and q black ball. Two players A and, B alternately draw a ball from the bag, replacing the balls, each time after the draw till one of them draws a white ball, and wins the game. If A begins the game and the probability, of A winning the game is three times that of B, then the ratio, p : q is :, (a) 3 : 4, (b) 4 : 3, (c) 2 :1, (d) 1 : 2, 15. From a pack of 52 playing cards; half of the cards are randomly, removed without looking at them. From the remaining cards,, 3 cards are drawn randomly. The probability that all are king., (a), , 1, 25, 17, ( )( )(13), , (b), , (c), , 1, 52, 17, ( )( )(13), , (d), , ANSWER KEYS, Exercise 1 : Topic-wise MCQs, (c), 25 (b) 33 (c) 41 (b) 49, (a), 26 (b) 34 (d) 42 (d) 50, (c), 27 (b) 35 (a) 43 (a) 51, (c), 28 (d) 36 (b) 44 (c) 52, (a), 29 (d) 37 (d) 45 (a) 53, (c), 30 (c) 38 (b) 46 (d) 54, (d), 31 (a) 39 (b) 47 (a) 55, (d), 32 (a) 40 (c) 48 (b) 56, Exercise 2 : Exemplar & Past Year MCQs, (c), 19 (c) 25 (d) 31 (a) 37, (d), 20 (b) 26 (b) 32 (c) 38, (a), 21 (b) 27 (c) 33 (d) 39, (b), 22 (d) 28 (c) 34 (d) 40, (c), 23 (c) 29 (a) 35 (a) 41, (b), 24 (a) 30 (b) 36 (b) 42, Exercise 3 : Try If You Can, (d) 9 (b) 11 (d) 13 (a), 7, (b) 10 (c) 12 (c) 14, (c), 8, , 1, , ( 25)(15)(13), 1, (13)( 51)(17 ), , 57, 58, 59, 60, 61, 62, 63, 64, , (b), (c), (c), (c), (b), (a), (d), (a), , 65, 66, 67, 68, 69, 70, 71, 72, , (d), (b), (b), (c), (c), (d), (d), (d), , 43, 44, 45, 46, 47, 48, , (c), (a), (b), (b), (b), (b), , 49, 50, 51, 52, , (b), (b), (a), (c), , 15, , (a), , 73, 74, 75, , (b), (c), (d)