Sequence & Series PYQs Set 2 (PDF)

Set 2 of 10 · Questions 31-60 of 284

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sequence & Series · Set 2 (Q31-60)
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31. $If three distinct numbers a, b, c are in G . P . and the equations a x^{2} + 2 bx + c = 0 and d x^{2} + 2 ex + f = 0 have a common root, then which one of the following statements is correct?$ (1) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c} are in A.P.$ (2) $d, e, f are in A . P .$ (3) $d, e, f are in G . P .$ (4) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c} are in G.P.$ 32. $If 3^{2} sin 2 α - 1, 14 and 3^{4 - 2} sin 2 α are the first three terms of an A . P . for some α, then the sixthterm of this A . P is:$ (1) $65$ (2) $78$ (3) $81$ (4) $66$ 33. $If the 10^{th} term of an A . P . is \frac{1}{20} and its 20^{th} term is \frac{1}{10}, t hen the sum of its first " 200 term is$ (1) $201 \frac{1}{2}$ (2) $101 \frac{1}{2}$ (3) $301 \frac{1}{2}$ (4) $100 \frac{1}{2}$ 34. $Let \sum a_{k} = α n^{2} + βn . If a_{10} = 59 and a_{6} = 7 a_{1}, then α + β is equal to$ (1) $3$ (2) $12$ (3) $7$ (4) $5$ 35. $Let T_{r} be the r^{th} term of an A.P. If for some m, T_{m} = \frac{1}{25}, T_{25} = \frac{1}{20}, and 20 \sum_{r = 1}^{25} T_{r} = 13, then 5 m \sum_{r = m}^{2 m} T_{r} is equal to$ (1) $98$ (2) $126$ (3) $142$ (4) $112$ 36. $If S_{n} = 4 + 11 + 21 + 34 + 50 + ... to n terms, then \frac{1}{60} (S_{29} - S_{9}) is equal to$ (1) $223$ (2) $226$ (3) $220$ (4) $227$ 37. $If 7 = 5 + \frac{1}{7} (5 + α) + \frac{1}{7 ^{2}} (5 + 2 α) + \frac{1}{7 ^{3}} (5 + 3 α) + \infty, then the value of α is :$ (1) $\frac{6}{7}$ (2) $6$ (3) $\frac{1}{7}$ (4) $1$ 38. $If the sum of the series 20 + 19 \frac{3}{5} + 19 \frac{1}{5} + 18 \frac{4}{5} + \dots upto n^{th} term is 488 and the n^{th} term is negative, then$ (1) $n = 41$ (2) $n^{th} term is - 4 25$ (3) $n = 60$ (4) $n^{th} term is - 4$ 39. $If α, β are natural numbers such that 100^{α} - 199 β = (100) (100) + (99) (101) + (98) (102) + \dots \dots + (1) (199), then the slope of the line passing through (α, β) and origin is :$ (1) $540$ (2) $550$ (3) $530$ (4) $510$ 40. $If a, b and c be three distinct real numbers in G . P . and a + b + c = x b, then x cannot be:$ (1) $- 2$ (2) $- 3$ (3) $4$ (4) $2$ 41. $Let S_{k} = \frac{1 + 2 + 3 + \dots . + k}{2} . If S_{1}^{2} + S_{2}^{2} + \dots .. + S_{10}^{2} = \frac{5}{12} A Then A is equal to$ (1) $283$ (2) $301$ (3) $303$ (4) $156$ 42. $Let a_{1}, a_{2}, a_{3}, a_{4} be an A.P. of four terms such that each term of the A.P. and its common difference l are integers. If a_{1} + a_{2} + a_{3} + a_{4} = 48 and a_{1} a_{2} a_{3} a_{4} + l^{4} = 361, then the largest term of the A.P. is equal to$ (1) $23$ (2) $21$ (3) $27$ (4) $24$ 43. $A person is to count 4500 currency notes. Let an denote the number of notes he counts in the nth minute. If a1 = a2 = .....= a10 = 150 and a10, a11,....are in an AP with common difference -2, then the time taken by him to count all notes is$ (1) $34 minutes$ (2) $125 minutes$ (3) $135 minutes$ (4) $24 minutes$ 44. $If the average of the first n numbers in the sequence 148 , 146, 144, \dots, is 125 , then n is equal to$ (1) $18$ (2) $24$ (3) $30$ (4) $36$ 45. $If the sum and product of the first three terms in an A . P . are 33 and 1155, respectively, then a value of its 11^{th} term is:$ (1) $- 35$ (2) $25$ (3) $- 36$ (4) $- 25$ 46. $Let A = 1, 6, 11, 16, \dots and B = 9, 16, 23, 30, \dots be the sets consisting of the first 2025 terms of two arithmetic progressions. Then n (A \cup B) is$ (1) $3814$ (2) $4003$ (3) $4027$ (4) $3761$ 47. $The sum of the series \frac{1}{1 - 3 \cdot 1 ^{2} + 1 ^{4}} + \frac{2}{1 - 3 \cdot 2 ^{2} + 2 ^{4}} + \frac{3}{1 - 3 \cdot 3 ^{2} + 3 ^{4}} + \dots up to 10 terms is$ (1) $\frac{45}{109}$ (2) $- \frac{45}{109}$ (3) $\frac{55}{109}$ (4) $- \frac{55}{109}$ 48. $In an increasing, geometric series, the sum of the second and the sixth term is \frac{25}{2} and the product of the third and fifth term is 25 . Then, the sum of 4^{th}, 6^{th} and 8^{th} terms is equal to:$ (1) $35$ (2) $30$ (3) $26$ (4) $32$ 49. $Which of the following statements is a tautology?$ (1) $p \to (p \land (p \to q))$ (2) $(p \land q) \to ((p) \to q))$ (3) $(p \land (p \to q)) \to q$ (4) $p \lor (p \land q)$ 50. $If x = \sum_{n = 0}^{\infty} (- 1)^{n} tan^{2 n} θ and y = \sum_{n = 0}^{\infty} cos^{2 n} θ, for 0 < θ < \frac{π}{4} then$ (1) $y (1 - x) = 1$ (2) $y (1 + x) = 1$ (3) $x (1 + y) = 1$ (4) $x (1 - y) = 1$ 51. $1 + 3 + 5^{2} + 7 + 9^{2} + \dots upto 40 terms is equal to$ (1) $40870$ (2) $41880$ (3) $43890$ (4) $33980$ 52. $Find \sum_{t = 1}^{39} f (t) if f : R \to R is defined as f (x + y) = f (x) + f (y), x, y \in R and f (1) = 7$ (1) $5187$ (2) $5460$ (3) $5740$ (4) $5407$ 53. $Let X be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11, \dots., and Y be the set consisting of the first 2018 terms of the arithmetic progression 9, 16, 23, \dots.. Then, the number of elements in the set X \cup Y is _____.$ 54. $Suppose a, b, c are in A.P. and a^{2}, 2 b^{2}, c^{2} are in G.P. If a < b < c and a + b + c = 1, then 9 (a^{2} + b^{2} + c^{2}) is equal to____.$ 55. $If 1, lo g_{10} (4^{x} - 2) and lo g_{10} (4^{x} + \frac{18}{5}) are in arithmetic progression for a real number x, then the value of the determinant 2 (x - \frac{1}{2}) 1 x x - 1 01 x^{2} x 0 is equal to:$ 56. $The 4^{t h} term of GP is 500 and its common ratio is \frac{1}{m}, m ϵ N . Let S_{n} denote the sum of the first n terms of this GP. If S_{6} > S_{5} + 1 and S_{7} < S_{6} + \frac{1}{2}, then the number of possible values of m is _______$ 57. $Let S_{n} (x) = lo g_{a^{1/2}} x + lo g_{a^{1/3}} x + lo g_{a^{1/6}} x + lo g_{a^{1/11}} x + lo g_{a^{1/18}} x + lo g_{a^{1/27}} x + \dots up to n -terms, where a > 1 . If S_{24} (x) = 1093 and S_{12} (2 x) = 265, then value of a is equal to$ 58. $The sum \sum_{n = 1}^{7} \frac{n ( n + 1 ) ( 2 n + 1 )}{4} is equal to$ 59. $Different A.P.'s are constructed with the first term 100, the last term 199, And integral common differences. The sum of the common differences of all such, A.P's having at least 3 terms and at most 33 terms is.$ 60. $Let \frac{1}{16}, a, and b be in G . P . and \frac{1}{a}, \frac{1}{b}, 6 be in A . P ., where a, b > 0 . Then 72 (a + b) is equalto$