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Std. XII, MATHEMATICS (40), Specimen, Question Bank, , 1, , PC1 latest correction

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Chapter 1 : Mathematial Logic, (2 - Marks), 1., , 2., , 3., , 4., , Write the truth values of the following statments., i), , 2 is a rational number and it is the only even prime number., , ii), ,  x  N such that x + 3 > 5, , iii), , 3 + 2i is a real number or it is a complex number., , iv), , It is false that New Delhi is not a capital of India., , v), , The cube roots of unity are in G.P., , If p, q, r are statments with truth value T, F T respectively, determine the truth values the of, following :, i), , (p v r)  q, , ii), , (p  q)  r, , iii), , (p  q)  (q  r), , iv), , (r   q) v (p   r), , v), , (r  q)   p, , Write the negations of the following statements., i), , He is rich and happy., , ii), iii), , If I beome a teacher, then I will open a school., A, x  N, x + 5 > 8, , iv), , A person is busy if and only if he is a doctor., , v), ,  p  (q v r), , vi), , All parents care for their children., , Prepare the truth table for each of the following statement pattern., i), , p  (q  p), , ii), , ( p v  q)  (p  q), , iii), , (p   p) v q, , iv), , (p  q) v (q  p), , v), ,  q   p, , vi), , pvp, 2, , PC1 latest correction

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5., , 6., , Write the following statments in symbolic form :, i), , Manisha does not live in Mumbai., , ii), , If a number n2 is even, then n is even., , iii), , Rohit is neither healthy nor wealthy., , iv), , If ABC is right angled at B, then AB2 + BC2 = AC2., , v), , It is raining if and only if the weather is humid., , Express the given circuits in symbolic form., i), , Lamp, , Battery, , ii), , Lamp, , Battery, , 7., , If p : The earth is round., q : The moon rotates around the earth., and r : The sun is hot., Write the following in verbal form., , 8., , i) p q, , (ii) p  q, , (iv) ( p  q) v r, , (v) q  r, , (iii) p (q v r), , If A = {4, 5, 7, 9}, determine the truth value of each of the following., quantified statements., , ii), ,  x  A, such that x + 2 = 7, A, x  A , x + 3 < 10, , iii), ,  x  A , such x + 5 > 9, , iv), ,  x A, such that x is even., A, x  A, 2x < 17, , i), , v), , 3, , PC1 latest correction

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9., , 10., , Write duals of the following statements., i), , (p v q) V r, , ii), , (p v q)  T, , iii), , (p v q)  [p v  (q   r)], , iv), , Sohan and Kavita can not read french., , v), , ( p   q)   ( p v q), , vi), , (p T) V (F   q), , Prove the following results ; using truth tables., i), , pqpVq, , ii), , p  q  (p  q)  (q  p), , iii), ,  (p  q)   p V  q, , iv), , p  q   (p   q), , v), ,  (p  q)  p   q, , 3 Marks, 1., , 2., , 3., , State the converse, inverse and contrapositive of the following conditional statement., i), , If the teacher is absent, then the students are happy., , ii), , If 2 + 3 < 7 then 7 + 3 > 2, , iii), , If f (x) is differentiable function then it is continuous, , iv), , [p  (p  q)]  q, , v), , A family becomes literate if the woman in it are literate., , Prepare the truth table of the following statement patterns., i), , (p  q)  ( p V  q), , ii), , [(p q) v r] [  r v (p  q)], , iii), , ( p  q)  (p   q), , iv), , (p  r)  (q  p), , Using truth tables. Prove the following logical equivalenes., i), , (p  q)  (p  q) V ( p   q), , ii), , (p  q)  r  p  (q  r), , iii), ,  ( p  q)  (p V q)  p, 4, , PC1 latest correction

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4., , 5., , 6., , Using truth tables, examine whether each of the following statement patterns is a tautology or, a contradiction or a contingency., i), , [(p  q)  p]  q, , ii), , (p  ~ p)  (~ p  p), , iii), , ( p  q)  (q  p), , iv), , (p V q) V r  p V (q V r), , v), , (p V q)  (p V r), , Using the rules of negation, write the negation of the following., i), , p  (q  r), , ii), , ( p  q) V (p   q), , iii), , (p  q) r, , iv), , If 10 > 5 and 5 < 8 then 8 < 7, , v), , It is false that the sky is not blue., , Express the following circuits in symbolic form and write input output table., i), , Battery, , Lamp, , Battery, , Lamp, , ii), , 5, , PC1 latest correction

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iii), , Lamp, , Battery, , 7., , 8., , 9., , 10., , Construct the swiching circuits of the following statement patterns., i), , [p  (q V r) V ( p V s), , ii), , (p  q) V ( P) V (p   q), , iii), , [ (p V q)   p] V [ r  ( q V s)], , iv), , p  [q V (r   p)]  s, , Write the following compund statement in symbolic form and write their negations :, i), , Mahesh is fat but not lazy., , ii), , It is neither cold nor raining., , iii), , Some countries are digital and all people are technosavy., , iv), , If I drive fast and do not follow traffic rules, then I will meet with an accident., , If p, q are true statements and r, s are false statements, then find the truth values of the following, compound statements., i), , (~ p V q)  (s   r), , ii), , [(p  q)  r]  s, , iii), , (p  s)  (p  q), , iv), , [p  (q V r)] V [s   q], , v), , p  [q  (  p  r) V  s] V  r, , i), , Write the contrapositive of the inverse of the statement “If two numbers are, not equal, then their squares are not equal”., , ii), , If (p q)  r is false, then find the ruth value of the negation of the statement., (p v  r)  (q   p), , iii), , Show that the dual of (p   q) V q is a contradiction., , 6, , PC1 latest correction

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(D) for 4 marks, 1., , Simplify the following so that the new circuit has minimum number of switches. Also, draw the, simplified circuit., , 2., , Without using truth table, prove that, , Battery, , 3., , 4., , Lamp, , i), , ( p  q) V ( p   q) V (p   q)   p V  q, , ii), , [ p  (q V r) ] V [  r   q  p]  p, , iii), , p  q   (p   q)   (q   p), , Identify the pairs of following statements having same meaning., i), , If a person is a social, then he is happy., , ii), , If a person is not social, then he is not happy., , iii), , If a person is unhapyy, then he is not social., , iv), , If a person is happy, then he is social., , Write the following statement in four different ways, conveying the same meaning., “If you drive over 80 km per hour, then you will get a fine.”, , 5., , Show that, the following, circuit can be simplified and reconstructed so as to reduce its number, of switches from 7 to 4., , Lamp, , Battery, , 7, , PC1 latest correction

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6., , State the dual of the following statement by applying the principle of duality. Also, prove that, both sides of the dual are equivalent., p  (q  r)   (p  q)  r, , 7., , Simplify the following circuit and reconstruct an alternative circuit having minimum switches :, , Lamp, , Battery, , 8., , Write the following circuit symbolically and construct its switching table. What conclusion would, you draw from the table ?, , Lamp, , Battery, , ****, , 8, , PC1 latest correction

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Chapter 2 : Matries, (2 - Marks), 1., , Find the adjoint of the following matrices., i), , 2., , iv), , –2, , 4, , ii), , 5, , 6, , 3, , 4, , iii), , 1, , –2, , 4, , 3, , 2, , –3, , 5, , 7, , cos , , sin , , –sin , , cos , , ii), , v), , iii), , 1, , 3, , 2, , 7, , 3, , 2, , 6, , 1, , 1, , 2, , 2, , 2, , 5, , 4, , 5, , 2, , 1, , Find the inverse of the following Matrices using elementary column transformations., i), , iv), , 4., , 3, , Find the inverse of the following Matrices using elementary row transformations., i), , 3., , 1, , 1, , 2, , 2, , –1, , 5, , 4, , 3, , 2, , ii), , v), , iii), , 2, , –3, , –1, , 2, , 1, , 3, , 3, , 1, , 4, , 3, , 1, , 3, , 4, , cosec , , cot , , –cot , , cosec , , Find the inverse of the following Matrices using adjoint method., i), , iv), , 1, , 2, , 2, , 3, , 2, , –2, , 4, , 3, , ii), , v), , iii), , 1, , 2, , 3, , 4, , 1, , 0, , 0, , 2, , 2, , 0, , 3, , 4, , 5, , sin , , 1, , 0, , cos , , 9, , PC1 latest correction

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5., , 6., , Express the following equations in matrix form and solve them by using., a), , reduction method, , b), , Inversion method., , i), , x + y = 2,, , 3x + 2y = 5, , ii), , 2x + y = 5,, , 3x + 5y = –3, , iii), , x + 3y = 4,, , 4x – y = 3, , iv), , 2x – y = –2, 3x + 4y = 3, , v), , 4x + 3y = 1, 2x + y = 1, , i), , Find the matrix X such that AX = B, where A =, , –1, , 2, , 2, , –1, , and, , 3, B=, , 1, , ii), , Find the matrix X such that AX = I where A =, , iii), , If A =, , C=, , iv), , If A =, , 1, , 1, , 1, , 2, , 24, , 7, , 31, , 9, , 3, , 0, , 0, , 4, , ,B=, , 4, , 1, , 3, , 1, , 1, , 2, , 3, , 4, and, , , then find the matrix X such taht AXB = C, , ,B=, , 2, , ,, , then find the matrix X such, , 1, , that A–1X = B, , 7., , i), , If A =, , 1, , 1, , 0, , 1, , and B =, , 2, , 4, , 1, , 3, , then find matrix (AB)–1., , ii), , iii), , If A =, , If A =, , 1, , 3, , 4, , 5, , 2, , 3, , 5, , –2, , , then find adj (adj A), , and A–1 = KA, then find the value of K., , 10, , PC1 latest correction

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8., , i), , If A =, , 6, , 5, , 5, , 6, , and B =, , 11, , 0, , 0, , 11, , find A'B'., , –1, ii), , If A =, , , B = [3, , 2, , 1, , –2 ] verity that (AB)' = B'A'., , 3, , iii), , iv), , If A =, , 3, , –5, , –4, , 2, , 2, , If f (x) = x – 2x – 3,, , show that A2 – 5A – 14 I = 0, , find f (A) when A =, , 1, , 2, , 2, , 1, , 9., , If A is invertible matrix of order 3 and | A | = 5, then find the value of | adj A|., , 10., , If A =, , 2, 1, , 3, ,B=, , 5, , 2, , 4, , 6, , 19, , verify that adj (AB) = (adj B) (adj A), , 3 Marks, 1., , i), , If A =, , ii), , If A =, , iii), , If A =, , 3, , 1, , –1, , 2, , 4, , 5, , 2, , 1, , 1, , 2, , 1, , 2, , 1, , 0, , , show that A2 – 5A + 7 I = 0 and hence find A–1., , , show that A–1 =, , 1, , (A–5I), , 6, ,B=, , 0, , 1, , 2, , 3, , 1, , –1, , , then, , –1, , find (AB), , iv), , If A =, , 0, , 4, , 3, , 1, , –3, , –3, , –1, , 4, , 4, , then prove that A2 = I. Hence show that A–1 = A, , 11, , PC1 latest correction

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2., , 3., , 4., , Find adj A, if A =, , 1, , 2, , 3, , 2, , 3, , 2, , 3, , 3, , 4, , Nina and Meena want to buy pens and books. Nina wants 2 pens and 5 books while Meena, wants 6 pens and 8 books. They both go to a shop and buy them. When the shopkeeper gives, them the pens and the books. Nina pays him Rs. 110 and Meena pays Rs. 190. Find the prices, of one pen and one book using matrices., , If A =, , 1, , 0, , 1, , 0, , 2, , 3, , 1, , 2, , 1, , and B =, , 1, , 2, , 3, , 1, , 1, , 5, , 2, , 4, , 7, , then find the matrix X such that XA = B., 5., , Find the adjoint of the matrix, A=, , 6., , 1, , 2, , 3, , –5, , and verify that A (adj A) = (adj A) A = | A | I, , Find the inverse of the matrix, using adjoint method., , 7., , 1, , 0, , 0, , 0, , cos , , sin , , 0, , sin , , –cos , , If A and B are two invertible matrices of the same order, then prove that, (AB)–1 = B–1A–1, , 8., , 9., , 10., , If A =, , If A =, , –4, , –3, , –3, , 1, , 0, , 1, , 4, , 4, , 3, , 1, , 2, , 1, , 3, , 1, , 4, , 0, , –1, , 2, , , show that adj A = A, , with usual notations verify that, , i), , a31 C31 + a32 C32 + a33 C33 = | A |, , ii), , a21 C31 + a22 C32 + a23 C33 = 0, Where Cij the co factor of aij, , Using elementary transformations show that the inverse of the matrix., a, , –b, , 0, , b, , a, , 0, , 0, , 0, , 1, , is, , a, , b, , 0, , –b, , a, , 0, , 0, , 0, , 1, , if a2 + b2 = 1, , 12, , PC1 latest correction

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4 Marks, 1., , Find the inverse of the following matrices using, (a) elementary row transformation. (b) elementary column transformation., , i), , iii), , v), , 2, , i), , 2, , 1, , 3, , 2, , 0, , 1, , 5, , 3, , 1, , 5, , 1, , 0, , 3, , 2, , 3, , 0, , 1, , 3, , 3, , 1, , 2, , 0, , 1, , 2, , 5, , 2, , 4, , 1, , 2, , 3, , –2, , 3, , 9, , 3, , 1, , 1, , 1, , 1, , 3, , 1, , 2, , –2, , –1, , 3, , 0, , 0, , –2, , 1, , iv), , vii), , 5, , 2, , 6, , 2, , –1, , –4, , If the matrix A =, , B–1 =, , ii), , ii), , 1, , 1, , 2, , 0, , 2, , –3, , 3, , –2, , 4, , 1, , 2, , 0, , 0, , 3, , –1, , 1, , 0, , 2, , For the matrix A =, , and, , then compute (AB)–1., , 2, , –1, , 1, , –1, , 2, , –1, , 1, , –1, , 2, , ,, , verify that A3 – 6 A2 + 9 A – 4 I = 0. Hence find A–1., iii), , Find the matrix A such that, , A, , 2, , –1, , 0, , –5, , 4, , 1, , 2, , –2, , 1, , =, , –7, , 5, , 6, , 7, , –5, , 4, , –7, , 7, , 4, , 13, , PC1 latest correction

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3., , Express the following equations in matrix form and solve them by (a) Reduction method (b), Inversion method, i), , 2x–y+z=1,x+2y+3z=8,3x+y–4z=1, , ii), , x + y + z = 3 , 3 x – 2 y + 3 z = 4 , 5 x + 5 y + z = 11, , iii), , x+y+z=3, 2x–y+z=2,, , iv), , 2 x + 3 y + 3 z = 5 , x – 2 y + z = –4 , 3 x – y – 2 z = 3, , v), , 2, , 3, , +, , x, , 5., , 10, , y, , 6, , 4., , +, , 9, , z, –, , 4, , –, , x, , 6, , +, , y, , 5, , =1, , z, , 20, , =2, x, y, z, The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs. 60. The cost of 2 kg onion, 4kg wheat, and 6 kg rice is Rs. 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs. 70. Find cost, of each item per kg by matrix method., , If A =, , +, , =4,, , x–2y+3z=2, , 2, , –1, , 3, , 1, , 3, , 2, , 3, , –4, , –1, , , find A–1., , Using A–1 , solve the following system of linear equations., 2 x – y + 3 z = 13,, 6., , x + 3 y + 2 z = 1, 3 x – 4 y – z = 8, , A salesman has the following record of sales during the past three months for three items A, B, and C which have the different rates of commission., Months, , Sales of Units, , Total commission, , A, , B, , C, , (in Rs.), , Janaury, , 90, , 100, , 20, , 800, , February, , 130, , 50, , 40, , 900, , March, , 60, , 100, , 30, , 850, , Find out the rates of commission on items A, B and C, , 7., , If f (x) =, , cos x, , –sin x, , 0, , sin x, , cos x, , 0, , 0, , 0, , 1, , show that ( f (x) )–1 = f (–x)., , 14, , PC1 latest correction

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8., , Ramesh buyes half a dozen pencils, 2 erasers and 2 sharpeners from a shop and pays Rs. 14, from the same shop, Suresh buys 15 pencils, 5 erasers and 3 sharpeners and pays Rs. 35,, whereas their friend Mahesh, who accompanied them to the shop, buys as a token 1 pencil,, 1 eraser and 1 sharpener for the payment of Rs. 3. Find the price of each item at the shop,, by using matrices., , 9., , Three cricket fans, nick named as Soni, Moni and Dhoni, went to play for a country match., Their individual scores being x , y and z respectively. Find x, y, z using inversion method, from the following data :, i), , the sum of their scores is a centrury., , ii), , if we subtract the sum of Soni and Moni’s score from twice of Dhoni’s score it, is still a half centuary., , iii), 10., , four times Moni's score minus Soni's score equal to Dhoni's score., , Solve the following equations by using Reduction method., logxe + ey + z2 = 3, logxe + 2 ey + 3 z2 = 6, 2 logxe + 3 ey + 4 z2 = 1, , ****, , 15, , PC1 latest correction

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8., , Find the principal value of cosec (x) = 2, , 9., , In right angled triangle ABC, right angled at C. Show that tan A + tan B =, , c2, , 10), , In ABC, , ab, , sin (A/2) sin (C/2) = sin (B/2) and ‘2s’ is the perimeter of the triangle then find ‘s’., , 4 Marks, b–c, a, , tan (B/2) – tan (C/2), tan (B/2) + tan (C/2), , 1., , In ABC prove that, , 2., , In ABC , a cos2 (C/2) + c . cos2 (A/2) =, , =, , 3b, then prove that a, b and c, 2, , are in A.P., 3., , In ABC, if a2, b2, c2 are in A.P. then prove that cot A , cot B, cot C are in A.P., , 4., , Find the value of the expression tan {, , 5., , 6., , 1, , 2x, , –1, , sin, , 2, , + cos, , 1 + x2, , Where x > 0, y > 0 Such that xy < 1, , then prove that sin (A – B) =, In ABC if C =, 2, , 1 + x – 1 – x, 1 + x + 1 – x, , Show that tan–1, , =, , , , –, , 4, , 1, , –1, , 1 – y2, 1 + y2, , {, , a2 – b2, a2 + b2, cos–1x,, , 2, , –1, for, , 7., , 2, , < x < 1., , Prove the following sin–1, , 3, , 12, , + cos–1, , 5, x–1, 8., , If tan–1, , 9., , Prove that sin–1, , 10., , Show that, , x–2, , sin–1, , 13, , –1, 2, 8, 17, , x+1, + cos–1, , + sin–1, , 65, , , , x+2, + cot–1, , 56, , = sin–1, , =, , 4, , – 3, 2, 3, , , then find x., , = sin–1, , 5, , –1, , = cos–1, , 2, 77, 85, , ****, 17, , PC1 latest correction

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Chapter 4 : Pairs of Straight lines, (2 - Marks), 1., , Find the condition that the lines joining origin to the points of intersection of the line, y = mx + c and the curve x2 + y2 = a2 will mutually perpendicular., , 2., , Find the distance between pair of parallel lines given by x2 + 2xy + y2 – 8ax – 9a2 = 0, , 3., , The lines represented by x2 + xy + 2y2 = 0 and the lines represented by, (1 + ) x2 – 8xy + y2 = 0 are equally inclined, then find ., , 4., , Show that the equations, (y – mx)2 = a2 (1 + m2) and (y – nx)2 = a2 (1 + n2) form a rhombus., , 5., , For what value of ‘k’ the sum of the slopes of the lines given by 3x2 + kxy – y2 = 0, is zero., , 6., , Show that the equation 2x2 – xy – 3y2 – 6x + 19y – 20 = 0 represents a pair of lines., , 7., , Find the equations of angle bisectors between the lines 3x + 4y – 7 = 0 and, 12x + 5y + 17 = 0, , 8., , If the angle between the pair of straight lines represented by the equation., x2 – 3xy + y2 + 3x – 5y + 2 = 0 is tan–1 (1/3) where ‘’ is non negative real number, then, find ‘’., , 9., , The orthocentre of the triangle formed by the lines x + y = 1, 2x + 3y = 6 and, 4x – y + 4 = 0 lie in which quadrant ?, , 10., , The slopes of the lines represented by x2 + 2hxy + 2y2 = 0 are in the ratio 1 : 2 then find ‘h’., , 3 Marks, 1., , Find the joint equation of pair of lines through the origin which are perpendicular to the lines, represented by 5x2 + 2xy – 3y2 = 0, , 2., , Find the joint equation of the pair of lines which bisects anlges between the lines given by, x2 + 3xy + 2y2 = 0, , 3., , OAB is formed by the lines x2 – 4xy + y2 = 0 and the line AB. The equation of the line AB, is 2x + 3y – 1 = 0. Find the equation of the median of the triangle drawn from the origin., , 18, , PC1 latest correction

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4., , Show that the lines x2 – 4xy + y2 = 0 and x + y =  6 form an equilateral., Also find its area., , 5., , If the lines represented by the equation 2x2 – 3xy + y2 = 0 makes angle  and  with X - axis,, find the value of cot2 + cot2., , 6., , Two lines are given by (x – 2y)2 + k (x – 2y) = 0 then find the value of k, so that the distance, between them is 3., , 7., , Find the difference between slopes of the lines represented by equation, x2 (sec2  – sin2 ) – 2xy tan  + y2 sin2  = 0, , 8., , Find the condition of slope of one of the lines represented by ax2 + 2hxy + by2 = 0 is the square, of the other., , 9., , Find the number of lines that are parallel to 2x + 6y + 7 = 0 and have intercept of length 10, between the co-ordinate axes., , 10., , If the lines px2 – qxy – y2 = 0 makes the angles  and  with X - axis then find the value of, tan ( + ), , 4 Marks, 1., , Find the condition that the pair of lines ax2 + 2 (a + b) xy + by2 = 0 lie among diameters of, a circle and divide the circle into four sectors such that the area of one of the sector is thrice, the area of the another sector., , 2., , Prove that the product the lengths of perpendicular form P(x1, y1) to the line representd by, ax2 + 2hxy + by2 = 0 is, ax21 + 2hx1y1 + by12, ,  (a – b)2 + 4 h2, 3., , Find the equation of the bisectors of the angles between the lines., (  a + c ) x2 + 2  d xy + (  b +  c ) y2 = 0, , 4., , Find the measure of the acute angle between the lines., (a2 – 3b2) x2 + 8 ab xy + (b2 – 3a2) y2 = 0, , 5., , Find the condition that the equation ax2 + by2 + cx + cy = 0 may represents a pair of lines., , 6., , Show that the equation (x – 3)2 + (x – 3) (y – 4) – 2 (y – 4)2 = 0 reprsents a pair of lines, also find the acute angle between them., , 19, , PC1 latest correction

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7., , Find the joint eqation of pair of lines passing through the origin and making an angle of 30º with, the lines x + y = 5., , 8., , Find the combined equation of the lines, throught the origin forming an equilateral triangle with, the line x + y =  3, , 9., , Find the condition that the equation hxy – gx – fy + c = 0 represents a pair of lines., , 10., , Find ‘k’ if sum of the slopes of the lines represented by x2 + kxy – 3y2 = 0 is twice their, product., , ****, , 20, , PC1 latest correction

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Chapter 5 : Vectors, (2 - Marks), 1., , Find vector c if | c | = 3  6 and c is directed along the angle bisectors of the vectors, OA = 7 i – 4 j – 4 k and OB = – 2 i – j + 2 k., , 2., , If a and b are non collinear vectors then find the value of x for which vectors,  = (x – 2) a + b and  = (3 + 2x) a – 2 b are collinear., , 3., , Let a, b, c are three non-zero vectors such that any two of them are non-collinear., If a + 2 b is collinear with c and b + 3 c is collinear with a then prove that, a+2b+6c=O, , 4., , DA = a ; AB = b ; CB = k a ; k > 0 and X, Y are mid points of DB and AC repectively such, that | a | = 17 and | XY | = 4. Find value of k., , 5., , If a + b + 3 c ; – 2 a + 3 b – 4 c ; a – 3 b + 5 c are coplanar then find the value of ., , 6., , Find the value of, , a . (b  c), b . (c  a), , 7., , b . (c  a), c . (a  b), , +, , c . (a  b), a . (b  c), , In ABC, M is mid-point of side BC. If BAM = then using vector method prove that, cos  =, , 8., , +, , sin C + sin B cos A, ,  sin2B + sin2C + 2sinB sinC cos A, , a , b , c represent three concurrent edges of a rectangular parallelepiped whose lengths are 4,, 3, 2 units respectively then find value of, ( a + b + c) . ( a  b + b  c + c  a ), , 9., , If D, E, F are three points on the sides BC, CA, AB repectively of a ABC, such that AD,, BE, CF are concurrent, then using vector method prove that, BD, , , , CD, 10., , CE, AE, , , , AF, BF, , = 1, , a , b are perpendicular vectors, find projection of the vector, l, , a, |a|, , + m, , b, | b|, , +n, , ab, |ab|, , along the angle bisector of the vectors a and b, , 21, , PC1 latest correction

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(3 - Marks), 1., , If a, b, c are non co-planer non-zero vectros in the plane and r is any vector in the space, then show that [ b c r ] a + [ c a r ] b + [ a b r ] c = [ a b c ] r, , 2., , A parallelogram is constructed on the vector a = 3  –  , b =  + 3  and, c, |  | = |  | = 2 and angle between  and  is, then find lengths of the diagonals., 3, , 3., , a, b, c are three vectors such that | a | = | b | = | c | = 4 and angles between a and, c, b ; b and c ; c and a are equal to, . Find volume of parallelopiped whose adjacent sides, 3, a, b, c., , 4., , a, b, c are three vectors and vectors a, b, c are three vectors such that, a . a = b . b = c . c = 1 and a . b = a . c = b . c = c . a = c . b = 0, 1, Then prove that [ a b c ] =, , 5., , a, |a|, , ca, , ab], , +, , b, |b|, , A transversal cuts the sides OL, OM and diagonal ON of the parallelogram at A, B, C, repectively. Prove that, , OL, , +, , OA, 7., , [bc, , If OA = a and OB = b then show that the vector along the angle bisector of AOB is, given by d = , , 6., , [a b c]3, , OM, OB, , =, , ON, , using vector method., , OC, , Find all values of  for which, ( i + j + 3 k) x + ( 3 i – 3 j + k ) y + ( – 4 i + 5 j ) z =  ( x i + y j + z k) where, x, y z are not all equal to 0., , 8., , A straight line intersects sides AB, AC and AD in point B1, C1, D1., If AB1 = 1 AB ; AD1 = 2 AD ; AC1 = 3 AC then prove that, , 9., , 1, 1, =, 3, 1, , +, , 1, 2, , If cos   1 ; cos   1 ; cos   1 Prove that vectors, a = cos i + j + k ; b = i + cosj + k ; c = i + j + cosk can never coplanar., , ****, 22, , PC1 latest correction

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(4 - Marks), , 1., , 2., , 3., , If a , b , c, u , v , w are vectors prove that [ a b c ] [ u v w ] =, , If a , b , c are vectors prove that [ a b c ]2 =, , a.a, , b.a, , c.a, , a.b, , b.b, , c.b, , a.c, , b.c, , c.c, , a.u, , b.u, , c.u, , a.v, , b.v, , c.v, , a.w, , b.w, , c.w, , a, , b, , c, , If a , b , c , l , m are vectors prove that [ a b c ] ( l  m ) = a.l, , b.l, , c.l, , b.m, , c.m, , a.m, 4., , Prove that a  ( b  c ) = ( a . c ) b – ( a . b ) c, , 5., , If a , b , c are non-coplanar unit vectors each including the angle of measure 300 with the other, then find the volume of tetrahedron whose co-terminal edges are a , b , c., , 6., , In OAB, E is the mid-point of OB and D is a point on AB such that AD : DB = 2 : 1. If, OD and AE intersect at P, determine ratio OP : PD using vector method., , 7., , a and b are two non-collinear vectors, show that the points having positions vectors., l1a + l2b ; m1a + m2b ; n1a + n2b are co-linear if, ( l1m2 – m1l2 ) + ( m1n2 – n1m2 ) + ( n1l2 – l1n2 ) = 0, , 8., , Let the perpendicular lines B B and CC intersect at A and position vector of A w.r.t. O, be a . AB and AC are parallel to b and c respectively. If P is any point on the bisector, of CAB then prove that position vector of P is given by a + , , b, |b|, , 9., , +, , c, |c|, , a , b , c are position vectors of points A, B, C and P, Q, R are points BC, CA, AB, respectively such that BP : PC = CQ : QA = AR : RB = 1 : 2, Find position vector of vertices of XYZ formed by lines AP, BQ and CR., , ****, , 23, , PC1 latest correction

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Chapter 6 : Co-ordinate Geometry, (2 - Marks), 1., , Find the values of  for which the triangle with vertices A (6, 10, 10) ; B (1, 0, –5) ;, C (6, –10, ) is a right angled triangle at B., , 2., , Find direction ratios of the line which bisects the angle between the lines whose direction, consines are l1, m1, n1 and l2, m2, n2., , 3., , The equation of motion of a particle in space is given by x = 2t ; y = – 4t ; z = 4t , where, t is measured in second and co-ordinates of particle in kilometre. Then find the distance, covered from the starting point by the particle in 15 seconds., , 4., , The equation of motion of a particle in space is given by x = 2t ; y = – 4t ; z = 4t , where, t is measured in second and co-ordinates of particle in kilometre. Then find the speed of particle, in km / sec., , 5., , If distance of the point P (4, 3, 5) from the Y- axis is , then find the value of 72., , 6., , A (3, 2, 0) ; B (5, 3, 2) ; C (–9, 6, –3) are the vertices of ABC. If the bisector of BAC, meets BC at D then find the ratio in which C divides BD., , 7., , Planes are drawn parallel to the co-ordinate planes through the point (1, 2, 3) and, (3, –4, –5). Find the lengths of edges of the parallelepiped so formed., , 8., , Find the ratio in which the plane ax + by + cz + d = 0 divides the join the points, (x1, y1, z1) and (x2, y2, z2) ., , 9., , Find r, if direction ratio of vector r are 2, –3 6 and | r | = 21 and r makes obtuse angle, with the x -axis., , 10., , Let PM be the perpendicular drawn from the point P (x, y, z) on XY plane and OP makes an, angle with the positive direction of Z - axis, OM makes an anlge  with positive direction of, X - axis then Prove that, x = r sin cos ; y = r sin sin ; z = r cos , , 24, , PC1 latest correction

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(3 - Marks), 1., , The points (0, 1, –2) ; (3, , –1) ; (, –3, –4) are collinear show that the point, (12, 9, 2) lies on the same line., , 2., , If  is the angle between the lines having direction cosines l1, m1, n1 and l2, m2, n2., Then prove that sin =, ,  (l m – m l ), 1, , 2, , 1 2, , 2, , + (m1n2 – n1m2)2 + (n1l2 – l1n2)2, , 3., , l1 , m1 , n1 and l2 , m2 , n2 are direction cosines of perpendicular lines. Find the direction ratios, of the line perpendicular to both these lines., , 4., , If the diretion cosines of the line in two adjacent positions are, l, m, n and l + l, m + m, n + n then show that the small angel  between two position, is given by 2 = l2 + m2 +n2, , 5., , Find the angle included between the lines whose direction cosines are given by the equations, 3l + m + 5n = 0 and 6mn – 2nl + 5lm = 0, , 6., , If direction cosines of the line satisfy the relation (l + m) = n and mn + nl + lm = 0 then, find the value of  for which the two lines are perpendicular., , 7., , If l1 , m1 , n1 ; l2 , m2 , n2; and l3 , m3 , n3 are direction cosines of mutually perpendicular vectors, OA, OB, OC. respectively then prove that the line having direction cosines proportional to, l1 + l2 + l3 , m1 + m2 + m3 and n1 + n2 + n3 make equal angles with OA , OB , OC., , 8., , Let PM be the perpendicular drawn from the point P (1, 2, 3) on XY plane. OP makes an angle,  with the positive direction of z - axis. OM makes an angle  with positive direction of, X - axis find  and ., , 9., , A (2, 3, 5) ; B (–1, 3, 2) ; C (, 5, ) are vertices of ABC and median through vertex A, is equally inclined to the axes then find area of ABC., , ****, , 25, , PC1 latest correction

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Chapter 7 : Line, (2 - Marks), 1., , Find the vector equation of line passing through the point A (5, 3, 8) and parallel to vector, 3 i + 4 j + 5 k., , 2., , A line passes through the point with position vecter 3 i – 4 j + k and is in the direction, of 2 i + j – 2 k, find the equation of the line in vector and cartesian form., , 3., , Find the equation of line in symmetric form passes through the point (8, 3, 7) and, (–2, 5, –3), , 4., , Find the equation of line in cartesian form passing through the point (2, 1, –2) and, perpendicular to the vector 2 i – 3 j + 4 k and 2 i – 2 j + 3 k., , 5., , Find the vector equation of line perpendicular to the line, x, y–2, z–3, and x = 5,, =, =, 2, 3, 4, , z–2, y–3, =, 3, 2, , , and passing through (3, –1, 11), , 6., , Write symmetric form of the equation of the line 3x – 1 = 4y + 8 = 3z – 3, , 7., , Find the angle between the pair of line, r = ( 4 i + 7 j – 4 k ) +  ( i + 2 j + 2 k ) and, r = ( 5 i – 4 j + 3 k ) +  ( 3 i + 2 j + 6 k ), , 8., , Find the distane between the parallel lines, r = ( i + 2 j – 4 k ) +  ( 2 i + 3 j + 6 k ) and, r = ( 3 i + 3 j – 5 k ) +  ( 2 i + 3 j + 6 k ), , 9., , Find the direction cosines of the line, 2x – 1, 3, , 10., , = 3y =, , 4z + 3, 2, , Find the vecter equation of a line passing through the point with position vector, 2 i – j + k and parallel to the line joining the points – i + 4 j + k and i + 2 j + 2 k., , 26, , PC1 latest correction

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(3 - Marks), 1., , Show that the lines r = ( 2 i – j + k ) +  ( 2 i + j + k ) and, r = ( i + j + k ) +  ( i + 3 j + 2 k ) intersect find their point of intersection., , 2., , Find the value of k, if the points A (1, 2, –1), B (4, –2, 4) and C (0, 0, k) form a triangle right, angled, at C., , 3., , Find the foot of the perpendicular drawn from the point A (1, 0, 3) to the line joining the points, B (4, 7, 1) and C (3, 5, 3), , 4., , Find the vector equation of the line passing through the point (2, 3, –4) and perpendicular to, XZ - Plane, Hence find the equation in cartesian form., , 5., , Find the distance of P (1, 2, –2) from the line, , 6., , A line makes the same angle  with each of X and Z - axis. If the angle  which it makes, with Y - axis is such that sin2 = 3sin2, then find the value of cos2., , 7., , If the line x – 1 = y + 1 = z – 1, 2, 3, 4, , z, x–1, y+2, =, =, –1, 2, 1, , and x – 3 = y – k = z, 1, 2, –1, , interect then find the value of k., 8., , Find the shortest distance between the line 1 + x = 2y = – 12z and x = y + 2 = 6z – 6, , 9., , Find the distance of P (2, –1, 3) from the line, , 10., , Find the two points on the line, , x–1, y–3, z–2, =, =, 2, –1, 2, , x–2, y+3, z–5, on either side (2, –3, –5), =, =, 1, –2, 2, , which are at a distance of 3 units from it., , ****, , 27, , PC1 latest correction

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Chapter 8 : Plane, (2 - Marks), 1., , Find the vector equation of the plane passing through the point A (4, –2, 3) and parallel, to the plane x + 2y – 5z + 8 = 0, , 2., , Find the vector equation of plane which passes through the point A (1, –1, 1) and, perpendicular to the vector 4 i + 2 j + 2 k ., , 3., , Find the acute angle between the planes r . (2 i + j + k ) = – 5 and, r.(i–j+2k)=8, , 4., , Find the value of p, if the planes x – y + pz + 7 = 0 and 3x + y – z = 4 are perpendicular, to each other., , 5., , Find the vector equation of the plane passing through the points (5, 2, –1) (2,2, 3) and, origin., , 6., , Find the equation of the plane through the point (2, –3, 1) and perpendicular to the line, whose d.r’s are 3, –1, 2., , 7., , Find the angle between the line, , 8., , Find the equation of a plane whose distance from the origin is 5 units and normal in the, direction of n =, , 9., , If the lines, , 2, 3, , i –, , x–1, =, 5, , y, z, and plane 12x + 4y – 3z = 25, =, 2, 14, , 1, 2, j+, k, 3, 3, , x – 2, y–3, z–4, =, =, 1, 1, k, , and, , x–1, y–4, z–5, =, =, k, 2, 1, , are co-planar then find the value of K., 10., , A plane makes intercept 1, 2, 3 on the co-ordinate axes. If the distance from origin is p, then find the value of p., , 28, , PC1 latest correction

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3 - Marks, 1., , Find the equation of a plane which bisects the line joining the point A (2, 3, 4) and, B (4, 5, 8) at right angles., , 2., , Find the equation of the plane through the intersection of the planes, r . ( i + 3 j ) – 6 = 0 and r . ( 3 i – j – 4 k) = 0 whose perpendicualr distance from origin, is unity., , 3., , Find the equation of plane containing the line 2x – 5y + z = 3 ; x + y + 4z = 5 and parallel, to the plane x + 3y + 6z = 1, , 4., , A variable plane, , x, y, z, = 1 at a unit distance from the origin cuts the, +, +, a, b, c, , co-ordinate axes at A, B and C. Centroid (x, y, z) of, 1, 1, 1, + 2 + 2= k then find the value of k, 2, x, y, z, , equation, 5., , If the angle between the line, –1, cos, , ABC satisfies the, , 5, , y–1, z–3, x, and the plane x + 2y + 3z = 4 is, =, = , 2, 1, , then find the value of , , 14, 6., , Find the distance of the plane passes throught (1, –2, 1) and perpendicular to planes, 2x – 2y + z = 0 and x – y + 2z = 4 from the point (1, 2, 2), , 7., , Find the vector and cartesian equation of the plane passing through the points (2, 3, 1),, (4, –5, 3) and parallel to the X - axis., , 8., , Find the equation of the plane passing through the point (3, –2, –1) and parallel to the lines, whose direction ratios are 1, –2, 4 and 3, 2, –5., , 9., , Find the angle between the planes x – 2y + 2z = 7 and x – y – 3z = 5, , 10., , Find the equation of plane in vector form and cartesian form if the plane is at a distance of 3, units from the origin and has i + j – 3 k as a normal vector., , ****, , 29, , PC1 latest correction

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8., , A dietician whishes to mix two types of food in such a way that the vitamin contents of the, mixture contain atleast 8 units of Vitamin A and 10 units of Vitamin C. Food ‘P’ Contains, 2 units per kg. of Vitamin A and 1 unit per kg of Vitamin C while food ‘Q’ contains 1 unt per, kg of Vitamin A and 2 units per kg of Vitamin C. It costs Rs. 50/- per kg to purchase food, ‘P’ and Rs. 70/- per kg to purchase food ‘Q’. Formulate the above linear programming problem, to minimize the cost of such a mixture., , 9., , A rubber company is engaged in producing three types of tyres A, B and C. Each type requires, processing in two plants. Plant I and Plant II. The Capacities of the two plants in number of, tyres per day are as follows., Plant, , A, , B, , C, , I, , 50, , 100, , 100, , II, , 60, , 60, , 200, , The monthly demand for tyre A, B and C is 2500, 3000 and 7000 respectively. If plant I costs, Rs. 2500/- per day and plant II costs Rs. 3500/- per day to operate. How many days should, each be run per month to minimize cost while meeting the damand ? Formulate the problem as, LPP., 10., , A firm is engaged in breeding goats. The goats are fed on various products grown on the, farm. They need certain nutrients named as X, Y and Z. The goats are fed on two products, A and B. One unit of product A contains 36 units of X, 3 units and Y and 20 units Z. while, one unit of product B contains 6 units of X, 12 units of Y and 10 units of Z. The, minimum requirement of X, Y and Z is 108 units, 36 units and 100 units repectively. Product, A costs Rs. 20/- per unit and peroduct B costs Rs. 40/- per unit. Formulate the LPP to minimize, the cost., , 32, , PC1 latest correction

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should he invest his money in order to maximize his profit? Translate this problem mathematically, and then solve it., 9., , A farm is engaged in breeding pigs. The pigs are fed on various products grown on the farm., In view of the need to einsure certain nutrient constituents (call then X, Y and Z), it is necessary, to buy two additional products say A and B. One unit of product A contains 36 units of X, 3, units of Y and 20 units of Z. One unit of product B contains 6 units of X, 12 units of Y and, 10 units of Z. The minimum requirement of X, Y and Z is 108 unis, 36 units and 100 units, respectively. Product A costs Rs. 20/- per unit and product B costs Rs. 40 per unit formulae, linear programming. Problem to minimize the total cost and solve the problem by using graphical, method., , 10., , If a young man rides his motocylce at 25 km / hour. He had to spend Rs. 2/- per km on petrol., If he rides at a faster speed of 140 km / hour the petrol cost increases at Rs. 5/- per km. He, has Rs. 100 to spend on petrol and wishes to find what is the maximum distance he can travel, within one hour. Express this as an LPP and solve it graphically., , ****, , 34, , PC1 latest correction

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Chapter 10 : Continuity, (2 - Marks), 1., , sin x, , Prove that the function, , is discontinuous at x = 0., , |x|, 2., , Let f (x) be a continuous function and g (x) be a discontiuous function, prove that, f (x) + g (x) is discontinuous function., , 3., , x5  x – 32  2, , Discuss the continuity of f (x) =, , x3  x – 8  2, 44, , =, , x – 1, n, , 7, , n, , If f(x) =, , 5., , Discuss the continuity of the function, f (x), , 6., , m, , , at x = 2, , for x  1 is continuous at x = 1 , find f (1), , 4., ,  xn – 1, , ,x2, , =, , x2 / a – a, , ,, , x<a, , =, , 0, , ,, , x=a, , =, , a – x2 / a, , x>a, , ,, , at x = a, , Find the point (s) in the interval [–1, 2] where the function f (x) = x for x 0 and, f (0) = 1 is discontinuous, sin [ 4 (x –3)], , 7., , If f (x) =, , 8., , If the function f (x), , x2 – 9, =, , =, , , x  3 is continuous at x = 3 then find f (3)., cos kx – cos 4x, x2, 6, , ,x  0, , ,x=0, , is continuous at x = 0 , find k., log x – 1, 9., , If the function f (x) =, , x–e, , , for x  e, , is continuous at x = e find f (e)., 10., , Show that the function f (x) = 2x – | x | is continuous at x = 0., 35, , PC1 latest correction

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3 - Marks, a, , 1., , If f (x), , ( 1 + | sin x | ( | sinx |, , =, =, , ,, , b, , ,, , – / < x < 0, 6, x = 0, , 0 < x < /, 6, Find a and b so that the function is continuous at x = 0, etan2x / tan3x, , =, , 2., , Find the value of f (0) so that the function, ,  a2 – ax + x2 –  a2 + ax + x2, , f (x) =, , 3., , ,, ,  a + x – a – x, , If the function f (x), , is continuous for all x., , x + x2 + x3 + x4 + x5 – 62, , =, , , x  2, , x–2, =, , 3k, , ,x=2, , is continuous at x = 2, find k., , 4., , If f (x), , =, , sin x – sin a, cos x – cos a, , =, , 1, , ,, , x  a, , ,, , x = a is continuous, at x = a find a, , (a + x)2 sin (a + x) – a2sin x, 5., , If the function f (x) =, , x, , , x  0, , is continuous at x = 0 find f (0), , 6., , If f (x) =, , =, , x ax – x, ,  1 + x2 –  1 – x2, k, , , x  0, , , x = 0 is continuous at x = 0 , find k., (5x – 1)4, , 7., , If the function f (x) =, , , x  0, x tan (x/5) log (1 + x2/5), , is continuous at x = 0 , find f (0)., , 36, , PC1 latest correction

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8., , Discuss the continuity of f (x), , 22x – 2 – 2x + 1, , =, , , x  1, , tan2(x – 1), =, , 9., , If the function f (x), , 2 log 2, , x + 1 –  x + 13, , =, , x–3, , = k, , , x = 1 at x = 1, , x  3, , x=3, , is continuous at x = 3, find k., (ax – 1)3, 10., , If f (x) =, , sin (x log a) log (1 + x2 log a2), , , x  0, , is continuous at x = 0 , find f (0)., , 4 - Marks, 1., , Find the value of A so that the function, 2x + 2 – 16, , f (x) =, , , x 2, , 4x – 24, =, , 2., , If f (x) =, , A, , , x = 2 is continuous at x = 2, , 2x – + 23–x – 6,  2–x – 21–x, , , x  2 is continuous, , at x = 2, find f (2), 1 – cos (x2 – x – 6), (x – 3)2, , 3., , If f (x) =, , 4., , If the function, f (x), , =, , 1, x8, , =, , k2, 4, , , x  3 is continuous at x = 3 find f (3), , 2, 2, 2, 2, [1 – cos (x /2) – cos (x /4) + cos (x /2) cos (x /4)],, , x  0, , , x = 0 is continuous at x = 0 find k., 37, , PC1 latest correction

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Chapter 11 : Differentation, (2 - Marks), 1., , 2., , If y = x [ (cos x/2 + sin x/2) (cos x/2 – sin x/2) + sin x] +, If y = tan–1, , (ax – b), , 1, 2 x, , find, , dy, dx, , then find dy/dx., , (bx + a), , x + 1, , + sin–1, , x – 1, , 3., , If y = sec– 1, , 4., , If y = cos–1, , 5., , If f (x) = | x – 1 | + | x – 3 | then find f (2), , 6., , Find the deivative of f (x) = [ cos–1 sin, , x – 1, 1+x, 2, , then find dy/dx., , x + 1, , find dy/dx., , 1+x, , + xx ] w.r.t.x at x=1, , 2, 7., , If xyyx = 16 then find, , 8., , If f (x) =, , x, , dy, at (2, 2), dx, , for x  R then find f  (0), , 1 + |x|, 9., , If f  (x) =  3x2–1 and y = f (x2) then find, , 10., , If y = tan–1, , log (e/x3), log (ex3), , dy, dx, , log (e4x3), , + tan–1, , log (e/x12), , show that, , d2y, =0, dx 2, , 39, , PC1 latest correction

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3 - Marks, 1., , Let f (x) = ex, g (x) = sin–1x and h (x) = f (g (x)) then find, , h(x), h(x), , 2., , Find the deivative of f (tan x) w.r. to g (sec x) at x = /4 where f (1) = 2, and g (  2 ) = 4, , 3., , 1, , If y = tan–1, , + tan–1, , 1 + x + x2, , 1, x2 + 3x + 3, , + tan–1, , 1, x2 + 5x + 7, , + .................. + n terms then find y(0), , 4., , If f (x) = cos x cos 2x cos 4x cos (8x) . cos 16x then find f  (/4), , 5., , If f be twice differentiable function such that f11 (x) = – f (x) and f (x) = g (x),, h (x) = [ f (x)]2 + [ g (x)]2 if h (5) = 10 then find h (10)., , 6., , 7., , If y = sin2, , cot–1, , 1, then find, , 1+x, 1–x, , If y = x cos y, show that, , dx, , dy, , 8., , If y =, , 9., , If y = ef(x) where f (x) =, , 1 + x + x2, , find, , cos2y, , =, , dx, x sinx, , dy, , cos y + y sin x, , dy, dx, x–1, , 10., , x+1, , then show that, , dy, dx, , y log y, =, , x2 – 1, , If y = tan x + 2/3 tan3x + 1/5 tan5x then find dy/dx in terms of sec x., , 40, , PC1 latest correction

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4 - Marks, 1., , Each side of an eqilateral triangle is increasing at the rate of  3 cm/sec. Find the rate of which, its area is increasing when its side 2 meters., , 2., , A circular blot of ink increases in area in such a way that the radius ‘r’ cm at a time ‘t’ sec., is given by r = 2t2 – t3/4 what is the rate of increase of the area when t = 4., , 3., , Water is being poured at the rate 36m2/min in to cylindrical vessel whose base is a circle of, radius 3 meters. Find the rate at which the level of water is rising., , 4., , 5., , The height of cone is 30 cm and it is constant, the radius of the base is increasing at the rate, of 2.5 cm/sec. Find the rate of increase of volume of the cone when the radius is 10cm., 2 3, A particle moves along the curve y =, x + 1. Find the point on the curve where the, 3, y - coordinate is changing twice as fast as the x - co-ordinate., , 6., , An edge of a veriable cube is increasing at the rate of 10 cm/sec. How fast is the volume, of the cube increasing when the edge is 5 cm long ?, , 7., , Find the point on the curve y2 = 8x for which the abscissa and ordinate change at the same, rate., , 8., , Find the slope of the tangent and the normal when x = a (–sin  ); y = a (1 – cos ), at = /2, , 9., , Find a point on the curve xy = – 4 where the tangents are inclined at an angle 450 with x, - axis., , 10., , Show that the tangents to the curve y = 2x2 – 3 at a point x = 2 and x = – 2 are parallel., , 11., , FInd the point on the curve y = 6x – x2 where the tangents has slope –4. Also find the, equation of the tanent at that point., , 12., , Find the second degree polynomial f (x) satisfying f (0) = 0, f (1) = 1, f 1(x) > 0 for all, x  (0, 1), , 13., , If the function f (x) = 2x3 – 9 ax2 + 12 a2x + 1 attains its maximum and minimum at p and, q respectively such that p2 = q then find the value of a., , 14., , If x + y = 8 then show that maximum value of x2y is, , 2048, 27, , ****, 44, , PC1 latest correction

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Chapter 13 : Indefinite Integeals, SECTION B, (2 - Marks), 1., , ex – 1, , Evaluate, , dx, , ex + 1, , 2., , sec2x – 7, , Evaluate, , dx, , sin7x, 3., , Evaluate, , 4., , Evaluate, , 5., , sin (101 x) sin99x dx, ,  cot x, sin x . cos x, , dx, , log (ex + 1), , Evaluate, , dx, , ex, , 6., , x4 + 1, , Evaluate, , dx, , x6 + 1, 7., , cos x – cos3x, , Evaluate, , dx, , 3, , 1 – cos x, 8., , 1+x, , Evaluate, , dx, , 3, , 1 + x, , 9., , Evaluate, , 10., , Show that, , x . (x x) x (2 log x + 1) dx, , dx, , is a polynomial of degree three in cot x., , sin4x, 45, , PC1 latest correction

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SECTION C, 3 - Marks, 1., , Evaluate, , 2., , Evaluate, , 3., , Evaluate, , 4., , Evaluate, , 5., , Evaluate, , 6., , Evaluate, , 7., , Evaluate, , 8., , Evaluate, , 9., , Evaluate, , 1–x, 1+x, , tan–1, , dx, , dx, cos3x  sin 2x, dx, x2 (x4 + 1)3/4, 1, 1 + x + x2 + x3, 1, x, , 1–x, 1+x, , dx, , dx, , (3 sin  – 2) cos , 5 – cos2 – 4 sin , , d, , x sin x . sec3x dx, sin x,  1 + sin x, , dx, , cos 4 x – 1, , dx, , cot x – tan x, , 10., , Show that, , (  tan x +  cot x ) dx =  2 tan–1, , tan x – 1, ,  2 tan x, , +c, , 46, , PC1 latest correction

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SECTION D, 4 - Marks, , 1., , Evaluate, , 2., , Evaluate, , tan–1 (1 +  x ) dx, 1, , dx, , sin3x + cos3x, , 3., , Evaluate, , 4., , Evaluate, , log |  1 – x +  1 + x | dx, , 1, , dx, , cos2x + cot2x, 1, 5., , Evaluate, , 6., , Evaluate, , sec x + cosec x, sin x, , dx, , dx, , sin 4 x, 7., , If, , [ log (logx) + (log x)–2 ] dx = f (x) + c then find f (x)., , Also find f (x) when graph of y = f (x) passes through the point (e, e), , 8., , Evaluate, , 9., , Evaluate, , x, a+x, , sin–1, , (x + 1)  x + 2, , x – 2, , dx, , dx, , dx, 10., , Evaluate, ,  2x + 3 +  x + 2 dx, , ****, , 47, , PC1 latest correction

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Chapter 14 : Definite Integrals and It’s Applications, SECTION B, 2 - Marks, /2, 1., , 3 sin  + 4 cos , , Evaluate, , sin  + cos , , , , 1, , 1, 2., , (x – x3) 3, , Evaluate, , dx, , x4, , , /2, 3., , cos99x dx = 0, , Show that, , , 4., , Find the area of the region enclosed by the lines y = x, x = e and the curve y =, and posiitve x - axis., e37, , 5., , 6., ,  sin ( log x), , Evaluate, , x, , , , 1, x, , dx, , 3, , Evaluate, , f (x – [x]) dx where f is signum function and [ . ] denotes greates, , –3, integer function., , 7., , x | x | dx where  < 0 < ., , Evaluate, , , 8., , /2, , Show that, , –, , (2 log sin x – log sin 2x) dx =, , , 2, , log 2, , e, 9., , 10., , Evaluate, , 1, e, , Show that, , , | log x | dx, , , /2, x f (sin x) dx =       f (cos x) dx., , ***, , 48, , PC1 latest correction

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SECTION C, 3 - Marks, /2, 1., , 1+e, , , 2., , 1, , Evaluate, , dx, sinx, , 1, , Evaluate, , [ 5x ] dx. Where [ . ] denotes greatest integer function., , , 3., , Find the area bounded between the parablas, y = 2., , x2 =, , /4, 4., , log (1 + tan2 + 2 tan ) d=, , Show that, , , y, , and x2 = 9y and the strainght line, , 4, , , log 2, , 4, 3, , 5., , Evaluate, , | x –2 | + [x]) dx, where [ . ] denotes greatest integer function., –1, x, , 6., ,  2 – t2 at then find the roots of the equation x2 – f  (x) = 0, , If f (x) =, 1, , 7., , Find the area of the region bounded by the curves y =, , x2, , , x<0, , and the line y = 4, , x, , , x>0, , =, , /2, 8., , (sin 2x . tan–1 (sin x)) dx., , Evaluate, , , 9., , If f (x) is a continuous function such taht f (2 – x) + f (x) = 0 for all x, then find, 2, 0, , 1, dx, , 1 + 2 f(x), 1, , 10., , Evaluate, , sin, 0, , 2 tan–1, , 1+x, 1–x, , dx, , ****, , 49, , PC1 latest correction

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SECTION D, 4 - Marks, 1, 1., , tan–1 (1 – x + x2) dx., , Evaluate, 0, , 2., , 1, 2, , Evaluate, , 3., , | x cos x | dx., , 1, 2, , –, , , , Evaluate, , x log x, (1 + x2)2, , 0, 1, 4., , Evaluate, , 5., , 6., , | log x |, , 1, e, , 2 – x2, , dx., , x2, , Find the area bounded by the curves x = y2 and x = 3 – 2y2., /3, , 7., , 2 – cos, , , , dx., , 3, ,  x2 – 4, , If I =, , dx then show that I =  3, , x4, , 2, 10., , |x|+, , Find the area enclosed within the curves | x | + | y | = 1, 4, , 9., ,  + 4x3, , Evaluate, –/3, , 8., , dx., , (1 + x)  1 – x2, , 0, e, , dx., , Let f (x) = maximum of { x + | x | , x – [ x ] } where [ . ] denotes greatest integer function, 2, then find, , f (x) dx., –2, ***, , ., 50, , PC1 latest correction

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Chapter 15 : Differential Equation, (2 - Marks), 1., , Find the differential equation of all circles passing through the origin and having their centres, on the X - axis., 1, , 2., , Verify y =, , 3., , Find the equation of the curve whose slope, , 4, , e, , –2x, , + cx + d, is the solution of the differential equation, dy, dx, , 2y, =, , d2y, 2, , dx, , = e–2x, , , x, y > 0 which passes, , x, , through the point (1, 1), 4., , Find the equation of curve passing through (1, /4) and having slope, , x + tan y, , at (x, y), 5., , sin 2y, , Find the integrating factor (I.F.) of differential equation, , dy, , = ex–y (1 –ey), , dx, 6., , If sin x is an integrating factor of the differential equation, , + Py = Q then, , dx, , find the value of P., 7., , dy, , Find the integrating factor of the differential equation (xy – 1), , dy, , + y2 = 0, , dx, , 8., , State order and degree of differential equation, , 1, 6, , d3y, 3, , dx, , 9., , By eliminating arbitary constant of equation y = c2 +, , dy, , 1, 3, , = 5, , dx, c, , find differential equation., , x, 10., , If y = sin–1 x then show that (1 – x2), , d2y, dx2, , = x dy, dx, , ****, , 51, , PC1 latest correction

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3 - Marks, 1., , Find the differential equation associated to the primitive y = ae4x – be–3x + c, , 2., , Find A if x = 4t3, y = 4t2 – t4 constitute a solution of the differential equation, d2y, , 36, 3., , [ y – (2x)2/3 ] = A +, , dx 2, , x, , 2/3, , 4, , Find the particular solution of the differenital equation given as, dy, , e dx = x + 1 at y (0) = 3, 4., , dy, Find y (/2) if (2 + sin x), + (y + 1) cos x = 0 and y (0) = 1, dx, , y+1, , 5., , Find the equation of the curve passing through (2, –2) and having slope x2 – x, , 6., , If curve y = f (x) passes through the point (1, –1) and satisfies the differenital, quation y (1 + xy) dx = x dy, then find f (–, , 7., , 1, , ), , 2, , Find the general solution of the differenital equation, 2, , 2, , (ex + ey ) y, , dy, , 2, , + ex (xy2 – x) = 0, , dx, 8., , The differential equation, , dy, = ex–y (ex – ey) whose general solution is given by, dx, , x, , f (x, y) ee = constant, then find f (0, 0), 9., , Find the general solution of the differential equation, cosx dy = y (sinx – y) dx, 0 < x < /, 2, , 10., , If f  (x) = f (x) and f (–1) = 1 then find f (5)., , 11., , Form and differential equation satisfying  1 – x4 + 1 – y4 = a (x2 – y2) and find the degree., , 12., , If the population of a country doubles in 50 years. Then find the number of years in which, population will be triple under the assumption that the rate of increase of population, proportional to the number of inhabitants., , 13., , A ray of light coming from origin after reflecting at the point P (x, y) of any curve become, parallel to x - axis find the equation of curve., ****, 52, , PC1 latest correction

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4 - Marks, 1., , The rate at which radioactive substance decay is known to be proportional to the number of, such nuclei that are present at that time in a given sample. In a certain sample 10% of the original, number of radioactive nuclei have undergoan disintegration in a period of 100 years. Find what, percentage of the original radioactive nuclei will remain after 1000 years., , 2., , In a college hostel accomodating 1000 students, one of them came in carrying a flue virus, then, the hostel was isoleted. If the rate of which the virus spreads is assumed to be proportional to, the product of the number N of infected students and the number of non infected students and, if the number of infected students is 50 after 4 days. Show that more than 95% of the students, will be infected after 10 days., , 3., , Water flows from the base of rectangular tank of depth 16 meters. The rate of flowing the water, is proportional to the square root of depth at any time ‘t’. After 2 hours depth of water is 4, meter, find the depth of water after 4 hours., , 4., , Water at 1000C cools in 10 minutes to 880C, in the room temperature of 250C. Find the, temperature of water after 20 minutes., , 5., , A tank of 100 m3 capacity is full with pure water. Begining at t = 0 brine containing, 1 kg/m3 of salt runs in at the rate 1 m3/min. The mixture is kepts uniform by stirring. It runs out, at the same rate when will there by 50 kg. of dissolved salt in the tank ?, , 6., , Show that the singular solution of the differential equation y = mx + m – m3 where, m=, , 7., , dy, , passes through the point (–1, 0), dx, Let the population of rabiits surviving at a time ‘t’ be governed by the differential equation, dp (t), dt, , 8., , 9., 10., , = –½ P (t) – 200. If p (0) = 100 then find p (t)., , A hemispherical tank (radius = 1m) is initially full of water and has an out let of 12 cm2 at the, bottom. When the out let is opened the flow of water is according to the law, v (t) =    h(t) where v (t) is the velocity of flow in cm/sec. h (t) is the water level in, cms. and is constant. Find the time taken to empty of tank is sec., A mothball evaporate at a rate proportional to the instantaneus surface area. Its radius to half, , in value in one day. Find the requaired for the ball to disapper compleletly., An inverted conical tank of 2 m radius and 4n height is initially full of water has an out let, at bottom. The outlet is opened at some instant. The rate of flow through the outlet at any, time t is 6 h3/2, where h is height of water level above the out let at time t. Then fine the time, it takes to empty the tank., 53, ***, , PC1 latest correction

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Chapter 16 : Probability Distribution- VSA Qns, (2 - Marks), 1., , 2., , Two cards are drawn from a pack of 52 cards., Prepare the probability distribution of the random variable defined as, number of black, cards., State with reasons whether the following represent the p.m.f of a random variable., x:, p(x) :, , 0, , 1, , 2, , 3, , y, , 0.1, , 0.2, , -0.1, , 0.7, , P(y), , 0, , 1, , 2, , 0.1, , 0.2, , 0.5, , 3., , Verify whether the following function is p.m.f. of continuous r.v.X., f (x) =, x/2, -2 < x< 2, =, 0, , otherwise, , 4., , Verify whether the following function can be regarded as the p . m. f for the given values, of X, P (X = x), , =, , x - 2, , X = 1,2,3,4,5, , 5, =, 5., , = l/5, , for x = 0,1,2,3,4, , = 0, , otherwise, , = kx ,, , x = 1,2,3,4,5, , = 0 ,, , otherwise, , Determine k such that the following function is a p.m.f., P (X = x), , 8., , otherwise, , Determine k such that the following function is a p.m.f., P (X = x ), , 7., , ,, , Verify whether the following function can be regarded as the p.m.f. for the given values of, X, P (X = x), , 6., , 0, , = k(.2x/x!) , x =, , 0,1,2,3, , = 0, , otherwise, , In a Pizza Hut , the following distribution is found for a daily demand of Pizzas. Find the, expected daily demand., No. of Pizzas :, Probability, , :, , 5, , 6, , 7, , 8, , 9, , 10, , 0.07, , 0.2, , 0.3, , 0.3, , 0.07, , 0.06, , 54, , PC1 latest correction

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18., , Find the c.d.f F (x) associated with the following pdf. f (x), f (x), , = 12x2 (1 - x), 0 < x < 1,, = 0 , otherwise. Also, find P (, , 1, 1, < x <, ) by using c.d.f. and, 2, 3, , sketch the graph of F(x)., 19., , A die is tossed twice. Getting a number greater than 4 is considered a “success”. Find the, mean and variance of the probability distribution of the number of successes., , ****, , 57, , PC1 latest correction

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Chapter 17 : Binomail Distribution -VSA, (2 - Marks), 1., 2., 3., 4., 5., , 6., , Write the Binomial distribution if mean for the distribution is 3 and the standard deviation, is 3/2., If X ~ B (6, p) and 2 P (X = 3) = P (X = 2) then find the value of p., If a die is thrown twice, then find the probability of occurence of 4 atleast once., A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn one-by-one, with replacement then find the variance of the number of yellow balls., India plays two matches each with West Indies and Australia. In any match the probabilities, of India getting point 0, 1, 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the, outcomes are independent. Find the probability of India getting at least 7 points ., 100 identical coins, each with probability p, of showing up heads are tossed once. If, 0 < p < l and the possibility of heads showing on 50 coins is equal to that of heads showing, on 51 coins, then find the value of p ., , 3 Marks, 1., 2., , 3., 4., 5., , 6., , If the mean and the variance of the binomial varite X are 2 and 1 respectively, then find, the probability that X takes a value greater than one., Numbers are selected at random, one at a time, from the two-digit numbers 00, 01, 02,, 03,....99 with replacement. An event E occurs if and only if the product of the two digits, of a selected number is 18. If four numbers are selected, find probability that the event E, occurs at least 3 times., A man takes a step forward with probability 0.4 and backwards with probability 0.6. Find, the probability at the end of eleven steps he is one step away from the starting point., Find the minimum number of times a fair coin needs to be tossed, so that the probability, of getting at least two heads is at least 0.96., A lot of 100 pens contains 10 defective pens. 5 pens are selected at random from the lot, and sent to the retail store. What is the probability that the store will receive at least one, defective pen?, In a production process, producing bulbs, the probability of getting a defective bulb remains, constant and it is 0.3. If we select a sample of 10 bulbs, what is the probability of getting, 3 defective bulbs?, , 58, , PC1 latest correction

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7., 8., 9., 10., , 11., , In a bag containing 100 eggs, 10 eggs are rotten. Find the probability that out of a sample, of 5 eggs none are rotten, if the sampling is with replacement., A fair coin is tossed six times. What is the probability of obtaining four or more heads?, Each of two persons A and B toss 3 fair coins. Find the probability that both get the same, number of heads., The probability of India winning a test match against England is 2/3. Assuming independence, from match to match, find the probability that in a 7 match series India’s third win occurs at, the 5th match.., Let x denotes the number of times heads occur in n tosses of a fair coin, if P(X = 4),, P (X = 5) and P(X = 6) are in A.P., find n., , ****, , 59, , PC1 latest correction