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Production Team, , Anil Kumar Sharma, , Published at Delhi Bureau of Text Books , 25/2 Institutional Area, Pankha, Road, New Delhi-110058 by Anil Kaushal, Secretary, Delhi Bureau of, Text Books and Printed by Supreme Offset Press , New Delhi-110017
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DIRECTORATE OF EDUCATION, Govt. of NCT, Delhi, SUPPORT MATERIAL, (2019-2020), , MATHEMATICS, Class : X, (English Medium), , NOT FOR SALE, , PUBLISHED BY : DELHI BUREAU OF TEXTBOOKS
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Content, S.No., , Chapter Name, , Page No., , 1., , Real Numbers, , 01–13, , 2., , Polynomials, , 14–23, , 3., , Pair of Linear Equations in Two Variables, , 24–33, , 4., , Quadratic Equations, , 34–49, , 5., , Arithmetic Progression, , 50–65, , 6., , Similar Triangles, , 66–90, , 7., , Co-ordinate Geometry, , 8., , Trigonometry, , 103–112, , 9., , Some Applications of Trigonometry, , 113–121, , 91–102, , (Heights and Distances), 10., , Circles, , 122–139, , 11., , Constructions, , 140–147, , 12., , Areas Related to Circles, , 148–170, , 13., , Surface Areas and Volumes, , 171–191, , 14., , Statistics, , 192–207, , 15., , Probability, , 208–224, , 16., , Practice Test, , 225–256
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CHAPTER, , 1, , Real Numbers, , KEY POINTS, , Decimal form of Real Numbers, , Real Number (Q), , Terminating decimal, (3/5, 5/4, …), n m, denominator = 2 5 ,, where, n m ∈, +ve integers, , Irrational Number (I), , Non-terminating, but, repeating, decimal, (2/3, 1/7, …), , Non terminating &, non repating, (1.010010001..,, 1.232232223), , PROPERTIES OF REAL NUMBERS, , Euclid division Lemma, given +ve integers a & b, there, exist unique integers q & r, satisfying a = bq + r, 0 ≤ r < b., , Fundamental Theorem of Airthmatic, Every composite number can be expressed, factorised as a product of primes and this, factorisation is unique apart from the, order in which the prime factors occur., , Euclid division algorithm, The HCF of any two +ve integers a & b with a > b, is obtained, as follows:, Step 1 : Apply Euclid division lemma to a & b to find q & r,, where a = bq + r , 0 ≤ r < b., Step 2 : If r = 0, the HCF is b If r ¹ 0 then apply Euclid 's lemma, to b & r to find b1, and r1, where b = b1 r + r1, Step 3 : If r1 = 0, HCF is b, If r1 ¹ 0, then continue the process., till rn = 0 then bn at this stage will be HCF., , Mathematics-X, , 1
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VERY SHORT ANSWER TYPE QUESTIONS, , 1., 2., 3., 4., , A number N when divided by 16 gives the remainder 5 ______ is the remainder, when the same number is divided by 8., HCF of 33 × 54 and 34 × 52 is ________ ., If a = xy2 and b = x3y5 where x and y are prime numbers then LCM of (a, b) is, _____ ., In factor tree find x and y, y, x, , 2, , 7, , 5, , 5., , If n is a natural number, then 252n – 92n is always divisible by :, (i) 16, (ii) 34, (iii)both 16 or 34, (iv) None of these, , 6., , The decimal expansion of the rational number, , 7., , (a) One decimal place, (b) Two decimal place, (c) Three decimal place, (d) More than three decimal place, Which of the following rational numbers have terminating decimal?, (i), , 16, 225, , (ii), , 5, 18, , (iii), , 2, 21, , 327, 23 × 5, , will terminate after, , (iv), , 7, 250, , (a) (i) and (iii), (b) (ii) and (iii), (c) (i) and (iii), (d) (i) and (iv), 8. Euclid’s division Lemma states that for two positive integers a and b, there exist, unique integers q and r such that a = bq + r, where r must satisfy., (a) 1 < r < b, (b) 0 < r ≤ b, (c) 0 ≤ r < b, (d) 0 < r < b, n, n, n, 9. p = (a × 5) For p to end with the digit zero a = _____ for natural number n., (a) any natural number, (b) even number, (b) odd number, (d) none of these, Mathematics-X, 2
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10. HCF is always, (a) multiple of LCM, (b) Factor of LCM, (c) divisible by LCM, (d) a and c both, 11. All decimal numbers are, (a) rational number, (b) irrational numbers, (c) real numbers, (d) integers, 12. Which of these numbers always end with the digits 6., (a) 4n, (b) 2n, (c) 6n, (d) 8n, 13. Write the general form of an even integer, 14. Write the form in which every odd integer can be written taking t as variable., 15. What would be the value of n for which n2–1 is divisible by 8., 16. What can you say about the product of a non-zero rational and irrational number?, 13497, 17. After how many places the decimal expansion of, will terminate?, 1250, 18. Find the least number which is divisible by all numbers from 1 to 10 (both, inclusive)., 19. The numbers 525 and 3000 are divisible by 3, 5, 15, 25 and 75 what is the HCF, of 525 and 3000?, 20. What will be the digit at unit’s place of 9n?, SHORT ANSWER TYPE QUESTIONS-I, , 21. If n is an odd integer then show that n2 – 1 is divisible by 8., 22. Use Euclid’s division algorithm to find the HCF of 16 and 28., 23. Show that 12n cannot end with the digit 0 or 5 for any natural number n., (NCERT Exemplar), 395, will have terminating or, 24. Without actual performing the long division, find if, 10500, non terminating (repeating decimal expansion.), 25. A rational number in its decimal expansion is 327. 7081. What can you say about the, p, ? Give reasons., q, 26. What is the smallest number by which 5 – 2 is to be multiplied to make it a, rational number? Also find the number so obtained?, , prime factors of q, when this number is expressed in the form of, , Mathematics-X, , 3
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27. Find one rational and one irrational no between 3 and 5 ., 28. If HCF of 144 and 180 is expressed in the form 13m – 3, find the value of m., (CBSE 2014), n, 2n, 2n + 1, 4n+2, 29. Find the value of : (–1) + (–1) + (–1), + (–1) , where n is any positive and, integer., (CBSE : 2016), 30. Show that any positive add integer is of the form 4q + 1 or 4q + 3, where q is some, integer., (CBSE : 2012), 31. Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maximum, capacity of a container which can measure the petrol of either tanker in exact number, of times., (CBSE : 2016), SHORT ANSWER TYPE QUESTIONS-II, , 32. Show that the cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3 for, some integer m., 33. Prove that 3 is an irrational number.., 34. State fundamental theorem of Arithmetic and hence find the unique factorization, of 120., 35. Prove that 3 + 5 is irrational, 36. Prove that 5 –, 37. Prove that, , 3, 3 is an irrational number.., 7, , 1, is an irrational number.., 2– 5, , 38. Find HCF and LCM of 56 and 112 by prime factorization method., 39. Explain why:, (i) 7 × 11 × 13 × 15 + 15 is a composite number, (ii) 11 × 13 × 17 + 17 is a composite number., (iii) 1 × 2 × 3 × 5 × 7 + 3 × 7 is a composite number., 40. On a morning walk, three perosns steps off together and their steps measure 40 cm,, 42 cm, and 45 cm respectively. What is the minimum distance each should walk, so, that each can cover the same distance in complete steps?, (NCERT Exemplar), , 4, , Mathematics-X
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41. During a sale, colour pencils were being sold in the pack of 24 each and crayons in, the pack of 32 each. If you want full packs of both and the same number of pencils, and crayons, how many packets of each would you need to buy? (CBSE : 2017), 42. Find the largest number that divides 31 and 99 leaving remainder 5 and 8 respectively., 43. The HCF of 65 and 117 is expressible in the form 65 m – 117. Find the value of m., Also find the LCM of 65 and 117 using prime factorisation method., 44. Using Euclid’s division algorithm, find the largest number that divides 1251, 9377, and 15628 leaving remainder 1, 2 and 3 respectively., (NCERT Exemplar), 45. Show that square of any odd integer is of the form 4m + 1, for some integer m., 46. Find the HCF of 180, 252 and 324 by Euclid’s Division algorithm., 47. Find the greatest number of six digits exactly divisible by 18, 24 and 36., 48. Three bells ring at intervals of 9, 12, 15 minutes respectively. If they start ringing, together at a time, after what time will they next ring together?, 49. Show t hat only one of the number of n, n + 2 and n + 4 is divisble by 3., 50. Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of two, given number., (CBSE : 2018), LONG ANSWER TYPE QUESTIONS, , 51. Find the HCF of 56, 96, 324 by Euclid’s algorithm., 52. Show that any positive odd integer is of the form 6q + 1, 6q + 3 or 6q + 5, where, q is some integer., 53. Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for, some integer, q., 54. Prove that the product of three consecutive positive integers is divisible by 6., 55. For any positive integer n, prove that n3–n is divisible by 6. (NCERT Exemplar), 56. Show that one and only one of n, n + 2, n + 4 is divisible by 3., 57. Aakriti decided to distribute milk in an orphanage on her birthday. The supplier, brought two milk containers which contain 398 l and 436 l of milk. The milk is to be, transferred to another containers so that 7 l and 11 l of milk is left in both the containers, respectively. What will be the maximum capacity of the drum?, 58. Find the smallest number, which when increased by 17, is exactly divisible by both, 520 and 468., 59. A street shopkeeper prepares 396 Gulab jamuns and 342 ras-gullas. He packs, them, in combination. Each containter consists of either gulab jamuns or ras-gullab, but have equal number of pieces., Mathematics-X, , 5
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Find the number of pieces he should put in each box so that number of boxes are, least., (CBSE 2016), 60. Show that the square of any positive integer cannot be of the form 5q + 2 or, 5q + 3 for integer q., 61. Express the HCF of numbers 72 and 124 as a linear combination of 72 and 124., 62. Show that there is no positive integer n for which n − 1 + n + 1 is rational., 63. Find the HCF of numbers 134791, 6341 and 6339 by Euclid’s div ision algorithm., 64. In a seminar, the no. of participants in Hindi, English and Mathematics are 60, 84 and, 108 respectively. Find the minimum number of rooms required if in each room the, some the same number of participants are to be seated and all of the them being of, the the same subject., (HOTS), 65. State fundamental theorem of Arithmetic. Is it possible that HCF and LCM of two, numbers be 24 and 540 respectively. Justify your answer., ANSWERS AND HINTS, , 1. 5, , 2. 33 × 52, , 3. x3 × y5, , 4. x = 35, y = 70, , 5. (iii) 252n – 92n is of the form a2n – b2n which is divisible by both a – b and, a + b so, by both 25 + 9 = 34 and 25 – 9 = 16., 6. (c) three decimal place, , 7. (d) (i) and (iv), , 8. (c) 0 ≤ r < b, , 9. (b) even number, , 10. (b) Factor of LCM, , 11. (c) real numbers, , 12. (c) 6n, , 13. 2m, , 14. 2t + 1, , 15. An odd integer, , 16. Irrational, , 17. 4, , 18. 2520, , 19. 75, , 20. 1 and 9, 21. Any +ve odd integer is of the form 4q + 1 or 4q + 3 for some integer q so if, n = 4q + 1., n2 – 1 = (4q + 1)2–1 = 16q2 + 8q = 8q(2q + 1) ⇒ n2 – 1 is divisible by 8., 6, , Mathematics-X
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If n = 4q + 3, n2 – 1 = (4q + 3)2 – 1 = 16q2 + 24q + 8 = 8 (2q2 + 3q + 1) ⇒ n2 – 1 is divisible, by 8., 22. 4, 23. As 12 has factors 2, 2, 3 it doesnot has 5 as its factor so 12n will never end with 0, or 5., 24. Non-terminating repeating., 25. Denominator is the multiple of 2’s and 5’s., 26., , 5+ 2 , 3, , 28. By Euclid’s division lemma, 180 = 144 × 1 + 36, 144 = 36 × 4 + 0, HCF of 180 and 144 is 36., 29. Given that n is a positive odd integer, ⇒ 2n and 4n + 2 are even positive integers and n and 2n + 1 are odd positive, integers., ∴ (–1)n = – 1, (–1)2n = + 1, (–1)2n + 1 = – 1, (–1)2n + 2 = + 1, ∴ (–1)n + (–1)2n + (–1)2n + 1 + (–1)4n + 2 = – 1 + 1 – 1 + 1 = 0, 30. By applying Euclid division algorithm to a and b such that a = 4q + r, where, b = 4, Now r = 0, 1, 2, 3., where,, r = 0, a = 4q, which is even number., where,, r = 1, a = 4q + 1 an odd number., where,, r = 2, a = 4q + 2 =2 (2q + 1), an even number., where,, r = 3, a = 4q + 3 an odd number., 31. HCF of 850 and 680 is 2 × 5 × 17 = 170 litres., 32. Let n be any psoitve integer. Then it is of the form 4q, 4q + 1, 4q + 2 and, 4q + 3., When n = 4q, n3 = 64q3 = 4 (16q3) = 4 m, where m = 16q3, When n = 4q + 1, n3 = (4q + 1)3 = 64q3 + 48q2 + 12q + 1, = 4(16q3 + 12q2 + 3q) + 1 = 4 m + 1., where m = 16q3 + 12q2 + 3q, Similarly discuss for n = 4q + 2 and 4q + 3., Mathematics-X, , 7
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34. 2 × 2 × 2 × 3 × 5, 35. Prove that 3 and 5 is irrational number separately and sum of two irrational, number is an irrational number., 3, 3 is an irrational number. Difference of a rational number, 7, and irrational number is an irrational number., HCF : 56, LCM : 112, (1) 15 × (7 × 11 × 13 + 1) as it has more than two factors so it is composite no., LCM of 40, 42, 45 = 2520, Minimum distance each should walk 2520 cm., LCM of 24 and 32 is 96, , 36. 5 is rational no. and, 38., 39., 40., 41., , 96 crayons or, 96 pencils or, , 96, = 3 packs of crayons, 32, 96, = 4 packs of pencils., 24, , 42. Given number = 31 and 99, 31 – 5 = 26 and 99 – 8 = 91, Prime factors of 26 = 2 × 13, 91 = 7 × 13, HCF of (26, 91) = 13., ∴ 13 is the largest number which divides 31 and 99 leaving remainder 5 and, 8 respectively., 43. HCF of 117 and 65 by Euclid division algorithm., 117 = 65 × 1 + 52, 65 = 52 × 1 = 52, 52 = 13 × 4 + 0, HCF (117, 52) = 13., Given that 65 m – 117 = 13 ⇒ 65 m = 130 ⇒ m = 2., LCM (65, 117) = 13 × 32 × 5 = 585, , 8, , Mathematics-X
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44. 1251 – 1 =1250, 9377 – 2 = 9375, 15628 – 3 = 15625, HCF of (15625, 9375) = 3125, HCF of (3125, 1250) = 625, ⇒ HCF of (1250, 9375, 15625) = 625, 45. By Euclid’s division algorithm, we have a = bq + r, where 0 ≤ r < 4. On putting b, = 4 we get a = 4q + r where, r = 0, 1, 2, 3., If r = 0,, a = 4q which is even, If r = 1,, a = 4q + 1 not divisible by 2, If r = 2,, a = 4q + 2 = 2(2q + 1) which is even, If r = 3,, a = 4q + 3 not divisible by 2., So, for any +ve integer q, 4q + 1 and 4q + 3 are odd integers., How,, a2 = (4q + 1)2 = 16q2 + 1 + 8q = 4(4q2 + 2q) + 1 = 4m + 1, where m = 4q2 + 2q similarly for 4q + 3., 46. HCF (324, 252, 180) = 36, 47. LCM of (18, 24, 36) = 72., Greatest six digit number = 999999, 72, , 999999, – 72, 279, – 216, 639, – 576, 639, – 576, 639, 576, , 13888, , Require six digit number, 999999, – 63, 999936, , 63, , 48. LCM of (9, 12, 15) = 180 minutes., 49. Let the number divisible by 3 is of the form 3k + r, r = 0, 1, 2, a = 3k, 3k + 1 or 3k + 2, (i) When, a = 3k, n = 3k ⇒ n is divisible by 3., n + 2 = 3k + 2 ⇒ n + 2 is not divisible by 3., n + 4 = 3k + 4 = 3k + 3 + 1 = 3(k + 1) + 1 ⇒ n + 4 is not, divisible by 3., Mathematics-X, , 9
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(ii), , 50., , 51., 52., , 54., , 10, , So, only one out of n, n + 2 and n + 4 is divisible by 3., When, a = 3k + 1, n = 3k + 1 ⇒ n is not divisible by 3., n + 2 = 3k + 1 + 2 = 3k + 3 = 3(k + 1), ⇒ n + 2 is divisible by 3., n + 4 = 3k + 1 + 4 = 3k + 5 = 3(k + 1) + 2, ⇒ n + 2 is not divisible by 3., So, only one out of n, n + 2 and n + 4 is divisible by 3., Similarly do for a = 3k + 2., HCF (404, 96) = 4, LCM (404, 96) = 9696, HCF × LCM = 38, 784, Also,, 404 × 96 = 38,784, 4, Let a be +ve odd integer, divide it by 6 then q is the quotient and r is the remiander., ⇒, a = 6q + r where r = 0, 1, 2, 3, 4, 5, If,, a = 6q + 0 = 2(3q) is an even integer so not possible, If,, a = 6q + 1 is an odd integer, If,, a = 6q + 2 = 2(3q + 1) is an even integer so not possible, If,, a = 6q + 3 is an odd integer, If,, a = 6q + 4 = 2(3q + 2) is an even integer so not possible, If,, a = 6q + 5 is an odd integer., Let the three consecutive integers be a, a+1, a + 2,, Case I : If a is even,, ⇒ a + 2 is the also even, a(a + 2) is divisible by 2, a(a + 2) (a + 1) is also divisible by 2, Now a, a + 1, a + 2 are three consecutive numbers, ⇒ a (a + 1) (a + 2) is a multiple by 3, ⇒ a (a + 1) (a + 2) is divisible by 3, as it is divisible by 2 and 3 hence divisible by 6., , Mathematics-X
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55., , 57., 58., 59., , Case II : If a is odd, ⇒ a + 1 is even, ⇒ a + 1 is divisible by 2, ⇒ a(a + 1) (a + 2) is also divisible by 2, Again a, a + 1, a + 2 are three consecutive numbers, ⇒ a (a + 1) (a + 2) is a multiple by 3, ⇒ a (a + 1) (a + 2) is divisible by 3, as it is divisible by 2 and 3 hence divisible by 6., n3 – n = n(n2 – 1) = n (n – 1) (n + 1), = (n – 1) (n) (n + 1), = Product of three consecutive +ve integers, Now to show that produce of three consecutive +ve integers is divisible by 6., Any +ve integer a is of the form 3q, 3q + 1 or 3q + 2 for some integer q., Let a, a + 1, a + 2 be any three consecutive integers., Case I : a = 3q, (3q) (3q + 1) (3q + 2) = 3q (2m) [as (3q + 1) and (3q + 2) are consecutive, integers so their product is also even], = 6q m, which is divisible by 6., Case II : If a = 3q + 1, a (a + 1) (a + 2) = (3q + 1) (3q + 2) (3q + 3), = 2m3(q + 1), (as (3q + 1) (3q + 2) = 2m), = 6 m (q + 1), which is divisible by 6., Case III : If a = 3q + 2, a (a + 1) (a + 2) = (3q + 2) (3q + 3) (3q + 4), = (3q + 2) 3(q + 1) (3q + 4), = 6m, which is divisible by 6., 17, 4663, HCF (396, 342) = 18, , Mathematics-X, , 11
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61. HCF (124, 72) = 4, 4 = 124 × 7 + 72 × (– 12), x = 7, y = – 12, 62. Let, , n −1 + n +1 =, , p, (1)q ≠ 0, p, q, co-prime., q, , q, =, p, , n −1 – n +1, 1, ×, n −1 + n +1, n −1 − n +1, , q, =, p, , n −1 – n +1, −2, , n −1 + n +1 = −, , 2q, or, p, , n +1 − n −1 =, , Adding (1) & (2) we get 2 n + 1 =, , 2q, p, , …(2), , p 2q p 2 + 2q 2, +, =, q, p, pq, , 2, 2, Subtracting (1) & (2) we get 2 n − 1 = p − 2q, pq, , …(3), …(4), , From (3) & (4) we get n + 1 + n − 1 are rational numbers., But, ∴, , n − 1 + n + 1 is an irrational number., These exist no positive integer n, for which, , n − 1 + n + 1 is rational., , 63. HCF (134791, 6341, 6339) = 1., 64. HCF of 60, 84 and 108 is 22 × 3 = 12, Total number of participants, No. of rooms required =, 12, 60 + 84 + 108, =, = 21 rooms, 12, 65., HCF = 24, LCM = 540, LCM, 540, =, = 22.5, not an integer., HCF, 24, Hence two numbers cannot have HCF and LCM as 24 and 540 respectively., , 12, , Mathematics-X
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PRACTICE-TEST, Real Number, Time : 1 Hr., , M.M. : 20, SECTION A, , 1., 2., 3., 4., , 51, will terminate., 1, 150, In Euclid’s Division Lemma, when a = bq + r where a, b are positive integers then, what values r can take?, 1, 4 5, 8 3, HCF of x y and x y ., 1, LCM of 14 and 122 ., 1, , After how many decimal places the decimal expansion of, , SECTION B, 5., , Show that 9n can never ends with unit digit zero., , 2, , 6., , Without actual division find the type of decimal expansion of, , 7., , Show that the square of any odd integer is of the form 4m + 1, for some integer, m., 2, , 935, 10500, , 2, , SECTION C, 1, is an irrational number., 3–2 5, , 8., , Prove that, , 9., , Find the HCF of 36, 96 and 120 by Euclid’s Lemma., , 3, 3, , SECTION D, 10. Once a sports goods retailer organized a campaign “Run to remember” to spread, awareness about benefits of walking. In that Soham and Baani participated., There was a circular path around a sports field. Soham took 12 minutes to drive, one round of the field, while Baani took 18 minutes for the same. Suppose they, started at the same point and at the same time and went in the same direction., After how many minutes have they met again at the starting point?, 4, ppp, , Mathematics-X, , 13
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CHAPTER, , 2, , Polynomials, , KEY POINTS, , 1., , Polynomial : If x is a variable, n is a natural number and a0, a1, a2, a3, ………., an are real numbers, then p(x) = an xn + an–1 xn–1 + ......... + a1 x + a0, (an 0) is, called a polynomial in x., 2. Polynomials of degree 1, 2 and 3 are called linear, quadratic and cubic polynomials, respectively., 3. A quadratic polynomial is an algebraic expression of the form ax2 + bx + c,, where a, b, c are real numbers with a 0., 4. Zeros of a polynomial p(x) are precisely the x – coordinates of the points where, the graph of y = p(x) intersects the x–axis, i.e., x = a is a zero of polynomial p(x), if p(a) = 0, 5. A polynomial can have at most the same number of zeros as the degree of the, polynomial., 6. (i) If one zero of a quadratic polynomial p(x) is negative of the other, then, coefficient of x is 0., (ii) If zeroes of a quadratic polynomial p(x) are reciprocal of each other, then, coefficient of x2 = constant term., 7. Relationship between zeros and coefficients of a polynomial, If and are zeros of p(x) = ax2 + bx + c (a 0), then, b, Sum of zeros = + = –, a, c, Product of zeros = =, a, 8. If are zeros of a quadratic polynomial p(x), then, p(x) = k[x2 – (sum of zeros) x + product of zeros], p(x) = k [x2 – (+ )x + ]; where k is any non-zero real number., 9. Graph of linear polynomial p(x) = ax + b is a straight line., 10. Division Algorithm states that given any polynomials p(x) and g(x), there exist, polynomial q(x) and r(x) such that:, 14, , Mathematics-X
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•, •, , p(x) = g(x). q(x) + r(x) ; g(x) ≠ 0,, [where either r(x) = 0 or degree r(x) < degree g(x)], Graph of different types of polynomials:, Linear Polynomial : The graph of a linear polynomial ax + b is a straight line,, intersecting x-axis at one point., Quadratic Polynomial:, (i) Graph of a quadratic polynomial p(x) = ax2 + bx + c is a parabola open, upwards like U, if a > 0 and intersect x-axis at maximum two distinct points., y, , x′, , 0, , x, , y', , (ii) Graph of a quodratic polynomial p(x) = ax2 + bx + c is a parabola open, downwards like ∩, if a < 0 and intersect x-axis at maximum two distinct, points., y, , x', , 0, , x, , (iii) Polynomial and its graph : In general a polynomial p(x) of degree n crosses, the x-axis at most n points., y, , x', , 0, , x, , y', , Mathematics-X, , 15
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VERY SHORT ANSWER TYPE QUESTIONS, , 1., , If one root of the polynomial P(x) = 5x2 + 13x + K is reciprocal of the other, then, value of k is, 1, (d) 6, 6, If α and β are the zeroes of the polynomial p(x) = x2 – p(x + 1) – c such that, (α + 1) (β + 1) = 0, the c = _______ ., If one zero of the quadratic polynomial x2 + 3x + k is 2, then the value of k is, (a) 10, (b) – 10, (c) 5, (d) – 5, 2, If the zeroes of the quadratic polynomial x + (a + 1)x + b are 2 and – 3, then, (a) a = – 7, b = – 1, (b) a = 5, b = – 1, (c) a = 2, b = – 6, (d) a = 0, b = – 6, What should be added to the polynomial x2 – 5x + 4, so that 3 is the zero of the, resulting polynomial:, (a) 1, (b) 2, (c) 4, (d) 5, If α and β are the roots of the polynomial, , (a) 0, 2., 3., 4., , 5., , 6., , (b) 5, , f (x) = x2 + x + 1, then, , (c), , 1 1, + =, α β, , 7., , If a quadratic polynomial f(x) is not factorizable into linear factors, then it has no real, zero. (True/False), 8. If a quadratic polynomial f(x) is a square of a linear polynomial, then its two zeros are, coincident. (True/False)., 9. The product of the zeros of x3 + 4x2 + x – 6 is, (a) – 4, (b) 4, (c) 6, (d) 6, 3, 10. Given that two of the zeros of the cubic polynomial ax + bx2 + cx + d are 0, the, third zero is, b, b, c, d, (b), (c), (d) −, a, a, a, a, 11. What will be the number of zeros of a linear polynomial p(x) if its graph (i) passes, through the origin. (ii) doesn’t intersect or touch x-axis at any point?, 12. Find the quadratic polynomial whose zeros are, , (a) −, , (5 + 2 3) and (5 – 2 3), 16, , Mathematics-X
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13. If one zero of p(x) = 4x2 – (8k2 – 40k) x – 9 is negative of the other, find values of, k., 14. What number should be added to the polynomial x2 – 5x + 4, so that 3 is a zero of, polynomial so obtained., 15. How many (i) maximum (ii) minimum number of zeroes can a quadratic polynomial, have?, 16. What will be the number of real zeros of the polynomial x2 + 1?, 17. If α and β are zeros of polynomial 6x2 – 7x – 3, then form a quadratic polynomial, where zeros are 2α and 2β, (CBSE), 1, 18. If α and, are zeros of 4x2 – 17x + k – 4, find the value of k., α, 19. What will be the number of zeros of the polynomials whose graphs are parallel to (i), y-axis (ii) x-axis?, 20. What will be number of zeros of the polynomials whose graphs are either touching or, intersecting the axis only at the points:, (i) (–3, 0), (0, 2) & (3, 0) (ii) (0, 4), (0, 0) and (0, –4), SHORT ANSWER TYPE (I) QUESTIONS, , 21. If –3 is one of the zeros of the polynomial (k – 1)x2 + k x + 1, find the value of k., 22. If the product of zeros of ax2 – 6x – 6 is 4, find the value of a. Hence find the sum of, its zeros., 23. If zeros of x2 – kx + 6 are in the ratio 3 : 2, find k., 24. If one zero of the quadratic polynomial (k2 + k)x2 + 68x + 6k is reciprocal of the, other, find k., 25. If α and β are the zeros of the polynomial x2 – 5x + m such that α – β = 1, find m., (CBSE), 26. If the sum of squares of zeros of the polynomial x2 – 8x + k is 40, find the value of k., 27. If α and β are zeros of the polynomial t2 – t – 4, form a quadratic polynomial whose, zeros are, , 1, 1, and ., β, α, , 28. What should be added to the polynomial x3 – 3x2 + 6x – 15, so that it is completely, divisible by x – 3 ?, (CBSE 2016), Mathematics-X, , 17
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m n, + ., n m, (CBSE, 2012), , 29. If m and n are the zeros of the polynomial 3x2 + 11x – 4, find the value of, , 30. Find a quadratic polynomial whose zeros are, , 3+ 5, 3− 5, and, ., 5, 5, , (CBSE, 2013), SHORT ANSWER TYPE (II) QUESTIONS, , 31. If (k + y) is a factor of each of the polynomials y2 + 2y – 15 and y3 + a , find the, values of k and a., 32. Obtain zeros of 4 3 x 2 + 5 x – 2 3 and verify relation between its zeroes and, coefficients., 33. If x4 + 2x3 + 8x2 + 12x + 18 is divided by (x2 + 5) , remainder comes out to be (px, + q) , find values of p and q., 34. –5 is one of the zeros of 2x2 + px – 15, zeroes of p(x2 + x) + k are equal to each, other. Find the value of k., 35. Find the value of k such that 3x2 + 2kx + x – k – 5 has the sum of zeros as half of, their product., 36. If α and β are zeros of y2 + 5y + m, find the value of m such that (α + β)2 – αβ =, 24, 37. If α and β are zeros of x2 – x – 2, find a polynomial whose zeros are (2α + 1) and, (2β + 1), 38. Find values of a and b so that x4 + x3 + 8x2 + ax + b is divisible by x2 + 1., 39. What must be subtracted from 8x4 + 14x3 – 2x2 + 7x – 8 so that the resulting, polynomial is exactly divisible by 4x2 + 3x – 2 ?, 40. What must be added to 4x4 + 2x3 – 2x2 + x – 1 so that the resulting polynomial is, divisible by x2 – 2x – 3 ?, , 18, , Mathematics-X
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LONG ANSWER TYPE QUESTIONS, , 41. Find all zeros of the polynomial 2x3 + x2 – 6x – 3 if two of its zeroes are, 3 and – 3 ., 42. If, , 3, 2, 2 is a zero of (6 x + 2 x – 10 x – 4 2) , find its other zeroes., , 43. If two zeros of x4 – 6x3 – 26x2 + 138 x – 35 are (2 3) , find other zeroes., 44. On dividing the polynomial x3 – 5x2 + 6x – 4 by a polynomial g(x), quotient and, remainder are (x –3) and (– 3x + 5) respectively. Find g(x), 45. Obtain all zeros of the polynomial 2x4 – 2x3 – 7x2 + 3x + 6 if two factors of this, , 3, ., polynomial are x ±, 2 , , , 46. If the polynomial x4 – 3x3 – 6x2 + kx – 16 is exactly divisible by x2 – 3x + 2, then find, the value of k., (CBSE, 2014), 47. If the polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by x2 – 2x + k, then find the, vlaue of k and a., (CBSE), 2, 48. If α and β are zeros of the polynomial x + 4x + 3, find the polynomial whose zeros, are 1 +, , β, α, and 1 + ., β, α, , (CBSE), , 49. Find K, so that x2 + 2x + K is a factor of 2x4 + x3 – 14x2 + 5x + 6. Also find all the, zeros of the two polynomials:, (Exempler, HOTS), 50. If x − 5 is a factor of the cubic polynomial x3 − 3 5x2 + 13x − 3 5 , then find, all the zeros of the polynomial., ANSWERS AND HINTS, , 1. (b) 5, , 2. –1, , 3. (b) –10, , 4. (d) a = 0, b = –6, , 5. (b) 2, 7. True, , 6. – 1, 8. True, , 9. (c) 6, , 10. (a) −, , Mathematics-X, , b, a, 19
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11. (i) 1 (ii) 0, 13. k = 0, 5, , 12. x2 – 10x + 13, 14. 2, , 15. (i) 2 (ii) 0, , 16. 0, , 2, , 17. 3x – 7x – 6, , 18. k = 8, , 19. (i) 1 (ii) 0, , 20. (i) 2 (ii) 1, , 21. 4/3, , 3, 22. a = − , sum of zeroes = − 4, 2, , 23. – 5, 5, , 24. 5, , 25. 6, 26. 12, 2, 27. 4t + t – 1, 28. On dividing x3 – 3x2 + 6x – 15 by x – 3, remainder is + 3, hence – 3 must be, added to x3 – 3x2 + 6x – 15., 2, , 11, 4, − − 2 − , 2, 2, 2, 3, 3 = − 145, m n m +n, (m + n) − 2mn, +, =, =, =, 29., 4, 12, n m, mn, mn, −, 3, 6, 4, 30. α + β = , αβ =, ,, 31. k = 3, – 5 and a = 27, – 125, 5, 25, 25x2 – 30x + 4, 32. −, 34., , 2, ,, 3, , 3, 4, , 7, 4, , 33. p = 2, q = 3, 35. 1, , 36. 1, 38. a = 1, b = 7, , 37. x2 – 4x – 5, 39. 14x – 10, , 40. 61x – 65, , 41., , 42. −, , 20, , 2 −2 2, ,, 2, 3, , 3, − 3, −, , 1, 2, , 43. – 5, 7, , Mathematics-X
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PRACTICE-TEST, Polynomials, Time : 1 Hr., , M.M. : 20, SECTION- A, , 1., 2., 3., , If α and β are zeros of a quadratic polynomial p(x), then factorize p(x)., 1 1, If α and β are zeros of x2 – x – 1, find the value of + ., α β, , 1, 1, , If one of the zeros of quadratic polynomial (K –1) x2 + kx + 1 is – 3 then the value, of K is,, (a), , 4., , 4, 3, , 1, (b) −, , 4, 3, , (c), , 2, 3, , (d) −, , 2, 3, , A quadratic polynomial, whose zeros are – 3 and 4, is, (a) x2 – x + 12, (b) x2 + x + 12, (c), , x2 x, − −6, 2 2, , 1, , (d) 2x2 + 2x – 24, , SECTION-B, 1, αβ ., 2, 2, , 5., , If α and β are zeros of x2 – (k + 6)x + 2(2k –1). find the value of k if α + β =, , 6., , Find a quadratic polynomial one of whose zeros is (3 + 2) and the sum, of its zeroes is 6., , 7., , 22, , If zeros of the polynomial x2 + 4x + 2a are α and, , 2, 2, then find the value of a., α, , 2, , Mathematics-X
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SECTION-C, 8., 9., , Find values of a and b if (x2 + 1) is a factor of the polynomial x4 + x3 + 8x2 + ax, + b., 3, If truth and lie are zeros of the polynomial px2 + qx + r, (p ≠ 0) and zeros are, reciprocal to each other, Find the relation between p and r., 3, , SECTION-D, 10. On dividing the polynomial x3 + 2x2 + kx + 7 by (x – 3), remainder comes out to be, 25. Find quotient and the value of k. Also find the sum and product of zeros of the, quotient so obtained., 4, ppp, , Mathematics-X, , 23
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CHAPTER, , Pair of Linear Equations, in Two Variables, , 3, KEY POINTS, , Linear equation in, two variables, a1x + b1y + c1 = 0 … (1), a2x + b2y + c2 = 0 … (2), , Equation reducible, to a pair of linear, equation, , Equation of a, straight line, Method to solve, ax + by + c = 0, Solution (x, y) → Points, lying on straight line,, , Graphical, method, , 2, x, , +, , 3, y, , = 13,, 1, , 5, x, 1, , +, , 4, y, , =2, , Put x = 0, y = b, 2a + 3b = 13, 5a + 4b = 2, , x', , ax, , +, , by, , +, , c=, , 0, , y, , Algebraic, method, , 0, y', , Substitution Method, a1x + b1y + c1 = 0 …(1), a2x + b2y + c2 = 0 …(2), Substituting the value of, x from eq. (1) in eq. (2), and finding y and, vice versa., , Intersecting lines, , x, Elimination Method, a1x + b1y = c1 …(1), a2x + b2y = c2 …(2), Multiplying eq. (1) by a2, and eq. (2) by a1 making, coefficient of x same and, hence eliminating x and, finding y., , Cross multiplication, method, b1 x c1 y a1 1 b1, b2, x, b1c2 – b2c1, , a1, a2, , b1, b2, , +, , (intersect at 1 point) one solution, a, b, c, Coincident a = b = c, Coincide, Infinite solution, a, b, c, Parallel lines a = b = c, (Not Intersecting) No solution, 1, , 1, , 2, , 2, , 1, 2, , 1, , 1, , 1, , 2, , 2, , 2, , c2, a, b2, y 2, = c a – a c = a 1b – b a, , x=, y=, , 1 2, , 1 2, , 1 2, , 1 2, , b1c2 – b2c1, a1b2 – b1a2, c1a2 – a1c2, a1b2 – b1a2, , VERY SHORT ANSWER TYPE QUESTIONS, , 1., 2., , 24, , If the lines given by 3x + 2ky = 2 and 2x + 5y = 1 are parallel, then the value of, k is _______ ., If x = a and y = b is the solution of the equation x – y = 2 and x + y = 4, then the, values of a and b are respectively _______ ., , Mathematics-X
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3., , 4., 5., 6., , 7., 8., , A pair of linear equations which has a unique solution x = 2 and y = – 3 is, (a) x + y = 1 and 2x – 3y = – 5, (b) 2x + 5y = – 11 and 2x – 3y = – 22, (c) 2x + 5y = – 11 and 4x + 10y = 22, (d) x – 4y – 14 = 0 and 5x – y – 13 = 0, The area of the triangle formed by the lines x = 3, y = 4 and x = y is _____ ., The value of K for which the system of equations 3x + 5y = 0 and kx + 10y = 0, has a non-zero solutions is ____ ., If a pair of linear equations in two variables is consistent, then the lines represented, by two equations are:, (a) Intersecting, (b) Parallel, (c) always coincident (d)intersecting or coincident, For 2x + 3y = 4, y can be written in terms of x as _______ ., One of the common solution of ax + by = c and y axis is, c, (a) 0, , b, , b, (b) 0, , c, , c , (c) , 0, b, , c, , (d) 0, − , b, 9. If ax + by = c and lx + my = n has unique solution then the relation between the, coefficient will be:, (a) am ≠ lb, (b) am = lb, (c) ab = lm, (d) ab ≠ lm, 10. In ΔABC, ∠C = 3∠B, ∠C = 2(∠A + ∠B) then, ∠A, ∠B, ∠C are respectively., (a) 30°, 60°, 90°, (b) 20°, 40°, 120°, (c) 45°, 45°, 90°, (d) 110°, 40°, 50°, 11. If x = 3m –1 and y = 4 is a solution of the equation x + y = 6, then find the value, of m., 12. What is the point of intersection of the line represented by 3x – 2y = 6 and the, y-axis?, 13. For what value of p, system of equations 2x + py = 8 and x + y = 6 have no, solution., 14. A motor cyclist is moving along the line x – y = 2 and another motor cyclist is, moving along the line x – y = 4 find out their moving direction., 15. Find the value of k for which pair of linear equations 3x + 2y = –5 and x – ky =, Mathematics-X, , 25
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2 has a unique solution., 16. Express y in terms of x in the expression 3x – 7y = 10, 17. If 2x + 5y = 4, write another linear equation, so that lines represented by the pair are, coincident., 18. Check whether the graph of the pair of linear equations x + 2y – 4 = 0 and 2x +, 4y – 12 = 0 is intersecting lines or parallel lines., 19. If the lines 3x + 2 ky = 2 and 2x + 5y + 1 = 0 are parallel, then find value of k., 20. If we draw lines of x = 2 and y = 3 what kind of lines do we get?, SHORT ANSWER TYPE (I) QUESTIONS (2 MARKS QUESTIONS), , 21. Form a pair of linear equations for: The sum of the numerator and denominator, of the fraction is 3 less than twice the denominator. If the numerator and, denominator both are decreased by 1, the numerator becomes half the, denominator., 22. For what value of p the pair of linear equations (p + 2)x – (2p + 1)y = 3(2p – 1), and 2x – 3y = 7 has a unique solution., 23. ABCDE is a pentagon with BE || CD and BC || DE, BC is perpendicular to CD, If the perimeter of ABCDE is 21 cm, find x and y., A, 3 cm, B, , 3 cm, 5 cm, , E, , x–y, C, , x+y, , D, , 24. Solve for x and y, x–, , x 2y 2, y, =, = 3 and –, 2 3, 3, 2, , 25. Solve for x and y, 3x + 2y = 11 and 2x + 3y = 4, , 26, , Mathematics-X
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Also find p if p = 8x + 5y, 26. Solve the pair of linear equations by substitution method x – 7y + 42 = 0 and, x – 3y – 6 = 0, 27. Ram is walking along the line joining (1, 4) and (0, 6), Rahim is walking along the line Joining (3, 4) and (1, 0), Represent on graph and find the point where both of them cross each other, 28. Given the linear equation 2x + 3y – 12 = 0, write another linear equation in these, variables, such that. geometrical representation of the pair so formed is, (i) Parallel Lines (ii) Coincident Lines, 29. The difference of two numbers is 66. If one number is four times the other, find the, numbers., 30. For what value of k, the following system of equations will be inconsistent, kx + 3y = k – 3, 12x + ky = k, SHORT ANSWERS TYPE (II) QUESTIONS, , 31. Solve graphically the pair of linear equations 5x – y = 5 and 3x – 2y = – 4, Also find the co-ordinates of the points where these lines intersect y-axis, 32. Solve for x and y, 5, 1, +, =2, x+ y x – y, 15, 5, –, = –2, x+ y x – y, 33. Solve by Cross – multiplication method, (CBSE), x y, +, =a+b, a b, x, y, + 2 =2, 2, a, b, 34. For what values of a and b the following pair of linear equations have infinite number, of solutions?, (CBSE), 2x + 3y = 7, a(x + y) – b(x – y) = 3a + b – 2, , Mathematics-X, , 27
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35. Solve the pair of linear equations, 152x – 378y = – 74, – 378x + 152y = – 604, 36. Pinky scored 40 marks in a test getting 3 marks for each right answer and losing 1, mark for each wrong answer. Had 4 marks been awarded for each correct answer, and 2 marks were deducted for each wrong answer, then pinky again would have, scored 40 marks. How many questions were there in the test?, 37. A two digit number is obtained by either multiplying sum of digits by 8 and adding 1, or by multiplying the difference of digits by 13 and adding 2. Find the number, 38. Father’s age is three times the sum of ages of his two children. After 5 years his age, will be twice the sum of ages of two children. Find the age of the father., 39. On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gain ` 2000. But, if he sells the T.V. at 10% gain and fridge at 5% loss, he gains ` 1500 on the transaction., Find the actual price of the T.V. and the fridge, 40. Sunita has some ` 50 and ` 100 notes amounting to a total of ` 15,500. If the total, number of notes is 200, then find how many notes of ` 50 and ` 100 each, she has., LONG ANSWER TYPE QUESTIONS, , 41. Solve graphically the pair of linear equations 3x – 4y + 3 = 0 and 3x + 4y – 21 = 0, Find the co-ordinates of vertices of triangular region formed by these lines and, x-axis. Also calculate the area of this triangle., 42. Solve for x and y, 1, 12, 1, +, =, 2(2 x + 3 y ) 7(3 x – 2 y ), 2, 7, 4, +, =2, (2 x + 3 y ) (3 x – 2 y, For 2x + 3y ≠ 0, 3x – 2y ≠ 0., 43. Solve the pair of equations by reducing them to a pair of linear equations, 4x – 2 y, 3x + 2 y, = 13, = 1 and, xy, xy, hence find a for which y = ax – 4, 44. A man travels 600 km to his home partly by train and partly by bus. He takes 8, 28, , Mathematics-X
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hours, if he travels 120 km by train and rest by bus. Further, it takes 20 minute, longer, if he travels 200 km by train and rest by bus. Find the speeds of the train and, the bus., 45. A and B are two points 150 km apart on a highway. Two cars start with different, speeds from A and B at same time. If they move in same direction, they meet in, 15 hours. If they move in opposite direction, they meet in one hour. Find their, speeds, 46. A boat covers 32 km upstream and 36 km downstream, in 7 hours. Also it Covers, 40 km upstream and 48 km downstream in 9 hours. Find the speed of boat in still, water and that of the stream., (CBSE), 47. The sum of the numerator and denominator of a fraction is 4 more than twice the, numerator. If the numerator and denominator are increased by 3, they are in the, ratio 2 : 3. Determine the fraction., 48. 8 Women and 12 men can complete a work in 10 days while 6 women and 8 men, can complete the same work in 14 days. Find the time taken by one woman, alone and that one man alone to finish the work., 49. The ratio of incomes of two persons A and B is 3 : 4 and the ratio of their, expenditures is 5 : 7. If their savings are ` 15,000 annually find their annual, incomes., 50. Vijay had some bananas and he divided them into two lots A and B. He sold the first, lot at the rate of ` 2 for 3 bananas and the second lot at the rate of ` 1 per banana, and got a total of ` 400. If he had sold the first lot at the rate of ` 1 per banana and, the second lot at the rate of ` 4 for 5 bananas, his total collection would have been `, 460. Find the total number of bananas he had., (HOTS, Exampler), 51. A railway half ticket cost half the full fare but the reservation charges are the, same on a half ticket as on a full ticket. One reserved first class ticket costs, ` 2530. One reserved first class ticket and one reserved first class half ticket, from stations A to B costs ` 3810. Find the full first class fare from stations A to, B and also the reservation charges for a ticket., (Exemplar), 52. Solve the following pair of equations., , Mathematics-X, , 29
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2, 3, 4, 9, , 2 and, , 1, x, y, x, y, , (CBSE, 2015), , 53. Determine graphically, the vertices of the triangle formed by the times y = x,, 3y = x and x + y = 8., (NCERT Exemplar)., 54. Draw the graphs of the equations x = 3, x = 5 and 2x – y – 4 = 0. Also find the area, of the quadrilateral formed by the lines and the x-axis., (NCERT Exemplar, HOTS), 55. The area of a rectangle gets reduced by a 9 square units, if its length is reduced by 5, units and the breadth is increased by 3 units. The area is increased by 67 sqaure units, if length is increased by 3 units and breadth is increased by 2 units. Find the perimeter, of the rectangle., (CBSE), ANSWERS AND HINTS, , 1. K , , 15, 4, , 2. a = 3 and b = 1, , 3. (b) 2x + 5y = – 11 and 4x + 10y = – 22, 4., , 1, sq. unit, 2, , 6. (d) intersecting or coincident, c, 8. (a) 0, , b, 10. (b) 20°, 40°, 120°, 12. (0, –3), , 14. move parallel, 16. y , , 3 x – 10, 7, , 18. Parallel lines, 20. Intersecting lines, 30, , 5. 6, 7. y , , 4 2x, 3, , 9. (a) am lb, 11. m = 1, 13. p = 2, –2, 15. k , 3, 17. 4x + 10y = 8, 15, 4, 21. x – y = – 3, 2x – y = 1, , 19. k , , Mathematics-X
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12., 24., 26., 28., 29., 31., 33., 35., 37., 39., 40., 41., 42., 44., 46., 48., 49., 50., , p≠4, 23. x = 5, y = 0, 4, 2, 25. x = 5, y = – 2, p = 30, 42, 12, 27. (2, 2), (i) 4x + 6y + 10 = 0, (ii) 4x + 6y – 24 = 0, 88, 22, 30. k = – 6, (2, 5) (0, – 5) and (0, 2), 32. (3, 2), 34. a = 5, b = 1, x = a2, y = b2, 2, 1, 36. 40 questions, 41, 38. 45 years, T.V. = ` 20,000 Fridge = ` 10,000, ` 50 notes = 90, ` 100 notes = 110, Solution (3, 3), Vertices (– 1, 0) (7, 0) and (3, 3), Area = 12 square unit, –2, 1, –45, , y= ,a=, (2, 1), 43. x =, 5, 2, 4, 60 km/hr, 80 km/hr, 45. 80 km/hr , 70 km/hr, 5, 10 km/hr, 2 km/hr, 47., 9, 1 woman in 140 days, 1 man in 280 days, ` 90,000, ` 1,20,000, Let the no. of bananas in lots A be x and in lots B be y, 2, x + y = 400 ⇒ 2x + 3y = 1200, 3, 4, Case 2 : x + y = 460 ⇒ 5x + 4y = 2300, 5, x = 300, y = 200, Total bananas = 500., , Case I :, , 51. Let the cost of full and half ticket be ` x & `, , x, and reservation charge by, 2, , ` y per ticket., Case I : x + y = 2530, , x, + y = 3810, 2, x = 2500, y = 3810, Full first class fare is ` 2500 and reservation charge is ` 30., , Case 2 : x + y +, , Mathematics-X, , 31
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52. x = 4, y = 9, 53. Vertices of the triangle are (0, 0) (4, 4) (6, 2)., 54. Area of quadrilateral ABCD where,, A(3, 0), B(5, 0), C(5, 6), D(3, 2), =, , 1, × AB × ( AD + BC ), 2, , 1, × 2 × (6 + 2) = 8 sq. units., 2, 55. Length of rectangle is 17 units., Breadth of rectangle is 9 units., Perimeter of rectangle is 52 units., , =, , rrr, , 32, , Mathematics-X
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PRACTICE-TEST, Pair of Linear Equations In Two Variables, Time : 1 Hr., , M.M. : 20, SECTION-A, , 1., 2., 3., , 4., , For what value of k system of equations, x + 2y = 3 and 5x + ky + 7 = 0 has a unique solution., 1, Does the point (2, 3) lie on line of graph of 3x – 2y = 5., 1, The pair of equations x = a and y = b graphically representes lines which are:, 1, (a) Parallel, (b) Intersecting at (b, a), (c) Coincident, (d) Intersecting at (a, b), For what value of K, do the equation 3x – y + 8 = 0 and 6x – Ky = –16 represent, coincident lives?, 1, , 1, 2, (c) 2, (a), , 1, 2, (d) –2, , (b) –, , SECTION-B, 5., , 6., 7., , For what values of a and b does the pair of linear equations have infinite number, of solutions, 2x – 3y = 7, ax + 3y = b, 2, Solve for x and y, 0.4x + 0.3y = 1.7, 0.7x – 0.2y = 0.8, 2, If the system of equations 6x + 2y = 3 and kx + y = 2 has a unique solution, find the, value of k., 2, , SECTION-C, 8., , 9., , Solve for x and y by cross multiplication method, x+y =a+b, ax – by = a2 – b2, 3, Sum of the ages of a father and the son is 40 years. If father’s age is three times that, of his son, then find their ages. 3, , SECTION-D, 10. Solve the following pair of equations graphically., 3x + 5y = 12 and 3x – 5y = –18., , 4, , Also shade the region enclosed by these two lines and x-axis., Mathematics-X, , 33
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CHAPTER, , 4, , Quadratic Equations, Basic Concepts, Quadratic Polynomial, 2, , A polynomial of the form ax + bx + c is called a Quadratic polynomial,, a ≠ 0. Degree of this polynomial is 2, Quadratic Equation, 2, , A equation of the form ax + bx + c = 0, a ≠ 0 is called a Quadratic equation,, in one variable x, where a, b, c are real numbers., , Methods for solving, Quadratic Equations, , Roots of a Quadratic Equations, , By Using Quadratic, Formula (Given by, Sridhar Acharya), , By factorisation, , (a) By using identities, (b) By splitting the middle term, 2, , Quadratic equation ax + bx + c = 0, has two roots α and β given by, , Discriminant, 2, D = b – 4ac is, called discriminant, , α=, , −b + b 2 − 4ac, − b − b2 − 4ac, ,β=, 2a, 2a, , Nature of roots, 2, , Case I : D > 0, b – 4ac > 0, Roots are real and distinct, , 2, , Case II : D = 0, b – 4ac = 0, Roots are real and equal, , 2, , Case III : D < 0, b – 4ac < 0, Roots are not real, , Real roots can be, rational or irrational, 34, , Mathematics-X
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NOTES:, 1., 2., 3., , −b ± b 2 − 4ac, 2a, −b −b, Real and equal roots are, ,, 2a 2a, There are quadratic equation which donot have any real roots e.g. x2 + 1 = 0, Real and distinct roots are, , VERY SHORT ANSWER TYPE QUESTIONS, , Multiple Choice Questions:, 1. Which of the following is not a Quadratic Equation?, (b) 3x – x2 = x2 + 6, (a) 2(x – 1)2 = 4x2 – 2x + 1, (c), 2., , 3., , (, , 2, 3 x + 2 ) = 2x – 5x, 2, , (d) (x2 + 2x)2 = x4 + 3 + 4x2, , Which of the following equation has 2 as a root, (a) x2 + 4 = 0, (b) x2 – 4 = 0, (c) x2 + 3x – 12 = 0, (d) 3x2 – 6x – 2 = 0, 1, 5, is a root of x2 + px – = 0 then value of p is, 2, 4, (a) 2, (b) –2, , If, , 1, 1, (d), 4, 2, Every Quadratic Equation can have at most, (a) Three roots, (b) One root, (c) Two roots, (d) Any number of roots, 2, Roots of Quadratic equation x – 7x = 0 will be, (a) 7, (b) 0, –7, (c) 0, 5, (d) 0, 7, Fill in the blanks:, (a) If px2 + qx + r = 0 has equal roots then value of r will be ______ ., (b) The qaudratic equation x2 – 5x – 6 = 0 if expressed as (x + p) (x + q) = 0 then, value of p and q respectively are ______ and _______ ., (c) The value of k for which the roots of qaudratic equations x2 + 4x + k = 0 are, real is ______ ., , (c), 4., , 5., , 6., , Mathematics-X, , 35
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7., , 8., , (d) If roots of 4x2 – 2x + c = 0 are reciprocal of each other then the value of c is, ________ ., (e) If in a quadratic equation ax2 + bx + c = 0, value of a is zero then it become, a _____ equation., Write whether the following statements are true or false. Justify your, answers., (a) Every quadratic equation has atleast one real roots., (b) If the coefficient of x2 and the constant term of a quadratic equation have, opposite signs, then the quadratic equation has real roots., (c) 0.3 is a root of x2 – 0.9 = 0., (d) The graph of a quadratic polynomial is a straight line., (e) The discriminant of (x – 2)2 = 0 is positive., Match the following :, (i) Roots of 3x2 – 27 = 0, (a) 169/9, 5, x–2=0, 3, (iii) Sum of roots of 8x2 + 2x – 3 = 0, (iv) A quadratic equation with roots a and b, , (ii) D of 2x2 +, , (v) The product of roots of x2 + 8x = 0, , (b) 0, (c) x2 – (a + b)x + ab = 0, (d) 3, – 3, (e), , −1, 4, , SHORT ANSWER TYPE QUESTIONS-I, , If the Quadratic equation Px2 – 2 5 Px + 15 = 0 has two equal roots then find, the value of P., 10. Solve for x by factorisation, (a) 8x2 – 22x – 21 = 0, 9., , (b) 3 5 x 2 + 25 x + 10 5 = 0, (c) 3x2 – 2 6 x + 2 = 0, (d) 2x2 – ax + a2 = 0, (e) 3 x 2 + 10 x + 7 3 = 0, , (CBSE 2010), (CBSE 2014), , (f ) 2 x 2 + 7 x + 5 2 = 0, (g) (x – 1)2 – 5(x – 1) – 6 = 0, 11. If – 5 is a root of the quadratic equation 2x2 + px – 15 = 0 and the quandratic, equation p(x2 + x) + k = 0 has equal roots find the value of k. (CBSE 2014, 2016), 36, , Mathematics-X
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2, and x = – 3 are roots of the quadratic equation ax2 + 7x + b = 0. Find the, 3, value of a and b., (CBSE 2016), 13. Find value of p for which the product of roots of the quadratic equation px2 + 6x, + 4p = 0 is equal to the sum of the roots., 14. The sides of two squares are x cm and (x + 4) cm. The sum of their areas is 656, cm2 Find the sides of these two squares., 15. Find K if the difference of roots of the quadratic equation x2 – 5x + (3k – 3) = 0 is 11., , 12. If x =, , SHORT ANSWER TYPE QUESTIONS-II, , 16. Find the positive value of k for which the quadratic equation x2 + kx + 64 = 0, and the quadratic equation x2 – 8x + k = 0 both will have real roots., 17. Solve for x, (a), , (b), , 1, 1 1 1, = + +, a b x, a+b+ x, , a + b + x ≠ 0,, , 1, 1 1 1, + +, =, 2a + b + 2 x, 2a b 2 x, , 2a + b + 2x ≠ 0,, , (CBSE 2005), , a, b, x ≠ 0, , a, b, x ≠ 0, , 2x, 1, 3x + 9, −3, +, +, = 0, x ≠ 3,, x − 3 2 x + 3 ( x − 3)(2 x + 3), 2, 1, 1, 6, −, = , x ≠ 1, 5, (e), x −1 x + 5 7, (d) 4x2 + 4bx – (a2 – b2) = 0, (f) 4x2 – 2(a2 + b2) x + a2b2 = 0, (c), , (g), , (CBSE 2010), , 2, 3, 23, +, =, , x ≠ 0, – 1, 2, x + 1 2( x − 2), 5x, 2, , 10 x, 2x , − 24 = 0, x ≠ 5, (h) , +, x−5, ( x − 5), (i) 4x2 – 4a2x + a4 – b4 = 0, (j) 2a2x2 + b(6a2 + 1)x + 3b2 = 0, , 3 −1, 7 x + 1, 5x − 3, − 4 , = 11, x ≠ ,, (k) 3 , 5x − 3, 7x +1, 5 7, , Mathematics-X, , 37
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(l), , 1, 1, 11, −, = , x ≠ − 4, 7, x + 4 x − 7 30, , (NCERT), , x − 4 x − 6 10, (m) x − 5 + x − 7 = 3 , x ≠ 5, 7, 1, 2, 4, +, =, ,, x +1 x + 2 x + 4, 1, 1, +, = 1,, (o), 2x − 3 x − 5, , (n), , (CBSE 2014), x ≠ –1, –2, – 4, x≠, , 3, ,5, 2, , (p) x2 + 5 5x – 70 = 0, 16, 15, −1 =, , x ≠ 0, – 1, (CBSE 2014), (q), x, x +1, 18. Solve by using quadratic formula abx2 + (b2 – ac) x – bc = 0., (CBSE 2005), 2, 19. If the roots of the quandratic equation (p + 1)x – 6(p + 1) x + 3(p + 9) = 0 are equal, find p and then find the roots of this quadratic equation., LONG ANSWER TYPE QUESTIONS, , 20. A train travels at a certain average speed of 54 km and then travels a distance of 63, km at an average speed of 6 km/hr more than the first speed. If it takes 3 hours to, complete the total journey, what is its first speed?, 21. A natural number, when increased by 12, equals 160 times its reciprocal. Find the, number., 22. A theif runs with a uniform speed of 100 m/minutes. After one minute a policeman, runs after the thief to catch him. He goes with a speed of 10 m/minute in the first, minute and increases his speed by 10 m/minute every succeeding minute. After, how many minutes the policemen will catch the thief?, 23. Two water taps together can fill a tank in 6 hours. The tap of larger diameter, takes 9 hours less than the smaller one to fill the tank separately. Find the time in, which each tap can separately fill the tank., 24. In the centre of a rectangular lawn of dimensions 50 m × 40 m, a rectangular, pond has to be constructed, so that the area of the grass surrounding the pond, would be 1184 m2. Find the lenght and breadth of the pond., 25. A farmer wishes to grow a 100 m2 recangular garden. Since he has only 30 m, barbed wire, he fences three sides of the rectangular garden letting compound wall of, this house act as the fourth side fence. Find the dimensions of his garden., 26. A peacock is sitting on the top of a pillar, which is 9 m high. From a point 27 m, away from the bottom fo a pilar, a snake is coming to its hole at the base of the, 38, , Mathematics-X
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27., 28., 29., 30., 31., 32., 33., 34., , 35., 36., 37., 38., 39., , 40., 41., , pillar. Seeing the snake the peacock pounces on it. If their speeds are equa, at what, distance from the hole is the snake caught?, If the price of a book is reduced by ` 5, a person can buy 5 more books for, ` 300. Find the original list price of the book., ` 6500 were divded equally among a certain number of persons. Had there been, 15 more persons, each would have got ` 30 less. Find the original number of, persons., In a flight of 600 km, an aircraft was slowed down due to bad weather. Its, average speed was reduced by 200 km/hr and the time of flight increased by 30, minutes. Find the duration of flight., A fast train takes 3 hours less than a slow train for a journey of 600 km. If the, speed of the slow train is 10 km/hr less than the fast train, find the speeds of the, two trains., The speed of a boat in still water is 15 km/hr. It can go 30 km upstream and, return downstream to the orignal point in 4 hrs 30 minutes. Find the speed of the, stream., Sum of areas of two squares is 400 cm2. If the difference of their perimeter is 16, cm. Find the side of each square., The area of an isoscles triangle is 60 cm2. The length of equal sides is 13 cm, find length of its base., The denominator of a fraction is one more than twice the numerator. If the sum of, 16, the fraction and its reciprocal is 2 . Find the fraction., 21, A girl is twice as old as her sister. Four years hence, the product of their ages (in, years) will be 160. Find their present ages., A two digit number is such that the product of its digits is 18. When 63 is, subtracted from the number, the digit interchange their places. Find the number., CBSE 2006, Three consecutive positive integers are such that the sum of the square of the, first and the product of other two is 46, find the integers., CBSE 2010, A piece of cloth costs ` 200. If the piece was 5 m longer and each metre of cloth, costs ` 2 less than the cost of the piece would have remained unchanged. How, long is the piece and what is the original rate per metre?, A motor boat whose speed is 24 km/hr in still water takes 1 hour more to go 32, km upstream than to return downstream to the same spot. Find the speed of the, stream, (CBSE 2016), If the roots of the quadratic equation (b – c)x2 + (c – a)x + (a – b) = 0 are equal,, prove 2b = a + c., If the equation (1 + m2)n2x2 + 2mncx + (c2 – a2) = 0 ha equal roots, prove that c2, = a2 (1 + m2)., , Mathematics-X, , 39
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ANSWERS AND HINTS, , 1., 2., 3., 4., 5., 6., , (d) [x4 + 4x2 + 4x3 = x4 + 3 + 4x2 4x3 = 3 degree = 3], (b) [Check by substituting x = 2 in the equation.], 1, 5, in x2 + Px – = 0.], 2, 4, (c) [ A quadratic polynomial is of degree 2 and it has atmost two zeroes.], (d) [x(x – 7) = 0 x = 0, x = 7.], , (a) [Substitute x =, , q2, (a) [r =, (D = 0 q2 – 4pr = 0)], 4p, (b) p = – 6, q = 1 [x2 – 5x – 6 = 0 (x – 6) (x + 1) = 0], (c) K < 4, C, C, =1, =1), A, 4, (e) Linear equation (x = 0 ax2 + bx + c = 0 reduces to bx + c = 0), (a) False (A quadratic equation has atmost two real root)., (b) True (Coefficient of x2 = a, Constant = – c, D = b2 – 4ac = b2 – 4(a) (– c) = b2, + 4ac > 0), , (d) c = 4 ( product = 1 , 7., , 8., , 9., , 40, , (c) False (x2 = 0.9 x = 0.9 ), (d) False (Degree of quadratic polynomial is 2 not 1 Not a straight line), (i) d, (ii) a, (iii) e, (iv) c, (v) b, D=0, 20p2 – 60p = 0, p 0, 20p (p – 3) = 0, p= 3, , Mathematics-X
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10. (a) x =, , 7, 3, ,x= −, 2, 4, , (c) x =, , 2, 2, ,x=, 3, 3, , −2 5, 3, , (b) x =, , 5, x =, , (d) x =, , a, ,x=–a, 2, , −7 3, −5 2, (f) x = − 2 , x =, 3, 2, (g) Take (x – 1) = y, y2 – 5y – 6 = 0 ⇒ (y + 1) (y – 6) = 0, y = –1, y = 6, x – 1 = – 1, x – 1 = 6, x = 0, x = 7, 11. 2(– 5)2 + p(– 5) – 15 = 0 ⇒ p = 7, ∴ 7x2 + 7x + k = 0, D = 49 – 28 k = 0, , (e) x = − 3 , x =, , 49, 7, =, 28, 4, 2, 12. Sub, x = to get, 3, Sub, x = – 3 to get, Solve (1) and (2) to get a = 3, b = – 6., , ⇒k=, , 13. Product =, , 4p, c, =, = 4,, p, a, , −6, =4 ⇒, p, 14. x2 + (x + 4)2 = 656, x2 + 4x – 320 = 0, ATQ =, , D = 1296, , x=, , P=, , 4a + 9b = – 42, , ...(1), , 9a + b = – 21, , ...(2), , sum =, , −6, −b, =, p, a, , −6, −3, =, 4, 2, , −4 + 36 −4 − 36, −4 ± 1296, =, ,, 2, 2, 2, , 32, = 16, (rejecting –ve value), 2, Sides are 16 cm, 20 cm, 15. ATQ α – β = 11, , x=, , Solve to get α = 8, β = 3, Mathematics-X, , 41
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Sum of roots α + β =, Product of roots =, , −b, =5, a, , c, a, , 24 = 3k – 3, 27 = 3k, , ⇒ k=9, , Ans., , 16. x2 + kx + 64 = 0 → D1 = k2 – 256 ≥ 0,, ⇒ k ≥ 16, , k2 ≥ 256, , ...(1), , k ≤ – 16, x2, , – 8x + k = 0 → D2 = 64 – 4k ≥ 0, , ⇒ k ≤ 16, , ...(2), , (1) and (2) gives k = 16, 17. (a), , 1, 1, 1 1, − = +, a+b+ x x, a b, , x−a−b− x, a+b, =, (a + b + x) x, ab, – (a + b) ab = (a + b) (a + b + x) x, x2 + xa + bx + ab = 0, (x + a) (x + b) = 0, x = – a, x = – 6, (b), , 1, 1, 1 1, − = +, a+b+ x x, a b, , x−a−b− x, a+b, =, (a + b + x) x, ab, – (a + b) ab = (a + b) (a + b + x) x, x2 + xa + bx + ab = 0, (x + a) (x + b) = 0, x = – a, x = – 6, (c) Take LCM to get 2x2 + 5x + 3 = 0, x = – 1, x ≠, , 42, , −3, ., 2, , Mathematics-X
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7x +1, =y, 5x − 3, 4, ∴ 3y –, = 11 ⇒ 3y2 – 11y – 4 = 0. Solve to get, y, , (k) Let, , 1, y= − ,y=4, 3, Sub y and get x = 0, 1, (l) Take LCM to get 9x2 + 3x – 12 = 0, , 4, 3, 2, (m) Take LCM to get 2x – 27x + 88 = 0, , Solve to get x = 1, x = −, , 11, 2, (n) Take LCM to get x2 – 4x – 8 = 0 (Use quadratic formula), , x = 8,, , Ans. x = 2 ± 2 3, (o) Take LCM to get 2x2 – 16x + 23 = 0, Solve using Quadratic formula, Ans. x =, , −8 ± 3 2, 2, , (p) x 2 + 7 5 x − 2 5 x − 70 = 0, , ( x + 7 5 )( x − 2 5 ), , =0, , x = 2 5, −7 5, (q), , 16 − x, 15, =, x, x +1, x2 – 16 = 0, , x=±4, 54, 63, +, = 3, x → speed of train at first, x + 6 → Increased speed., x x+6, Ans. x = 36, x ≠ – 3., , 20. Equation, , 44, , Mathematics-X
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21. Let the natural number be x., ATQ x + 12 =, , 160, to get, x, , x2 + 12x – 160 = 0, (x + 20) (x – 8) = 0, x = 8,, , x ≠ – 20, , 22. Let total time to be n minutes., Policeman will catch the theif in (n – 1) minutes., Total distance covered by thief = (100 x) metres, (as distance covered in 1 min = 100 min), Distance covered by policemen, 100 + 110 + 120 + .... + to (n – 1) tan, , ...(1), , ...(2), , (n − 1), [2 × 100 + (n – 2) 10], 2, Solve and get, n2 – 3n – 18 = 0, n = 6, n ≠ – 3, Policeman will catch the thief in 5 minutes., 23. Time taken by top of smaller diameter = x hrs, Time taken by larger tap = (x – 9) hrs, , (1) and (2) ⇒ 100 n =, , 1, 1, 1, +, = and get x 2 – 21x + 54 = 0, x x−9, 6, Ans. x = 3, x = 18, x = 3 rejeced as x – 9 = – 6 < 0, ∴ x = 18 hrs x – 9 = 18 – 9 = 9 hrs, , ATQ, , 24., , Pond, , x, , 40, , x, 50, , Length of rectangular lawn = 50 m, Breadth of rectangular lawn = 40 m, Length of pond = 50 – 2x, Breadth of pond = 40 – 2x, Area of lawn – Area of pond = area of grass, 50 × 40 – (50 – 2x) (40 – 2x) = 1184, Mathematics-X, , 45
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get x2 – 45x + 296 = 0, x = 37, x = 8, x = 37 rejected 40 – 2x = 40 – 2(37) < 0, Ans. Length of pond = 34 m, Breadth of pond = 24 m, 25. x + y + x = 30, xy = 100, House, Solve x = 5m, 10 m,, y = 20 m, 10 m, x, 26., , x, , 27, , –, , x, , yA, , x, , 27 – x, C, , D, , B, , In ΔABD, pythogorus theorem 92 + x2 = (27 – x)2. Solve it to get x = 12 m., 27. Let original list price = ` x, 300 300, −, =5, x−5, x, Solve and get x = 20, x = – 15 → rejected, Ans. ` 20, 28. Let original number of persons be x, , ATQ, , 6500 6500, −, = 30, x, x + 15, Solve and get x = 50, x ≠ – 65., , ATQ, , 600, 600, 1, −, =, x − 200, x, 2, Solve to get x = 600, x ≠ – 400, , 29. ATQ, , Duration of flight, , [Speed of slow train = x km/hr], , 600, = 1hr., 600, , 600 600, −, = 3 (Speed of slow train x km/hr), x, x + 10, Solve to get x = 40, x ≠ – 50, Ans. 5 km/hr, , 30. ATQ, , 46, , Mathematics-X
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30, 30, 9, , , 15 x 15 x 2, Solve to get x = 5, x –5, Ans. 5 km/hr, 32. x2 + y2 = 400, ...(1), 4x – 4y = 16 x – y = 4, ...(2), y–x=4, ...(3), Solve (1) and (2) to get x = 16, x –12, Solve (1) and (3) to get x = 12, x –16, Ans. x = 16 m, y = 12 m from (1) and (2), x = 12 m, y = 16 m from (1) and (3), 33. BC = 2x, BD = x, Use pythagoreas to get, 31. ATQ, , (Speed of stream x km/hr), , AD = 169 x 2 60, 1, 2 x 169 x 2 60, 2, Solve to get x2 = 144, x2 = 25, x = 12 or x = 5, x –12, –5, base 2x = 24, 10 cm, , A=, , x, 2x 1, x, 2x 1, 16 58, , ATQ, =2, =, 2x 1, x, 21 21, 7, Solve to get x = 3, x , 11, 3, Ans. Fraction = ., 7, 35. Age of sister = x years, Age of girl = 2x, ATQ (x + 4) (2x + 4) = 160, Solve to get x2 + 6x – 72 = 0, Ans. x = 6 years, x – 12, 2x = 12 years, Mathematics-X, , 34. Fraction is, , 47
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36. Let tens place digit = x, then units digits =, , 18, ., x, , 18, x, 18 10 × 18, , , + x = 63, ATQ 10 x + − , x x, , , Solve to get x = 9, x ≠ – 2., Ans. No. 92, 37. Let no. be x, x + 1, x + 2, ATQ (x)2 + (x + 1) (x + 2) = 46, To get 2x2 + 3x – 44 = 0, , No, 10x +, , Use quadratic formula to solve q get x = 4, x ≠ –, , 22, 4, , ∴ No.s are 4, 5, 6., 38. Let length of piece be x metre., 200 200, −, =2, x, x+5, Solve to get x2 + 5x – 500 = 0, Solve to get x = 20, x ≠ –25, , ATQ, , 200, 200, =, = ` 10, 20, x, 39. Let speed of boat = x, , Rate per meter =, , 32, 32, −, =1, 24 − x 24 + x, x2 – 64x – 576 = 0, (x – 72) (x + 8) = 0, x≠–8, x = 72 km/hr, 40. Find D and let D = 0, (c – a)2 – 4(b – c) (a – b) = 0, Solve to get (a + c – 2b)2 = 0, ∴ a + c = 2b, 41. D = 0, (2 mnc)2 – 4 (1 + m2) n2 (c2 – a2) = 0, to get 4n2c2 = 4n2a2 (1 + m2), ∴ c2 = a2 (1 + m2), , ATQ, , 48, , Mathematics-X
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Practice Test, Quadratic Equations, Time: 1 Hour, , M.M : 20, SECTION-A, , 1., , The value of k is .................. if x = 3 is one root of x2 – 2kx – 6 = 0., , 2., , If the discriminant of 3x + 2x + = 0 is double the discriminant of x – 4x + 2 = 0, then value of is, 1, , 3., , If discriminant of 6x2 – bx + 2 = 0 is 1 then value of b is ............... ., , 4., , (x – 1)3 = x3 + 1 is quadratic equation. (T/F), , 2, , 1, 2, , 1, 1, , SECTION-B, 5., , If roots of x2 + kx + 12 = 0 are in the ratio 1 : 3 find k., , 6., , Solve for x : 21x2 – 2x +, , 7., , Find k if the quadratic equation has equal roots : kx (x – 2) + 6 = 0., , 2, , 1, =0, 21, , 2, 2, , SECTION-C, 8., , Solve using quadratic formula, , 3, , 4 3x 2 5 x 2 3 0, 9., , For what value of k, (4 – k)x2 + (2k + 4)x + (8k + 1) = 0 is a perfect square., , 3, , SECTION-C, 7, hours. The tap with longer diameter, 8, takes 2 hours less than the tap with smaller one to fill the tank separately. Find the, time in which each tap can fill the tank separately., (CBSE 2018), , 10. Two water taps together can fill a tank in 1, , 4, , Mathematics-X, , 49
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CHAPTER, , 5, , Arithmetic Progression, , Points to, ponder, , a + 3d, , 50, , Mathematics-X
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VERY SHORT ANSWER TYPE QUESTIONS, , 1., 2., 3., 4., 5., 6., , FInd 5th term of an A.P. whose nth term is 3n – 5, Find the sum of first 10 even numbers., Write the nth term of odd numbers., Write the sum of first n natural numbers., Write the sum of first n even numbers., Find the nth term of the A.P. – 10, – 15, – 2, – 25, ..........., , 7., , 1 2 1, Find the common difference of A.P. 4 , 4 , 4 , ............., 9 9 3, , Write the common difference of an A.P. whose nth term is an = 3n + 7, What will be the value of a8 – a4 for the following A.P., 4, 9, 14, ............., 254, 10. What is value of a16 for the A.P. – 10, – 12, – 14, – 16, ......., 11. 3, k – 2, 5 are in A.P. find k., 8., 9., , 12. For what value of p, the following terms are three consecutive terms of an A.P., p, 2., 13. In the following A.Ps, find the missing terms in the boxes :, , 26, (b), , 13,, ,3, (a) 2,, (c) 5,, , ,, , ,9, , 1, 2, , (d) – 4,, , ,, , ,, , 4, ,, 5, , (NCERT), , ,, , ,6, , (e), , 38,, ,, ,, , – 22, 14. Multiple Choice Questions:, (a) 30th term of the A.P. 10, 7, 4 .... is, (A) 97, (B) 77, (C) –77, (D) –87, 1, (b) 11th term of an A.P. – 3, − , , ... is, 2, , (A) 28, , (B) 22, , (C) –38, , (D) − 48, , Mathematics-X, , 1, 2, 51
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(c) In an A.P. if d = – 4, n = 7, an = 4, then a is, (A) 6, (B) 7, (C) 120, (D) 28, (d) The first three terms of an A.P. respectively are 3y – 1, 3y + 5 and 5y + 1, then y equals:, (CBSE 2014), (A) –3, (B) 4, (C) 5, (D) 2, (e) The list of numbers – 10, – 6, – 2, 2, ... is, (A) An A.P. with d = – 16, (B) An A.P. with d = 4, (C) An A.P. with d = – 4, (D) Not an A.P., (f) The 11th term from the last term of an A.P. 10, 7, 4, ...., – 62 is (NCERT), (A) 25, (B) –32, (C) 16, (D) 0, (g) The famous mathematician associated with finding the sum of the first 100, natural numbers is, (A) Pythagoras, (B) Newton, (C) Gauss, (D) Euclid, (h) What is the common difference of an A.P. in which a18 – a14 = 32 ?, (A) 8, (B) – 8, (C) – 4, (D) 4, 15. Match the following :, Column A, Column B, (a) a = – 18, n = 10, d = 2 then an of A.P., , (a), , a+c, 2, 0, – 41, 8, A.P., , (b) a, b and c in A.P. then their Arithmetic mean is, (b), (c) If 2, 4, 6, are in A.P. then 4, 8, 12 will also be an, (c), (d) If an = 9 – 5 n of an A.P. then a10 will be, (d), (e), (e) If d = – 2, n = 5 and an = 0 in A.P. then a is, 16. State True/False and justify, (a) 301 is a term of A.P. 5, 11, 17, 23 ...., (NCERT), (b) Difference of mth and nth term of an A.P. = (m – n) d., (c) 2, 5, 9, 14, .... is an A.P., (d) Sum of first 20 natural numbers is 410., (e) nth term of A.P. 5, 10, 15, 20 .... n terms and nth term of A.P. 15, 30, 45, 60,, ... n terms are same., 52, , Mathematics-X
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SHORT ANSWER TYPE QUESTIONS-I, , 17., 18., 19., 20., 21., 22., , 23., 24., 25., 26., 27., 28., 29., , Is 144 a term of the A.P. 3, 7, 11, ......... ? Justify your answer., Find the 20th term from the last term of the A.P. 3, 8, 13, ...., 253, Which term of the A.P. 5, 15, 25, ....... will be 130 more than its 31st term?, The first term, common difference and lat term of an A.P. are 12, 6 and 252, respectively, Find the sum of all terms of this A.P., Find the sum of first 15 multiples of 8., Is the sequence formed in the following situations an A.P., (i) Number of students left in the school auditorium from the total strength of 1000, students when they leave the auditorium in batches of 25., (ii) The amount of money in the account every year when Rs. 100 are deposit, annually to accumulate at compound interest at 4% per annum., Find the sum of even positive integers between 1 and 200., If 4m + 8, 2m2 + 3m + 6, 3m2 + 4m + 4 are three consecutive terms of an A.P. find, m., How many terms of the A.P. 22, 20, 18, ....... should be taken so that their sum is, zero., If 10 times of 10th term is equal to 20 times of 20th term of an A.P. Find its 30th, term., Find the middle term of the A.P. 6, 13, 2, ...... 216., Find whether (– 150) is a term of A.P. 11, 8, 5, 2, ..... ?, (NCERT), Find how many two digit numbers are divisible by 6?, (CBSE 2011), , 30. If, , 1, 1, 1, ,, and, are in A.P. find x., x+2 x+3, x+5, , (CBSE 2011), , 31. Find the middle term of an A.P. – 6, – 2, 2, .... 58., , (CBSE 2011), , 32. In an A.P. find Sn, where an = 5n – 1. Hence find the sum of the first 20 terms., (CBSE 2011), 33. Which term of A.P. 3, 7, 11, 15 .... is 79? Also find the sum 3 + 7 + 11 + ... + 79., (CBSE 2011C), 34. Which term of the A.P. : 121, 117, 113 ... is the first negative terms ?(NCERT), 35. Find the 20th term from the last term of the A.P. 3, 8, 13, ... 253., , Mathematics-X, , (NCERT), , 53
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SHORT ANSWER TYPE QUESTIONS-II, , 36. Find the middle terms of the A.P. 7, 13, 19, ......., 241., 37. Find the sum of integers between 10 and 500 which are divisible by 7., 38. The sum of 5th and 9th terms of an A.P. is 72 and the sum of 7th and 12th term is 97., Find the A.P., 39. If the mth term of an A.P. be, , 1, 1, and nth term be , show that its (mn)th is 1., n, m, , 40. If the pth of term A.P. is q and the qth term is p, prove that its nth term is (p + q – n)., 41. Find the number of natural numbers between 101 and 999 which are divisible by, both 2 and 5., 42. The sum of 5th and 9th terms of an A.P. is 30. If its 25th term is three times its 8th, term, find the A.P., 43. If Sn, the sum of first n terms of an A.P. is given by Sn = 5n2 + 3n, then find its nth, term and common difference., 44. Which term of the A.P. 3, 15, 27, 39 .... wil be 120 more than its 21st term?, (CBSE 2018), 2, 45. If Sn, the sum of first n terms of an A.P. is given by Sn = 3x – 4x, find the nth, term., (CBSE 2018), 46. In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in, the third and so on. There are 5 rose plants in the last row. How many rows are, there in the flower bed?, (NCERT), 47. For what value of n, are the nth term of two A.P’s 63, 65, 67 ......... and 3, 10, 17, ..... are equal ?, (NCERT), 48. Which term of an A.P. 3, 15, 27, 39 .... will be 132 more than its 54th term?, (NCERT), 49. If the sum of the first 14 terms of an A.P. is 1050 and its first term is 10, find the, 20th term., (NCERT), 50. Find the sum of odd numbers between 0 and 50., (NCERT), 2, 51. If Sn = 4n – n in an A.P. find the A.P., (NCERT), 52. How many terms of the A.P. 9, 17, 25, ..... must be taken to give a sum of 636?, (NCERT), LONG ANSWER TYPE QUESTIONS, 54, , Mathematics-X
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53. The sum of third and seventh terms of an A.P. is 6 and their product is 8. Find the sum, of first 16th terms of the A.P., 54. Determine the A.P. whase 4th term is 18 and the difference of 9th term from the 15th, term is 30., 55. The sum of first 9 terms of an A.P. is 162. The ratio of its 6th term to its 13th term is, 1:2. Find the first and fifteenth terms of the A.P., 56. If the 10th term of an A.P. is 21 and the sum of its first 10 terms is 120, find its nth, term., 57. The sum of first 7 terms of an A.P. is 63 and the sum of its next 7 term is 161., Find the 28th term of this A.P., 58. The sum of first 20 terms of an A.P. is one third of the sum of next 20 term. If first, term is 1, find the sum of first 30 terms of this A.P., 59. If the sum of the first four terms of an AP is 40 and the sum of the first fourteen, terms of an AP is 280. Find the sum of first n terms of the A.P. (CBSE 2018), 60. Ramkali required Rs. 2500 after 12 weeks to send her daughter to school. She, saved ` 100 in the first week and increased her weekly savings by ` 20 every, week. Find wheather she will be able to send her daughter to school after 12, weeks., (CBSE 2015), 61. In an AP of 50 terms, the sum of first 10 terms is 210 and the sum of last 15 terms, is 2565. Find the A.P., (CBSE 2014), 62. The sum of first n terms of an A.P. is 5n2 + 3n. If the mth term is 168, find the, value of m. Also find the 20th term of the A.P., (CBSE 2013), 63. If the sum of the first seven terms of an A.P. is 49 and the sum of its first 17 terms, is 289. Find the sum of first n terms of an A.P., (CBSE 2016), 64. If the 4th term of an A.P. is zero, prove that the 25th term of the A.P. is three, times its 11th term., (CBSE 2016), 65. In an A.P. if S5 + S7 = 167 and S10 = 235. Find the A.P., where Sn denotes the sum, of its first n terms., (CBSE 2015), 66. In an AP prove S12 = 3 (S8 – S4) where Sn represent the sum of first n terms of an, A.P., (CBSE 2015), , Mathematics-X, , 55
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ANSWERS AND HINTS, VERY SHORT ANSWER TYPE QUESTIONS-I, , 1., , an = 3x – 5, , 2., , Sn =, , a5 = 10, , 3., , 10, [2 × 2 + 9 × 2] = 110, 2, 1, 3, 5, ......, an = 1 + (n – 1)2 = 2n – 1., , 4., , 1 + 2 + ........ + n =, , 5., 6., , n, [1 + n], 2, n, 2 + 4 + 6 + ... + 2n = [2 + 2n] = n(n + 1), 2, an = a + (n – 1)d = – 5(n + 1), , 1, 9, 8. a1 = 3 + 7 = 10, a2 = 6 + 7 = 13, d = 3, 9. (a + 7d) – (a + 3d) = 4d = 20, 10. a16 = a + 15d = – 40, 11. 3, k – 2, 5 are in A.P., , 7., , d = a2 – a1 =, , ∴ K–2=, 12. P =, , 3+ 5, =4, 2, , K=6, , 7, (same as Q.11), 5, , 13. (a) 14, 1, (c) 6 , 8, 2, , (b) 18 , 8, (d) −2 , 0 , 2 , 4, , (e) 53 , 23 , 8 , −7, 14. (a), (c), (e), (g), 56, , C, D, B, C, , (b), (d), (f), (h), , B, C, B, A, Mathematics-X
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→ (b), (b) → (a), → (e), (d) → (c), → (d), False, 301 = 5 + (n – 1) 6, 151, Solving we get n =, which is not a natural number., 3, ∴ 301 is not a term of this A.P., (b) True [a + (m – 1) d] – [a + (n – 1) d] = (m – n) d, (c) False a2 – a1 = 5 – 2 = 3, a3 – a2 = 9 – 5 = 4, , 15. (a), (c), (e), 16. (a), , n( n + 1), 20 × 21, =, = 210, 2, 2, (e) True (If a, b, c, d ... are in AP then ka, kb, kc, kd ..... are in AP), k≠0, (f) 144 = 3 + (n – 1) 4, , (d) False Sn =, , 141, + 1 = n which is not possible, 4, 18. No, use l – (n – 1) d, Ans. 158, 19. Let an = 130 + a31, Solve to get n = 44, Ans. 44th term, 20. a = 12, d = 6, an = 252 ⇒ n = 41, , Find S41 = 5412, use Sn =, , n, [2a + (n – 1) d], 2, , 15, [2a + 14d], 2, where a = 8, d = 8, Ans. 960, 22. (i) Yes (ii) No, 23. 2 + 4 + 6 + .... + 198, a = 2, d = 2, an = 198 ⇒ n = 99, , 21. S15 =, , Sn =, , n, [a + l ] = 9900, 2, , Mathematics-X, , 57
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24. b =, , a+c, 2, , 4m + 8 + 3m 2 + 4m + 4, 2, Solve to get m2 – 2m = 0, m = 0, 2, ∴ 2m2 + 3m + 6 =, , n, [44 + (n – 1) (– 2)] = 0., 2, Solve n = 23, , 25. Sn = 0 ⇒, , 26. ATQ 10 a10 = 20 a20, ⇒ a10 = 2a20, a + 9d = 2a + 38d, a = – 29d ...(1), a30 = a + 29d, Substitute a from (1), Ans. a30 = 0, 27. 6, 13, 20, ..., 216, Find n from an = a + (n – 1) d, then use concept of median, Middle term = 111., 28. Let an = –150, 11 + (n – 1) (– 3) = – 150, Solve and get n is not a natural number., ∴ Ans. No., 29. Two digit No.s divisible by 6 are 12, 18, 24, .... 96., a2 – a1 = a3 – a2 = 6, ∴ A.P., an = 96 ⇒ n = 15, 30., , 2, 1, 1, +, =, (2b = a + c), x+3, x+2 x+5, , Solve to get x = 1., 58, , Mathematics-X
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31. an = a + (n – 1) d, 58 = – 6 + (n – 1) 4, find n = 17, Find Middle term using conceptof median, n + 1, = , , 2 , , th, , term = 9th term, , a9 = – 6 + 8(4) = 26, 32. an = 5n – 1, Find AP a1 = 4, a2 = 9, a3 = 14, 4, 9, 14, ...., a2 – a1 = 5 = a3 – a2, Sn =, , n, n, [2a + (n – 1)d] = [8 + (n – 1) 5], 2, 2, , n, [5n + 3], 2, 20, S20 =, [100 + 3] = 10 × 103 = 1030, 2, 33. 79 = 3 + (n – 1) 4, n = 26, , =, , 26, [3 + 79] = 13[82], 2, S26 = 1066, 34. Let an < 0, 121 + (n – 1) (– 4) < 0, 121 – 4n + 4 < 0, 125 < 4n, , S26 =, , 125, 4, ∴ n = 32, 32nd term will be first negative term., 35. 20th term from end using [l – (n – 1) d], = 253 – 19 × 5, = 253 – 95 = 158, , n>, , Mathematics-X, , 59
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SHORT ANSWER TYPE QUESTIONS-II, , 36. Same as Q.27., Ans. 121, 127, 37. No.s between 10 and 500 which are divisible by 7, 14, 21, 28 ..., 497, Find n, using an = a + (n – 1) d , then use Sn =, , n, [2a + (n – 1) d], 2, , Ans. Sn = 17885., 38. a5 + a9 = 72, a7 + a12 = 97, Solve these equations to get a and d., A.P., 6, 11, 16, 21, 26, ......., 1, n, 1, an =, m, , 39. am =, , 1, n, 1, ⇒ a + (n – 1)d =, m, – –, –, ––––––––––––––, , ⇒ a + (m – 1)d =, , (m – n) d =, , 1 1, m−n, − =, n m, mn, , 1, 1, , find a =, mn, mn, = a + (mn – 1) d, , ∴d=, amn, , 1, 1, + (mn − 1), mn, mn, amn = 1., 40. ap = q, aq = p, Solve to get a and d then find ap + q – n = 0, 41. No.s divisible by both 2 and 5, ⇒ No.s divisible by 10., No.s between 101 and 999 divisible by 2 and 5 both 110, 120, 130, 140, ...,, 990., Use an = 990 to get n = 89., , =, , 60, , Mathematics-X
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42. ATQ a5 + a9 = 30, a25 = 3 a8, Solve to get a = 3, d = 2, A.P. 3, 5, 7, 9, ..., 43. Sn = 5n2 + 3n, Find an = Sn – Sn – 1 = 10 n – 2, Use it to get d = 10, 44. Let an = 120 + a21, 3 + (n – 1)d = 120 + [3 + 20d], 3 + (n – 1)12 = 120 + [3 + 20 × 12], = 120 + 243, (n – 1)12 = 363 – 3 = 360, n = 31, 45. Sn = 3n2 – 4n, an = Sn – Sn–1, = (3n2 – 4n) – [3(n – 1)2 – 4(n – 1)], = (3n2 – 4n) – [3n2 + 3 – 6n – 4n + 4], = – [7 – 6n], an = 6n – 7, 46. 23, 21, 19, ... 5, an = a + (n – 1) d, S = 23 + (n – 1) (– 2), n = 10, 47. 63, 65, 67, ....., an = 63 + (n – 1) 2, = 61 + 2n, 3, 10, 17, ...., an = 3 + (n – 1) 7, = 7n – 4, 61 + 2n = 7n – 4, 65 = 5n, n = 13, Mathematics-X, , 61
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equation (2) will give d = 5, Substitute d = 5 in (1) to get a = 3, A.P. 3, 8, 13, ...., 55. ATQ S9 = 162 ⇒, ATQ, , 9, [2a + 8d] = 162, 2, , ...(1), , 1, a6, = solve and get a = 2d, 2, a13, , Sub a = 2d in (1) to get d = 3, a = 6, a15 = a + 14d, Ans. a15 = 48, 56. a10 = 21, S10 = 120. Solve these to get a and d then find, an = a + (n – 1)d, Ans. an = 2n + 1, 57. ATQ S7 = 63,, ...(1), Sum of next 7 terms = S14 – S7 = 161, Use Sn =, , ...(2), , n, [2a + (n – 1) d], 2, , Solve (1) and (2) to get a and d then find a28 using an = a + (n – 1) d., Ans. a28 = 57, 58. ATQ S20 =, , 1, (S – S20), a = 1, 3 40, , n, [2a + (n – 1) d] and a = 1 to find d, 2, then find S30., Ans. 900, , Use Sn =, , 59. S4 = 40 ⇒, , 4, [2a + 3d] = 40, 2, , 14, [2a + 13d] = 280, 2, Solve to get a = 7, d = 2, , S14 = 280 ⇒, , Mathematics-X, , 63
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Practice Test, Arithmetic Progression, Time: 1 Hr., , M.M. : 20, Section-A, , 1., , Find the sum of first 10 natural numbers., , 1, , 2., , 1 2 3, What is the common difference of an A.P. 8 ,8 ,8 ,..........., 8 8 8, , 1, , 3., , If k, 2k – 1 and 2k + 1 are in A.P. them value of k is ..................., , 1, , 4., , The 10th term from the end of the AP 8, 10, 12, ...., 126 is ..................., , 1, , Section-B, 5., , How many 2 digit number are there in between 6 and 102 which are divisible, by 6., 2, , 6., , The sum of n terms of an A.P. is n2 + 3n. Find its 20th term., , 2, , 7., , Find the sum (–5) + (–8) + (–11) + ...+(–230), , 2, , Section-C, 8., , Find the five terms of an A.P. whose sum is 12, is 2 : 3., , 9., , 1, and first and last term ratio, 2, 3, , Find the middle term of an A.P. 20,16,12,.......,– 176., , 3, , Section-D, 10. The sum of three numbers in A.P. is 24 and their product is 440. Find the, numbers., 4, , Mathematics-X, , 65
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CHAPTER, , 6, , Triangles, , s, , A., A., A., , S.S.S., S.A, .S., , Tr, , gl e, ian, , S, Tri imila, an g r, les, , s, lem, b, o, Pr, THE, , Areas of, Triangles, Converse of BPT, , TH, , EO, RE, M, , S, , MS, ORE, , BPT, , Pythagoras, Theorem, , 66, , Converse, of Pythagoras, Theorem, , Mathematics-X
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Key Points:, 1. Similar Triangles: Two triangles are said to be similar if their corresponding angles, are equal and their corresponding sides ar proportional., 2. Criteria for Similarity:, in ΔABC and ΔDEF, (i) AAA Similarity : ΔABC ~ ΔDEF when ∠A, ∠D, ∠B = ∠E and ∠C = ∠F, (ii) SAS Similarity :, AB BC, =, and ∠B = ∠E, DE, EF, AB AC BC, (iii) SSS Similarity : ΔABC ~ ΔDEF,, =, =, DE DF, EF, The proof of the following theorems can be asked in the examination :, (i) Basic Proportionality Theorem : If a line is drawn parallel to one side of a, triangle to intersect the other sides in distinct points, the other two sides are, divided in the same ratio., (ii) The rato of aras of two similar triangles is equal to the square of the ratio of their, corresponding sides., (iii) Pythagoras Theorem: In a right triangles the square of the hypotenuse is equal, to the sum of the squares of the other two sides., (iv) Converse of pythagoras theorem : In a triangle, if the square of one side is, equal to the sum of squares of other sides then the angle opposite to the first side, is a right angle., , ΔABC ~ ΔDEF when, , 3., , VERY SHORT ANSWER TYPE QUESTIONS, , 1., , Fill in the blanks :, (i) All equilateral triangles are __________ ., (ii) If ΔABC ~ ΔFED, then, , AB, , ., ED, (iii) Circles with equal radii are _________ ., (iv) If a line is drawn parallel to one side of a triangle to intersect the other two sides, in distinct points, the other two sides are divided in the _________ ratio., (v) In __________ triangle, the square of the hypotenuse is equal to the sum of the, squares of the other two sides., , Mathematics-X, , =, , 67
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2., , 3., , 4., , State True or False :, (i) All the similar figures are always congruent., (ii) The Basic Proportionality Theorem was given by Pythagoras., (iii) The mid-point theorem can be proved by Basic Proprotionality Theorem., (iv) Pythagoras Theorem is valid for right angled triangle., (v) If the sides of two similar triangles are in the ratio 4 : 9, then the areas of these, triangles are in the ratio 16 : 81., Match the following :, Column I, Column II, (a) If corresponding angles are equal in two, (i) SAS similarity criterion, triangles, then the two triangles are similar., (b) If sides of one triangle are proportional to, (ii) ASA similarity criterion, the sides of the other triangle, then the two, triangles are similar., (c) If one angle of a triangle is equal to one, (iii) AAA similarity criterion, angle of the other triangle and the sides, including these angles are proportional,, then the two triangles are similar., (iv) SSS similarity criterion, In the following figure, XY || QR and, , PX, PY 1, =, = , then, XQ YR 2, P, , X, Q, , 68, , R, , 1, QR, 3, 1, (c) XY2 = QR2, (d) XY = QR, 2, In the following figure, QA ⊥ AB and PB ⊥ AB, then AQ is, , (a) XY = QR, , 5., , Y, , (b) XY =, , Mathematics-X
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P, A, , 10 units, , 9 units, , O, 6 units, , B, , Q, , 6., , 7., , 8., , (a) 15 units, (b) 8 units, (c) 5 units, (d) 9 units, The ratio of areas of two similar triangles is equal to the, (a) ratio of their corresponding sides., (b) ratio of their corresponding altitudes., (c) ratio of the square of their corresponding sides., (d) ratio of their perimeter., The areas of two similar triangles are 144 cm2 and 81 cm2. If one median of the first, triangleis 16 cm, length of corresponding median of the second triangle is, (a) 9 cm, (b) 27 cm, (c) 12 cm, (d) 16 cm, In a right triangle ABC, in which ∠C = 90° amd CD ⊥ AB. If BC = a, CA = b,, AB = c and CD = p, then, A, (a), (c), , 1, 1, 1, = 2+ 2, 2, p, a, b, 1, 1, 1, < 2+ 2, 2, p, a, b, , (b), (d), , 1, 1, 1, ≠ 2+ 2, 2, p, a, b, 1, 1, 1, > 2+ 2, 2, p, a, b, , b, , C, , 9., , D, , c, , p, , a, , B, , AB 1, = , then ar(DABC) is, DE, 2, 2, (b) 25 cm, (d) 200 cm2, , If ΔABC ~ ΔDEF, ar(ΔDEF) = 100 cm2 and, (a) 50 cm2, (c) 4 cm2, , 10. If the three sides of a triangle are a, 3a and 2a , then the measure of hte, angle opposite to longest side is, (a) 45°, (b) 30°, (c) 60°, (d) 90°, 11. A vertical pole of length 3 m casts a shadow of 7 m and a tower casts a shadow, of 28 m at a time. The height of tower is, Mathematics-X, , 69
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12., , 13., , 14., 15., 16., , (a) 10 m, (b) 12 m, (c) 14 m, (d) 16 m, The lengths of the diagonals of a rhombus are 16 cm and 12 cm. Then, the length of, the side of the rhombus is, (NCERT Exempler), (a) 9 cm, (b) 10 cm, (c) 8 cm, (d) 20 cm, If ΔABC ~ ΔEDF and ΔABC is not similar to ΔDEF, then which of the following is, not true?, (NCERT Exempler), (a) BC.EF = AC.FD, (b) AB.EF = AC.DE, (c) BC.DE = AB.EF, (d) BC.DE = AB.FD, Write the statement of pythagoras theorem., Write the statement of Basic Proportionality Theorem., Is the triangle with sides 12 cm, 16 cm and 18 cm a right triangle?, , Area (Δ ABC) 9, = , AB = 18 cm, BC = 15 cm, then find the, Area (Δ PQR) 4, length of PR., (CBSE 2018), 18. In the given Fig., ∠M = ∠N = 46°, Express x in terms of a, b and c., 17. If ΔABC ~ ΔQRP,, , L, , P, , a, x, 46°, M, , 46°, , b, , c, , K, , 19. In the given Fig. ΔAHK ~ ΔABC. If AK = 10 cm, BC = 3.5 cm and HK = 7 cm,, find AC., (CBSE 2010), H, , C, , A, , K, , B, , 20. It is given that ΔDEF ~ ΔRPQ. Is it true to say that ∠D = ∠R and ∠F = ∠P?, 70, , Mathematics-X
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21. If the corresponding Medians of two similar triangles are in the ratio 5 : 7. Then find, the ratio of their sides., 22. An aeroplane leaves an airport and flies due west at a speed of 2100 km/hr. At the, same time, another aeroplane leaves the same place at airport and flies due south at, a speed of 2000 km/hr. How far apart will be the two planes after 1 hour?, 23. The areas of two similar ΔABC and ΔDEF are 225 cm2 and 81 cm2 respectively. If, the longest side of the larger triangle ΔABC be 30 cm, find the longest side of the, smaller triangle DEF., 24. In the given figure, if ΔABC ~ ΔPQR, find the value of x?, A, R, 6 cm, , B, , 5 cm, , 3.75 cm, , x, , Q, 4.5 cm, , C, , 4 cm, , P, , 25. In the given figure, XY || QR and, , PX PY 1, =, = , find XY : QR., XQ YR 2, P, , X, Q, , Y, R, , 26. In the given figure, find the value of x which will make DE || AB ?, (NCERT Exempler), , Mathematics-X, , 71
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A, , B, , 3x + 19, , 3x + 4, D, , E, x+3, , x, C, , 27. If ΔABC ~ ΔDEF, BC = 3EF and ar (DABC) = 117cm2 find area (ΔDEF)., 28. If ΔABC and ΔDEF are similar triangles such that ∠A = 45° and ∠F = 56°, then find, the ratio of their corresponding attitudes., 29. If the ratio of the corresponding sides of two similar triangles is 2 : 3, then find the, ratio of their corresponding attitudes., SHORT ANSWER TYPE QUESTIONS-I, , 30. In the given Fig. PQ = 24 cm, QR = 26 cm, ∠PAR = 90°, PA = 6 cm and AR = 8, cm, find ∠QPR., Q, , P, A, R, , 31. In the given Fig., DE || AC and DF || AE. Prove that, , FE EC, =, BF BE, , A, D, , B, , F, , E, , C, , 32. In ΔABC, AD ⊥ BC. Such that AD2 = BD × CD. Prove that ΔABC is right, angled triangle., 33. In the given Fig., D and E are points on sides AB and CA of ΔABC such that, ΔB = ∠AED. Show that ΔABC ~ ΔAED., , 72, , Mathematics-X
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A, D, E, B, , C, , 34. In the given fig., AB || DC and diagonals AC and BD intersects at O. If OA = 3x – 1, and OB = 2x + 1, OC = 5x – 3 and OD = 6x – 5, find the value of x., D, 6x –, 1, , 3x –, , C, 5 5x, O, , –3, , 2x +, , 1, , A, , B, , 35. In the given Fig. PQR is a triangle, right angled at Q. If XY || QR, PQ = 6 cm,, PY = 4 cm and PX : XQ = 1 : 2. Calculate the lengths of PR and QR., P, , X, , Y, R, , Q, , 36. In the given figure, AB || DE. Find the length of CD., B, 6 cm, 5 cm, A, , C, E, 3 cm, D, , 37. In the given figure, ABCD is a parallelogram. AE divides the line segment BD, in the ratio 1 : 2. If BE = 1.5 cm find BC., , Mathematics-X, , 73
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A, , C, O, , B, , D, , 44. If AD and PS are medians of ΔABC and ΔPQR respectively where ΔABC ~ ΔPQR,, Prove that, , AB AD, ., =, PQ PS, , 45. In the given figure, DE || AC. Which of the following is correct?, x=, , a+b, ay, , or, , x=, , B, , ay, a+b, , a, E, , D, , x, , b, C, , A, , y, , 46. Prove that the sum of the square of the sides of a rhombus is equal to the sum of the, squares of its diagonals., (NCERT, CBSE 2019), 47. A street light bulb is fixed on a pole 6 m above the level of the street. If a woman of, height 1.5 m casts a shadow of 3 m, find how for she is away from the base of the, pole., (NCERT Exempler), 48. Two poles of height a metrs and b metres are p metres apart. Prove that the height of, the point of intersection of the lines joining the top of each pole to the foot of the, ab, metres., a+b, 49. In the given figure AB || PQ || CD, AB = x, CD = y and PQ = z. Prove that, , opposite pole is gives by, , 1 1 1, + = ., x y z, A, C, x, , P, z, , B, , Mathematics-X, , Q, , y, D, , 75
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50. In the given figure, triangle., , PS PT, and ∠PST = ∠PRQ. Prove that PQR is an isoscles, =, SQ TR, (NCERT), P, , S, , T, , Q, , R, , 51. In the figure, a point O inside ΔABC is joined to its vertices. From a point D on, AO, DE is drawn parallel to AB and from a point E on BO, EF is drawn parallel, to BC. Prove that DF || AC., A, D, O, E, , F, C, , B, , 52. Two triangles BAC and BDC, right angled at A and D respectively are drawn on, the same base BC and on the same side of BC. If AC and DB intersect at P., PRove that AP × PC = DP × PB., (CBSE 2019), D, A, P, B, , C, , 53. Hypotenuse of a right triangle is 25 cm and out of the remaining two sides, one, is larger than the other by 5 cm, find the lenghts of the other two sides., (NCERT Exempler), 54. In the given figure DE || AC and, , 76, , BE BC, =, . Prove that DC || AP., EC CP, , Mathematics-X
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A, , D, , B, , E, , C, , P, , 55. In a quadrilateral ABCD, ∠B = 90°, AD2 = AB2 + BC2 + CD2. Prove that, ∠ACD = 90°., D, , C, , A, , B, , 56. In the given figure, DE || BC, DE = 3 cm, BC = 9 cm and ar (DADE) = 30 cm2., Find ar (BCED)., A, , D, , B, , 3cm, , 9 cm, , E, , C, , 57. In an equilateral ΔABC, D is a point on side BC such that BD =, Mathematics-X, , 1, BC. Prove, 3, 77
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that 9AD2 = 7AB2., (NCERT, CBSE 2018), 58. In ΔPQR, PD ⊥ QR such that D lies on QR. If PQ = a, PR = b, QD = c and DR = d, and a, b, c, d are positive units. Prove that (a + b) (a – b) = (c + d) (c – d)., (NCERT Exempler), 59. Prove that the ratio of the areas of two similar triangles is equal to the ratio of, the squars of their corresponding sides., (CBSE 2010, 2018, 2019), 60. In the given figure, the line segment XY is Parallel to AC of ΔABC and it, divides the triangle into two parts of equal areas. Prove that, , AX, =, AB, , 2 −1, ., 2, , A, X, , B, , Y, , C, , 61. Through the vertex D of a parallelogram ABCD, a line is drawn to intersect the, DA FB FC, =, =, ., AE BE CD, 62. Prove that if in a triangle, the square on one side is equal to the sum of the, squares on the other two sides, then the angle opposite to the first side is a right, angle., (CBSE 2019), 63. Prove that is a right angle triangle, the square of the hypotenuse is equal the sum, of the squares of other two sides., (CBSE 2018, 2019), 64. If a line is drawn parallel to one side of a triangle to intersect the other two, sides in distinct points, then prove that the other two sides are divided in the, same ratio., (CBSE 2019), , sides BA and BC produced at E and F respectively. Prove that, , 78, , Mathematics-X
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ANSWERS AND HINTS, VERY SHORT ANSWER TYPE QUESTIONS-I, , 1., , 2., 3., , AB BC, =, FE E D, (iv) Same, (v) Right, (i) False, (ii) False, (iv) True, (v) True, (a) (iii) AAA similarity criterion., (b) (iv) SSS similarity criterion., (c) (i) SAS similarity criterion., , (i) Similar, , (ii), , 5., 6., 7., , 1, QR, 3, (A) 15 units, (C) Ratio of the square of their corresponding sides., (C) 12 cm, , 8., , (A), , 9., 10., 11., 12., 13., 16., 17., 18., , (B) 25 cm2, (D) 90°, (B) 12 m, (B) 10 cm, (C) BC.DE = AB.EF, No, because (12)2 + (16)2 ≠ (18)2, 10 cm, ΔKPN ~ ΔKLM, , 4., , (iii) Congruent, , (iii) True, , (B) XY =, , 1, 1, 1, = 2+ 2, 2, p, a, b, , x, c, =, a b+c, , x=, , ac, b+c, , Mathematics-X, , 79
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AK HK, =, ⇒, AC BC, 20. ∠D = ∠R (True), ∠F = ∠P (False), 21. 5 : 7, , 10, 7, =, AC 3.5, , 19., , ⇒ AC = 5 cm, , N, , 2100 Km, , 22., , W, , E, , 0, , A, , B, , AB =, , (2100) 2 + (2000) 2 = 2900 km, , 2000 Km, , 5, B, , 23. Let longest side of the ΔDEF be x cm., 225 30 , = , 81, x, , 2, , x = 18 cm, 24., , AB, BC, =, PQ, QR, , ⇒, , 6, 4, =, x, 4.5, , ⇒ x = 3cm, , 25. ΔPXY ~ ΔPQR, PX, XY, 1, =, =, PQ, QR, 3, ∴ XY : QR = 1 : 3, 26., , x+3, x, =, 3x + 4, 3 x + 19, x=2, , (By B.P.T.), , ar (ABC) BC 2 3EF 2 3 2, 27., = , = , = , ar (DEF), EF , EF , 1, 80, , Mathematics-X
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∠A = ∠A, ∴ ΔABC ~ ΔAED, 34., , 3x − 1, 2x + 1, =, 5x − 3 6 x − 5, , (Common), [AA similarity criterion], ⇒ x=, , 1, or 2, 2, , 1, is neglected due (5x – 3) get negative value., 2, So, x = 2 is the required value., , But x =, , 35., , PX, PY, =, XQ, YR, , ⇒, , 1, 4, =, 2, YR, , ⇒ YR = 8 cm, , ∴ PR = 8 + 4 = 12 cm, QR =, , (12) 2 − (6) 2 = 6 3 cm, , 36. ΔABC ~ ΔEDC, , (AA Similarity criterion), , 6, 5, =, 3, CD, CD = 2.5 cm, 37. ΔBOE ~ ΔDOA, , (AA Similarity criterion), , BO, BE, =, DO, DA, 1, 1.5, =, 2, DA, DA = 3 cm, BC = DA = 3 cm, 38. (i) 65°, (ii) 45°, (iii) 45°, (iv) 70°, , (Opposite sides of a parallelogram), , ar (Δ ABC) 144 2 9, 39., =, =, ar (Δ PQR) 96 , 4, ∴ ar (ΔABC) : ar (ΔPQR) = 9 : 4, 40. In ΔPQR, ∠1 = ∠2, 82, , Mathematics-X
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∴ ∠PQR = ∠PST, (A.I.A), But ∠PST = ∠PRQ, So, ∠PQR = ∠PRQ, ∴ PQ = PR, So, ΔPQR is an isosceles triangle., 51. In ΔOAB,, , OD OE, =, .... (1), DA, EB, , OE, OF, =, .... (2), EB, FC, From (1) and (2), we get, , In ΔOBC,, , OD, OF, =, DA, FC, By converse of BPT, DF || AC., 52. ΔAPB ~ ΔDPC, , 53., , 54., 55., , 56., , ( BPT), ( BPT), , (AA Similarity criterion), , AP, PB, =, ( C.P.S.T.), DP, PC, AP.PC = DP.PB, Let sides of right angled triangle other than hypotenuse be x cm and (x + 5) cm., By Pythagoras theorem,, (x)2 + (x + 5)2 = (25)2, x = 15 or – 20, But side is always positive, So, x = 15., ∴ Length of two sides is 15 cm and 20 cm., Same as Q.31., In right angled ΔABC, AC2 = AB2 + BC2, ...(1), 2, 2, 2, 2, Given, AD = (AB + BC ) + CD, ⇒ AD2 = AC2 + CD2, [From (1)], By converse of Pythagoras theorem, ∠ACD = 90°., ΔADE ~ ΔABC, , ar (ΔADE) DE 2, =, ar (ΔABC) BC , , Mathematics-X, , 87
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PRACTICE-TEST, Triangles, Time : 1 Hrs., , M.M. : 20, SECTION - A, , 1., , If sides of two similar triangles are in the ratio of 8:10, then areas of these triangles, are in the ratio __________ ., 1, , 2., , If in two triangles ΔABC and ΔPQR,, , 3., , 4., , AB BC CA, =, =, , then, PQ, RP, QR, , 1, , (A) ΔPQR ~ ΔCAB, (B) ΔPQR ~ ΔABC, (C) ΔCBA ~ ΔPQR, (D) ΔBCA ~ ΔPQR, ΔABC is an isosceles right triangle, right angled at C, then AB2 = ........... ., (A) AC2, (B) 2 AC2, (C) 4 AC2, (D) 3 AC2, 1, A line DE is drawn parallel to base BC of ΔABC, meeting AB in D and AC at E., If, , AB, = 4 and CE = 2 cm, find the length of AE., BD, , SECTION B, 5., 6., 7., , The length of the diagonal of a rhombus field are 32 m and 24 m. Find the length, of the side of the field., 2, A man goes 24 m towards West and then 10 m towards North. How far is he, from the starting point?, 2, Using converse of Basic Proportionality Theorem, prove that the line joining, the mid-points of any two sides of a triangle is parallel to the third side., 2, , SECTION C, 8., 9., , E is a point on the side AD produced of a parallelogram ABCD and BE intersect, CD at F. Show that ΔABE ~ ΔDCB., 3, In an equilateral triangle, prove that three times the square of one side is equal, to four times the square of one of its altitude., 3, , SECTION D, 10. State and prove Basic Proportionality Theorem., 90, , 4, Mathematics-X
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CHAPTER, , 7, , Co-ordinate Geometry, , Key Points, 1. The system of geometry where the position of points on the plane is described using, an ordered pair of numbers., , 2., , Distance Formula, Finding distance betwen tow given points :, , A(x1, y1), , B(x2, y2), , AB (Distance between A and B) = ( x2 x1 )2 ( y2 y1 ) 2, Mathematics-X, , 91
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=, , 1, [x (y – y ) + x2 (y3 – y1) + x3 (y1 – y2)] sq. units, 2 1 2 3, A (x1, y1), , C (x3, y3), , B (x2, y2), , If area of a triangle is zero then points are collinear., VERY SHORT ANSWER TYPE QUESTIONS, , Fill in the blanks :, 1. The distance of a point from the y-axis is called its x-coordinate or ______ ., 2. The distance of a point from the x-axis is called its __________ or ordinate., 3. The point (5, 0) lies on ____ axis., 4. A point which lies on y-axis are of the form _______ ., 5. A linear equation of the form ax + by + c = 0 when represented graphically gives a, _______ ., 6. The distance of a point P(x, y) from the origin is ________, Multiple Choice Question :, 7. P is a point on x-axis at a distance of 3 unit from y-axis to its left. The co-ordinates, of P are :, (a) (3, 0), (b) (0, 3), (c) (– 3, 0), (d) (0, – 3), 8. The distance of P(3, – 2) from y-axis is, (a) 3 units, (b) 2 units, (c) – 2 units, , (d), , 13 units, 9. The co-ordinates of two ponts are (6, 0) and (0, – 8). The co-ordinates of the mid, points are, (a) 3, 4, (b) 3, – 4, (c) 0, 0, (d) – 4, 3, 10. If the distance between P(4, 0) and Q(0, x) is 5 units, the value of x will be, Mathematics-X, , 93
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(a) 2, (c) 4, , (b) 3, (d) 5, , x y, + = 7 intersects y-axis are, a b, (a) a, 0, (b) 0, b, (c) 0, 7b, (d) 2a, 0, 12. The area of triangle OAB, the co-ordinates of whose vertices are A(4, 0), B(0, – 7), and O origin, is :, (a) 11 sq. units, (b) 18 sq. units, (c) 28 sq. units, (d) 14 sq. units, , 11. The co-ordinates of the point where line, , 11 , 2 , 13. The distance between the points P − , 5 and Q − , 5 is, 3, 3, (a) 6 units, (b) 4 units, (c) 3 units, (d) 2 units, 14. The distance between the points (5 cos 35°, 0) and (0, 5 cos 55°) is, (a) 10 units, (b) 5 units, (c) 1 unit, (d) 2 units, 15. The co-ordinates of vertex A of ΔABC are (– 4, 2) and a point D which is mid point, of BC are (2, 5). The coordinates of centroid of ΔABC are, (a) (0, 4), , 7, , (b) −1, , 2, , 7, , (c) −2, , 3, , (d) (0, 2), , 16. The distance between the line 2x + 4 = 0 and x – 5 = 0 is, (a) 9 units, (b) 1 unit, (c) 5 units, (d) 7 units, 17. The perimeter of triangle formed by the points (0, 0), (2, 0) and (0, 2) is, (a) 4 units, (b) 6 units, (c) 6 2 units, (d) 4 + 2 2 units, 18. If the centroid of the triangle formed by (9, a), (b, – 4) and (7, 8) is (6, 8), then the, value a and b are :, , 94, , Mathematics-X
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(a) a = 4, b = 5, (b) a = 5, b = 4, (c) a = 5, b = 2, (d) a = 3, b = 2, State True or False, 19. The point P(– 4, 2) lies on the line segment joining the points A(– 4, 6) and, B(– 4, –6), 20. The points (0, 5), (0, – 9) and (3, 6) are collinear., 21. For what value of P, the points (2, 1), (p, – 1) and (– 1, 3) are collinear., 22. Find the area of ΔPQR, whose vertices are P(– 5, 7), Q (– 4, – 5) and R (4, 5)., 23. Find the point of trisection of the linear segment joining the points (1, – 2) and, (– 3, 4)., 24. The midpoints of the sides of a triangle are (3, 4), (4, 1) and (2, 0). Find the, vertices of the triangle., 25. Find the value of x if the points A(4, 3) and B(x, 5) lie on a circle whose centre, is O(2, 3)., 26. Find the ratio in which x-axis divides the line segment joining the points (6, 4), a, n, d, (1, – 7)., 27. Show that the points (– 2, 3), (8, 3) and (6, 7) are the vertices of a right angle, triangle., 28. Find the point on the y-axis which is equidistant from the points (5, – 2) and (–, 3, 2)., 29. Find the ratio in which y-axis divides the line segment joining the points A(5, –, 6) and B(– 1, – 4)., 30. Find the co-ordinates of a centroid of a triangle whose vertices are (3, – 5), (–, 7, 4) and (10, – 2)., 31. Find the relation between x and y such that the points (x, y) is equidistant from, the points (7, 1) and (3, 5)., 32. Find the ratio in which the line segment joining the points (1, – 3) and (4, 5) is, divided by x-axis. Also find the co-ordinates of this point on x-axis., 33. What is the value of a if the points (3, 5) and (7, 1) are equidistant from the point, (a, 0) ?, 34. Find a relation between x and y if the prints A(x, y), B(– 4, 6) and C(– 2, 3) are, collinear., 35. Find the area of a triangle whose vertices are given as (1, – 1), (– 4, 6) and (– 3,, – 5)., , Mathematics-X, , 95
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36. Name the type of triangle formed by the points A(– 5, 6), B(– 4, – 2) and C(7, 5)., (NCERT Exempler), 37. Find the points on the x-axis which are at a distance of 2 5 from the point (7, –, 4). How many such points are there?, (NCERT Exempler), 38. What type of quadrilateral do the points A(2, – 2), B(7, 3), C(11, – 1) and D(6, –, 6), taken in that order, form ?, 39. Find the co-ordinates of the point Q on the x-axis which lies on the perpendicular, bisector of the line-segment joining the points A(– 5, – 2) and B(4, – 2). Name, the type of triangle formed by the points Q, A and B., 40. Let P and Q be the points of trisection of the line segment joining the points A(2,, – 2) and B(– 7, 4) such that P is nearer to A. Find the co-ordinates of P and Q., SHORT ANSWER TYPE QUESTIONS-II, , 41. The line segment joining the points A(2, 1) and B(5, – 8) is trisected at the point, P and Q such that P is nearer to A. If P also lies on the line given by 2x – y + k =, 0, find the value of k., 42. Find the ratio in which the line x – 3y = 0 divides the line segment joining the, points (– 2, – 5) and (6, 3). Find the co-ordinates of the point of intersection., HOTS, 43. Point A lies on the line segment XY joining X(6, – 6) and Y(– 4, – 1) in such a, XA, 2, = . If point A also lies on the line 3x + k(y + 1) = 0, find the, XY, 5, value of k., HOTS, 44. Find the area of the triangle formed by joining the mid points of the sides of the, triangle ABC, whose vertices are A(0, – 1), B(2, 1) and C(0, 3)., 45. Find the value of k so that the area of triangle ABC with A(k + 1, 1), B(4, – 3), and C(7, – k) is 6 square units., 46. Point P divides the line segment joining the points A(2, 1) and B(5, – 8) such that, , way that, , 96, , Mathematics-X
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AP 1, = . If P lies on the line 2x – y + k = 0. Find the value of k., PB 3, 47. A point P on the x-axis divides the line segment joining the points (4, 5) and (1, – 3), in certain ratio. Find the co-ordinates of point P., , 48. In right angled ΔABC, ∠B = 90° and AB = 34 units. The co-ordinates of points, B, C are (4, 2) and (– 1, y) respectively. If ar ΔABC = 17 sq. units, then find the, value of y., 49. If A(– 3, 2), B(x, y) and C(1, 4) are the vertices of an isosceles triangle with AB =, BC. Find the value of (2x + y)., 50. If the point P(3, 4) is equidistant from the points A(a + b, b – a) and B(a – b, a + b), then prove that 3b – 4a = 0., LONG ANSWER TYPE QUESTIONS-III, , 51. If A(–5, 7), B(– 4, – 5), C(– 1, – 6) and D(4, 5) are vertices of a quadrilateral, ABCD. Find the area of quadrilateral ABCD., 52. If P(x, y) is any point on the line joining A(a, 0) and B(0, b) then show that, x y, + = 1., a b, 53. If the points (x, y), (– 5, – 2) and (3, – 5) are collinear, prove that 3x + 8y + 31 =, 0., 54. Find the relation between x and y if A(x, y), B(– 2, 3) and C(2, 1) form an, isosceles triangle with AB = AC., , (, , ), , 55. Prove that the point x, 1 − x 2 is at a distance of 1 unit from the origin., 56. If R(x, y) is point on the line segment joining the points A(a, b) and B(b, a), then, prove that x + y = a + b., 57. If the points (a, b), (c, d) and (a – c, b – d) are collinear show that bc = ad., 58. Find the co-ordinates of the circumcenter of the triangle whose vertices are (3,, 7), (0, 6) and (– 1, 5). Find the circumradius., (HOTS), 59. In a triangle PQR, the co-ordinates of points P, Q and R are (3, 2), (6, 4) and (9,, 3) respectively. Find the co-ordinates of centroid G. Also find the areas of ΔPQG, and ΔPRG., 60. If the points (5, 4) and (x, y) are equidistant from the point (4, 5), prove that, x2 + y2 – 8x – 10y + 39 = 0., , Mathematics-X, , 97
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ANSWERS AND HINTS, VERY SHORT ANSWER TYPE QUESTIONS-I, , 1., 3., , abscissa, x-axis, , 2. y-coordinate, 4. (0, y), , 5., , straight line, , 6., , 7., 9., 11., 13., 15., , (iii), (ii), (iii), (c), (a), , 17. (d), , (– 3, 0), (3, – 4), (0, 7b), 3 units, (0, 4), , 8., 10., 12., 14., 16., , ( 4 + 2 2 ) units, , 18. (d) a = 20, b = 2, , 19. False, 21. P = 3, , A, (1,–2), , (i) 3 units, (ii) 3, (iv) 14 sq. units, (b) 5 units, (d) 7 units, , 20. False, 22. 25 sq. units, 1:, , 23., , x2 + y 2, , 1:, P, , 1, Q, , A, (–3, 4), , AP : PB = 1 : 2, AQ : QB = 2 : 1, 1 , P = − , 0, 3, 5 , Q = − , 2, 3, 24., , 98, , Mathematics-X
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39. Use distance formula and midpoint formula., 1 , Q − , 0, 2 , Δ is isosceles., 40. P(– 1, 0), Q (– 4, 2), 41. P(3, – 2), Put value of x = 3, y = – 2 is equation, then k = – 8., 42. Let P(x, y) be the point and m : n is the ratio, then x =, , 6n − 2m, ,, m+n, , y=, , 3n − 5m, m+n, , From equation of line x = 3y ⇒, By putting x = 3y or, , ...(1), , x, =3, y, , x, = 3 is (1), y, , m : n = 3 : 13, 9 3, Then P(x, y) = , , 2 2, XA 2, = ., AY 3, Let A(x , y) is the point., x = 2, y = – 4, A (2, – 4), Put x = 2 and y = – 4 in equation., ∴K=2, 44. 1 sq. unit, 45. K = 3, , 43. Find, , −17, 4, 47. m : n = 5 : 3, , 46. K =, , 100, , Mathematics-X
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17 , P , 0, 8 , 48., 49., 50., 51., 52., 53., 54., 55., 56., 58., , y = –1, y = 5, 2x + y = 1, 3b – 4a = 0 proved by using distance formula., Area of quadrilateral ABCD = Area of ΔABC + Area of ΔADC, Ar (ABCD) = 72 sq. units., Prove by section formula., Prove by area of Δ = 0 if points are collinear., Prove by distance formula., Prove by distance formula., Prove by using area of triangle = 0 if points are collinear., Find co-ordinates of mid points of AB, BC, CA, then DO = OE = OF, 13 , then (circumcentre) O(x, y) = 1, , 2, circumradius AO =, , ar Δ PQG =, , F, , D, , 17, ., 2, , 59. G(x, y) = (6, 3), , A, , O, B, , E, , C, , 3, sq. units, 2, , 3, sq. units, 2, 60. Use distance formula, , ar Δ PRG =, , Mathematics-X, , 101
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PRACTICE-TEST, Coordinate Geometry, Time : 1 Hr., , M.M. : 20, SECTION - A, , 1., , 1 15 , Find the value of m in which the points (3, 5), (m, 6) and , are collinear. 1, 2 2, , 2., 3., 4., , What is the distance between the points A(c, 0) and B(0, –c), 1, The distance of point P(– 6, 8) from the origin is _______ ., 1, Find the value of ‘a’ so that the point (3, a) lies on the line segment 2x – 3y = 5. 1, , SECTION B, 5., 6., 7., , For what value of p, the points (– 3, 9), (2, p) and (4, – 5) are collinear., 2, If the points A(8, 6) and B(x, 10) lie on the circle whose centre is (4, 6) then find the, value of x., 2, Find the perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0)., 2, , SECTION C, 8., 9., , Show that the points A(–3, 2), B(– 5, –5), C(2, –3) and D(4, 4) are the vertices of, a rhombus., 3, Find the ratio in which the point (2, y) divides the line segment joining the points A(–, 2, 2) and B(3, 7). Also find the value of y., 3, , SECTION D, 10. If the point P divides the line segment joining the points A(–2, –2) and B(2, – 4) such, that, , 102, , AP 3, = , then fidn the coordinate of P., AB 7, , 4, , Mathematics-X
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CHAPTER, , Introduction to, Trigonometry, , 8, KEY POINTS, , A, , • A branch of mathematics which deals, with the problems related to right angled, H (hypotenuse), triangles. It is the study of relationship (Perpendicular) P, between the sides and angles of a right, angled triangel., C, B, B, Note : For ∠A — Perpendicular is BC, (base), base is AB., For ∠C, Perpendicualr is AB Base is BC., Trigonometric Rations of an acute angle in a right angled triangle express the, relationship between the angle and the length of its sides., , Tangent, P, B, , Sine, P, H, , Cosine, B, H, , Trigonometric, Ratios, Co-tangent, B, P, , Secant, H, B, Cosecant, H, P, , Mind Trick: To learn the relationship of sine, cosine and tangent follow this, sentences., Some People Have Curly Brown Hair Through Proper Brushing, sin A =, , P, H, , Mathematics-X, , cos A =, , B, H, , tan A =, , P, B, 103
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5., , Trigonometric ratios of some specific angles, ∠A, , 0°, , 30°, , 45°, , 60°, , 90°, , sin A, , 0, , 1, 2, , 1, 2, , 3, 2, , 1, , cos A, , 1, , 3, 2, , 1, 2, , 1, 2, , 0, , tan A, , 0, , 1, 3, , 1, , 3, , Not defined, , cot A, , Not defined, , 3, , 1, , 1, 3, , 0, , sec A, , 1, , 2, 3, , 2, , 2, , Not defined, , Not defined, , 2, , 2, , 2, 3, , 1, , cosec A, 6., , Trigonometric ratios of complimentary angles, sin (90° – θ), =, cos θ, cos (90° – θ), =, Sin θ, tan (90° – θ), =, cot θ, cot (90° – θ), =, tan θ, sec (90° – θ), =, cosec θ, cosec (90° – θ) =, sec θ, VERY SHORT ANSWER TYPE QUESTIONS, , 1., 2., 3., , If Sin θ = cos θ, find the value of θ, If tan θ = cot (30° + θ), find the value of θ, If Sin θ = cos (θ – 6°), find the value of θ, , 4., , If cos A =, , 7, , find the value of tan A + cot A, 25, , 5., , If tan θ =, , 4, sin θ +cos θ, then find the value of, 3, sin θ – cos θ, , Mathematics-X, , 105
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22. If A and B are complementary angles, then, (a) sin A = sin B(b) cos A = cosB(c) tan A = tan B, (d) sec A = cosec B, 23. In Fig. if AD = 4 cm, BD = 3 cm and CB = 12 cm. then cot θ =, (a), , 12, 5, , (b), , 5, 12, , (c), , 13, 12, , (d), , 12, 13, , A, , θ, , D, , 24. The value of tan 1°, tan 2°, tan 3° ______ tan 89° is. C, B, (a) 1, (b) – 1, (c) 0, (d) None of these, 25. If θ and 2θ – 45° are acute angles such that sin θ = cos (2θ – 45°) then tan θ is, (a) 1, , (b) – 1, , (c), , (d), , 3, , 1, 3, , SHORT ANSWER TYPE (I) QUESTIONS, , Prove that :, 26. sec4 θ – sec2 θ = tan4 θ + tan2 θ, 27., , 1 + sin θ, = tan θ + Sec θ, 1 – sin θ, , 28. If x = p sec θ + q tan θ & y = p tan θ + q sec θ then prove that x2 – y2 = p2 – q2, 29. If 7 sin2 θ + 3 cos2 θ = 4 then show that tan θ =, 30. If Sin (A – B) =, , 1, 3, , 1, 1, , cos (A + B) = then find the value of A and B., 2, 2, , cos 2 20° + cos 2 70°, 31. Find the value of, ., sin 2 59° + sin 2 31°, , 32. Prove that : tan 1° tan 11° tan 21° tan 69° tan 79° tan 89° = 1, 33. If sec 4 A = cosec (A – 20°) then find the value of A., , Mathematics-X, , 107
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5., , 7, , 7., , a, , 9., , 1, 5, , 1, 3, 8. 0, , 6., , 10. 9, , 1 + Cos2 θ, Cot θ, , 11., , 12. 0°, , 13., , 50°, , 14. tan2 θ, , 15., , tan θ, , 16., , 17., 19., 21., 23., 25., 27., 29., 31., , 2, –6, (c), (a), (a), —, —, 1, , 18., 20., 22., 24., 26., 28., 30., 32., , 33., , 22°, , 34., , 35., , 20°, , 36. Hint : A + B + C = 180°, , 37., , AC = 10 cm, BC = 5 3 cm, , 38. 30°, , 40., , 60°, , 49., , 51., , –1, , 53. 2, , 56., , 20 + 9 3, 4+3 3, , 60., , 2+2 3, 2, , 71., , 2, , 72., , 2, 3, , Mathematics-X, , 1, 2, 0, sec 42° + cot 2°, (d), (a), —, —, A = 45°, B = 15°, —, 17, 8, , 3, , 111
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PRACTICE-TEST, Introduction to Trigonometry, Time : 1 Hrs., , M.M.: 20, SECTION-A, 4, what is the value of cos θ., 5, , 1, , 1., , If Sin θ =, , 2., 3., , Write the value of Sin (45° + θ) – Cos (45° – θ)., If cos 9α = sin α and 9α < 90°, then the value of tan 5α is, , 1, 1, , 1, (b) 3, (c) 1, (d) 0, 3, If sin A + sin2 A = 1, then the value of (cos2 A + cos4 A) is, , 1, , (a), 4., , (a) 1, , (b), , 1, 2, , (c) 2, , (d) 3, , SECTION-B, 5 Sin θ – 3Cos θ, 5 Sin θ + 2Cos θ, , 5., , If 5 tan θ = 4 then find the value of, , 6., , Find the value of tan 35° tan 40° tan 45° tan 50° tan 55°, , 2, , 7., , Prove that (sin α + cos α) (tan α + cot α) = sec α + cosec α, , 2, , 2, , SECTION-C, 8., , Prove that, , Sin θ, 1 + Cos θ, +, = 2 Cosec θ, 1 + Cos θ, Sin θ, , 3, , 9., , Cos A, Sin 2 A, –, = Sin A+ Cos A, Prove that, 1 – tan A Cos A – Sin A, , 3, , SECTION-D, 10. Prove that, , Cos θ, tan θ + Sec θ – 1, =, ., tan θ – Sec θ + 1 1 – Sin θ, , 4, ppp, , 112, , Mathematics-X
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CHAPTER, , Some Applications, of Trigonometry, , 9, KEY POINTS, , •, , •, , •, , Applications of trigonometry involve finding, heights of the objects and distance between, them. Without actual measurement., Angle of Elevation: Let AB be an object, standing vertically on a plane CB. C is the C, observer looking upto to A (the top of AB)., AC is called the line of sight and ∠ACB is, angle is elevations., , A, si, e of, n, i, L, , ght, , angle of elevation, Observer, , Object, B, , Distance between object, and observer, , Angle of Depression : Let A is the observer looking at C (the object) from a height, BC. AC is line of sight and ∠BAC is angle of depression., A (Observer), , B, ght, f si, o, e, Lin, , C, , •, , If the observer moves towards the object the angle of elevation increases and if the, observer moves away from the object, the angle of depression decreases., , •, , Numerically, angle of elevation is equal to angle of depression (both are measured), with the same horizontal parallel planes)., VERY SHORT ANSWER TYPE QUESTIONS, , 1., , The length of the shadow of a tower on the plane ground is 3 times the height of, the tower. The angle of elevation of sun is :, (CBSE 2017), (a) 45°, , Mathematics-X, , (b) 30°, , (c) 60°, , (d) 90°, 113
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2., , The tops of the poles of height 16 m and 10 m are connected by a wire of length l, metres. If the wire makes an nagle of 30° with the horizontal, the l =, (a) 26 m, , 3., , 4., , 5., , 6., , 7., 8., , (b) 16 m, , (c) 12 cm, , (d) 10 m, , A pole of height 6 m casts a shadow 2 3 m long on the ground. the angle of, elevation of the sun is, (CBSE 2017), (a) 30°, (b) 60°, (c) 45°, (d) 90°, A ladder leaning aginast a wall makes an angle of 60° with the horizontal. If the, foot of the ladder is 2.5 m away from the wall, then the length of the ladder is —, (CBSE 2016), (a) 3 m, (b) 4 m, (c) 5 m, (d) 6 m, If a tower is 30 m hight, costs a shadow 10 3 m long on the ground, then the, angle of elevation of the sun is:, (CBSE, 2017), (a) 30°, (b) 45°, (c) 60°, (d) 90°, A tower is 50 m high. When the sun’s altitude is 45° then what will be the length, of its shadow?, 50, m. find the sun’s altitude., 3, Find the angle of elevation of a point which is at a distance of 30 m from the, The length of shadow of a pole 50 m high is, , base of a tower 10 3 m high., 9., , A kite is flying at a height of 50 3 m from the horizontal. It is attached with a, string and makes an angle 60° with the horizontal. Find the length of the string., , 10. In the given figure find the perimeter of rectangle ABCD., D, , C, , 10 m, 30°, A, , 11. The length of the shadow of a pillar is, , B, , 3 times its height. Find the angle of, , elevation of the source of light., 114, , Mathematics-X
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12. In the figure, find the value of DC., A, 10 m, D, 45°, , C, , B, , SHORT ANSWER TYPE QUESTIONS, , 13. In the figure, find the value of BC., A, , D, 100 m, 80 m, 45°, , 60°, , B, , C, , E, , 14. In the figure, two persons are standing at the opposite direction P & Q of the, tower. If the height of the tower is 60 m then find the distance between the two, persons., A, , 60 m, 30°, , 45°, , P, , Q, , B, , 15. In the figure, find the value of AB., A, , B, 60°, D, , Mathematics-X, , 45°, 1000 m, , C, , 115
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16. In the figure, find the value of CF., C, , B, , 45°, , D, , 5m, A, , 20 m, , F, , 17. If the horizontal distance of the boat from the bridge is 25 m and the height of the, bridge is 25 m, then find the angle of depression of the boat from the bridge., 18. State True/False., If the length of the shadow of a tower is increasing, then the angle of elevation, of the sun is also increasing., 19. If a man standing on the deck of a ship 3 m above the surface of sea observes a cloud, and its reflection in the sea, then the angle of elevation of the cloud is equal to the, angle of depression of its reflection., 20. The angle of elevation of the top of the tower is 30°. If the height of the tower is, doubled, then the angle of elevation of its will also bed doubled., 21. From the top of a hill, the angles of depression of two consecutive stones due east, are found to be at 30° and 45°. Find the height of the hill., 22. The string of a kite is 150 m long and it makes an angle 60° with the horizontal. Find, the height of the kite above the ground. (Assume string to be tight), 23. The shadow of a vertical tower on level ground increases by 10 m when the altitude, of the sun changes from 45° to 30°. Find the height of the tower., 24. An aeroplane at an altitude of 200 m observes angles of depression of opposite, points on the two banks of the river to be 45° and 60°, find the width of the, river., 25. The angle of elevation of a tower at a point is 45°. After going 40 m towards the, foot of the tower, the angle of elevation of the tower becomes 60°. Find the, height of the tower., 116, , Mathematics-X
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26. The upper part of a tree broken over by the wind makes an angle of 30° with the, ground and the distance of the root from the point where the top touches the, ground is 25 m. What was the total height of the tree?, 27. A vertical flagstaff stands on a horizontal plane. From a point 100 m from its, foot, the angle of elevation of its top is found to be 45°. Find the height of the, flagstaff., 28. The length of a string between kite and a point on the ground is 90 m. If the, , 29., , 30., , 31., , 32., , 3, string makes an angle with the level ground and sin α = . . Find the height of, 5, the kite. There is no slack in the string., An aeroplane, when 3000 m high, passes vertically above another plane at an, instant when the angle of elevation of two aeroplanes from the same point on, the ground are 60° and 45° respectively. Find the vertical distance between the, two planes., A 7 m long flagstaff is fixed on the top of a tower on the horizontal plane. From, a point on the ground, the angle of elevation of the top and the bottom of the, flagstaff are 45° and 30° respectively. Find the height of the tower., From the top of a 7 m high building, the angle of elevation of the top of the, tower is 60° and the angle of depression of the foot of the tower is 30°. Find the, height of the tower., Anand is watching a circus artist climbing a 20m long rope which is tightly, stretched and tied from the top of vertical pole to the ground. Find the height of, the pole if the angle made by the rope with the ground level is 30°., LONG ANSWER TYPE QUESTIONS, , 33. The angle of elevation of a cloud from a point 60 metres above a lake is 30° and the, angle of depression of its reflection of the cloud in the lake is 60°. Find the height of, the cloud., 34. A man standing on the deck of a ship, 10 m above the water level observes the angle, of elevation of the top of a hill as 60° and angle of depression the bottom of a hill as, 30°. Find the distance of the hill from the ship and height of the hill., 35. From a window 60 m high above the ground of a house in a street, the angle of, elevation and depression of the top and the foot of another house on the opposite, side of the street are 60° and 45° respectively. Show that the height of opposite, , Mathematics-X, , 117
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house is 60(1 + 3) metres., 36. The angle of elevation of an aeroplane from a point A on the ground is 60°. After, a flight of 30 seconds, the angle of elevation changes to 30°. If the plane is flying, at a constant height of 3600 3 m, find the speed in km/hour of the plane., 37. A bird is sitting on the top of a tree, which is 80 m high. The angle of elevation, of the bird, from a point on the ground is 45°. The bird flies away from the point, of observation horizontally and remains at a constant height. After 2 seconds,, the angle of elevation of the bird from the point of observation becomes 30°., Find the speed of flying of the bird., 38. The angles of elevation of the top of a tower from two points on the ground at, distances 9 m and 4 m from the base of the tower are in the same straight line with it, are complementary. Find the height of the tower., 39. A boy standing on a horizontal plane finds a bird flying at a distance of 100 m from, him at an elevation of 30°. A girl, standing on the roof of 20 m high building, finds the, angle of elevation of the same bird to be 45°. Both the boy and girl are on the, opposite sides of the bird. Find the distance of bird from the girl., 40. An observer from the top of a light house, 100 m high above sea level, observes, the angle of depression of a ship, sailing directly towards him, changes from, 30° to 60°. Determine the distance travelled by the ship during the period of, observation., 41. The angles of elevation and depression of the top and bottom of a light house from, the top of a building 60 m high are 30° and 60° respectively. Find, (i) The difference between the height of the light house and the building., (ii) distance between the light house and the building., 42. A fire in a building ‘B’ is reported on telephone in two fire stations P an Q, 20 km, apart from each other on a straight road. P observes that the fire is at an, angle of 60°, to the road, and Q observes, that it is at an angle of 45° to the road. Which station, should send its team to start the work at the earliest and how much distance will this, team has to travel?, 43. A 1.2m tall girl spots a balloon on the eve of Independence Day, moving with the, wind in a horizontal live at a height of 88.2 m from the ground. The angle of elevation, of the balloon from the of the girl at an instant is 60°. After some time, the angle of, elevation reduces to 30°. Find the distance travelled by the balloon., , 118, , Mathematics-X
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44. The angle of elevation of the cloud from a point 60 m above take is 30° and the, angle of depression of the reflection of the cloud in the take is 60°. Find the, height of the cloud., (CBSE, 2011 C), 45. The pillars of equal heights stand on either side of a roadway 150 m wide From, a joinj on the roadway between the pillars, the angles of elevation of the top of, the pillars are 60° and 30°. Find the height of pillars and the position of the, point., (CBSE, 2011), 46. The angle of elevation of the top of tower from certain point is 30°. If the, observer moves 20 m towards the tower the angle of elevation of the top increases, by 15°. Find the height of the tower., 47. A moving boat is observed from the top of a 150 m high diff moving away form, the cliff. The angle of depression of the boat changes form 60° to 45° in 2, minutes. Find the speed of the boat in m/w., (CBSE 2017), 48. From the top of a 120 m hight tower a man observes two cars on the opposite, sides of the tower and in straight line with the base of tower with angles of, depression as 60° and 45°. Find the distance between the cars., (CBSE, 2017), 49. From the top of a tower h metre high, the angles of depression of two objects,, which are in the line with the foot of the tower are α & β (β > α). Find the, distance between the two objects., (NCERT, Exampler), 50. A window of a house is h metres above the ground. From the window the angles, of elevation and depression of the top and bottom of another house situated on, the opposite side of the lane are found to be α & β respectively. Prove that the, height of the house is h (1 + tan α × tan β) metres., (NCERT Exampler), ANSWERS AND HINTS, , 1. (b), , 2. (c), , 3. (b), , 4. (c), , 5. (c), , 6. 50 m, , 7. 60°, , 8. 30°, , Mathematics-X, , 119
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9. 100 m, , 10. 20, , (, , 11. 30°, , 12. 60 m, , 13. 130 m, , 14. 60, , 15. 1000, , (, , ), , 3 –1 m, , (, , ), , 3 +1 m, , ), , 3 +1 m, , 16. 25 m, , 17. 45, , 18. False, , 19. False, , 20. False, , 21. 1.37 km., , 22. 75 3 m, , 23. 13.65 m, , 24. 315.8 m, , 25. 94.8 m, , 26. 43.3 m, , 27. 100 m, , 28. 120 m, , 29. 1268 m, , 30. 9.6 m, , 31. 28 m, , 32. 10 m, , 33. 120 m, , 34. 40 m, 17.32 m, , 36. 864 km/hour, , 37. 29.28 m, , 38. 6 m, , 39. 30 2 m, , 40. 115.5 m, , 41. 20 m, 34.64 m, , 42. Station P, 14.64 km, , 43. 58 3m, , 44. 120 m, 45. height = 64.95 m, distance (Position) = 112.5 m, 46. 10( 3 + 1) m, , 47. 1902 m/h, , 48. 189.28 m, , 49. h(cot α – cot β) m, , 120, , Mathematics-X
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PRACTICE-TEST, Heights and Distances, , Time : 1 Hr., , M.M.: 20, SECTION-A, , 1., 2., 3., , A pole which is 6 m high cast a shadow 2 3 on the ground. What is the sun’s, angle of elevation., 1, The height of a tower is 100 m. When the angle of elevation of sun is 30°, then, what is the shadow of tower?, 1, The angle of elevation of the sun, when the shadow of a pole h meters high is, 3 h is., , 4., , (a) 30°, (b) 45°, (c) 60°, (d) 90°, An observer 1.5 metre tall is 20.5 metre away from a tower 22 metres high. The, angle of elevation of the top of the tower from the eye of the observer is,, (a) 30°, , (b) 45°, , (c) 60°, SECTION-B, , (d) 0°, , 1, , 5., , From a point on the ground 20 m away from the foot of a tower the angle of, elevation is 60°. What is the height of tower?, 2, , 6., , The ratio of height and shadow of a tower is 1:, , 7., , 1, .What is the angle of elevation, 3, of the sun?, 2, The angle of elevation of the top of a tower is 30°. If the height of the tower is tripled,, then prone that the angle of elevation would be doubled., 2, SECTION-C, , 8., , 9., , The tops of the two towers of heigth x and y standing on level ground, subtend angles, of 30° and 60° respectively at the centre of the line joining their feet, then find x : y., 3, The angle of elevation of the top of a rock from the top and foot of a 100 m high, tower are 30° and 45° respectively. Find the height of the rock., 3, SECTION-D, , 10, , A man standing on the deck of a ship, 10 m above the water level observes the, angle of elevation of the top of a hill as 60° and angle of depression of the base, of the hill as 30°. Find the distance of the hill from the ship and height of the, hill., 4, , Mathematics-X, 121
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CHAPTER, , 10, , Circles, Mind–Maping, Cirele, , Tangent to a circle, , De?nition, , Property of Tangent to a circle, , Theorem, and its, Application, , Number of Tangents, from a point on a circle, , Theorem and its, Application, , Length of the, Tangent, , KEY POINTS, , 1., , A circle is a collection of all those points in a plane which are at a constant, distance from a fixed point. The fixed point is called the centre and fixed distance, is called the radius., , 2., , Secant: A line which intesects a circle in two distinct points is called a secant, of the circle., , P, , 3., , Q, , Tangent: It is a line that intersects the circle at only one point. The point where, tangent touches the circle is called the point of contact., Here A is the poin of contact., , 122, , Mathematics-X
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P, , A, , B, , 4., , Number of Tangent: Infinitely many tangents can be drawn on a circle., , 5., , Number of Secant: There are infinitely many secants which can be drawn on a, circle., , 6., , The proofs of the following theorems can be asked in the examination:–, (i) The tangent at any point of a circle is perpendicular to the radius through, the point of contact., (ii) The lengths of tangents drawn from an external point to a circle are equal., , 7., , There is only one tangent at a point of the circle., , 8., , The tangent to a circle is a special case of the secant., , 9., , There is no tangent to a circle passing through a point lying inside the circle., , 10. There is one and only one tangent to a circle passing through a point lying on the, circle., 11. There are exactly two tangents to a circle through a point lying outside the, circles., VERY SHORT ANSWER TYPE QUESTIONS, , 1., , In fig., ΔABC is circumscribing a circle. Find the length of BC., A, , 9 cm, , 3 cm, , M, , N, , 6 cm, , 4 cm, B, , Mathematics-X, , L, , C, , 123
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2., , The length of the tangent to a circle from a point P, which is 25 cm away from the, centre, is 24 cm. What is the radius of the circle., , 3., , In fig., ABCD is a cyclic quadrilatreral. If ∠BAC = 50° and ∠DBC = 60°, then, find ∠BCD., C, , D, , 60 °, 50°, , A, , 4., , B, , In figure, O is the centre of a circle, PQ is a chord and the tangent PR at P makes, an angles of 50° with PQ. Find ∠POQ., P, , R, 50°, , O, Q, , 5., , If two tangents inclined at an angle 60° are drawn to a circle of radius 3 cm, then, find the length of each tangent., , 6., , If radii of two concentric circles are 4 cm and 5 cm, then find the length of the, chord of one circle which is tangent to the other circle., , 7., , In the given figure, PQ is tangent to outer circle and PR is tangent to inner, circle. If PQ = 4cm, OQ = 3 cm and QR = 2 cm then find the length of PR., Q, , P, , O, R, , 124, , Mathematics-X
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8., , In the given figure, O is the centre of the circle, PA and PB are tangents to the circle, then find ∠AQB., (CBSE 2016), A, , o, , Q, , P, , 40°, , B, , 9., , In the given figure, If ∠AOB = 125° then find ∠COD., B, , A, 125°, O, , D, , C, , 10. If two tangent TP and TQ are drawn from an external point T such that, ∠TQP = 60° then find ∠OPQ., P, , T, , O, 60°, Q, , 11. How many tangents can a circle have?, , (NCERT), , 12. A tangent to a circle intersects it in _________ points., , (NCERT), , Q, , 13., , O, , P, , If PQ is a tangent then find the value of POQ + QPO ., Mathematics-X, , 125
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14. Choose the correct Answer., A tangent PQ at a point P of a circle of radius 5 cm meets a line through the, centre O at a point Q so that OQ = 12 cm. Length PQ is :, (a) 12 cm, , (b) 13 cm, , (c) 8.5 cm, , (d), , 119, , cm (NCERT), , 15. A circle can have _______ parallel tangents at the most., , (NCERT), , 16. The common point of a tangent to a circle and the circle is called ________ ., (NCERT), SHORT ANSWER TYPE-I QUESTIONS, , 17. If diameters of two concentric circle are d1 and d2 (d2 > d1) and c is the, length of chord of bigger circle which is tangent to the smaller circle. Show, that d22 = c2 + d12., 18. The length of tangent to a circle of radius 2.5 cm from an external point P is 6, cm. Find the distance of P from the nearest point of the circle., 19. TP and TQ are the tangents from the external point T of a circle with centre O. If, ∠OPQ = 30° then find the measure of ∠TQP., 20. In the given fig. AP = 4 cm, BQ = 6 cm and AC = 9 cm. Find the semi perimeter, of ΔABC., A, m, 4c, , R, , C, , P, , Q, , 6 cm, , B, , 21. A circle is drawn inside a right angle triangle whose sides are a, b, c where c is, , a+b–c, 2, (NCERT Exampler, 2012), , the hypotenuse, which touches all the sides of the triangle. Prove r =, where r is the radius of the circle., 126, , Mathematics-X
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22. Prove that the tangent at any point of a circle is perpendicular to the radius, through the point of contact., 23. Prove that in two concentric circles the chord of the larger circle which is, tangent to the smaller circle is bisected at the point of contact., 24. In the given Fig., AC is diameter of the circle with centre O and A is point of, contact, then find x., C, x, B, , O, 40°, , P, , Q, , A, , 25. In the given fig. KN, PA and PB are tangents to the circle. Prove that:, KN = AK + BN., A, , K, C, , O, , P, , N, B, , 26. In the given fig. PQ is a chord of length 6 cm and the radius of the circle is 6 cm., TP and TQ are two tangents drawn from an external point T. Find ∠PTQ., P, , T, , O, , Q, , Mathematics-X, , 127
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SHORT ANSWER TYPE-II QUESTIONS, , 27. In the given figure find AD, BE, CF where AB = 12 cm, BC = 8 cm and, AC = 10 cm., C, , E, , F, , A, , B, , D, , 28. Two tangents PA and PB are drawn to a circle with centre O from an external, point P. Prove that ∠APB = 2 ∠OAB, (NCERT, Exemplar-2), A, , O, P, , B, , 29. In the given fig. OP is equal to the diameter of the circle with centre O. Prove, that ΔABP is an equilateral triangle., A, , o, , P, , B, , 128, , Mathematics-X
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30. In the given fig., find PC. If AB = 13 cm, BC = 7 cm and AD = 15 cm., A, , R, B, O, , S, C, , 4 cm, , P, , Q, , D, , LONG ANSWER TYPE QUESTIONS, , 31. In the given fig. find the radius of the circle., m, , 32. In the given fig. PQ is tangent and PB is diameter. Find the value of x and y., P, y, o, , A, , x, , y, , Q, , 35°, B, , ANSWERS AND HINTS, , 1. Since length of both the tangents from a point outside the circle is equal, So, BN = BL, CM = CL, BL + CL = BC = 10 cm, Q, 24 m, , 2. P, , 25 m, , O, , By Pythagorous Rule, QR = 7 cm., Mathematics-X, , 129
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3. Angle is the same segment are euqal., • DC is the chord so DAC = 60°., • The sum of the opposite angles of a cyclic quadrilateral is 180°., So BCD = 70°, 4. The tangent at any point of a circle is perpendicular to the radius through the point, of contact., So,, , RPO, , = 90°, , OPQ = OQP = 40°, POQ = 100°, Q, 3 cm, 60°, , O, 3 cm, , 5., , R, , P, , ΔQPO ≅ ΔRPO, 60°, = 30°, 2, , ⇒, , QPO = RPO =, , In ΔQPO,, , OQP = 90°(Tangent is perpendicular at the point of contact)., OQ, , tan 30° = QP, , O, , 6., A, , ⇒, , QP = 3 3 cm, , B, , P, , In ΔAOP, right angled at P., OA2 = AP2 + OP2, , 130, , ⇒, , AP = 3, , In ΔPQO,, , AB = 6 cm, , ⇒ (5)2 = AP2 + 42, , ⇒, , AP2 = 9, , Mathematics-X
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(4)2 + (3)2 = (OP)2, , 7., , 5 = OP, (5)2 = (2)2 + (PR)2, , In PRO,, , PR =, , 21 cm, , A, 4, , 8., , Q, , 5, , O, , 3, , 40°, , 1, , P, , 2, B, , In Quadrilateral PROQ, 1 2 3 4 = 360°, 1 3 = 180°, 3 = 140°, , Now,, , 3 = 25, 5 = 70° or, , 9., , AQB = 70°, , 1 2, 3 4, 5 6 (CPCT) of their corresponding triangles., , 7 8, , °, , 2( 2 3 6 7 ) = 360°, or AOB COD = 186°, or COD = 55°, 10., , OQT = 90° (Angle between tangent & radius), PQO = 30° (90° – 60°), PQO = OPQ = 30°, , 11. Infinity many, 12. One, 13. 90° as OQP = 90° (Angle between tangent and radius of the circle), 14. D( 119 cm), Mathematics-X, , 131
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15. Two, 16. Point of Contact, d2, , d1, O, , 17., A, , P, , C, , B, , AO2 = OP2 + AP2, 2, , 2, d2 , d , = 1 + AP 2, 2, 2, 2, , 2, , d2 , d1 , − = AP2, 2, 2, , 1, 2, 2, (d 2 ) − (d1 ) , 4, 2, , = AP, , 1, 2, 2, (d 2 ) − (d1) = AB, 4, , (d 2 )2 − (d1)2 = C, 2, (d 2 )2 − (d1)2 = C, , d 22 = C2 – d 2, 1, O 2.5 m Q, , 2.5 m, , 18., , P, , 6m, , T, , (OP)2 = (OT)2 + (PT)2, (OP)2 = (2.5)2 + (6)2, = 42.25, (OP)2 = (6.5)2, QP = 4 cm, , 132, , Mathematics-X
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P, , 19., , O, , 30°, , T, , Q, , OQP = OPQ = 30°, , OQT = 90° (Angle between radius tangent), TQP = OQT − OQP, , = 90° – 30° = 60°, 20., , AP = AR = 4 cm, CR = CQ = (9 – 4) cm = 5 cm, =, , 1, [AC + AB + BC], 2, , =, , 1, [ 9 + 10 + 11] = 15 cm, 2, A, , c, b, , F, o, , 21., B, , or,, This gives,, , Mathematics-X, , a, , D, , r, , E, , C, , b – r = AF, a – r = BF, AB = C = AF + BF = b – r + a – r, r=, , a+b–c, 2, , 133
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O, , 22., , (Theorem 10.1, NCERT), Y, , P, , Q, , X, , 23. Join OP, AB is tangent to C1 at P and OP is radius, , C1, , C2, , O, , OP ⊥ AB, , A, , AB is chord of circle C2 and OP ⊥ AB., , P, , B, , Therefore OP is the bisector of the chord AB as the perpendicular from the centre, bisects the chord i.e,, AP = BP, 24., , ∠OAB = 50°, , x + B + OAB = 180°, x + 90° + 50° = 180°, x = 40°, 25., , AK = KC, , …(1), , BN = NC, , …(2), , KN = KC + NC =AK + BN, 26., , POQ + PTQ = 180°, , [from (1) & (2)], , 6 cm, , 60° + PTQ = 180°, , O, , PTQ = 120°, , 27., , P, , AC = AF + FC = 10 cm …(1), AB = AD + DB = 12 cm …(2), BC = BE + CE = 8 cm, , …(3), , BD = BE , AD = AF , CF = CE , , , 2AD, 2FC, 2BD are obtained, , …(4), , 60°, , 6 cm, , T, , 6 cm, Q, , Replace from (4) in (1), (2), (3) (So that in (5) + (6) + (7)). 2AD, 2FC, 2BD, are obtained., 134, , Mathematics-X
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AC = AD + FC = 10 cm …(1), AB = AD + DB = 12 cm …(2), BC = BD + CE = 8 cm …(3), Add (5, 6, 7), 2(AD + FC + DB) = 30, AD + FC + DB = 15, Substitute values from (1), (2) & (3), A, and find. AD = 7 cm, BE = 5 cm, CF = 3 cm., 28., PA = PB, 2, O, 4, 1, 3, 1, So,, ∠2 = ∠3 = (180° − ∠1), P, 2, B, 1, 90, °, −, ∠, 1, ∠2 = ∠3 =, 2, ∠4 = 90° (Angle between tangent & Radius), ∠OAB = ∠4 – ∠2, 1 , 1, , = 90° − 90° − ∠1 = 90° − 90° + ∠1, 2, 2, 1, ∠OAB = ∠APB, 2, 2∠OAB = ∠APB, 29., OP = 2r, ⇒, QP = QP = r, A, , O, , Q, , P, , B, , Consider ΔAOP is which OA ⊥ AP and OP is the hypotenuse., OQ = AQ = OA, (Mid point of hy potenuse is equidistance from the vertices)., ⇒ OAQ is an equilitateral triangle., ⇒, , AOQ = 60°, , Consider right angled triangle OAP, AOQ = 60°, , Mathematics-X, , 135
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= 90°, , OAP, , ⇒ ∠APO = 30°, , ∠APB = 2∠APO = 2 × 30° = 60°, PA = PB (tangents), ⇒, , ∠PAB = ∠PBA, In ΔAPB = 60°, , each angle of DPAB = 60°., , 2 cm, , R, , 4 cm, , O, , cm, B5, , Q, , 31., , S, , R, , O, , A, , m, 18 c, , 4 cm, , Q, 11 cm, , 4 cm, , S, 5 cm, , 30., , 7 cm, , cm, B2, , m, 23 c, , A, , m, 11 c, , 4 cm, 15 cm, , m, 13 c, , Hence Proved., , 5 cm, , ⇒, , 180° − 60°, = 60°, 2, , 18 cm, 29 cm, , =, , PAB = PBA, , C 5 cm P 4 cm D, , C, , PC or CP = 5 cm, , P 11 cm D, , r = 11 cm, , B, 2, , y, , x, , 32., , A, , 1, , y, , O, , Q, , 35°, C, , In ΔABC,, ∠1 =, ∠1 + 35° + y =, 90° + 35° + y =, y=, ΔOBQ,, ∠2 =, ∠2 + ∠x + ∠y =, 90° + ∠x + 55° =, x=, 136, , 90°, 180°, 180°, 55°, 90°, 180°, 180°, 35°, , (Angle in semi-circles), , (Angle between tangent and radius), , Mathematics-X
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PRACTICE-TEST, CIRCLES, , Time : 1 Hr., , M.M.: 20, SECTION-A, , 1., , In the given figure find x, where ST is the tangent., , 1, , T, x-40°, , x, , S, , O, , 2., , In the given figure if AC = 9, find BD., , 1, B, , A, , C, , D, , 3., , In the given figure, ΔABC is circumscribing a circle, then find the length of BC. 1, , 3c, m, , A, , M, , N, 4c, m, , 8 cm, , B, , 4., , L, , C, , From the external point P tangents PA and PB are drawn to a circle with centre O. If, ∠PAB = 50°, then find ∠AOB., (Delhi-2016, CBSE) 1, SECTION-B, , 5., , If the angle between two tangents drawn from an external point P to a circle of radius, a and centre O is 60° then find the length of OP., (All India 2017) 2, , Mathematics-X, , 137
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6., , In the following figure find x., , 2, A, x-1, , X, x+ 1, , O, , P, , B, , 7., , Two concentric circle with centre O are of radii 6 cm and 3 cm. From an external, point P, tangents PA and PB are drawn to these circle as shown in the figure. If, AP = 10 cm. Find BP, 2, A, , o, , P, , SECTION-C, 8., , In the given figure, AB is a tangent to a circle with centre O. Prove ∠BPQ =, ∠PRQ., 3, R, , O, , A, , 9., , 138, , Q, , P, , B, , In the given figure ΔABC is drawn to circumscribe a circle of radius 3 cm,, such that the segment BD and DC into which BC is divided by the point of, Mathematics-X
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contact D are of length 6 cm and 8 cm respectively, find side AB if the ar(ΔABC), = 63 cm², 3, A, , E, , F, , B, 6 cm, , D, , C, , 8 cm, , SECTION-D, 10. AB is a diameter of a circle with centre O and AT is a tangent. If ∠AOQ = 58°, find ∠ATQ., 4, B, , O, , 58°, , A, , Q, T, , ppp, , Mathematics-X, , 139
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CHAPTER, , 11, , Constructions, , TOPICS, , • Division of a line segment., • Construction of a Triangle., • Construction of Tangents of a Circle., MIND MAPING, Construction, , Division of, Line segment, , To divide a, line segment, in the given, ratio, , Construction of a, triangle similar to, a given triangle, as per the given, scale factor, , Construction of, a tangent to a circle, , Construction of a, tangent in a circle, at a point lies, on it, , Construction of, tangents to a, circle from, a, point out, side the circle, , By using centre, of circle, , With out using, centre of circle, , KEY POINTS, , 1., , Construction should be neat and clean and there should be no donbling., , 2., , Construction should be as per a given scale factor which may be less than 1 or, greater than 1 for a triangle similar to a given triangle., , 3., , Step of construction should be provided only when it is mentioned in the, question., , 140, , Mathematics-X
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4., , We make use of compass and ruler only but in case of non-standard angles, protractor, can be used., , 5., , Divide a line segment in the given ratio means to determine a point on the given line, segment which divides it in the the given ration., , 6., , A tangent to a circle is a straight line which touches the circle at a point. This point is, called the point of content and the radius through the point of contact is perpendicular, to the tangent., , 7., , Tangents drawn from an external point to a circle are equal., VERY SHORT ANSWER TYPE QUESTIONS, , 5, of the corresponding, 3, sides of ΔABC, a ray BX is drawn such that CBX is an acute angle and X is on the, opposite side od A with respect to BC. What is the minimum no. of points to be, located at equal distances on ray BX., , 1., , Construct a triangle similar to a given ΔABC with its sides, , 2., , Draw a pair of tangents to a circle which are inclined to each other at an angle of 30°., What should be the angle between two radii?, , 3., , 2, of the corresponding, 5, sides of ΔABC , firstly a ray BX is drawn such that CBX is an acute angle and X lies, on the opposite side of A with respect to BC then points B1, B2, B3, are located on, BX at equal distances Which two points will be joined in the next step., , Constract a triangle similar to a given ΔABC with its sides, , 4., , Divide a line segment AB in the ratio 3:7, What is the minimum number of points, marked on a ray AX at equal distances?, , 5., , How many tangents can be drawn from a point lying inside a circle?, , 6., , Divide a line segment AB in the ratio 4:5, a ray AX is drawn first such that ∠BAX is, an acute angle and then points A1, A2, A3, ........ are located at equal distances on the, ray AX which should be joined to B?, , 7., , Divide a line segment AB in the ratio 4:5, the points A1, A2, A3,....and B1, B2, B3,...., are located at equal distances on the ray AX and BY respectively. Which two points, should be joined to divide a line segment?, , 8., , Draw a line segment of length 6 cm. Find a point P on it which divides it in the ratio 3, : 4., (Delhi-2011), Mathematics-X, 141
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9., , Draw a line segment AB = 8 cm and divide it internally in the ration 3 : 2., , 10. Draw a line segment AB of length 6.5 cm. Find a point P on it such that, , AP 3, =, AB 5, , 11. Geometrically divide a line segment of length 8.4 cm in the ratio 5 : 2. (forign–2011), CBSE – 2015, 12. Is it possible to div ide a line segment in the ration 5 :, , 1, by geometrical construction?, 5, , 13. Draw a line segment of length 7.6 cm and divide it in t he ratio 3 : 2. (Foreign – 2011), 14. Write True or False., By geometrical construction, it is possible to divide a line segment in the ratio, 3:, , 1, ., 3, , (NCERT Exampler), , 15. Is it possible to construct a pair of tangents from point P to circle of rarius 5 cm, situated at a distance of 4.9 cm from the centre?, 16. Is it possible to construct a pair of tangents from point A lying on the circle of, radius 4 cm and centre O., 17. Compare the length of the tangents drawn from the external point to circle., LONG ANSWER TYPE QUESTIONS, , 18. AB is a line segment of length 8 cm. Locate a point C on AB such that AC =, , 1, CB., 3, , 19. Construct a ΔABC in which AB = 6.5 cm, ∠B = 60° and BC = 5.5 cm. Also, 3, construct a triangle AB’C’ similar to ΔABC, whose each side is, times the, 2, corresponding sides of ΔABC., 20. Construct a ΔABC in which BC = 5 cm, CA = 6 cm and AB = 7. Construct a, ΔA’BC’ similar to ΔABC, each of whose side are times, , 7, the corresponding, 5, , sides of ΔABC., 21. Construct a triangle with side 4 cm, 5 cm, 7 cm. Then construct a triangle similar to, 2, it whose sides are of the corresponding sides of the given triangle., 3, Mathematics-X, 142
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22. Construct a right triangle in which sides (other than hypotenuse) are of lengths 8, cm and 6 cm. Then construct another triangle similar to this triangle whose sides, are times the corresponding sides of the first triangle., 23. Construct a ΔABC in which BC = 8 cm, ∠B = 45° cm and ∠C = 30°. Construct, another triangle similar to ΔABC such that each side are, , 3, of the corresponding, 4, , sides of ΔABC, 24. A triangle ABC is given such that AB = 4 cm, BC = 7 cm and ∠BAC = 50°., Draw another triangle A’BC’ similar to ΔABC with sides BA’ and BC’ equal to, 6 cm and 10.5 cm respectively. Find the scale factor., 25. Draw a pair of tangents to a circle of radius 6 cm which are inclined to each, other at an angle of 60°. Also justify the construction., 26. Construct a triangle ABC in which AB = 5 cm, ∠B = 60° and attitude CD = 3, cm. Construct a ΔAQR ~ ΔABC such that each sides is 1.5 times that of the, corresponding sides of ΔABC., 27. Draw an isosceles ΔABC with AB=AC and base BC=7cm, vertical angle is, 1, 120°. Construct ΔAB´C´ ~ ΔABC with its sides 1 times of the corresponding, 3, , sides of ΔABC., 28. Draw a circle of radius 3 cm. From a point 5 cm from the centre of the circle,, draw two tangents to the circle. Measure the length of each tangent., 29. Draw a circle of radius 4 cm with centre O. Draw a diameter POQ. Through P, or Q draw a tangent to the circle., 30. Draw two circle of radius 5 cm and 3 cm with their centres 9 cm apart. From, the centre of each circle, draw tangents to other circles., 31. Draw two circles of radii 6 cm and 4 cm. From a point on the outer circle, draw, a tangent to the inner circle and measure its length., 32. Draw a circle of radius 3 cm. Take two points P and Q on one of its extended, diameter each at a distance of 7 cm from its centre. Draw tangents to the circle, from these two points., Mathematics-X, , 143
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33. Draw a line segment PQ = 10 cm. Take a points A on PQ such that, , PA, 2, =, PQ, 5, , Measure the length of PA and AQ, 34. Draw an equilateral triangle PQR with side 5cm. Now construct ΔPQ´R´ ~, PQ, 1, ΔPQR such that, = ., PQ ' 2, 35. Draw a line segment of length 8 cm and divided it in the ratio 5:8. Meeasure the, two parts., 36. Construct a triangle ABC with sides AB = 7 cm, BC = 7.5 cm and CA = 6.5 cm., 3, of the corresponding sides, Construct a Δ similar to ΔABC whose sides are, 2, of ΔABC., ANSWERS AND HINTS, 5, 3, , 1. Since the ratio is , 5 is the larger number so Answer is 5., , 30°, , O, , 2., , Sum of both the angles shown in figure is 180° if one is 30° the other will be, 150°., A, , C, , 3. B, , B5 to C, B1 B, 2, , B3, , B4 B, 5, , X, , 4. 3 + 7 = 10, 5. 0, 144, , Mathematics-X
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11. As above Question-9., 12. Yes, as 5 :, , 1, =5:1, 5, , 13. As above question No. 9., 14. True as 3 :, , 1, can be simplified as 3 : 1., 3, , 15. No, 16. No, 17. Equal., Questions No. 18 to 36., Questions are similar to examples given in NCERT. Please refer NCERT example., , 146, , Mathematics-X
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PRACTICE-TEST, CONSTRUCTIONS, , Time : 1 Hrs., , M.M.: 20, SECTION-A, , 1., , Draw a perpendicular bisector of line segment AB = 8cm., , 1, , 2., , Draw a line parallel to a given line., , 1, , 3., , Draw the tangent to a circle of diameter 4 cm at a point P on it., , 1, , 4., , Draw two tangents to a circle of radius 4 cm from a point T at a distance of 6 cm, from its centre., 1, SECTION-B, , 5., , Draw a pair of tangents to a circle of radius 5 cm, which are inclined to each other at, an angle of 60°., (Foreign - 2014) 2, , 6., , Draw an angle bisectorof 75°., , 2, , 7., , Draw a line segment of 5.6cm. Divide it in the ratio 2:3., , 2, , SECTION-C, 8., , Draw two tangents to a circle of radius 3.5cm from a point P at a distance of, 5.5cm from its centre. Measure its length., 3, , 9., , Draw a circle of radius 3.5cm. Draw two tangents to the circle such that they include, an angle of 120°., 3, SECTION-D, , 10. Construct a ΔABC of sides AB = 4cm, BC = 5cm and AC = 7 cm.Construct, another triangle similar to ΔABC such that each of its sides is, corresponding sides of ΔABC., , 5, of the, 7, 4, , ppp, , Mathematics-X, , 147
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CHAPTER, , 12, , Areas Related to Circles, , TOPICS, , Perimeter and Area of a circle., Area of sector and semgnet of a circle., MIND MAPING, , KEY POINTS, , Circle: A circle is the locus of a point which moves in a plane in such a way that its, distance from a fixed point always remains the same. The fixed point is called the, 148, , Mathematics-X
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centre and the constant distance is known as the radius of the circle., If r is radius of a circle, then, (i), , Circumference = 2r or d where d = 2r is the diameter of the circle, , (ii), , Area = r2 or, , (iii), , Area of semi circle =, , (iv), , Area of quadrant of a circle =, , d 2, 4, , r2, 2, , r2, , 4, Area enclosed by two concentric circles: If R and r are radii of two concentric, circles, then area enclosed by the two circles =R2 – r2, , r, , R, , = (R2 – r2), = (R + r) (R – r), , (i) If two circles touch internally, then the distance between their centres is equal, to the difference of their radii., (ii) If two circles touch externally, then distance between their centres is equal to, the sum of their radii., (iii) Distance moved by rotating wheel in one revolution is equal to the, circumference of the wheel., (iv) The number of revolutions completed by a rotating wheel in, one minute =, , Distance moved in one minute, Circumference of the wheel, , Segment of a Circle: The portion (or part) of a circular region, enclosed between a chord and the corresponding arc is called a, segment of the circle. In adjacent fig. APB is minor segment and, AQB is major segment., , Q, Major, segment, O, , B, or nt, in e, M egm, P, s, , A, , Mathematics-X, , 149
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Area of segment APB = Area of the sector OAPB – Area of OAB, =, , =, , , 360°, , 1 2, r sin or, 2, , × r2 –, , , , , r 2 r 2 sin cos, 360, 2, 2, Q, , O, , , A, , B, , P, , Sector of a circle: The portion (or part) of the circular region enclosed by the two, radii and the corresponding arc is called a sector of the circle., In adjacent figure OAPB is minor sector and OAQB is the major sector., Q, Major, sector, O, , A, , Area of the sector of angle , , Minor, scetor, , P, , B, , =, , , × r2, 360°, , =, , 1, 1, × length of arcc × radius = lr, 2, 2, , , × 2r, 360, (i) The sum of the arcs of major and minor sectors of a circle is equal to the, circumference of the circle., , Length of an arc of a sector of angle =, , (ii) The sum of the areas of major and minor sectors of a circle is equal to the, area of the circle., (iii) Angle described by minute hand in 60 minutes = 360°, 150, , Mathematics-X
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Angle described by minute hand in one minute =, , 360°, = 6°, 60°, , Thus minute hand rotates through an angle of 6° in one minute, (iv) Angle described by hour hand in 12 hours = 360°, Angle described by hour hand in one hour =, , 360°, = 30°, 12°, , Angle described by hour hand in one minute =, , Thus, hour hand rotates through an angle of, , 30°, 1, =, 60°, 2, , ( ), , ( 12 ) in one minute., , VERY SHORT ANSWER QUESTIONS, , 1., , If the diameter of a semi circular protactor is 14 cm, then find its perimeter., , 2., , If circumference and the area of a circle are numerically equal, find the diameter of, the circle., , 3., , Find the area of the circle ‘inscribed’ in a square of side a cm., , 4., , Find the area of a sector of a circle whose radius is r and length of the arc is l., , 5., , The radius of a wheel is 0.25 m. Find the number of revolutions it will make to travel, a distance of 11 kms., , 6., , If the area of circle is 616 cm², then what is its circumference?, , 7., , What is the area of the circle that can be inscribe in a square of side 6 cm?, , 8., , What is the diameter of a circle whose area is equal to the sum of the areas of two, circles of radii 24 cm and 7 cm?, , 9., , A wire can be bent in the form of a circle of radius 35 cm. If it is bent in the form of, a square, then what will be its area?, , 10. What is the angle subtended at the centre of a circle of radius 6 cm by an arc of length, 3π cm?, 11. Write the formula for the area of sector of angle θ (in degrees) of a circle of radius r., 12. If the circumference of two circles are in the ratio 2:3, what is the ratio of their areas?, Mathematics-X, , 151
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13. If the difference between the circumference and radius of a circle is 37 cm, then find, 22, ), the circumference of the circle. ( Use π =, 7, 14. If diameter of a circle is increased by 40%, find by how much percentage its area, increases?, 15. The hour hand of a clock is 6 cm long. Find the area swept by it between 11:20 am, and 11:55 am., 16. What is the diameter of a circle whose area is equal to the sum of areas of two circles, of radii 24 cm and 7 cm., (NCERT Exemplar), 17. What is the area of the circel that can be incresed in a square of side 6 cm., (NCERT Exemplar), 18. The length of the minute hand of a clock is 14 cm. Find the area swept by the, minute hand in one minute., 19. Tied the correct Answer, If the peremeter and the area of a circle are numercally equal, then the radius of, the circle is:, (a) 2 units, (b) 11 units, (c) 4 units, (d) 7 units, 20. Circumference of a circle of radius r is ___________ ., 21. Area of a circle of radius s is _______ ,, 22. Length of an arc of a sector of a circle with radius r and angle θ is ______ ., 23. Area of a sector with radius r and angle with degrees measure θ is ______ ., 24. Area of segment of a circle = Area of the corresponding sector ________ ., SHORT ANSWER TYPE I QUESTIONS, , 25. Find the area of a quadrant of a circle whose circumference is 22 cm., 26. What is the angle subtended at the centre of a circle of radius 10 cm by an arc of, length 5π cm?, 27. If a square is inscribed in a circle, what is the ratio of the area of the circle and, the square?, 28. Find the radius of semicircle if its perimeter is 18 cm., 29. If the perimeter of a circle is equal to that of square, then find the ratio of their, areas., 152, , Mathematics-X
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30. What is the ratio of the areas of a circle and an equilateral triangle whose diameter, and a side are respectively equal?, 5, 31. In fig., O is the centre of a circle. The area of sector OAPB is, of the area of the, 18, circle. Find x., , O, x, A, , B, P, , 32. Find the perimeter of a given fig, where AED is a semicircle and ABCD is a, rectangle., (CBSE, 2015), 20 cm, , 14 cm, , B, , A, , E, , C, , D, , 20 cm, , 33. In fig. OAPBO is a sector of a circle of radius 10.5 cm. Find the perimeter of the, sector., P, B, , A, , 60°, O, , 34. In the given fig, APB and CQD are semi circles of diameter 7 cm each, while, ARC and BSD are semicircles of diameter 14 cm each. Find the perimeter of, 22, the shaded region. (Use π =, ), (Delhi, 2011), 7, , Mathematics-X, , 153
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R, , P, 7 cm, , A, , 7 cm, , 7 cm, D, , C, , B, , Q, , S, , SHORT ANSWER TYPE II QUESTIONS, , 35. Area of a sector of a circle of radius 36 cm is 54π cm2 . Find the length of the, corresponding arc of the sector., 36. The length of the minute hand of a clock is 5 cm. Find the area swept by the, minute hand during the time period 6:05 am to 6:40 am., 37. In figure ABCD is a quadrant of a circle of a radius 28 cm and a semi circles, BEC is drawn with BC as diameter find the area of shaded region:, , D, , A, , E, , C, , 38. In fig, OAPB is a sector of a circle of radius 3.5 cm with the centre at O and, ∠AOB = 120°. Find the length of OAPBO., P, , O, 120°, A, , B, , 39. Circular footpath of width 2 m is constructed at the rate of ` 20 per square, meter, around a circular park of radius 1500 m. Find the total cost of construction, of the foot path. (Take π = 3.14 ), 40. A boy is cycling such that the wheels of the cycle are making 140 revolutions, per minute. If the diameter of the wheel is 60 cm. Calculate the speed of cycle., 154, , Mathematics-X
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41. In a circle with centre O and radius 4 cm, and of angle 30°. Find the area of, minor sector and minor sector AOB. (π = 3.14), 42. Find the area of the largest triangle that can be inscribed in a semi circle of radius r, unit., (NCERT Exempler), 43. Figure ABCD is a trapezium of area 24.5 cm in it AD||BC, DAB = 90°, AD = 10, cm, BC= 4cm. If ABE is a quadrant of a circle. Find the area of the shaded region (π, 22, =, ), 7, E, , D, , B, , C, 44., , A, , From each of the two opposite corners of a square of side 8 cm, a quadrant of a, circle of radius 1.4 cm is cut. Another circle of radius 4.2 cm is also cut from, the centre as shown in fig. Find the area of the shaded portion. (Use π =, , 22, )., 7, , 45. A sector of 100° cut off from a circle contains area 70.65 cm². Find the radius of the, circle. (π = 3.14 ), 46. In fig. ABCD is a rectangle with AB= 14 cm and BC= 7 cm. Taking DC, BC and AD, as diameter, three semicircles are drawn. Find the area of the shaded portion., D, , C, 7 cm, , A, , Mathematics-X, , 14 cm, , B, , 155
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47. A square water tank has its each side equal to 40 m. There are four semi circular, grassy plots all around it. Find the cost of turfing the plot at Rs 1.25 per sq. m., (Use π = 3.14 ), 48. Find the area of the shaded region shown in the fig., , (NCERT – Exemplar), , 8m, 4m, , 6m, , 49. Find the area of the minor segment of a circle of radius 21 cm, when the angle of, the corresponding sector is 120°., 50. A piece of wire 11 cm long is bent into the form of an arc of a circle subtending, an angle of 45° at its centre. Find the radius of the circle., 51. Find the area of the flower bed (with semicircular ends). (NCERT Exampler), , 16 cm, 44 cm, , 52. In fig. from a rectangular region ABCD with AB= 20 cm, a right triangle AED, with AE= 9 cm and DE= 12 cm, is cut off. On the other end, taking BC as, diameter, a semi circle is added on outside the region. Find the area of the, shaded region., B, , A, 9 cm, 90°, , E, , 15 cm, , 12 cm, D, , C, , 53. The circumference of a circle exceeds the diameter by 16.8 cm. Find the radius, of the circle., 54. Find the area of the shaded region., 156, , (NCERT Exampler), Mathematics-X
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4m, 3m, , 3m, , 12 m, , 4m, 26 m, , LONG ANSWER TYPE QUESTIONS, , 55. Two circles touch externally. The sum of their areas is 130π sq. cm and the, distance between their centres is 14 cm. Find the radii of the circles., 56. Three circles each of radius 7 cm are drawn in such a way that each of their, touches the other two. Find the area enclosed between the circles. (All India, 2010), 57. Find the number of revolutions made by a circular wheel of area 6.16 m² in, rolling a distance of 572 m., 58. All the vertices of a rhombus lie on a circle. Find the area of the rhombus, if, area of the circle is 2464 cm²., 59. With vertices A, B and C of a triangle ABC as centres, arcs are drawn with, radius 6 cm each in fig. If AB= 20 cm, BC= 48 cm and CA= 52 cm, then find the, area of the shaded region., A, 52 cm, 20 cm, , B, , 48 cm, , C, , 60. ABCDEF is a regular hexagon. With vertices A, B, C, D, E and F as the centres,, circles of same radius ‘r’ are drawn. Find the area of the shaded portion shown, in the given figure., , Mathematics-X, , 157
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A, , B, , C, , F, , E, , D, , 61. ABCD is a diameter of a circle of radius 6 cm. The lengths AB, BC and CD, are equal. Semicircles are drawn on AB and BD as diameter as shown in the, fig. Find the perimeter and area of the shaded region., , A, , B, , C, , D, , 62. A poor artist on the street makes funny cartoons for children and earns his, living. Once he made a comic face by drawing a circle within a circle, the, radius of the bigger circle being 30 cm and that of smaller being 20 cm as, shown in the figure. What is the area of the cap given in this figure?, , 63. In a given figure ABCD is a trapzium with AB || DC,, AB = 18 cm, DC = 32 cm and distance between, AB and DC = 14 cm. If arc of equal radii 7 cm with, centres A B, C and D have been drawn, then find, the area of shaded region., , A, , D, , 158, , B, , C, , Mathematics-X
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64. Find the area of the shaded region in the given figure., , 20 cm, 7 cm, , ANSWERS AND HINTS, , 22, × 7 + 14 = 36 cm, 7, , 1., , πr + d =, , 2., , 2πr = πr2 ⇒ 4 units., , 3., , Side of the square is equal to diameter of the circle,, πr2 = π ×, , a, a2, (side = a, radius = ), 2, 4, , θ, θ, × 2 πr , Area =, × πr 2, 360°, 360°, , ⇒, , l × πr 2 lr, =, sq. units, 2, 2πr, , 4., , l=, , 5., , distance, 11 × 1000 × 7 × 100, =, = 7000, circumference, 2 × 22 × 25, , 6., , πr2 = 616 ⇒ r = 14 cm, , 7., , Side of the square is equal to the diameter of the circle, ⇒, , or 2πr = 88 cm, , r = 3 cm or πr2 = π(3)2 = 9π cm2., , 8., , πR 2 = πr12 + πr22 ⇒ R = 25 and diameter = 50 cm., , 9., , 2πr = 2 ×, , 22, 220, × 35 = 220 cm , Side of square, = 55 cm, 7, 4, Area of square = 55 × 55 = 3025 cm2, , Mathematics-X, , 159
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26. l =, , θ, θ, × 2πr 5π =, × 2π × 10 θ = 90°, 360, 360, , 27., , If side of square is 1 unit by Pythagoras, Diameter or dianotla = 2 unit., Area of square = 1 × 1 = 1 sq units., 2, Area of Circle = πr = π ×, , 2, 2 π, ×, =, 2, 2, 2, , 22, 2, 2 11, ×, ×, =, 7, 2, 2, 7, or 11 : 7, πr + 2r = 18 cm, , =, , So,, , 4:π, , 28., , 22, r + 2r = 18, 7, , 22, , r, + 2 = 18, 7, , r=, 29. 2πr = 4 unit, , or, , 7, 2, , 2πr, Perimeter of circle, =, 4 unit Perimeter of square, r=, , 7, unit, 11, , πr 2 22 7 7, 14, =, × ×, =, 1, 7 11 11, 11, , 30. Area of equilateral triangle =, , Mathematics-X, , or 3.5 cm, , or 14 : 11, 3 2, a, 4, , 161
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a, Area of circle required = π , 2, , 2, , 3 2, a, 4, ratio =, 2 =π:, a, π , 2, , 3, , 5, θ, πr 2 = πr 2 ×, 360, 18, θ = 100°, 32. 20 cm + 14 cm + 20 cm + πr, , 31., , 20 cm + 14 cm + 20 cm +, 33., , 22, × 7 = 76 cm, 7, , θ, 60 × 2 × 22 × 105, × 2πr =, = 11 cm, 360, 360 × 7 × 10, , Perimeter = 10.5 + 10.5 + 11 cm = 32 cm, 34. Perimeter of shaded region = Perimeters of semi circles,, = ARC + APB + BSD + CQD, = π (r1 + r2 + r3 + r4), =, , 22 , 7, 7 22, 7+ +7+ =, × 21 = 66 cm, , 7 , 2, 2 7, , θ × π × 36 × 36, 360, θ = 15°, , 54 π =, , 35., , l=, 36. Area =, , 15 × 2 × π × 36, θ, × 2 πr =, =3π, 360, 360, , θ, 210 × 22 × 5 × 5 1650, 5, × πr 2 =, =, = 45 ⋅ cm2, 360, 360 × 7, 36, 6, , (θ = 210° in 35 minutes), 37. AC = 28 cm, BC = 28 2 cm (by Pythagoras)., 162, , Mathematics-X
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radius = 14 2 cm, Shaded region = Area of semicircle – Area of segment BCD, =, , 1, 90°, 1, π (14 2) 2 −, × π (28)2 + × 28 × 28, 2, 360°, 2, , = 392 cm2, 38., , l=, , 240 × 2 × 22 × 35, = 14.6, 360 × 7 × 10, , Length of OAPBO = 14.6 + 3.5 + 3.5, = 21.6 cm, , π (r12 − r12 ) = π[(1502)2 – (1500)2 ] × 20, , 39., , = 3.14 [(1502)2 – (1500)2] × 20, = ` 3770.51.2, 40., , Circumference of cycle = 2πr, 22, × 30 cm, 7, = 188.57 cm, , = 2×, , Speed of cycle =, , 18857 × 140 × 60, 100 × 100000, , = 15.84 km/h, 41., , Area of Minor sector =, , θ, × πr 2, 360, , 30, × 3.14 × 4 × 4 cm 2, 360, A, = 4.19 cm2, , =, , Area of major sector =, =, , Mathematics-X, , C, , r, O, , B, , θ, × πr 2, 360, 330, × 3.14 × 4 × 4, 360, , 163
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= 46.1 cm2 (approx), Area of Δ =, , 42., , =, , 1, base × height, 2, , 1, AB × OC, 2, , 1, 2r × r, 2, = r2 square unit, , =, 43. Let AB = h cm, , Area of tropezium =, , 1, ( AD + BC ) × AB, 2, , 1, (10 + 4) × 4, 2, h = 3.5 cm, , 24.5 =, , Area of quadrant ABE =, , 90°, × π (3.5) 2 sq.m, 360°, , = 9.625 sq.m, Area of shaded region = 24.5 – 9.625, = 14.875 sq. m, 44. Area of shaded portion =, Area of square – Area of circle – (Area of 2 quadrauts) or Area of Semicircle., = 64 −, , 22 × 42 × 42 22 × 14 × 14 × 1, −, 7 × 10 × 10, 7 × 10 × 10 × 2, , = 64 – 55.44 – 3.08, = 5.48 cm2, 45., , 100 × 314 × r 2, 7065, =, 360 × 100, 100, 7065 × 360, = r2, 100 × 314, 9= r, r = 9 cm., , 164, , Mathematics-X
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46. Area of shaded portion is = One circle and Area of rectangle – semicircle of diamter, DC,, 2, , DC , π, , Area of shaded portion =, 2 , πr 2 AB × BC −, , , 2, , =, , 22, 22 × 7 × 7 , , × (3.5) 2 + 98 −, 7, 7 × 2 , , , = 38.5 + [98 – 77], = 38.5 + 21, = 59.5 cm2, 47. Four semicircluar means 2 circles ,, Area of 2 circles = 2πr 2, = 2 × 3.14 × 20 × 20, = 2512, = 2512 × 1.25, = ` 3140, 48. Redraw the figure and decide in into well known shapes,, One semi circle + Rectangle, Area of shaded region = l × b +, , πr 2, 2, , 2× 2, 2, 2, = (32 + 2π) cm, Area of the segment = Area of sector – Area of Δ, , = 8× 4 + π×, , 49., , Area of sector =, Area of Δ =, , 120 22, ×, × 21 × 21 = 462 cm2, 360 7, , 441, 3 cm 2 (NCERT example – 3), 4, , 441 , , 3 cm2, Area of segment = 462 −, , 4, Mathematics-X, , 165
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=, 50., , l=, , 21, (88 − 21 3 ) cm2, 4, θ, × 2 πr, 360, , 45 2 × 22 × r, 360, 7, 14 = r, r = 14 cm, 51. Flower bed has two semi-circular shapes and one rectangular shape., Area = l × b + πr2, = (44 × 16 + π × 8 × 8), = (704 + 64π) cm2, 52., Area of shaded region = Rectangle + Semicircle – Triangle, , 11 =, , = 20 × 15 + 28.12 π –, , 1, × 12 × 9, 2, , = 334.39 cm2, 53., , 2πr = 2r + 16.8, 2×, , or,, , 22, 168, r − 2r =, 7, 10, , or, , 22 168, 2r − 1 =, 7, 10, , 168, 15 , 2r =, 7, 10, , or, , 168 × 7, 1176, =, = 3.92 cm, 10 × 2 × 15, 300, , 54. Area of shaded region = Area of rectangle – [Area of 2 semicircles + Area of, rectangle], πr 2, , + l × b, = L × B − 2, 2, , = 26 × 12 − [π × 2 × 2 + 16 × 4], = 312 – 4π – 64 = (248 – 4π) m2, 2, 2, πr12 + πr22 = 130 π ⇒ r1 + r2 + 130, , 55., ⇒, , r1 + r2 = 14, , ...(1), , …(2), , Substitude the value of r1 from (2) in (1) and solve., 166, , Mathematics-X
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2r22 – 28 r2 + 66 = 0, , r22 – 14r + 33 = 0, , (Neglecting – ve), , r = 11 cm and r = 3 cm, 56., , Area of shaded region = Area of Δ – Area of 3 sectors., area Δ =, , 3, 3, × 14 × 14 =, ×196 = 49 3, 4, 4, , Area of 3 Sectors = 3 ×, , 60 22, × × 7 × 7 = 77, 360 7, , = (49 3 − 77), πr2 =, , 57., , 616, 100, , 2πr = 2 ×, Number of revolution =, , or, , r 2 = 1.96, , or, , r = 1.4 m, , 22 14 616, ×, =, = 8.8 m, 7 10 100, , 572, = 65, 8.8, , πr2 = 2464 cm2, , 58., , r = 28 cm, Area of rhombus =, =, 59., , Ans., , or d = 28 + 28 = 56 cm, , 1, 1, d1d 2 or d 22 (d1 = d 2 ), 2, 2, 1, × 56 × 56 = 1568 cm2, 2, , Area of shaded region = Area of Δ – Area of 3 sectors., 1, πr 2, ×, 20, −, (θ1 + θ 2 + θ3 ), =, 48, 360, 22 × 6 × 6, (180°), 7 × 360, = 480 – 56.57, , = 480 −, , = 423.43, Mathematics-X, , 167
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60. 2r2 (Area is equal to 2 circles.), 2r1 2r2 2r3, , , 2, 2, 2, 22 6, 22 4, 22 2 , , 2, 2, , = 2 , 7 2, 7 2, 7 2 , , 22, 264, = 2 3 2 1 =, = 37.71 cm, 7, 7, r12, 22, r22, r32 , (18 8 2), Area = =, 7, 2, 2, 2, , 61., , Perimeter =, , = 31.71 cm2, 62. Radius of bigger circle O = 30 cm, Radius of Smaller O = 20 cm, Difference of their radii = (30 – 20) = 10 cm, AB is tangent to small circle, Radius = OC i.e. OD AB, , , O, , OCA = 90° = OCB, , In OCA by Phythagoras, , , , , AC =, AC =, AB =, AB =, , 20 2 cm, CB, AC + CB, AC + AC = 2 AC, , , , AB = 2 × 20 2 cm, = 40 2 cm, CD = Radius of bigger circle-OC, = 30–10 = 20 cm, Area of cap =, , 1, AB × CD, 2, , =, , 1, 40 2 20 cm 2, 2, , = 400 2 cm 2, 168, , Mathematics-X
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63., , Area of trapezium =, =, Area of four sectors =, , 1, × h ( a + b), 2, 1, × 14 × (18 + 32) = 350 cm2, 2, , πr 2, × ( ∠A + ∠B + ∠C + ∠D ), 360, , π×7×7, × 360, 360, = 49 π cm2, , =, , 64., , πr12 πr22 πr32 , +, +, Area of shaded region = , , 2, 2, 2 , 17 × 17 10 × 10 7 × 7 , +, +, , = π , 7, 2, 2 , = 688.28 cm2, rrr, , Mathematics-X, , 169
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PRACTICE-TEST, AREAS RELATED TO CIRCLES, , Time : 1 Hr., , M.M.: 20, SECTION-A, , 1., , If the circumference of two circles are equal, then what is the ratio between, their areas?, 1, , 2., , If the diameter of a protactor is 21 cm, then find its perimeter., , 1, , 3., , Area of a circle of radius P is ___________ ., , 1, , 4., , Tick the correct answer., If the perimeter and the area of a circle are numerically equal then the radius of, the circle is, 1, (b) π units, , (a) 2 units, , (c) 4 units, , (d) 7 units, , SECTION-B, 5., 6., , The length of minute hand of a clock is 14 cm. Find the area swept by the mixutre, hand in 8 minutes., 2, Find the area of a circle whose circumference is 22 cm., 2, , 7., , Find the area of a quadrant of a circle whose circumference is 44 cm., , 2, , SECTION-C, 8., , A horse is tied to a pole with 28 cm long string. Find the area, where the horse can graze., 3, , 9., , In fig. two concentric circles with centre O, have radii 21 cm, and 42 cm. If ∠AOB = 60° find the area of the shaded region., (Use π =, , 22, ), 7, , O, 60°, , 3, A, , B, , SECTION-D, 10. A chord AB of a circle of radius 10 cm makes a right angle at the centre of the circle., Find the area of the minor and major segments., 4, ppp, , 170, , Mathematics-X
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CHAPTER, , Surface Areas and Volumes, , cu., , bh, , 13, , Mathematics-X, , 171
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KEY POINTS, , 1., , Cuboid: 3-D shapes like a book, a metch box , an almirah, a room etc. are called, Cuboid., , h, b, , l, , For cuboid length = l, breadth = b, height = h, Volume = l × b × h, Lateral surface area of solid cuboid = 2h( l + b), Total surface area of solid cuboid = 2(lb + bh + hl), 2., , Cube: 3-D shapes like ice-cubes, dice, etc. are called cube., , a, a, a, , In cube, length = breadth = height = a, Volume = a³, Lateral surface area of solid cube = 4a², Total surface area of solid cube = 6a², 3., , Cylinder: 3-D shapes like jars, circular pillars, circular pipes, rood rollers etc. are, called cylinder., , h, r, 172, , Mathematics-X
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(a) For right circular cylinder solid, base radius = r, height = h, Volume = πr2h, πrh, Lateral surface area of solid cylinder = 2π, πr (r + h), Total surface area of solid cylinder = 2π, (b) For right circular cylinder (Hollow), external radius = R, internal radius = r, height = h, Volume = π(R² – r²)h, π(R + r)h, Curved surface area = 2π, π(R + r) h + 2π, π(R² – r²), Total surface area = 2π, 4., , Cone: 3-D shapes like conical tents, ice-cream cone are called Cone., , l, , h, r, , For right circular cone,, base radius = r, height = h, slant height = l, l=, , h2 + r 2, , Volume =, , 1 2, πr h, 3, , Curved surface area of solid cone = πrl, Total surface area of solid cone = πr (r + l), It may be noted that if radius and height of a cone and cylinder are same then, 3 × volume of a cone = volume of right circular cylinder, Mathematics-X, , 173
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5., , Sphere: 3-D shapes like cricket balls, footballs etc. are called sphere., , r, , (a) For sphere : Radius = r, Volume =, , 4 3, πr, 3, , π r2, surface area = 4π, (b) For Hemisphere (solid): Radius = r, Volume =, , 2 3, πr, 3, , r, , πr2, Curved surface area = 2π, πr2, Total surface area = 3π, 6., , Frustum: When a cone is cut by a plane parallel to the base of the cone, then the, portion between the plane and the base is called the frustum of the cone., Example = Turkish Cap, For a frustum of cone:, , r, , Base radius = R, , L, , h, , Top radius = r, , R, , Height = h, slant height = l, l=, volume =, , h 2 + (R – r ) 2, 1, πh(r2 + R2 + Rr), 3, , Curved surface area (solid frustum) = πl(R + r), Total surface area (solid frustum) = πl(R + r) + π(R² + r²), , 174, , Mathematics-X
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VERY SHORT ANSWER TYPE QUESTIONS, , 1., , Match the following:, Column I, , 2., , Column II, , (a) Surface area of a sphere, , (i) 2πrh, , (b) Total surface area of a cone, , (ii), , (c) Volume of a cuboid, , (iii) 2πr{r + h), , (d) Volume of hemisphere, , (iv), , (e) Curved surface area of a cone, , (v) πr (r + 1), , (f) Total surface area of hemisphere, , (vi) l × b × h, , (g) Curved surface area of a cylinder, , (vii), , (h) Volume of a cone, , (viii) πrl, , 1 2, πr h, 2, , 1, πh( r 2 + R 2 + rR ), 3, , 2 3, πr, 3, , (i) Total surface area of a cylinder, , (ix) 3 πr2, , (j) Volume of a frustum of a cone, , (x) 4πr 2, , Fill in the Wanks:, (i) The total surface area of cuboid of dimension a × a × b is.____________., (ii) The volume of right circular cylinder of base radius r and height 2r is, ______________., (iii) The total surface area of a cylinder of base radius r and height h, is_____________., (iv) The curved surface area of a cone of base radius r and height h, is_______________., (v) If the height of a cone is equal to diameter of its base, the volume of cone, is_________________ ., (vi) The total surface area of a hemisphere of radius r is____________., (vii) The lateral surface area of a hollow cylinder of outer radius R, inner, radius r and height h is, , Mathematics-X, , 175
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(viii) If the radius of a sphere is doubled, its volume becomes__________times the, volume of original sphere., (ix) If the radius of a sphere is halved, its volume becomes__________times the, volume of original sphere., (NCERT Exemplar), 3., , Write ‘True’ or ‘False’ in the following:, (i) Two identical solid hemispheres of equai base radius r are stuck together along, their bases. The total surface area of the combination is 6πr2., (ii) A solid cylinder of radius r and height h is placed over other cylinder of same, height and radius. The total surface area of the shape so formed is 4πrh +, 4πr2., (iii) A solid cone of radius r and height h is placed over a solid cylinder having, same base radius and height as that of a cone. The total surface area of the, combined. πr ( r 2 + h2 + 3r + 2h ), , 4., , (iv) A solid ball is exactly fitted inside the cubical box of side a. The volume of the, 4 2, ball iis πa ., 3, 1, 2, 3, (v) The volume of the frustum of a cone is πh(r1 + r2 + r 1r2 ) . where h is, 3, vertical height of the frustum and r1; r2 are the radii of the ends., The total surface area of a solid hemisphere of radius r is, , 5., , (a) πr2, (b) 2πr2, (c) 3πr2, (d) 4πr2, The volume and the surface area of a sphere are numerically equal, then the, radius of sphere is, , 6., , (a) 0 units, (b) 1 units, (c) 2 units, (d) 3 units, A cylinder, a cone and a hemisphere are of the same base and of the same height., The ratio of their volumes is, , 7., , (a) 1:2:3, (b) 2:1:3, (c) 3:1:2, (d) 3:2:1, A solid sphere of radius ‘r’ is melted and recast into the shape of a solid cone of, height ‘r’. Then the radius of the base of cone is, , 8., , (a) 2r, (b) r, (c) 4r, (d) 3r, Three solid spheres of diameters 6 cm, 8 cm and 10 cm are melted to form a, single solid sphere. The diameter of the new sphere is, (a) 6 cm, , 176, , (b) 4.5 cm, , (c) 3 cm, , (d) 12 cm, Mathematics-X
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9., , The radii of the ends of a frustum of a cone 40 cm high are 38 cm and 8 cm. The slant, height of the frustum of cone is, (a) 50 cm, , (b) 10 7 cm, , (c) 60.96 cm, , (d) 4 2 cm, 10. A metallic spherical shell of internal and external diameters 4 cm and 8 cm, respectively, is melted and recast into the form of a cone of base diameter 8 cm. The height of the, cone is:, (a) 12 cm, (b) 14 cm, (c) 15 cm, (d) 18 cm, 11. A solid piece of iron in the form of a cuboid of dimensions 49 cm × 33 cm, × 24 cm, is moulded to form a solid sphere. The radius of the sphere is, (a) 21 cm, (b) 23 cm, (c) 25 cm, (d) 19 cm, 12. A shuttle cock used for playing badminton has the shape of the combination of, (NCERT Exemplar), (a) A cylinder and a sphere, (b) a cylinder and a hemisphere, (c) a sphere and a cone, (d) frustum of a cone and hemsphere, 13. The radii of the top and bottom of a bucket of slant height 45 cm are 28 crn and, 7 cm, respectively. The curved surface area of the bucket is, (NCERT, Exemplar), (a) 4950 cm2, (b) 4951 cm2, (c) 4952 cm2, (d) 4953 cm2, 14. What geometrical shapes is a “FUNNEL” combination of?, , 15. What geometrical shapes is a cylindrical “PENCIL” sharped at one edge, combination of?, , Mathematics-X, , 177
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16. What geometrical 3-D shapes is a “GLASS (tumbler)”?, , 17. What geometrical shapes is a “GILLI” in gilli-danda game combination of?, , 18. A solid shape is converted from one form to another. What is the change in its, volume?, 19. What cross-section is made by a cone when it is cut parallel to its base?, 20. Find total surface area of a solid hemi-sphere of radius 7cm., 21. Volume of two spheres is in the ratio 64 : 125. Find the ratio of their surface, areas., 22. A cylinder and a cone are of same base radius and of same height. Find the ratio, of the volumes of cylinder to that of the cone., 23. A solid sphere of radius r is melted and recast into the shape of a solid cone of, height r. Find radius of the base of the cone., 24. If the volume of a cube is 1331 cm³, then find the length of its edge., SHORT ANSWER TYPE QUESTION (TYPE-I), , 25. How many cubes of side 2 cm can be cut from a cuboid measuring, (16cm×12cm×10cm)., 26. Find the height of largest right circular cone that can be cut out of a cube whose, volume is 729 cm³., 27. Two identical cubes each of volume 64 cm³ are joined together end to end. What, is the surface area of the resulting cuboid?, 28. Twelve solid spheres of the same sizes are made by melting a solid metallic, cylinder of base diameter 2 cm and height 16cm. Find the radius of each sphere., 29. The diameters of the two circular ends of the bucket are 44 cm and 24 cm. The, height of the bucket is 35cm. Find the volume of the bucket., 178, , Mathematics-X
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SHORT ANSWER TYPE QUESTION (TYPE-II), , 30. A bucket is in the form of a frustum of a cone and hold 28.490 litres of water., The radii of the top and bottom are 28 cm and 21 cm respectively. Find the, height of the bucket., 31. Three cubes of a metal whose edge are in the ratio 3:4:5 are melted and converted, into a single cube whose diagonal is 12 3 cm. Find the edge of three cubes., 32. Find the depth of a cylindrical tank of radius 10.5 cm, if its capacity is equal to, that of a rectangular tank of size 15 cm × 11 cm × 10.5 cm., 33. A cone of radius 8cm and height 12cm is divided into two parts by a plane, through the mid-point of its axis parallel to its base. Find the ratio of the volumes, of the two parts., 34. A petrol tank is a cylinder of base diameter 28cm and length 24cm filted with, conical ends each of axis length 9cm. Determine the capacity of the tank., 35. Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/, hour. How much area will it irrigate in 30 minutes; if 8 cm standing water is, needed?, (NCERT CBSE 2019), 36. A solid is in the form of a cylinder with hemispherical ends. The total height of, the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume, 22, of the solid., (Use F =, ) CBSE 2019, 7, 37. Two spheres of same metal weight 1 Kg and 7 Kg. The radius of the smaller, sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the, diameter of the new sphere. CBSE 2019, 38. A cone of height 24 cm and radius of base 6 cm is made up of modeling clay, A, child reshapes it in the form of a sphere. Find the radius of the sphere and hence, find the surface area of this sphere., (NCERT CBSE 2019), 39. A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical, tank in his field which is 10 m in diameter and 2 m deep. If water flows through, pipe at the rate of 3 Km/hr, in How much time will the tank be filled?, (NCERT CBSE 2019), Mathematics-X, , 179
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40. A juice seller was serving his customers using glasses as shown in figure. The inner, diameter of the cylindrical glass was 5 cm but bottom of the glass had a hemispherical, raised portion which reduced the capacity of the glass. If the height of a glass was 10, cm, find the apparent and actual capacity of the glass. { Use π] = 3.14}, (NCERT CBSE 2019, 2009), , 41. A girl empties a cylindrical bucket full of sand, of base radius 18 crn and height 32 cm, on the floor to form a conical heap of sand. If the height of this conical heap is 24 cm,, then find its slant height correct to one place of decimal., (CBSE 2019), 42. Water is flowing at the rate of 5 km/hour through a pipe of diameter 14 cm into a, tank with rectangular base which is 50 m long and 44 m wide. Find the time in, 22, which the level of water tank rises by 7 cm. (Use π =, }, (CBSE 2019), 7, 43. A field is in the form of rectangle of length 20 m and width 14 m, A 10 m deep, well of diameter 7 m is dug in one corner of the field and the earth taken out of, the well is spread evenly over the remaining part of the field. Find the rise in the, 22, level of the field. ( Use π =, ), (CBSE 2019), 7, LONG ANSWER TYPE QUESTIONS, , 44. A bucket open at the top is in the form of a frustum of a cone with a capacity of, 12308.8 cm2. The radii of the top and bottom of the circular ends of the bucket, are 20 cm and 12 cm respectively. Find the height of the bucket and also the area, of the metal sheet used in making it. ( Use π = 3.14 ), (CBSE 2019), 45. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm,, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the, mass of the pole, given that 1 cm3 of iron has approximately 8 gm mass. ( Use Π =, 3.14 ), (NCERT CBSE 2019), Mathematics-X, 180
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46. A right cylindrical container of radius 6 cm and height 15 cm is full of ice-cream,, which has to be distributed to 10 children in equal cones having hemispherical shape, on the top. If the height of the conical portion is four times its base radius, find the, radius of the ice-cream cone.(CBSE 2019), 47. A container opened at the top and made up of a meta! sheet, is in the form of a, frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm, and 20 cm respectively. Find the cost of milk which can completely fill the, container, at the rate of ? 50 per litre. Also find the cost of meta! sheet used to, make the container, if it costs ` 10 per 100 cm2 ( Take π = 3.14)., (NCERT CBSE 2019), 48. An open metallic bucket is in the shape of a frustum of a cone, If the diameters of the, two circular ends of the bucket are 45 cm and 25 cm and the vertical height of the, bucket is 24 cm, find the area of the metallic sheet used to make the bucket. Also find, 22, the volume of the water it can hold. { Use Π =, )., 7, 49. In the given figure, from the top of a solid cone of height 12cm and base radius 6cm, a, cone of height 4 cm is removed by a plane parallel to the base. Find the total surface area, of the remaining solid.(Use π =, , 22, and 5 = 2.236 ), 7, , (CBSE – 2015), , 4 cm, 12 cm, , 6 cm, , 50. A solid wooden toy is in the form of a hemi-sphere surmounted by a cone of, same radius. The radius of hemi-sphere is 3.5cm and the total wood used in the, 5, making of toy is 166 cm3. Find the height of the toy. Also, find the cost of, 6, painting the hemi-spherical part of the toy at the rate of ` 10 per cm²., 22, )., 7, Mathematics-X, , (use π =, , (CBSE, 2015), 181
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51. In the given figure, from a cuboidal solid metalic block of dimensions 15 cm × 10 cm, × 5 cm a cylindrical hole of diameter 7cm is drilled out. Find the surface area of the, remaining block. (Use π =, , 22, ).(CBSE – 2015), 7, 7 cm, , 5 cm, m, , 10 cm, 15 cm, , 52. A solid toy is the form of a right circular cylinder with a hemispherical shape at, one end and a cone at the other end. Their diameter is 4.2 cm and the heights of, the cylindrical and conical portions are 12 cm and 7 cm respectively. Find the, volume of the toy., 53. A tent is in the shape of a right circular cylinder upto a height of 3m and conical, above it. The total height of the tent is 13.5 m and radius of base is 14 m. Find, the cost of cloth required to make the tent at the rate of ` 80 per sq. m., 54. The rain water from a roof 22m × 20m drains into a cylindrical vessel having, diameter of base 2m and height 3.5m. If the vessel is just full, find the rainfall in cm., 55. The difference between outer and inner curved surface areas of a hollow right, circular cylinder, 14 cm long is 88cm2. If the volume of the metal used in making, the cylinder is 176cm3. Find the outer and inner diameters of the cylinder., ANSWERS AND HINTS, , 1. (a) (x) 4πr2, , (b) (v) πr (r + l), 2 3, πr, 3, , (c) (vi) l × b × h, , (d) (vii), , (e) (viii) πrl, , (f) (ix) 3πr2, , (g) (i) 2πrh, , (h) (ii), , 1 2, πr h, 3, , (i) (iii) 2πr(r + h), , (j) (iv), , 1, πh(r 2 + R 2 + rR), 3, , 182, , Mathematics-X
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2. (i) 2a2 + 4ab, , (ii) 2πr3, (iv) πr r 2 + h 2, , (iii) 2πr(r + h), (v), , 2 3, πr, 3, , (vi) 3πr2, , (vii) 2πh(R + r), (ix), , (viii) 8, , 1, 8, , 3. (i) False, , (ii) False, , (iii) False, , (iv) False, , (v) True, 4. (c) 3πr2, , 5. (d) 3 units, , 6. (c) 3 : 1 : 2, , 7. (a) 2r, , 8. (d) 12 cm, , 9. (a) 50 cm, , 10.(b) 14 cm, , 11. (a) 21 cm, , 12.(d) Frustem of a cone and a hemisphere 13. (a) 4950 cm2, 14., , Cylinder and frustem of a cone, , 15. Cylinder and cone, , 16., , Frustum of a cone, , 17. Cylinder with conical ends, , 18., , Remains unchanged, , 19. Circle, , 2, , 20., , 462 cm, , 21. 16 : 25, , 22., , 3:1, , 23. 2r, , 24., , 11 cm, , 25., , No. of cubes =, , 26., , Side of cube = 3 729 = 9cm, Height of largest cone = Side of cube = 9 cm, , 27., , Side of cube =, , 16 × 12 × 10, = 240, 2×2×2, , 3, , 64 = 4 cm, Length, breadth and height of new cuboid is 8 cm, 4 cm and 4 cm respectively., Surface area of cuboid = 2[8 × 4 + 4 × 4 + 4 × 8] = 160 cm2, , Mathematics-X, , 183
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28., , 29., , 30., , 31., , Volume of 12 solid sphere = Volume of solid cylinder, 4, 12 × πr 3 = π(1)2 × 16, 3, r3 = 1, r = 1 cm, 1 22, Volume of bucket = × × 35 [(22)2 + (12)2 + 22 × 12], 3 7, 2, = 32706 cm3, 3, Volume of bucket = 28490 cm3, 1 22, ×, × h [(28)2 + (21)2 + 28 × 21] = 28490, 3 7, h = 15 cm, Let the edges of three cubes be 3x cm, 4x cm and 5x cm., Volume of single cube = Sum of volume of three cubes, (Side)3 = (3x)3 + (4x)3 + (5x)3, Side = 6x cm, Diagonal of single cube = 12 3, 3 (6 x) = 12 3, , x=2, Hence edges of three cubes are 6 cm, 8 cm and 10 cm, 32. Capacity of cylindrical tank = Capacity of rectangular tank, 22, × (10.5) 2 × h = 15 × 11 × 10.5, 7, h = 5 cm, 33. ΔOAB ~ ΔOCD, , O, 6 cm, 4cm, A, , AB OA, =, CD OC, AB = 4 cm, , B, , 12 cm, , 6 cm, , C, , 1, π(4) 2 × 6, 3, , D, , 8 cm, , 1, Volume of conical part, =, =, 1, Volume of frustum part, π × 6[(8) 2 + (4) 2 + 8 × 4] 7, 3, ∴, 184, , required ratio is 1 : 7 or 7 : 1, Mathematics-X
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34. Capacity of tank = Volume of cylindrical part + 2 × Volume of conical part, = 18480 cm2, , 35. Length of canal covered in 30 mins = 5000 m, ∴ Volume of water flown in 30 mins, = 6 × 1.5 × 5000 m3, 6 × 1.5 × 5000, = 562500 m2, 0.08, 36. Height of cylinder = 20 – 3.5 – 3.5 = 13 cm, Volume of solid =Volume of cylindrical part + 2, × Volume of hemispherical part, , Area irrigated =, , =, , 22, 2 22, × (3.5) 2 × 13 + 2 × × (3.5)3, 7, 3 7, , 1, cm3, 6, 37. Radius of first sphere = 3 cm, Let density of metal be d kg/cm3, , = 680, , ∴, , 4, π (3)3 × d = 1, 3, , ...(1), , Let radius of second sphere be rcm., 4, π (r)3 × d = 7, ...(2), 3, From (1) and (2), we have, r3 = 7(3)3, Let the radius of new spnere by R cm., A.T.Q, , ∴, , Mathematics-X, , 4, 4, 4, π R3 = π (3)3 + π r 3, 3, 3, 3, 185
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R3 = (3)3 + 7(3)3, R = 6 cm, ∴ Diameter of new sphere = 2 × 6 = 12 cm., 38. Volume of sphere = Volume of cone, 4 3, 1, πr = π (6) 2 × 24, 3, 3, r = 6 cm, Surface area of sphere = 4 × π × (6)2 = 144 π cm2, , 39., , Time to fill tank =, =, , Volume of cylindrical tank, Volume of water flown in 1 hour, π(50)2 × 2, 2, , 1, π × 3000, 10 , , = 100 minutes or 1 hour 40 minutes., , 2, , 5, 40. Apparent capacity = 3.14 × × 10 = 196.25 cm3., 2, Actual capacity = Volume of cylindrical part – Volume of hemispherical part, 3, , 2, 5, = 196.25 − × 3.14 × , 2, 3, 3, = 163.54 cm approx, 41. Volume of conical heap = Volume of cylindrical bucket, 1 2, πr × 24 = π(18)2 × 32, 3, r = 36 cm, , Slant height,, , l=, , (36)2 + (24)2 = 43.2 cm, , 42. Volume of raised water in tank = 50 × 44 ×, , 7, = 154 m3, 100, 2, , 22 7 , Volume of water flown in 1 hr =, ×, × 5000 = 77 m3, 7 100 , Time taken =, , 186, , 154, = 2 hours, 77, , Mathematics-X
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43., , Rise in level =, , Earth taken out, Area of the remaining part of field, 2, , 22 7 , ×, × 10, 7 2, =, = 1.5 m approx., 22 7 7 , , 20 × 14 − 7 × 2 × 2 , , 44. Volume of bucket = 12308.8 cm3, 1, × 3.14 × h [(20)2 + (12)2 + 20 × 12] = 12308.8, 3, , h = 15 cm, l=, , (15)2 + (20 − 12)2 = 17 cm, , Surface area of metal sheet used, = 3.14 × 17 × (20 + 12) + 3.14 × (12)2, = 2160.32 cm2, 45., 8cm, 60 cm, , 60 cm, 12 cm, , Volume of solid = 3.14 × (12)2 × 220 + 3.14 × (8)2 × 60, = 111532.8 cm3, 8, kg, 1000, = 892.2624 kg, 46. Let radius of conical section be r cm., ∴ Height of conical section be 4r cm., , Mass of the pole = 111532.8 ×, , Mathematics-X, , 187
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According to the question, 10 × Volume of ice-cream in 1 cone = Volume of cylindrical container, 2, 1, , 10 × πr 2 × 4r + πr 3 = π(6)2 × 15, 3, 3, , r = 3 cm, 3.14 × 16, [(20)2 + (8)2 + 20 × 8], 3, = 10450 cm3 approx., = 10.45 litres, Cost of milk = 10.45 × 50 = ` 522.50, , 47. Volume of the container =, , Slant height =, , (16)2 + (20 − 8)2 = 20 cm, , Surface area of container, = 3.14 × 20 (20 + 8) + 3.14 × (8)2, = 1959.36 cm2, Cost of metal sheet =, , 10, × 1959.36 = ` 195.94, 100, , 45 25 , 48. Slant height = (24) 2 + − , 2, 2, , 2, , = 26 cm, , 22, 45 25 22 25 25, × 26 × + +, × ×, 2, 7, 2 7, 2, 2, = 3351.07 cm2 approx., , Surface area of bucket =, , Volume =, , 45 2 25 2 45 25 , 1 22, × × 24 × + + × , 3 7, 2, 2, 2 2 , , 49. Radii of frustum are 6 cm and 2 cm., Height of frustum = 12.4 = 8 cm, Slant height = (8)2 + (6.2)2 = 4 5 cm, Total surface area of frustum, 22, 22, 22, × 4 × 2.236 × [6 + 2] +, × (6) 2 +, × (2) 2, 7, 7, 7, = 350.592 cm2 approx., , =, , 188, , Mathematics-X
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50. Volume of toy =, , 1001, cm3, 6, , 3, , 2, , 1001, 2 22 7 , 1 22 7 , × × + × × ×h =, 6, 3 7 2, 3 7 2, h = 6 cm, Area of hemispherical part of toy, 2, , 22 7 , = 2 × × = 77 cm2, 7 2, Cost of painting =, 77 × 10 = ` 770, 51. Surface of the remaining block = TSA of cuboidal block + CSA of cylinder, Area of two circular bases, 2, , 22, 7, 22 7 , = 2(15 × 10 + 10 × 5 + 15 × 5) + 2 ×, × ×5–2×, × , 2, 7, 2, 7, = 583 cm2, 52. Volume of toy = Volume of cylindrical part + Volume of hemispherical part, + Volume of conical part, 22, 1 22, 2 22, × (2.1) 2 × 12 + ×, × (2.1) 2 × 7 + ×, × (2.1)3, 7, 3 7, 3 7, = 218.064 cm3, , =, , 53. Slant height =, , (14)2 + (10.5)2 = 17.5 m, , 22, 22, × 3 × 14 +, × 14 × 17.5, 7, 7, = 1034 m2, Cost of cloth = 1034 × 80 = ` 82720, , Surface area of tent = 2 ×, , 54., , Rainfall =, , Volume of cylindrical vessel, Area of roof, , 22, × (1) 2 × 3.5, 1, = 7, =, m, 22 × 20, 40, =, Mathematics-X, , 1, × 100 cm = 2.5 cm, 40, 189
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55. Let inner and outer radius of hallow cylinder be r cm and R cm respectively., Difference between Outer and Inner CSA = 88 cm2, 22, × 14 × [R − r] = 88, 7, R–r= 1, Volume of hollow cylinder = 176 cm3, 2×, , ...(1), , 22, × 14 × [R 2 − r 2 ] = 176, 7, R2 – r2 = 4, (R – r) (R + r) = 4, R+r= 4, ...(2) [ from (1)], From (1) and (2), we get, R = 2.5 cm and r = 1.5 cm, ∴ Outer and inner diameter are 5 cm and 3 cm respectively., , 190, , Mathematics-X
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PRACTICE-TEST, SURFACE AREAS AND VOLUMES, , Time : 1 Hr., , M.M.: 20, SECTION-A, , 1., , The total surface area of a hemisphere of radius r is .............., , 2., , Which two geometrical shapes are obtained by cutting a cone parallel to its, base? 1, , 3., , 4., , (a) a cylinder and a cone, , (b) a cone and a hemisphere, , (c) a sphere and a cone, , (d) frustum of a cone and a cone, , 1, , The radius (in cm) of the largest right circular cone that can be cut out from a, cube of edge 4.2 cm is, 1, (a) 4.2, , (b) 2.1, , (c) 8.4, , (d) 1.05, , The volume of a cube is 1000 cm3. Find the length of the side of the cube., , 1, , SECTION-B, 5., , The radii of the ends of a frustum of a cone 45 cm high are 28 cm and 7 cm. Find, its volume., 2, , 6., , A solid sphere of radius 10.5 cm is melted and recast into smaller solid cones,, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed. 2, , 7., , A cube and a sphere have equal total surface area. Find the ratio of the volume, of sphere and cube., 2, SECTION-C, , 8., , A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its, top, which is open, is 5 cm. It is filled with water up to brim. When lead shots,, each of which is a sphere of radius 0.5 cm are dropped in to the vessel, one-fourth, of the water flows out. Find the number of lead shots dropped in the vessel. 3, , 9., , A large right circular cone is made out of a solid cube edge 9 cm. Find the, volume of the remaining solid., 3, SECTION-D, , 10. In a hospital, used water is collected in a cylindrical tank of diameter 2 m and, height 5 m. After recycling, this water is used to irrigate a park of hospital, whose length is 25 m and breadth is 20 m. If tank is filled completely then what, will be the height of standing water used for irrigating the park?, 4, Mathematics-X, 191
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CHAPTER, , 14, , Statistics, , Statistics, , Measures of central Tendancy, Mean, , Median, , Mode, , Methods to find mean, median, mode, , Direct, method, , No Graphical, Representation, , Short cut, method, , Step, Deviation, method, , For, ungroup, deta, , For, Grouped, data, Graphical, Reporesentation, Histogram, , For, ungroup, Data, , For, Grouped, data, , Graphical, Represenation, Ogive, , Less than, ogive, , 192, , More, than, ogive, , Mathematics-X
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LIST OF FORMULES, , 1., , Mean x, (a) For raw data x =, , i. e. x =, , x + x + ... + xn, xi, = 1 2, n, n, , sum of observations, no of observations, , (b) For Grouped data, (i) If small calculation then we apply Direct method, f i xi, x = fi, , (ii) If calculations are tedius or observations are large then we apply short cut/, Assumed Mean method or step Deviation method, Short cut/Assumed Mean Method, fi di, x = a + f i , a → assumed mean, , di = xi – a, Step Deviation Method, Σf i ui, di, , h → class size, x = a + Σf × h , ui =, h, i, , 2., , Median, (a) For ungrouped data we first arrange data in ascending or descending order., th, , n + 1, Count number of times say n. If n is odd then Median = , observation, 2 , th, , n, n, , + + 1, 2, , It n is even then Median = 2 , 2, Mathematics-X, , th, , obsevation, , 193
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(b) For grouped data, n, , – cf , ×h., Median = l + 2, fi, , ( f1 – fo ), (3) Mode = l + 2 f – f – f × h (For grouped data), ( 1 o 2), For ungrouped data mode is the most frequent observation., NOTES:, , 1., , Empirical relationship between three measures of central tendency:, mode = 3 median – 2 mean., , 2., , It class interval is discontinuous then make it continuous by subtracting 0.5 from, Lower Limit and adding 0.5 to upper limit., , 3., , xi = class mark =, , 4., , h = class size = Upper Limit – Lower limit, , 5., , Modal class → A class interval having maximum frequency., , 6., , Median class → A class interval is which cumulative frequency is greater then and, nearest to, , 7., , Upper Limit + Lower Limit, 2, , n, ( n = Σfi ), 2, , The median of a group data can be obtained graphically as the x coordinate of the, point of intersection of more than and less than ogive., Y, , More than, ogive, Less than ogive, , X, Median, , 194, , Mathematics-X
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8., , It mean of x1, x2, ..... xn is x then, (a) Mean of kx1, kx2 ....., kxn is k x, (b) Mean of, , x1 x2, xn, x, ,, ....., is, k k, k, k, , (c) Mean of x1 + k, x2 + k, ......, xn + k is x + k, (d) Mean of x1 – k, x2 – k, ..... xn – k is x – k, 9., , It mean of n1 observation is x 1 and mean of n2 observation is x 2 then their combined, Mean =, , n1 x1 n2 x2, n1 n2, , 10. xi = n x, 11. Range of given data is given by, Highest observation – Lowest observation, 12. Graphical Representation of Mode is a Histogram, VERY SHORT ANSWER TYPE(I) QUESTIONS, , 1., 2., 3., 4., 5., 6., 7., 8., 9., (i), , What is the mean of first 12 prime numbers?, The mean of 20 numbers is 18. If 2 is added to each number, what is the new mean?, The mean of 5 observations 3, 5, 7, x and 11 is 7, find the value of x., What is the median of first 5 natural numbers?, What is the value of x, if the median of the following data is 27.5?, 24, 25, 26, x + 2, x + 3, 30, 33, 37, What is the mode of the observations 5, 7, 8, 5, 7, 6, 9, 5, 10, 6., The mean and mode of a data are 24 and 12 respectively. Find the median., Write the class mark of the class 19.5 – 29.5., Multiple Choice Question, If the class intervals of a frequency distribution are 1 – 10, 11 – 20, 21 – 30, ....., 51, – 60 then the size of even class is, , (a) 9, (b) 10, (c) 11, (d) 5.5, (ii) If the class intervals of a frequency distribution are 1 – 10, 11 – 20, 21 – 30 ...., 61, – 70, Then the upper limit of 21 – 30 is, Mathematics-X, , 195
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(a) 21, , (b) 30, , (c) 30.5, , (d) 20.5, , (iii) Consider the frequency distribution., Class, , 0–5, , 6 – 11, , 12 – 17, , 18 – 23, , 24 – 29, , 13, , 10, , 15, , 8, , 11, , (a) 17, (b) 17.5, (c) 18, (iv) Daily wages of a factory workers are recorded as:, , (d) 18.5, , Frequency, , The upper limit of median class is, , Daily wages in ` 121 – 126 127 – 132 133– 138 139 – 144 145 – 150, No. of workers, , 5, , 27, , 20, , 18, , 12, , The lower limit of Modal class is, (a) ` 127, (b) ` 126, (v) For the following distribution, Class, Frequency, , (c) ` 126.5, , (d) ` 133, , 0–5, , 5 – 10, , 10 – 15, , 15 – 20, , 20 – 25, , 10, , 15, , 12, , 20, , 9, , The sum of Lower limits of the median class and modal class is, (a) 15, (b) 25, 10. Fill in the blank, , (c) 30, , (d) 35, , (a) Mode = 3_________ – 2 _________, (b) An ogive curve is used to determine _________, (c) If the point of intersection of more than and less than ogiven is (20. 5, 30.7) then, the median is _________, (d) The mode of a frequency distribution is determined graphically by _________, (e) If the mode is 8 and mean is also 8 then median will be _________, (f), , The measure of central tendency which cannot be determined graphically is, _________, , (g) If the class marks of a continuous frequency distribution are 22, 30, 38, 46, 54, 62, then the class corresponding to class mark 46 is _________, (h) Construction of cumulative frequency distribution table is useful in determining, _________, 196, , Mathematics-X
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(i), (j), (k), (l), , The step deviation formula for finding mean is _________, The formula to find median of grouped data is _________, The formula to find mode of grouped data is _________, The Range of the observations 255, 125, 130, 160, 185, 170, 103 is _________, , 1, (________ + ________), 2, (n) The median of Ist ten prime numbers is ________, (o) The assumed mean method to find mean is ________, SHORT ANSWER TYPE QUESTIONS (I), , (m) Class mark is, , 11. The mean of 11 observation is 50. If the mean of first Six observations is 49 and, that of last six observation is 52, then find sixth observation., 12. Find the mean of following distribution, x, f, , 12, 5, , 16, 7, , 20, 8, , 24, 5, , 28, 3, , 32, 2, , 18, 4, , 20, 3, , 13. Find the median of the following distribution, x, f, , 10, 3, , 12, 5, , 14, 6, , 16, 4, , 14. Find the mode of the following frequency distribution., Class, Frequency, , 0–5 5–10 10 –15 15–20 20–25, 2, 7, 18, 10, 8, , 25–30, 5, , 15. Draw a ‘less than’ ogive of the following data, Less, Less, Less, Less, Less, Less, Less, Less, Less, , Marks, than 20, than 30, than 40, than 50, than 60, than 70, than 80, than 90, than 100, , Mathematics-X, , No. of students, 0, 4, 16, 30, 46, 66, 82, 92, 100, 197
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16. Write the following data into less than cummulative frequency distribution table., Marks, No. of students, , 0–10, 7, , 10–20, 9, , 20–30 30–40, 6, 8, , 40–50, 10, , 17. Find mode of the following frequency distribution., Class Interval, , 25 – 30, , Frequency, , 30 – 35 35 – 40, , 25, , 34, , 40 – 45 45 – 50, , 50, , 42, , 50 – 55, , 38, , 14, , (CBSE 2018 - 19), 18. What is the median of the following data?, , (CBSE 2011), , x, , 10, , 20, , 30, , 40, , 50, , f, , 2, , 3, , 2, , 3, , 1, , 19. Mean of a frequency distribution ( x ) is 45. It Σ f i = 20 find Σ f i xi, (CBSE 2011), SHORT ANSWER TYPE QUESTIONS (II), , 20. If the mean of the following distribution is 54, find the value of P., Class, , 0–20, , 20–40, , 40–60, , 60–80, , 80–100, , 7, , P, , 10, , 9, , 13, , Frequency, , 21. Find the median of the following frequency distribution., C.I., , 0–10, , 10–20, , 20–30, , 30–40, , 40–50, , 50–60, , f, , 5, , 3, , 10, , 6, , 4, , 2, , 22. The median of following frequency distribution is 24 years. Find the missing, frequency x., Age (In years), No. of persons, , 0–10, 5, , 10–20, 25, , 20–30, x, , 30–40, 18, , 40–50, 7, , 23. Find the median of the following data., Marks, No. of student, , 198, , Below 10, 0, , Below 20, 12, , Below 30, 20, , Below 40 below 50 Below 60, 28, 33, 40, , Mathematics-X
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24. Draw a ‘more than type’ ogive of the following data, Weight (In kg.), No. of Students, , 30–35, 2, , 35–40, 4, , 40–45, 10, , 45–50, 15, , 50–55, 6, , 55–60, 3, , 25. Find the mode of the following data., Height (In cm) Above 30 Above 40 Above 50 Above 60 Above 70 Above 80, No. of plants, 34, 30, 27, 19, 8, 2, 26. The following table represent marks obtained by 100 students in a test., Marks obtained, No. of students, , 30 – 35 35 – 40, 14, , 40 – 45, , 45 – 50, , 28, , 23, , 16, , 50 – 55 55 – 60 60 – 65, 18, , Find mean marks of the students., , 8, , 3, , (CBSE 2018 -19), , 27. The following table represent pocket allowance of children of a colony. The, mean pocket allowance is ` 18. Find missing frequency., Daily pocket, allowance, No. of children, , 11 – 13 13 – 15 15 – 17 17 – 19 19 – 21 21 – 23 23 – 25, 3, , 6, , 9, , 13, , k, , 5, , 4, , (CBSE – 2018), 28. Find mode of the following frequency distribution., Class Interval, No. of Students, , 0–20, 15, , 20–40, 18, , 40–60, 21, , 60–80, 29, , 80–100, 17, , The mean of above distribution is 53. Use Empirical formula to find approximate, value of median., LONG ANSWER TYPE QUESTIONS, , 29. The mean of the following data is 53, Find the values of f1 and f2., C.I, f, , 0–20, 15, , 20–40, f1, , 40–60, 21, , 60–80, f2, , 80–100, 17, , Total, 100, , 30. If the median of the distribution given below is 28.5, find the values of x and y., C.I, f, , 0–10, 5, , Mathematics-X, , 10–20, 8, , 20–30, x, , 30–40, 15, , 40–50, y, , 50–60, 5, , Total, 60, 199
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31. The median of the following distribution is 35, find the values of a and b., C.I, f, , 0–10 10–20, 10, 20, , 20–30, a, , 30–40, 40, , 40–50, b, , 50–60, 25, , 60–70, 15, , Total, 170, , 32. Find the mean, median and mode of the following data, C.I, f, , 11–15, 2, , 16–20, 3, , 21–25, 6, , 26–30, 7, , 31–35, 14, , 36–40 41–45 46–50, 12, 4, 2, , 33. The rainfall recorded in a city for 60 days is given in the following table., Raifall (In cm), No. of Days, , 0–10, 16, , 10–20, 10, , 20–30, 8, , 30–40, 15, , 40–50, 5, , 50–60, 6, , Calulate the median rainfall using a more than type ogive., 34. Find the mean of the following distribution by step- deviation method, Daily Expenditure 100–150, (in `), No. of Households, , 4, , 150–200, 5, , 200–250 250–300 300–350, 12, , 2, , 2, , 35. The distribution given below show the marks of 100 students of a class., Marks, , No. of students, , 0–5, , 4, , 5–10, , 6, , 10–15, , 10, , 15–20, , 10, , 20–25, , 25, , 25–30, , 22, , 30–35, , 18, , 35–40, , 5, , Draw a less than type and a more than type ogive from the given data. Hence, obtain the median marks from the graph., 36. The annual profit earned by 30 factories in an industrial area is given below., Draw both ogives for the data and hence find the median., , 200, , Mathematics-X
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` in lakh), Profit (`, No. of Factories, More than or equal to 5, 30, More than or equal to 10, 28, More than or equal to 15, 16, More than or equal to 20, 14, More than or equal to 25, 10, More than or equal to 30, 7, More than or equal to 35, 3, More than or equal to 40, 0, 37. Convert the following distribution into ‘Less than’ and then draw its ogive, (CBSE 2018 -19), Class Interval 30 – 40, Frequency, , 7, , 40 – 50, , 50 – 60, , 5, , 8, , 60 – 70 70 – 80, 10, , 80 – 90 90 – 100, , 6, , 6, , 8, , 38. If mean of the given distribution is 65.6 find the mining frequency. (CBSE 2017), Class Interval, Frequency, , 10 – 30, , 30 – 50, , 50 – 70, , 70 – 90, , 5, , 8, , f1, , 20, , 90 – 110 110 – 130, f2, , 2, , Total, 50, , ANSWERS AND HINTS, , 1. 16.4 approx., , 2. 20, , 3. 9, , 4. 3, , 5. x = 25, , 6. 5, , 7. Median = 20, , 8. 24.5, , 9. (i) B First make intervals continuous, Then find class size, (ii) C, (iii) C, (iv)C (Make continuous intervals Max. frequency is 27), Modal class 15 − 20, (v) B , Median class 10 − 15, , Mathematics-X, , 201
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10. (a) 3 Median – 2 mean, , (b) Median, , (c) 20.5, , (d) Histogram, , (e) 8, , (f) Mean, , (g) 42 – 50 (as difference b/w 2 consecutive observation is 8, 8, 8, form 46 for Lower Limit, Add to 46 for upper Limit), 2, 2, Σ f i ui, (h) Median, (i) x = a + Σf × h, i, , ∴ Subtract, , n, , 2 – Cf 0 , ×h, (j) Median = l + , f1 , , , , , ( f1 − fo ), (k) Mode = l + (2 f − f − f ) × h, 1, o, 2, , (l) Range = 255 – 103 = 152, , (m), , 1, (upper limit + Lower limit), 2, , Σf i d i, (o) x = a + Σf, i, , 11. 56, , 12. 20, , 13. 14.8, , 14. 12.89 approx., , 16., , 17., , 202, , Marks, , No. of students, , less than 10, , 7, , less than 20, , 16, , less than 30, , 22, , less than 40, , 30, , less than 50, , 40, , Class Interval, , Frequency, , 25 – 30, , 25, , 30 – 35, , 34 f0, , 35 – 40, , 50 f1, , 40 – 45, , 42 f2, , 45 – 50, , 38, , 50 – 55, , 14, Mathematics-X
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Mode = l +, , ( f1 – f0 ), , ( 50 – 34 ) × 5, 16 × 5, × h = 35 +, =, 35, +, (100 – 34 – 42 ), ( 2 f1 – f0 – f 2 ), 24, , = 35 + 3.33 = 38.33, 18., , xi, , fi, , Cfi, , 10, , 2, , 2, , 20, , 3, , 5, , 30, , 2, , 7, , 40, , 3, , 10, , 50, , 1, , 11, , Total, , 11, , N = 11 (odd), th, , N + 1, Median = , observation = 6th observation = 30, 2 , Σfi xi, Σfi xi, 19. x = Σf 45 = 20 Σfi xi = 900, i, , 20. 11, , 21. 27, , 22. 10, , 23. 30, , 25. 63.75 cm, 26., , Mark, , xi, , di, , ui, , fi, , fiui, , 30 – 35, , 32.5, , – 15, , –3, , 14, , – 42, , 35 – 40, , 37.5, , – 10, , –2, , 16, , – 32, , 40 – 45, , 42.5, , –5, , –1, , 28, , – 28, , 45 – 50, , 47.5 = a, , 0, , 0, , 23, , 0, , 50 – 55, , 52.5, , 5, , 1, , 18, , 18, , 55 – 60, , 57.5, , 10, , 2, , 8, , 16, , 60 – 65, , 62.5, , 15, , 3, , 3, , 9, , 110, , –59, , Mathematics-X, , 203
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Σfiui, 59, × 5 = 47.5 – 2.68 = 44.82, x = a + Σfi × h = 47.5 –, 110, 27. (Make Table just like Q 26), Σfiui, x = a + Σfi × h, 18 = 18 +, , ( k – 8) × 2, 40 + k, , 2k – 16 = 0, k=8, 28., , ( f1 – f0 ), Mole = l + 2 f – f – f × h, ( 1 0 2), ( 29 – 21), = 60 + 2 × 29 – 21 – 17 × 20 = 68, (, ), , Mode = 3 Median – 2 mean, 68 = 3 Median – 2 × 53, 68 × 106, = Median, 3, Median = 54, , 29. f1 = 18, f2 = 29, , 30. x = 20, y = 7, , 31. a = 35, b = 25, 32. Mean = 32.4, median = 33, mode = 34.39 approx., 33. Median = 25 cm, , 34. Mean = 211, , 35. Median = 24, , 36. Median = ` 17.5 lakhs., , 37. Less than, , 204, , f, , Cf, , Less than 40, , 7, , 7, , Less than 50, , 5, , 12, , Less than 60, , 8, , 20, , Less than 70, , 10, , 30, Mathematics-X
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Less than 80, , 6, , 36, , Less than 90, , 6, , 42, , Less than 100, , 8, , 50, , Plot (40,7), (50, 12), (60, 20), (70, 30) (80, 36), (90, 42), (100, 50), Join free hand to get ogive., 38., , C.I, , fi, , xi, , fixi, , 10 – 30, , 5, , 20, , 100, , 30 – 50, , 8, , 40, , 320, , 50 – 70, , f1, , 60, , 60f1, , 70 – 90, , 20, , 80, , 1600, , 90 – 110, , f2, , 100, , 100f2, , 110 – 130, , 2, , 120, , 240, , 35 + f1 + f2, 35 + f1 + f2 =, , 50 ⇒ f1 + f2 = 15, x =, , 65.6 =, , (1), , Σfixi, Σfi, 2260 + 60 f1 + 100 f 2, 50, , ⇒ 3 f1 + 5f2 = 51, Solve (1) & (2), , 2260 + 60 f1 + 100 f2, , (2), , f1 = 12, f2 = 3, rrr, , Mathematics-X, , 205
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PRACTICE-TEST, Statistics, Time : 1 Hr., , M.M. : 20, , 1., 2., 3., , SECTION-A, What is the class mark of a class a – b ., Find the mean of all the even numbers between 11 and 21., An ogive curve is used to detemine, , 1, 1, 1, , 4., , (a) Range, (b) Mean, State True/False, , 1, , (c) Mode, , (d) Median, , Mean can be determined graphically, 5., 6., 7., , SECTION-B, The mean of 50 observations is 20. If each observation is multiplied by 3, then, what will be the new mean?, 2, The mean of 10 observations is 15.3. If two observations 6 and 9 are replaced, by 8 and 14 respectively. Find the new mean., 2, Write the modal class for the following frequency distribution, 2, Classes, frequency, , 1–4, , 5–8, , 9 – 12, , 13 – 16, , 17 – 20, , 21 – 24, , 8, , 9, , 1, , 12, , 8, , 9, , SECTION-C, 8., , Find the mean:, , 3, , Marks, less than 20 less than 40 less than 60 less than 80 less than 100, No. of Students, 4, 10, 28, 36, 50, 9., , Find the value of x if the mode is given to be 58 years., Age (in years) 20–30, No. of patients, 5, , 30–40, 13, , 40–50, x, , 50–60, 20, , 3, 60–70, 18, , 70–80, 19, , SECTION-D, 10. The mean of the following frequency distribution is 57.6 and the number of, observations is 50. Find the missing frequencies f1 & f2., 4, 206, , Mathematics-X
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Class Interval 0–20, frequency, 7, , 20–40, f1, , 40–60, 12, , 60–80, f2, , 80–100, 8, , 100–120, 5, , OR, Following is the age distribution of cardiac patients admitted during a month in a, hospital:, Age (in years), No. of patients, , 20–30, 2, , 30–40, 8, , 40–50, 15, , 50–60, 12, , 60–70, 10, , 70–80, 5, , Draw a ‘less than type’ and ‘more than type’ ogives and from the curves, find the, median., ppp, , Mathematics-X, , 207
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CHAPTER, , 15, , Probability, Probabilty, , Experiment, , Experimental, probability, , Random, experiment, , Event, , Outcomes, , Equally likely, , Sample, space, , Complementary, event, Sure event, , Favourable, outcomes, , Impossible, event, , Theoritical, or classical, probabilty, , POINTS TO REMEMBER, , 1., , Probability is a quantitative measure of likelihood of occurrence of an event., , 2., , Probability of an event E =, , 3., , 0 ≤ P (E) ≤ 1, , 4., , If P(E) = 0 then it is an impossible event., , 5., , If P(E) = 1 then it is sure event., , 6., , If E is an event than not E( E ) is called complementary event., , 7., , P( E ) = 1 – P(E) ⇒ P(E) + P( E ) = 1, , 8., , Probability of an event is never negative., , 9., , Sample space : The collection of all possible outcomes of an event., , 208, , Number of outcomes favourable to E, Total number of outcomes, , Mathematics-X
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Examples of Sample space, 1., , When one coin is tossed then S = H, T, , 2., , When two coins are tossed then S = HH, TT, HT, TH, , 3., , When three coins are tossed than S = HHH, TTT, HTT, THT, TTH, THH, HTH,, HHT, , 4., , When four coins are tossed then S = HHHH, TTTT, HTTT, THTT, TTHT, TTTH,, HHHT, HHTH, HTHH, THHH, HTHT, THTH, TTHH, HHTT, THHT, HTTH., , , , , 2coins, , 3 coins, , 2 outcomes, , 2 × 2 outcomes, , 2×2×2=8, outcomes, , , , 1., 2., , 3., , , , , , 1 coin, , , , , , 4 coins, , , 2 × 2 × 2 × 2 = 16, outcomes, , When a die is thrown once then S = 1, 2, 3, 4, 5, 6, n(S) = 6, When two dice are thrown together or A die is thrown twice then, S = (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6), n(S) = 6 × 6 = 36, When 3 dice are thrown or a die is thrown thrice then, n(S) = 6 × 6 × 6 = 36,, n(S) no. of outcomes in sample space, Playing cards n(s) = 52, , Red Cards (26), Heart 13, , Diamond 13, , Black Cards (26), Spade 13, , Class 13, , Each suit contains 1 ace, 1 king, 1 Queen, 1 jack and nine, number cards 2, 3, 4, 5, 6, 7, 8, 9, 10, Face card 12, 4 king, 4 Queen & 4 Jack, Mathematics-X, , Non face card 40, 36 number cards + 4 aces, 209
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VERY SHORT ANSWER TYPE QUESTIONS, , 1., , Fill in the Blanks, (a) The probability of an event is greater than or equal to ............... and is less, than or equal to ..............., [NCERT], (b) The probability of an impossible event is ..............., (c) The probability of an event that is certain to happen is ............... and such an, event is called ..............., [NCERT], (d) The sum of probabilities of all the elementary events of an experiment is, ..............., [NCERT], (e) Probability of an event E + probability of the event not E is equal to ..............., [NCERT], (f) If probability of winning a game is 4/9, then the probability of its losing is, ..............., (g) If coin is tossed twice, then the number of possible outcomes is ..............., (h) If a die is thrown twice, then the number of possible outcomes is ..............., , 2., , State True/False, (a) The probability of an event can be negative., (b) The probability of an event is greater than 1., , 3., , Multiple Choice Questions, (a) Which of the following cannot be the probability of an event?, (A) 0.7, , (B), , 2, 3, , (C) – 1.5, , [NCERT], , (D) 15%, , (b) Which of the following can be the probability of an event?[NCERT Exemplar], 18, 8, (D), 23, 7, (c) An event is very unlikely to happen, its probability is closest to, , (A) – 0.04 (B) 1.004, , (C), , [NCERT Exemplar], (A) 0.0001 (B) 0.001, (C) 0.01, (D) 0.1, (d) Out of one digit prime numbers, one number is selected at random. The, probability of selecting an even number is:, 210, , Mathematics-X
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1, 1, 4, 2, (B), (C), (D), 2, 4, 9, 5, (e) When a die is thrown, the probability of getting an odd number less than3 is:, , (A), , 1, 1, 1, (B), (C), (D) 0, 6, 3, 2, (f) Rashmi has a die whose six faces show the letters as given below, , (A), , A, , B, , C, , D, , A, , C, , If she throws the die once, then the probability of getting C is, (A), , 1, 3, , (B), , 1, 4, , (C), , 1, 5, , (D), , 1, 6, , (g) A card is drawn from a well shuffled pack of 52 playing cards. The event E, is that the card drawn is not a face card. The number of outcomes favourable, to the event E is, 4., , (A) 51, (B) 40, (C) 36, (D) 12, Choose the correct answer from the given four options, (i) If the probability of an even is ‘p’ the probability of its complementary, event will be:, (A) p – 1, , (B) p, , (C) 1 – p, , (D) 1 −, , 1, p, , (ii) In a family of 3 children, the probability of having atleast one boy is:, [CBSE 2014], 7, 1, 5, 3, (B), (C), (D), 8, 8, 8, 4, (iii) The probability of a number selected at random from the numbers 1, 2, 3, .... 15, is a multiple of 4 is:, , (A), , 4, 2, 1, 1, (B), (C), (D), 15, 15, 5, 3, (iv) The probability that a non-leap year selected at random will contains 53 Mondays, is:, , (A), , (A), , 1, 7, , Mathematics-X, , (B), , 2, 7, , (C), , 3, 7, , (D), , 5, 7, 211
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(v) A bag contains 6 red and 5 blue balls. One ball is drawn at random. The, probability that the ball is blue is:, 5, 5, 6, 2, (A), (B), (C), (D), 6, 11, 11, 11, (vi) One alphabet is chosen from the word MATHEMATICS. The probability of, getting a vowel is:, 6, 5, 3, 4, (B), (C), (D), 11, 11, 11, 11, A card is drawn at random from a pack of 52 playing cards. Find the probability, that the card drawn is neither an ace nor a king., , (A), , 5., 6., , Out of 250 bulbs in a box, 35 bulbs are defective. One bulb is taken out at, random from the box. Find the probability that the drawn bulb is not defective., , 7., , Non Occurance of any event is 3:4. What is the probability of Occurance of this, event?, , 8., , If 29 is removed from (1, 4, 9, 16, 25, 29) then find the probability of getting a, prime number., , 9., , A card is drawn at random from a deck of playing cards. Find the probability of, getting a face card., , 10. In 1000 lottery tickets there are 5 prize winning tickets. Find the probability of, winning a prize if a person buys one ticket., 11. One card is drawn at random from a pack of cards. Find the probability that it is, a black card., 12. A die is thrown once. Find the probability of getting a perfect square., 13. Two dice are rolled simultaneously. Find the probability that the sum of the two, numbers appearing on the top is more than and equal to 10., 14. Find the probability of multiples of 7 in 1, 2, 3, .......,33, 34, 35., SHORT ANSWER TYPE QUESTIONS-I, 15. A card is drawn at random from a well shuffled pack of 52 playing cards. Find, probability of getting neither a red card nor a queen., [CBSE 2016], 16. Two different dice are rolled together. Find the probability (a) of getting a doublet,, (b) of getting a sum of 10, of the numbers on the two dice., [CBSE 2018], 212, , Mathematics-X
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17. A box contains 12 balls of which some are red in colour. If 6 more red balls are, put in the box and a ball is drawn at random, the probability of drawing a red, ball doubles than what it was before. Find the number of red balls in the box., [CBSE 2018], 18. An integer is chosen random between 1 and 100. Find the probability that (i) it, is divisible by 8, (ii) Not divisible by 8., [CBSE 2018], 19. Three different coins are tossed together. Find the probability of getting (i), exactly two heads, (ii) at least two heads, 20. Cards marked with number 3, 4, 5, .... 50 are placed in a box and mixed, thoroughly. A card is drawn at random from the box. Find the probability that the, selected cards bears a perfect square number., [CBSE 2016], SHORT ANSWER TYPE QUESTIONS-II, 21. A number x is selected at random from the numbers 1, 2, 3 and 4. Another, number y is selected at random from the numbers 1, 4, 9 and 16. Find the, probability that the product of x and y is less than 16., [CBSE 2016], 22. In a single throw of a pair of different dice, what is the probability of getting (a), a prime number on each dice, (b) a total of 9 or 11., [CBSE 2016], 23. A bag contains 15 white and some black balls. If the probability of drawing a, black ball from the bag is thrice that of drawing a white ball, find the number of, black balls in the bag., [CBSE 2017], 24. Two dice are rolled once. Find the probability of getting such numbers on the, two dice,, (a) whose produce is 12., (b) Sum of numbers on the two dice is atmost 5., 25. There are hundred cards in a bag on which numbers from 1 to 100 are written. A, card is taken out from the bag at random. Find the probability that the number on the, selected card., [CBSE 2016], (a) It is divisible by 9 and is a perfect square, (b) is a prime number greater than 80., 26. In a lottery, there are 10 prizes and 25 are empty. Find the probability of getting, a prize. Also verify P(E) + P( E ) = 1 for this event., Mathematics-X, , 213
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27. P(winning) =, , x, 1, , P(Losing) = . Find x., 12, 3, LONG ANSWER TYPE QUESTIONS, , 28. Cards marked with numbers 3, 4, 5, .........,50 are placed in a box and mixed, thoroughly. One card is drawn at random from the box, find the probability that the, number on the drawn card is, (i) divisible by 7 (ii) a two digit number., 29. A bag contains 5 white balls, 7 red balls, 4 black balls and 2 blue balls. One ball, is drawn at random from the bag. Find the probability that the balls drawn is, (i) White or blue, , (ii) red or black, , (iii) not white, , (iv) neither white nor black, , 30. The king, queen and jack of diamonds are removed from a pack of 52 playing, cards and the pack is well shuffled. A card is drawn from the remaining cards., Find the probability of getting a card of, (i) diamond, , (ii) a jack, , 31. The probability of a defective egg in a lot of 400 eggs is 0.035. Calculate the, number of defective eggs in the lot. Also calculate the probability of taking out a, non defective egg from the lot., 32. In a fair at a game stall, slips marked with numbers 3,3,5,7,7,7,9,9,9,11 are, placed in a box. A person wins if the mean of numbers are written on the slip., What is the probabilty of his losing the game?, 33. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn, at random from the box, find the probability that it bears, (i) a two digit number, , (ii) a perfect square number, , (iii) a number divisible by 5., 34. A card is drawn at randown from a well shuffled deck of playing cards. Find the, probability that the card drawn is, (i) a card of spade or an ace, , (ii) a red king, , (iii) neither a king nor a queen, , (iv) either a king or a queen, , 35. A card is drawn from a well shuffled deck of playing cards. Find the probability, that the card drawn is, 214, , Mathematics-X
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(i) a face card, , (ii) red colour face card, , (iii) black colour face card, 36. Ramesh got ` 24000 as Bonus. He donated ` 5000 to temple. He gave ` 12000, to his wife, ` 2000 to his servant and gave rest of the amount to his daughter., Calculate the probability of, (i) wife’s share, , (ii) Servant’s Share, , (iii) daughter’s share., 37. 240 students reside in a hostel. Out of which 50% go for the yoga classes early, in the morning, 25% go for the Gym club and 15% of them go for the morning, walk. Rest of the students have joined the laughing club. What is the probability, of students who have joined laughing club?, 38. A box contains cards numbered from 11 to 123. A card is drawn at random from, the box. Find the probability that the number on the drawn card is:[CBSE 2018], (i) A square number, , (ii) a multiple of 7., , 39. A die is thrown twice. Find the probability that:, (i) 5 will come up at least once, (ii) 5 will not come up either time, , [CBSE 2019], , 40. Cards marked 1, 3, 5 .... 49 are placed in a box and mixed thoroughly. One card, is drawn from the box. Find the probability that the number on the card is :, [CBSE 2017], (i) divisible by 3, , (ii) a composite number, , (iii), not a perfect square (iv), multiple of 3 and 5, 41. Red queens and black jacks are removed from a pack of 52 playing cards. Find, the probability that the card drawn from the remaining cards is: [CBSE 2015], (i) a card of clubs or an ace, , (ii) a black king, , (iii) neither a jack nor a king, , (iv) either a king or a queen, , 42. A box contain 100 red cards, 200 yellow cards and 50 blue cards. If a card is, drawn at random from the box, find the probability that it will be: [CBSE 2012], (a) a blue card, (b) not a yellow card, (c) neither yellow nor a blue card, Mathematics-X, , 215
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ANSWERS AND HINTS, , 1., , (a) 0 and 1, (e) 1, , (b) 0, (f) 5/9, , 2., , (a) False, because 0 ≤ P(A) ≤ 1, , (c) 1 and sure event(d) 1, (g) 4, (h) 36, , (b) False, because 0 ≤ P(A) ≤ 1, 3., , (a), (c), (e), (g), , (C), (b) (C), (A) (as unlikely to happen), (d) (B) (prime no. 2, 3, 5, 7), (A), (f) (A) (probability 2/6), (B) (Face card 12 Remaining cards 40), , 4., , (i) (C) (P + P = 1), (ii) (A) (Sample space = bbb, bbg, bgb, gbb, ggg, ggb, gbg, ggb), (iii)(C) (Probability 3/15), (iv)(A) (Total weeks 52, Remaining day 1, sample space = {S, M, Tu, W, Th, F,, Sat}), (v) (C), (vi) (D) (vowels A, A, E, I), , 5., , Total = 52, No. of Aces = 4, No. of kings = 4, P (neither ace nor king) =, , 6., , P(not defective) = 1 −, , 7., , Total case 3 + 4 = 7, P(occurrence) =, , 8., 9., , 35 43, =, 250 50, , 4, 7, , P(prime no.) = 0, Face card 12, P(face card) =, , 216, , 44 11, =, 52 13, , 12 3, =, 52 13, , Mathematics-X
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10. Probability of winning =, , 5, = 0.005, 1000, , 26 1, =, 52 2, 12. Sample space 1, 2, 3, 4, 5, 6, Perfect square 1, 4, , 11. Total black cards 26,, , 2 1, =, 6 3, 13. Favourable cases (4, 6), (5, 5), (6, 4), (5, 6), (6, 5), (6, 6), , P(perfect square) =, , 6 1, =, 36 6, 14. Multiples of 7 are 7, 14, 21, 28, 35, , P(sum of two numbers is ≥ 10) =, , Probability (multiple of 7) =, , 5 1, =, 35 7, , 15. No. of red cards = 26, No. of Queens = 04 – 2 = 02 (as 2 red queens are included already), No. of cards that are neither red nor queen = 56 – (26 + 2) = 24, 24 6, =, 52 13, 16. (i) Doublets are (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), , Required probability =, , 6 1, =, 36 6, (ii) Sum 10 cases (4, 6), (5, 5), (6, 4), , Required probability =, , Required probability =, 17., , 3, 1, =, 36 12, , x+6, x, = 2 ⇒ x = 3, 12 , 18, , 18. Total outcomes between 1 and 100 = 98, (i) Nos. divisible by 8 = 8, 16, 24, ...., 96, favourable cases = 12, Mathematics-X, , 217
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Required probability =, , 12 6, =, 98 49, , 6 43, =, 49 49, 19. Sample space HHH, TTT, HTT, THT, TTH, THH, HTH, HHT, , (ii) Probability (integer is not divisible by 8) = 1 −, , (i) P(exactly 2 heads) =, , 3, 8, , 4 1, = [Favourable cases HHT, HTH, HHT, HHH), 8 2, 20. Total cards = 50 – 3 + 1 = 48, perfect squares are 4, 9, 16, 25, 36, 49, , (ii) P(atleast 2 heads) =, , 6 1, =, 48 8, 21. Sample space, (1, 1), (1, 4), (1, 9), (1, 16), (2, 1), (2, 4), (2, 9), (2, 16), (3, 1), (3, 4), (3, 9), (3, 16), (4, 1), (4, 4), (4, 9), (4, 16), Favourable cases xy < 16, (1, 1), (1, 4), (1, 9), (2, 1), (2, 4), (3, 1), (3, 4), (4, 1), , Required probability =, , Required probability =, , 8 1, =, 16 2, , 22. Total outcomes = 36, (a) Favourable outcomes, (2, 2), (2, 3), (2, 5), (3, 2), (3, 3), (3, 5), (5, 2), (5, 3), (5, 5), Required probability, , 9 1, =, 36 4, , (b) Favourable outcomes, (3, 6), (4, 5), (5, 4), (6, 3), (5, 6), (6, 5), Required probability =, , 218, , 6 1, =, 36 6, , Mathematics-X
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23., , x, 15, , x = 45, = 3×, 15 + x, 15 + x, No. of black balls = 45, , 24. (a) S = (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) , (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) , , , (3,, 1), (3,, 2), (3,, 3), (3,, 4), (3,, 5), (3,, 6), , , (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) , , , (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) , Favourable outcomes (2, 6), (3, 4), (4, 3), (6, 2), 4 1, =, 36 9, (b) Favourable outcomes (sum ≤ 5), = (1, 1), (1, 2), (1, 3) (1, 4) (2, 1) (2, 2) (2, 3) (3, 1) (3, 2) (4, 1), , Required probability =, , Required probability =, , 10 5, =, 36 18, , 25. (i) Total number = 100, Number divisible by 9 and a perfect square = 9, 36, 81, 3, = 0.03, 100, (ii) Prime no. > 80 are 83, 89, 97, , Required probability, , Required probability, , 3, = 0.03, 100, , 26. Total tickets = 35, P(E) = P(getting a prize) =, , 10 2, =, 35 7, , P(E) = P(not getting a prize) =, , 25 5, =, 35 7, , 2 5 7, + = =1, 7 7 7, 27. P(winning) + P(losing) = 1, , P(E) + P (E) =, , x 1, + = 1, x = 8, 12 3, , Mathematics-X, , 219
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28. Total cards = 50 – 3 + 1 = 48, (i) No. divisible by 7 are 7, 14, 21, 28, 35, 42, 49, 7, 48, (ii) Two digit no. are 10, 11, 12, .... 50, No. of favourable outcomes = 50 – 10 + 1 = 41, , Required probability =, , Required probability =, 5+ 2 7, =, 18, 18, , 29. (i), , (ii), , 41, 48, , 7 + 4 11, =, 18, 18, , 7 + 4 + 2 13, 7+2 9 1, =, =, =, (iv), 18, 18, 18, 18 2, 30. (i) Remaining cards = 52 – 3 = 49, Remaining diamonds = 13 – 3 = 10, , (iii), , Required probability =, , 10, 49, , 3, (as 1 jack has been removed), 49, 31. Total eggs = 400, P(defective eggs) = 0.035, Let defective eggs = x, , (ii) P(jack) =, , x, = 0.035, 400, x = 400 × 0.035, x = 14, P(non defective) = 1 – 0.035 = 0.965, , 32. Mean =, , 3 + 3 + 5 + 7 + 7 + 7 + 9 + 9 + 9 + 11 70, =7, =, 10, 10, , P(he loses) = 1 −, , 7, 3, =, 10 10, , 33. Total no. = 90, (i) Two digit no.s 10, 11, 12, ...., 90, No. of favourable cases = 90 – 10 + 1 = 81, Required probability =, 220, , 81 9, =, 90 10, , Mathematics-X
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(ii) Perfect square no. = 1, 4, 9, 16, 25, 36, 49, 64, 81, Required probability =, , 9, 1, =, 90 10, , (iv) No.s divisible by 5, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, Required probability =, , 18 1, =, 90 5, , 34. (i) P(a card of spade or an ace) =, (ii) P(red king) =, , 13 + 3 16 4, =, =, 52, 52 13, , 2, 1, =, 52 26, , (iii)P(neither a king nor a queen) = 1 –, (iv)P(either a king or a queen) =, , 8, 2, =, 52 13, , (iii), , 6, 3, =, 52 26, , P(students who have joined laughing clubs) =, , 10, 1, =, 100 10, , 35. (i), , 12 3, =, 52 13, , 8, 2 11, = 1− =, 52, 13 13, , 36. (i) P(wifes share) =, , (ii), , 6, 3, =, 52 26, , 12000 1, =, 24000 2, , (ii) P(servant’s share) =, , 2000, 1, =, 24000 12, , 5000, 5, =, 24000 24, 37. 10% students joined laughing club, , (iii)P(Daughter’s share) =, , 38. Total cards = 123 – 11 + 1 = 113, (i) Square numbers 16, 25, 36, 49, 64, 81, 100, 121, 8, 113, (ii) Multiple if 7 are 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112,, 16, 119. Required Probality =, 113, , Required probability =, , Mathematics-X, , 221
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39. Total outcomes = 36, 11, 36, Favourable cases (1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (5, 1), (5, 2), (5,, 3), (5, 4), (5, 6), , (i) P(5 will come up at least once) =, , (ii) P(5 will not come up either time) = 1 −, , 11 25, =, 36 36, , 40. S = 1, 3, 5, ...., 49. Total outcome = 25, (i) No. divisible by 3 are 3, 9, 15, 21, 27, 33, 39, 45, 8, 25, (ii) Composite No.s 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, , Required probability =, , 10 2, =, 25 5, (iii)P(not a perfect square) = 1 – P(perfect square) {Perfect square 1, 9, 25, 49}, , Required probability =, , = 1−, , 4, 21, =, 25, 25, , (iv) Multiple of 3 and 5, ⇒ Multiple of 15 = 15, 45, Required probability =, 41. (i), , 2, 25, , 16 4, =, 52 13, , 8, 2 11, = 1− =, 52, 13 13, 50 1, =, 42. (a) P(blue card) =, 350 7, , (iii) 1 −, , (c) P(neither yellow nor blue) =, , (ii), , 2, 1, =, 52 26, , (iv), , 8, 2, =, 52 13, , (b) P(not yellow card) =, , 150 3, =, 350 7, , 100 2, =, 350 7, , rrr, , 222, , Mathematics-X
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PRACTICE-TEST, Probabiltiy, Time : 1 Hr., , M.M. : 20, SECTION-A, , 1., , A die is thrown once. find the probability of getting an odd number., , 2., , A bag contains 4 red and 6 black balls. one ball is drawn from the bag at random., Find the probability of getting a black ball., 1, A single letter is selected from the word PROBABILITY. The probability it is a, vowel = ..............., 1, The probability of selecting a rotten apple randomly from a heap of 900 apples is, 0.18. The number of rotten apples are ..............., (CBSE 2017)1, , 3., 4., , 1, , SECTION-B, 5., , Find the probability of having 53 friday in a year., , 6., , One card is drawn at random from the well shuffled pack of 52 cards. Find the, probability of getting a black face card or a red face card., 2, A coin is tossed twice. Find the probability of getting atleast one tail. (CBSE 2014) 2, , 7., , 2, , SECTION-C, 8., , A box contains 5 Red, 4 green and 7 white marbles. One marbles is drawn at random, from the box. What is the probability that marble is, (i) not white, , 8., , (ii) neither red nor white, , 3, , A die is thrown once. find the probability that the number., (i) is an even prime number, , (ii) is a perfect square, , 3, , SECTION-D, 10. A box contains cards numbered 1,3,5,........,35. Find the probability that the card, drawn is, (i) a prime number less than 15, , Mathematics-X, , (ii) divisible by both 3 and 15, , 4, , 223
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OR, From a deck of 52 playing cards, king, queen and jack of a club are removed and, a card is drawn from the remaining cards. Find the probability that the card, drawn is, (i) A spade, , (ii) a queen, , (iii) A club, ppp, , 224, , Mathematics-X
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PRACTICE TEST-I (With Solutions), Class : X, Mathematics (Basic), Time : 3 hours, , Max. Marks : 80, , General Instructions:, 1., 2., 3., , 4., 5., , All the questions are compulsory., The question paper consists of 40 questions and it is divided into four sections A, B,, C and D., Section A comprises of 20 questions carrying 1 mark each. Section B comprises of 6, questions carrying 2 marks each. Section C comprises of 8 questions carrying 3, marks each. Section D comprises of 6 questions carrying 4 marks each., There is no overall choice., Use of calculator is not permitted., SECTION A, , 1., 2., , If p and q are co-prime, then p2 and q2 are ............. ., If ΔABC ~ ΔDEF, BC = 3 cm, EF = 4 cm and ar(ΔABC) = 54 cm2, then ar(ΔDEF), = ............., , 3., , If 5 tan θ – 4 = 0, then the value of, (A), , 4., , 5, 3, , (B), , 5, 6, , 5sin θ − 4 cos θ, is, 5sin θ + 4 cos θ, , (C) 0, , (D), , 1, 6, , A die is thrown once. The probability of getting a prime number is:, (A), , 2, 3, , (B), , 1, 3, , (C), , 1, 2, , (D), , 1, 6, , 5., , If the equation x2 + 4x + k = 0 has real and distinct roots, then, , 6., , (A) k < 4, (B) k > 4, (C) k ≥ 4, (D) k ≤ 4, If the circumference and the area of a circle are numerically equal, then diameter of, the circle is, (A), , π, units, 2, , Mathematics-X, , (B) 2π units, , (C) 2 units, , (D) 4 units, 225
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7., , The next term of the A.P. : 7, 28, 63..., , 8., , (A) 70, (B) 84, (C) 97, (D) 112, The distance between the points (a cos θ + b sin θ, 0) and (0, a sin θ – b cos θ) is:, (A) a2 + b2, , 9., 10., 11., 12., , 13., 14., 15., , (B) a + b, , (C) a2 – b2, , (D), , a 2 + b2, If a quadratic polynomial f(x) is a square of a linear polynomial, then its zeros are, equal. (True/False), From a point lying on the circle, infinite number of tangents can be drawn. (True/, False), For what value of p, (– 4) is a zero of the polynomial x2 – 2x – (7p + 3)?, Find the number of solutions of the following pair of linear equations:, x + 2y – 8 = 0, 2x + 4y = 16, Find the area of a triangle with vertices (0, 4), (0, 2) and (3, 0), If A(1, 2), B(4, 3) and C(0, 0) are three vertices of parallelogram ABCD, find, the coordinates of D., In figure, PN || LM. Express x in terms of a, b and c, where a, b and c are lengths, of LM, MN and NK respectively., L, , a, , P, x, , M, , b, , N, , c, , K, , 16. State Basic Proportionality Theorem., 17. What is the probability that a non-leap year has 53 Mondays?, 18. If the total surface area of a solid hemisphere is 462 cm2, find its diameter., 19. A pole casts a shadow of length 2 3 m on the ground, when the sun’s elevation, is 60°, find the height of the pole., 20. If E be an event such that P(E) =, , 226, , 3, , what is P(not E) equal to?, 7, , Mathematics-X
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SECTION B, , 21. Given that 2 is irrational, prove that (5 + 3 2 ) is an irrational number., 22. For what value of ‘k’ the system of equation kx + 3y = 1, 12x + ky = 2 has no, solution., 23. The length of minute hand of a clock is 14 cm. Find the area swept by the minute, hand in 5 minutes., 24. Two cubes each of volume 27 cm3 are joined end to end to form a solid cuboid., Find the surface area of the resulting cuboid., 25. The following distribution table shows the marks scored by 140 students in an, examination:, Marks, 0–10, 10–20, 20–30, 30–40, 40–50, Number of students, 20, 45, 80, 55, 40, Calculate the mode of the distribution., 26. An integer is chosen at random between 1 and 100. Find the probability that it, is:, (i) divisible by 8., (ii) not divisible by 8., SECTION C, , 27. Find the HCF of 180, 252 and 324 by prime factorization method., 28. Find all zeros of the polynomial 2x4 – 9x3 + 5x2 + 3x – 1, if two of its zeros are, (2 + 3) and (2 − 3) ., 1, 1, 2, +, = ; x ≠ 1, 2, 3, (x − 1)(x − 2) (x − 2)(x − 3), 3, 30. The ninth term of an A.P. is equal to seven times the second term and twelth term, exceeds five times the third term by 2. Find the first term and the common difference., 31. Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the, squares of the other two sides., 32. Two tangents TP and TQ are drawn to a circle with centre O, from an external point, T. Prove that ∠PTQ = 2∠QPQ., , 29. Solve for x :, , 1 − cos θ, 1 + cos θ, 34. In ΔABC, ∠B = 90°, BC = 5 cm and AC – AB = 1 cm. Find the value of sin C and, cos C., , 33. Prove that (cot θ – cosec θ)2 =, , SECTION D, , Mathematics-X, , 227
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35. Draw the graph of the following equations and answer the following questions :, x + y = 5,, x–y=5, (i) Find the solution of the equations from the graph., (ii) Shade the triangular region formed by the lines and the y-axis., 36. If A and B are (– 2, – 2) and (2, – 4) respectively, find the coordinates of P such that, 3, AP = AB and P lies on the line segment AB., 7, 37. Construct ΔABC with BC = 7 cm, ∠B = 60° and AB = 6 cm. Construct another, 3, triangle whose sides are times the corresponding sides of ΔABC., 4, 38. As observed from the top of 100 m high light house from the sea level, the angles of, depression of two ships are 30° and 45°. If one ship is exactly behind the other on, the same side of the light house, find the distance between the two ships. (Use 3 =, 1.732), 39. A hollow sphere of internal and external diameter 4 cm and 8 cm respectively is, melted to form a cone of base diameter 8 cm. Find the height and the slant height of, cone., 40. Find the mean and median of the following distribution:, Class, 11–13 13–15 15–17 17–19 19–21 21–23 23–25, Frequency, 3, 6, 9, 13, 18, 5, 4, ANSWERS AND HINTS, , 1., 2., 3., 4., 5., 6., , Co-prime, 96 cm2, (C) 0, (C) 1/2, (A) k < 4, (D) 4 units, , 7., , (D), , 112, , 8., , (D), , a 2 + b2, , 9., , True, , 228, , Mathematics-X
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10. False, 11. (– 4)2 – 2 (– 4) – (7p + 3) = 0, p = 3, a1 b1 c1, 12. As a = b = c so, given pair of linear equations has infinitely many solutions., 2, 2, 2, 1, [0(2 – 0) + 0(0 – 4) + 3(4 – 0)] = 6 square units, 2, 14. Let coordinates of D be (x, y)., Coordinates of the mid-point of AC = Coordinates of the mid-point of BD, , 13. Δ =, , 1+ 0 2 + 0 4 + x 3 + y, ,, ,, , =, , 2, 2 2, 2 , ∴ x = – 3 and y = – 1, Hence, coordinate of D is (– 3, – 1), 15. ΔKLM ~ ΔKPN (AA similarity criterion), LM, KM, a, b+c, ac, =, ⇒ =, ⇒ x=, PN, KN, x, c, b+c, 16. Correct statement., , 17. P(53 Mondays) =, , 1, 7, , ⇒ r = 7 cm, Diameter = 14 cm, 19. AB be the pole and BC be its shadow., 18. 3πr2 = 462, , A, , AB, = tan 60°, In ΔABC,, BC, , AB = 2 3 × 3 = 6 m, 20. P(not E) = 1 −, , 3 4, =, 7 7, , C, , 60°, B, , 21. Let us assume that 5 + 3 2 be a rational number such that it can be written as 5, a, + 3 2 = ; b ≠ 0, a and b are co-prime numbers., b, , Mathematics-X, , 229
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2 =, , a − 5b, 3b, , RHS is rational. So, LHS is also rational which contradict that, , 2 is irrational., , So, our assumption is wrong. Therefore, 5 + 3 2 is irrational number., 22. For no solution,, k, 3 1, = ≠, 12 k 2, k = ± 6 or k ≠ 6, ∴, k= – 6, 23. Angle swept in 5 minutes = 30°, , Area swept in 5 minutes =, 24. Side of cube =, , 3, , 30° 22, 1, ×, × 14 × 14 = 51 cm 2, 360° 7, 3, , 27 = 3cm, , Length, breadth and height of cuboid is 3 + 3 = 6 cm, 3 cm, 3 cm respectively., Surface Area of cuboid = 2(6 × 3 + 3 × 3 + 3 × 6) = 90 cm2, 25. Modal class = 20 – 30, Mode = 20 +, , 40 − 24, × 10 = 28, 2 × 40 − 24 − 36, , 26. Number of integers between 1 to 100 is 98., (i) P(divisible by 8) =, , 12 6, =, 98 49, , (ii) P(not divisible by 8) = 1 −, , 6 43, =, 49 49, , 27. 180 = 22 × 32 × 5, 252 = 22 × 32 × 7, 324 = 22 × 34, HCF (180, 252, 324) = 22 × 32 = 36, 28. Let p(x) = 2x4 – 9x3 + 5x2 + 3x – 1, (2 +, 230, , 3 ) and (2 –, , 3 ) are zeros of p(x)., Mathematics-X
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x – (2 +, ∴, , 3 ) and x – (2 – 3 ) are factors of p(x)., , [x – (2 +, , 3 )] [x – (2 – 3 )] = x2 – 4x + 1 is also factor of p(x), By long division,, p(x) = (x2 – 4x + 1) (2x2 – x – 1), = [x – (2 +, ∴, , 29., , 3 )] [x – (2 –, , zeros of p(x) are 1,, , 3 )] (2x + 1) (x – 1), , −1, , (2 +, 2, , 3 ) and (2 –, , 3), , 1 1, 1 2, +, =, , x − 2 x − 1 x − 3 3, ⇒, , 1, 2(x − 2), 2, ×, =, x − 2 (x − 1)(x − 3) 3, , ⇒ x2 – 4x + 3 = 3, ⇒ x2 – 4x = 0, ⇒ x(x – 4) = 0, either x = 0 or x = 4, 30. a9 = 7a2, a + 8d = 7(a + d), ⇒, , d = 6a, , ...(1), , a12 = 5a3 + 2, ⇒ 4a – d + 2 = 0, , ...(2), , from (1) and (2), we have, a = 1 and d = 6, 31. Correct proof, 32. Join OP, OQ and PQ., Let ∠PTQ = x, , P, , In ΔPTQ, ∠TQP + ∠TPQ + ∠PTQ = 180°, ⇒, , ∠TQP + ∠TPQ = 180° – x, , ...(1), , Also, TP = TQ ( tangents from an external point), ∴, Mathematics-X, , ∠TQP = ∠TPQ, , T, Q, , ...(2), 231
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37. Correct construction, 38., A, 45° 30°, 100 m, 30°, , 45°, C, , In ΔABD,, , B, , D, , AB, = tan 45° ⇒, BD, , BD = 100 m, , 1, 100, AB, = tan 30° ⇒, =, CD + 100, 3, BC, ⇒, CD = 100(1.732 – 1) = 73.2m, 39. Let height of cone be h cm., Volume of cone = Volume of hollow sphere, In ΔABC,, , 1, 4, π (4) 2 h = π (43 − 23 ), 3, 3, l = 14 cm, , l=, 40., , Class, , fi, , 11–13, , (4) 2 + (14)2 = 2 53 cm, , 3, , xi, 12, , f ix i, 36, , c.f., 3, , 13–15, , 6, , 14, , 84, , 9, , 15–17, , 9, , 16, , 144, , 18, , 17–19, , 13, , 18, , 234, , 31, , 19–21, , 18, , 20, , 360, , 49, , 21–23, , 5, , 22, , 110, , 54, , 23–25, Total, , 4, 58, , 24, , 96, 1064, , 58, , Mean =, , 1064, = 18 ,, 58, , Mathematics-X, , Median = 17 +, , (29 − 18), × 2 = 18.69 approx., 13, 233
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PRACTICE TEST-II, Class : X, Mathematics (Basic), Time : 3 hours, , Max. Marks : 80, , General Instructions:, 1., 2., 3., , 4., 5., , All the questions are compulsory., The question paper consists of 40 questions and it is divided into four sections, A, B, C and D., Section A comprises of 20 questions carrying 1 mark each. Section B comprises, of 6 questions carrying 2 marks each. Section C comprises of 8 questions carrying, 3 marks each. Section D comprises of 6 questions carrying 4 marks each., There is no overall choice., Use of calculator is not permitted., SECTION A, , Question number 1 to 20 carry 1 mark each., 1., 2., 3., , Find the LCM of 96 and 360 by using Fundamental Theorem of Arithmetic., A line segment is of length 5 cm. If the coordinates of its one end are (2, 2) and, that of the other end are (– 1, x), then find the value of x., In figure, PA and PB are two tangents drawn from an external point P to a circle, with centre C and radius 4 cm. If PA ⊥ PB, then find the length of each tangents., A, C, , P, B, , 4., 5., 6., 234, , The first three terms of an A.P. respectively are 3y – 1, 3y + 5 and 5y + 1. Find, the value of y., A die is thrown once. What is the probability that it shows a number greater than, 4?, A solid sphere of radius r is melted and cast into the shape of a solid cone of height r., Find the radius of the base of cone., Mathematics-X
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7., , The graph of y = p(x) is given in the figure. The number of zeros of p(x) are:, (a) one, , (b) three, , (c) zero, , (d) two, , y, , x', , x, , 0, y', , 8., , In the figure : DE || BC then the value of EC is:, (a) 1 cm, , (b) 2 cm, , (c) 1.5 cm, , 1.5 cm, D, , (d) 3 cm, , A, 1 cm, E, , 3 cm, B, , 9., , C, , From a point Q the length of tangent to a circle is 24 cm and distance of Q from the, centre is 25 cm. The radius of the circle is:, , (a) 7 cm, (b) 12 cm, (c) 15 cm, (d) 24.5 cm, 10. The angle of elevation of the top of a 15 metres high tower from a point 15 metres, away from its foot is:, (a) 30°, (b) 45°, (c) 60°, (d) 90°, 11. The difference between the circumference and the diameter of a circle is 30 cm then, the radius of the circle is:, (a) 5 cm, (b) 7.7 cm, (c) 7 cm, (d) 6 cm, 12. Probability of event E + Probability of event ‘not E’ = ................, 13. A polynomial of degree two is called ................ polynomial., 14. The line x – y = 8 intersect y-axis at (0, – 8), (T/F), 15. Number of solution in the given pair of equation is infinitely many solutions., (T/F), x + 2y – 8 = 0, 2x + 4y = 16, Mathematics-X, 235
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16. 3 cot2 60° + sec245 = ................, 17. Cards marked with numbers 3, 4, 5 .... 50 are placed in a box and mixed thoroughly., A card is drawn at random from the box, find the probability that the selected card, bears a perfect square number., 18. In the figure ΔABC, DE||AB. If AD = 2x, DC = x + 3, BE = 2x – 1 and CE = x then, find the value of x., A, , D, , B, , C, , E, , 19. In the figure, l||m, ∠OAC = 80°, ∠ODB = 70°. Is ΔOCA ~ ΔODB?, A, , C, 80°, , l, , O, 70°, D, , B, , m, , 20. Find the value of k, for which one root of the quadratic equation Kx2 – 14x + 8, = 0 is six times the other., Question number 21 to 26 carry 2 mark each., 21. On a square handkerchief, nine circular designs each of radius 7 cm are made., Find the area of the remaining portion of the handkerchief., , 236, , Mathematics-X
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22. Write a rational number between 2 and 3 ., 23. For what value of k, will the following system of equations have no solutions?, (3k + 1)x + 3y = 2, 2, (k + 1)x + (k – 2)y = 5, 24. A cylindrical tub, whose diameter is 12 cm and height 15 cm is full of ice-cream. The, whole ice cream is to be divided into 10 children in equal ice-cream cones, with, conical base surmounted by hemispherical top. If the height of conical portion is, twice the diameter of base. Find the diameter of conical part of ice-cream cone., 25. Find the mean of the following frequency distribution:, Class, 0–5, 5–10, 10–15, 15–20, 20–25, 25–30, Frequency, 1, 2, 2, 6, 7, 2, 26. Cards are marked with the numbers from 2 to 151 are placed in a box and mixed, thoroughly. One card is drawn at random from this box. Find the probability, that the number on the card is:, (i) a prime number less than 75., (ii) an odd number., SECTION C, , Question number 27 to 34 carry 3 mark each., 27. Evaluate : (cos2 20° + cos2 70°) +, , cot 25°, + cot 5° cot 10° cot 60° cot 80° cot, tan 65°, , 85°., 28. QT and RS are medians of a triangle PQR right angled at P. Prove that 4(QT2 +, RS2) = 5QR2., 29. If α and β are zeroes of the polynomial p(x) = 2x2 + 11x + 5, find the value of, 1 1, + − 2αβ ., α β, 30. Prove that :, , sin θ, tan θ, = cos θ cosec θ + cot θ., +, 1 − cos θ 1 + cos θ, , 31. Find the roots of the equation, , 1, 1, 11, +, =, x ≠ − 4, 7 ., x + 4 x − 7 30, , 32. Show that one and only one out of n, n + 2, n + 4 is divisible by 3, where ‘n’ is any, positive integer., Mathematics-X, , 237
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33. The sum of first six terms of an A.P. is 42. The ratio of its 10th term to 30th term is 1, : 3. Calculate the first and 13th term of the A.P., 34. In figure, AB is a chord of a circle, with centre O, such that AB = 16 cm and radius of, circle is 10 cm. Tangents at A and B intersect each other at P. Find the length of PA., P, , B, R, , O, , A, Q, , C, , 35. Places A and B are 100 km apart on a highway. One car starts from A and, another from B at the same time. If the cars travel in the same direction at, different speeds, they meet in 5 hours. If they travel towards each other, they, meet in 1 hour. What are the speeds of the two cars?, 36. Determine the ratio in which the line 3x + y – 9 = 0 divides the line-segment, joining the points (1, 3) and (2, 7)., 37. The angle of elevation of the top of a building from the foot of the tower is 30°, and the angle of elevation of the top of the tower from the foot of the building is, 60°. If the tower is 60 m high, find the height of the building., 38. Due to sudden floods, some welfare “associations jointly requested the, government to get 100 tents fixed immediately and offered to contribute 50% of, the cost. If the lower part of each tent is of the form of a cylinder of diameter 4.2, m and height 4 m with the conical upper part of same diameter but of height 2.8, m, and the canvas to be used costs ` 100 per sq. m, find the amount, the, 22 , , associations will have to pay. Use π = , 7, , 39. The following distribution gives the daily income of 50 workers of a factory:, Daily income (in `) 200–250 250–300 300–350 350–400 400–450 450–500, Number of workers, , 10, , 5, , 11, , 8, , 6, , 10, , Convert the distribution to a less than type cumulative frequency distribution and, draw its ogive. Hence obtain the median daily income., , 238, , Mathematics-X
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40. Draw a circle of radius 5 cm. From a point P, 8 cm away from its centre, construct a, pair of tangents to the circle. Measure the length of each one of the tangents., ANSWERS, , 1., , 1440, , 1, 3, 9. 7 cm, 13. Quadratic, , 5., , 2. x = 6 or x = 2, , 3. PA = PB = 4 cm 4. y = 5, , 6. 2r, , 7. No, , 10. 45°, 14. True, , 11. 7 cm, 15. True, , 12. 1, 16. 3, , 19. Yes, , 20. K = 3, , 1, 3, 18., 8, 5, 21. Area of the remaining portion = 378 cm2, , 17., , 8. 2 cm, , 22. 1.5 is rational number lying between 2 and 3, 23. (–1), 26. (i), , 7, 50, , 6+ 3, 3, 31. x = 1, 2, , 27., , 34., 35., 36., 38., 39., , 24. – 6 cm, (ii), , 1, 2, , 29. −, , 25. Mean = 18, (iii), , 11, 150, , 36, 5, , 33. First term = 2, a13 = – 26, , 40, cm, 3, Speed of the two cars are 60 km/h and 40 km/h respectively, Ratio is 3 : 4 internally 37. Height of the building is 20 metres, The associations will have to pay the amount = ` 379500, Median daily income = ` 345, , Mathematics-X, , 239
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No, of workers (c.f.) ⎯ ⎯ →, , Y, (500, 50), , 50, , (450, 40), (400, 34), , 40, 30, , (350, 26), , 20, , (300, 15), (250, 10), , 10, O, , 250 300 350 400 450 500 550, , X, , Daily income (classes) ⎯ ⎯ →, Daily income (classes), , No. of workers (c.f.), , less than 250, , 10, , less than 300, , 15, , less than 350, , 26, , less than 400, , 34, , less than 450, , 40, , less than 500, , 50, , 40. Length of each tangent = 39 cm, , 240, , Mathematics-X
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PRACTICE PAPER- I (WITH SOLUTIONS), CLASS: X, Mathematics (Standard), Time : 3 hours, , Max. Marks : 80, , General Instructions:, 1., 2., 3., , 4., 5., , All the questions are compulsory., The question paper consists of 40 questions and it is divided into four sections A, B,, C and D., Section A comprises of 20 questions carrying 1 mark each. Section B comprises of, 6 questions carrying 2 marks each. Section C comprises of 8 questions carrying 3, marks each. Section D comprises of 6 questions carrying 4 marks each., There is no overall choice., Use of calculator is not permitted., SECTION A, , Question number 1 to 20 carry 1 mark each., 1., , The LCM of two numbers is 1200. Which of the following cannot be their HCF?, , 2., , (a) 4, (b) 5, (c) 6, (d) 3, The median of a given frequency distribution is found graphically with the help of, (a) histogram, , 3., , If the arithmetic mean of x, x + 3, x + 6, x + 9 and x + 12 is 10, then x = .........., (a) 1, , 4., , 1, 3, , (c) 6, , (d) 4, , 1, 6, , 1, 9, , 2, 3, A cylinder, a cone and a hemisphere are of same base and have same height. The, ratio of their volumes is, , (a) 3 : 1 : 2, 6., , (b) 2, , Two different dice are tossed together. The probability that the product of two numbers, on the top of dice is 6 is, (a), , 5., , (b) frequency curve(c) frequency polygon (d) ogive, , (b), , (b) 1 : 2 : 3, , (c), , (c) 3 : 2 : 1, , (d), , (d) 1 : 3 : 2, , Two isosceles triangles have equal angles and their areas are in the ratio 16 :, 25. The ratio of their corresponding heights is, , Mathematics-X, , 241
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(a) 4 : 5, 7., , (b) 5 : 4, , (c) 3 : 2, , In figure, DE||BC and AD =, , (d) 5 : 7, , 1, BD. If BC = 4.5 cm, find DE., 2, A, , D, , E, , C, , B, , 8., , If radii of two concentric circles are 4 cm and 5 cm find the length of each chord of, one circle which is tangent to the other circle., , 9., , If the diameter of a circle is increased by 40%, find by how much percentage its area, increases?, , 10. Find the discriminant of the quadratic equation 3 3x 2 + 10x + 3 = 0, 11. Write the nth term of the A.P. 1 , 1 + m , 1 + 2m ...., m m, m, 12. If (x + a) is a factor of 2x2 + 2ax + 5x + 10, find a., 13. What is the point of intersection of the line represented by 3x – 2y = 6 and the y-axis., 14. Find the coordinates of the point on y-axis which is nearest to the point (– 2, 5)., 15. If the ratio of the height of a tower and the length of its shadow is, , 3 :1 . What, , is the angle of elevation of the sun?, 16. In figure PQ is a tangent at a point C to a circle with centre O. If AB is a diameter, and ∠CAB = 30°, find ∠PCA., P, , A, , C, , O, , Q, B, , 17. If a quadratic polynomial f(x) is factorisable into linear distinct factors, then the total, number of real and distinct zeros of f(x) is ..........., , 242, , Mathematics-X
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18. The distance between the points A (sin θ – cos θ, 0) and B(0, sin θ + cos θ) is .........., 19. Sides of two similar triangles are in the ratio 4 : 9. The areas of these triangles are in, the ratio .........., 20. If tan A =, , 5, , then the value of (cos A – sin A) cosec A is .......... ., 12, SECTION B, , 21. In a single throw of a pair of different dice, what is the probability of getting (i) a, prime number on each dice (ii) a total of 9 or 11?, 22. A hemispherical tank of diameter 3 m is full of water. It is being empied by a pipe at, the rate of 3, , 4, litre per second. How much time will it take to make the tank half, 7, , empty?, 23. Cards marked with numbers 13, 14, 15, .... 60 are placed in a box and mixed, thoroughly. One card is drawn at random from the box. Find the probability that, number on the card drawn is:, (i) divisible by 5, , (ii) a number which is a perfect square., , 24. The length of the minute hand of a clock is 5 cm. Find the area swept by the minute, hand during the time period 6 : 05 am and 6 : 40 am., 25. Solve for x and y:, 4, 3, + 5y = 7 ; + 4y = 5, x, x, , 26. Show that any positive odd integer can be written in the form 6 m + 1, 6 m + 3 or 6, m + 5 where m is a positive integer., SECTION C, , 27. Prove that 2 + 3 is irrational., 28. If x = p sec θ + q tan θ and y = p tan θ + q sec θ, then prove that x2 – y2 = p2 – q2., 29. A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm. AP, and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a, point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the, Mathematics-X, , 243
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perimeter of the ΔABC., 30. Evaluate, without using trigonometric tables:, cot θ tan (90° – θ) – sec (90° – θ) cosec θ + sin2 65° + sin2 25° +, , 3 tan 5° tan, , 45° tan 85°, 31. If α and β are zeroes of the polynomial 6y2 – 7y + 2, find the quadratic polynomial, whose zeroes are, , 1, 1, and ., α, β, , 32. Find a natural number whose square diminished by 10 is equal to five times of 8, more than the given number., 33. Prove that the area of the semi circle drawn on the hypotenuse of a right angled, triangle is equal to the sum of the areas of the semi-circles drawn on the other, two sides of the triangle., 34. An AP consists of 45 terms. The sum of the three middle most terms is 546 and the, sum of the last three terms is 1050. Find the AP., SECTION D, , 35. On selling a tea set at 5% loss and a lemon set at 15% gain, a crockery seller gains `, 7. If he sells the tea-set at 5% gain and the lemon-set at 10% gain, he gains ` 13., Find the actual price of the tea-set and the lemon-set., 36. Point P divides the line segment joining the points A(2, 1) and B(5, – 8) such that, AP, 1, = . If P lies on the line 2x – y + k = 0, find the value of k. Also find the, AB 3, distance between AP., , 37. Draw an isosceles triangle ABC in which AB = AC = 6 cm and BC = 5 cm. Construct, a triangle PQR similar to ΔABC in which PQ = 8 cm. Also justify the construction., 38. A person observes the elevation of a cloud from a point 60 metres above a lake as, 30° and the angle of depression of its reflection in the lake as 60°. Find the height of, the cloud., 39. Water is flowing at the rate of 15 km/h through a pipe of diameter 14 cm into a, cuboidal pond which is 50 m long and 44 m wide. In what time will the level of, water in pond rise by 21 cm?, 244, , Mathematics-X
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40. If the median of the following frequency distribution is 525, in the table given below,, find the value of x and y. if total frequency is 100., Variable, , 0–100 100–200 200–300 300–400 400–500 500–600 600–700 700–800 800–900 900–1000 Total, , Frequency, , 2, , 5, , x, , 12, , 17, , 20, , y, , 9, , 7, , 4, , 100, , ANSWERS KEY, , 1., , (a) 4, , 2. (d) ogive, , 5., 7., , (a) 3 : 1 : 2, AD : AB = 1 : 3, , 6. (a) 4 : 5, , 3. (d) 4, , 4. (b), , 1, 6, , ΔADE ~ ΔABC, , ⇒, 8., , In ΔOPB, PB =, , DE, 1, =, 4.5, 3, DE = 1.5 cm, , (5)2 − (4)2 = 3 cm, , O, A, , 9., , P, , B, , AB = 2 × PB = 2 × 3 = 6 cm, Let diameter of circle be d units., πd 2, square units, Area of circle =, 4, , Diameter of circle after increasing 40% = d + 40% of d =, Increased area of circle =, , 14, d unit, 10, , 196πd 2, square units, 400, , 96πd 2, % increase in area = 4002 × 100 = 96%, πd, 4, Mathematics-X, , 245
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10. D = (10)2 – 4 × 3 3 ×, 11. a =, , 3 = 64, , 1, and d = 1, m, , 1, 1 + m(n − 1), + (n − 1) × 1 =, m, m, 2, 12. 2(– a) + 2a(– a) + 5(– a) + 10 = 0, a=2, 13. 3(0) – 2y = 6, y=–3, ∴ required point is (0, – 3), 14. (0, 5), , an =, , 3 ⇒, , 15. tan θ =, , θ = 60°, , 16. Join OC, OA = OC, ∠OCA = ∠OAC = 30°, ∠PCO = 90°, ∠PCA + ∠OCA = 90°, ∠PCA = 60°, , ⇒, , ∴, , ( sin θ − cos θ ) 2 + (sin θ + cos θ)2 = 2 units, , 17. 2, , 18., , 19. 16 : 81, , 20. (cos A – sin A) ×, , 21. (i), , 9 1, =, 36 4, , (ii), , 1, 12, 7, −1 = −, = cot A – 1 =, sin A, 5, 5, , 6 1, =, 36 6, , 1 2 22 3 3 3, 1, × × × × ×, × Volume of tank, 1, 22. Time = 2, = 2 3 7 2 2 2 = 16 minutes, 25, 2, Water flown in 1 second, 7000, 23. (i), , 246, , 10, 5, =, 48 24, , (ii), , 4, 1, =, 48 12, , Mathematics-X
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30. cot θ . cot θ – cosec θ + (sin265° + cos265°) +, = cot2θ – cosec2θ + 1 +, 31. α + β =, , 3 ×1×1=–1+1+, , 3 tan 5° . tan 45° . cot 5°, 3 =, , 3, , 7, 7, and α.β =, 6, 6, , S=, , 1 1 α+β 7, + =, =, α β, αβ, 2, , P=, , 1 1, 1, 6, × =, =, α β αβ 2, , 7, 6, ∴ required polynomial is k x 2 − x + , , 2, 2, Put k = 2, x2 – 7x + 6, 32. Let number be x, According to Question, x2 – 10 = 5(x + 8), x2 – 5x – 50 = 0, (x – 10) (x + 5) = 0, either x = 10 or x = – 5, but natural number is always positive. Hence, x = 10, 33. Area of semicircle with diameter AB + Area of semicircle with diameter BC, 2, , 2, , π AB , π BC , π AB2 + BC2 π AC2 , = , + , = , = , , 2 2 , 2 2 , 2, 4, 2 4 , = Area of semicircle with diameter AC, 34. a22 + a23 + a24 = 546, a + 22d = 182, ...(1), a43 + a44 + a45 = 1050, a + 43 d = 350, ...(2), From (1) and (2), a = 6 and d = 8, ∴ A.P. is 6, 14, 20,......., 35. Let actual price of the tea set be ` x, and actual price of the lemon set be ` y, 248, , A, , B, , C, , Mathematics-X
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According to the question,, −, , ⇒, , 5, 15, x+, y =7, 100, 100, – x + 3y = 140, , ...(1), , 5, 10, x+, y = 13, 100, 100, ⇒, x + 2y = 260, ...(2), From (1) and (2), x = ` 100 and y = ` 80, 36. P ↔ (3, – 2), 2(3) – (– 2) + k = 0, ⇒, k=–8, , AP =, , (3 − 2)2 + ( −2 − 1)2 = 10 units, , 37. Correct construction and justification., 38. In ΔADE,, , H − 60, = tan 30°, y, , ...(1), , In ΔADF,, , H + 60, = tan 60°, y, , ...(2), , From (1) and (2), H – 60 =, , 2 × 60 tan 30°, tan 60° − tan 30°, , H = 120 m, E, (H–60) m, A, , 30°, , ym, , Hm, , O, , 60m, B, , C, (H+60) m, F, , Mathematics-X, , Hm, , 249
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39. Time required =, , =, , 40., , 50 × 44 ×, , Volume of water at rise of level 21cm, water flown in 1 hour, 21, 100, , 22 7, 7, ×, ×, × 15000, 7 100 100, Variable, 0–100, 100–200, 200–300, 300–400, 400–500, 500–600, 600–700, 700–800, 800–900, 900–1000, Total, , = 2 hours, , Frequency, 2, 5, x, 12, 17, 20, y, 9, 7, 4, 100, , c.f., 2, 7, 7+x, 19 + x, 36 + x, 56 + x, 56 + x + y, 65 + x + y, 72 + x + y, 76 + x + y, , 76 + x + y = 100, x + y = 24, ...(1), 525 = 500 +, , 50 − (36 + x), × 100, 20, , x=9, from (1), y = 15, , 250, , Mathematics-X
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PRACTICE PAPER-II, CLASS: X, Mathematics (Standard), Time : 3 hours, , Max. Marks : 80, , General Instructions:, 1., , All the questions are compulsory., , 2., , The question paper consists of 40 questions and it is divided into four sections A, B,, C and D., , 3., , Section A comprises of 20 questions carrying 1 mark each. Section B comprises of, 6 questions carrying 2 marks each. Section C comprises of 8 questions carrying 3, marks each. Section D comprises of 6 questions carrying 4 marks each., , 4., , There is no overall choice., , 5., , Use of calculator is not permitted., SECTION A, , Question number 1 to 20 carry 1 mark each., 1., , If n is a natural number then 92n – 42n is always divisible by:, , 2., , (a) 5, (b) 13, (c) 5 and 13, (d) none of these, If the mean of the following distribution is 2.6 then the value of y is:, xi, , 1, , 2, , 3, , 4, , 5, , fi, , 4, , 5, , y, , 1, , 2, , (a) 3, 3., , (c) 13, , (d) 24, , If the difference between the circumference and radius of a circle is 37 cm then using, π=, , 22, , the circumference (in cm) of the circle is:, 7, , (a) 154, 4., , (b) 8, , (b) 44, , (c) 14, , (d) 7, , If am ≠ bl then the system of equations ax + by = c and lx + my = n, (a) has a unique solution, , (b) has no solution, , (c) has infinitely many solutions, , (d) may or may not have solution, , Mathematics-X, , 251
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5., , The value of k for which the quadratic equation x2 – kx + 4 = 0 have equal roots:, (a) 4, – 4, , 6., , (b) 16, , (c) – 4, , (d) 4, , The sum of three consecutive terms of an increasing A.P. is 51 and the product of 1st, and 3rd of these terms is 273 then the third term is:, (a) 13, , (b) 9, , (c) 21, , (d) 17, , 7., , If (k + 1) = sec2θ (1 + sin θ) (1 – sin θ), find k., , 8., , If (cosec θ + cot θ) = x find cosec θ – cot θ., , 9., , If a pole 6 m high casts a shadow of 2 3 long on the ground then what is the sun’s, elevation?, , 10. State true or false and justify, “If a die is thrown, there are two possible outcomes an odd number or an even, number. Therefore the probability of getting an odd number is, , 1, ”., 2, , 11. State true or false and justify, “A driver attempts to start a car. The car starts or doesnot start is an equally likely, outcome.”, 12. In an equilateral triangle, the lengths of the median is 3 cm, then find the length, of the side of this equilateral triangle., 13. In the given figure of ΔABC, D and E are points on CA and CB respectively, such that DE || AB, AD = 2x, DC = x + 3, BE = 2x – 1, CE = x find n., A, D, B, , E, , C, , 14. Find the altitude of an equilateral triangle of side 8 cm., 15. Fill in the blanks:, If P(2, 4), Q(0, 3), R (3, 6) and S(a, b) are vertices of a parallelogram then the, value of a + b is .........., 16. Find K if the point P(2, 4) is equidistant from A(5, K) and B(K, 7)., 252, , Mathematics-X
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17. Two tangents making an angle of 60° between them, are drawn to a circle of, radius 2 cm, then find the length of each tangent., 18. If the sum and product of the zeros of the polynomial ax2 – 5x + c is 10 find a and, c., 19. If α, β are zeros of 2x2 – 5x + 1 find a quadratic polynomial whose zeroes are, 2α and 2β., 20. If radii of two concentric circles are 4 cm and 5 cm, then find the length of the, chord of one circle, which is tangent to the other circle., SECTION B, , 21. Prove 3 − 5 is an irrational number., 22. Solve for x and y:, , 4, 3, + 5y = 7, + 4y = 5, x, x, , 23. A solid piece of iron is in the form of a cuboid of dimensions 4.4 m × 2.6 m × 10, m is melted to form a hollow cylinder of internal radius 30 cm and thickness 5, cm. Find the length of the pipe., 24. In the following data, find the values of p and q. Also find the median class and, modal class., C.I., , Frequency, , Cumulative frequency, , 100 – 200, 200 – 300, 300 – 400, 400 – 500, 500 – 600, 600 – 700, , 11, 12, 10, q, 20, 14, , 11, p, 33, 46, 66, 80, , 25. If 7 sin2θ + 3 cos2θ = 4, then find value of tan θ., 26. A box contains cards numbered from 13, 14, 15, ....., 60. A card is drawn at random, from the box. Find the probability that the number on the drawn card is, (i) divisible by 2 or 3, , Mathematics-X, , (ii) a prime number, , 253
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SECTION C, , 27. Show that the cube of any positive integer is of the form 9 m, 9 m + 1 or 9 m + 8., 28. Find all zeroes of the polynomial 2x4 – 10x3 + 5x2 + 15x – 12 when its two zeroes, are, , 3, 3, and −, ., 2, 2, , 29. Solve for x :, , x +1 x − 2, 2x + 3, , x ≠ 1, – 2, 2., +, = 4−, x −1 x + 2, x−2, , 30. Prove that the ratio of the areas of two similar triangles is equal to the square of the, ratio of their corresponding sides., 31. If an isosceles triangle ΔABC in which AB = AC = 6 cm is inscribed in a circle of, radius 9 cm, find the area of the triangle., 32. In an A.P. of 50 terms, the sum of first ten terms is 210 and the sum of last 15 terms, is 2565. Find the A.P., 2, , 3 tan 41° , sin 35° sec 55°, , , 33. Find the value of : , −, , cot 49° , tan10° tan 20° tan 60° tan 70° tan 80° , , 2, , 34. In the given figure ABCD is a trapezium with AB || DC and ∠BCD = 60°. If BFEC, is a sector of a circle with centre C and AB = BC = 7 cm and DE = 4 cm then find the, 22, , , , 3 = 1.732, area of the shaded region: π =, , 7, B, , A, F, D, , 60°, E, , C, , SECTION D, , 35. The angle of elevation of a cloud from a point 60 m above a lake is 30° and the angle, of depression of the reflection of cloud in the lake is 60°. Find the height of the cloud., 36. The height of a cone is 30 cm. A small cone is cut off at the top of a plane parallel to, , 254, , Mathematics-X
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1, of the volume of the given cone, at what height above, 27, the base is the section made?, , the base. If its volume is, , 37. Draw a Δ ABC, with side BC = 7 cm, ∠B = 45°, ∠A = 105°. Then construct a, triangle whose sides are, , 4, times the corresponding sides of ΔABC., 3, , 38. The distribution given below show the marks of 100 students of a class:, Marks, , No. of students, , 0–5, , 4, , 5–10, , 6, , 10–15, , 10, , 15–20, , 10, , 20–25, , 25, , 25–30, , 22, , 30–35, , 18, , 35–40, , 5, , 39. Find the value (s) of k for which the points (3k – 1, k – 2), (k, k–7) and (k – 1, –, k – 2) are collinear., 40. A motor boat whose speed is 18 km/hr in still water takes 1 hour more to go 24, km upstream than to return downstream to the same spot. Find the speed of the, stream., ANSWERS KEYS, , 1., 2., , (C) as a2n – b2n is divisible by both (a + b) and (a – b)., (B), 3. (B), 4. (A), , 5., , (A), , 9. 60°, 10. True, , Mathematics-X, , 6. (C), , 7. 0, , 8., , 1, x, , A = {1, 3, 5}, B = {2, 4, 6}, 255
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1, = P(B), 2, 11. False; Car will start or not is not always equally likely., , P(A) =, , 3, 5, , 12. a = 2, a ≠ – 2, , 13., , 16. K = 3, , 17. 3 cm, , 19. x2 + 5x + 1 = 0, , 20. 6 cm, , 1, , y = –1, 3, 24. P = 11 + 12 = 23, , 22. x =, , 14. 4 3cm, , 15. a + b = 12, , 1, ,c=5, 2, 21. Prove by method of contradiction, , 18. a =, , 23. 112 m, , q = 13, Median class 400–500, Modal class 500–600, 1, 32 2, 26. (i), =, 48 3, 3, 27. Use Euclid’s division lemma, 25. tan θ =, , 28. 4, 1, , 29. x = – 5, x =, , (ii), , 12 1, =, 48 4, , 6, 30. Proof of theorem, 5, , 31. 8 2 cm2, 32. S10 = 210, S50 – S35 = 2565, d = 4, a = 3, A.P. 3, 7, 11, ......, 26, 34. 28.89 cm2, 35. h = 120 m, 3, 36. h = 20 cm, 37. Construction, 38. Ogive median 24 (approximates) from graph, 39. K = 0, 3, 40. 6 km/hr, , 33., , rrr, , 256, , Mathematics-X
