Sequence & Series PYQs Set 3 (PDF)

Set 3 of 10 · Questions 61-90 of 284

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sequence & Series · Set 3 (Q61-90)
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61. $The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is :$ (1) $1356$ (2) $1256$ (3) $1365$ (4) $1465$ 62. $If α and β are the roots of the equation 375 x^{2} - 25 x - 2 = 0 then lim n \to \infty \sum α^{r} + lim n \to \infty \sum β^{r} then equal to :$ (1) $\frac{21}{346}$ (2) $\frac{29}{358}$ (3) $\frac{1}{12}$ (4) $\frac{7}{116}$ 63. $If 2^{10} + 2^{9} \cdot 3^{1} + 2^{8} \cdot 3^{2} + \dots + 2 \cdot 3^{9} + 3^{10} = S - 2^{11} then S is equal to :$ (1) $2 \cdot 3^{11}$ (2) $3^{11} - 2^{12}$ (3) $3^{11} 2+210$ (4) $3^{11}$ 64. $Let A_{1} and A_{2} be two arithmetic means and G_{1}, G_{2} and G_{3} be three geometric means of two distinct positive numbers. Then G_{1}^{4} + G_{2}^{4} + G_{3}^{4} + G_{1}^{2} G_{3}^{2} is equal to$ (1) $(A_{1} + A_{2})^{2} G_{1} G_{3}$ (2) $2(A_{1} + A) G_{1} G_{3}$ (3) $(A_{1} + A_{2}) G_{1}^{2} G_{3}^{2}$ (4) $2(A_{1} + A_{2}) G_{1}^{2} G_{3}^{2}$ 65. $The sum of the series \sum \frac{n ^{2} + 6 n + 10}{( 2 n + 1 ) !} Is equal to$ (1) $\frac{41}{8} e + \frac{19}{8} e^{- 1} - 10$ (2) $- \frac{41}{8} e + \frac{19}{8} e^{- 1} - 10$ (3) $\frac{41}{8} e - \frac{19}{8} e^{- 1} - 10$ (4) $\frac{41}{8} e + \frac{19}{8} e^{- 1} + 10$ 66. $If \frac{1}{1 + 2} + \frac{1}{2 + 3} + \dots + \frac{1}{99 + 100} = m and \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{99 \cdot 100} = n, then the point (m, n) lies on the line$ (1) $11 (x - 1) - 100 (y - 2) = 0$ (2) $11 x - 100 y = 0$ (3) $11 (x - 2) - 100 (y - 1) = 0$ (4) $11 (x - 1) - 100 y = 0$ 67. $The harmonic mean of two numbers is - \frac{8}{5} and their geometric mean is 2 . The quadratic equation whose roots are twice those numbers is$ (1) $x^{2} + 5 x + 4 = 0$ (2) $x^{2} + 10 x + 16 = 0$ (3) $x^{2} - 10 x + 16 = 0$ (4) $x^{2} - 5 x + 4 = 0$ 68. $Let A be the sum of the first 20 terms and B be sum of the first 40 terms of the series 1^{2} + 2. 2^{2} + 3^{2} + 2. 4^{2} + 5^{2} + 2. 6^{2} + ..... If B - 2A = 100 λ, then λ is equal to$ (1) $464$ (2) $496$ (3) $232$ (4) $248$ 69. $The positive integer n , for which the solutions of the equation x (x + 2) + (x + 2) (x + 4) + \dots + (x + 2 n - 2) (x + 2 n) = \frac{8 n}{3} are two consecutive even integers, is :$ (1) $3$ (2) $12$ (3) $9$ (4) $6$ 70. $Let three real numbers a, b, c be in arithmetic progression and a + 1, b, c + 3 be in geometric progression. If a > 10 and the arithmetic mean of a, b and c is 8 , then the cube of the geometric mean of a, b and c is$ (1) $128$ (2) $316$ (3) $120$ (4) $312$ 71. $The greatest positive integer k, for which 49^{k} + 1 is a factor of the sum 49^{125} + 49^{124} + \dots + 49^{2} + 49 + 1 is$ (1) $63$ (2) $65$ (3) $2$ (4) $5$ 72. $The sum of the first n terms of \frac{1 ^{2}}{1} + \frac{1 ^{2} + 2 ^{2}}{1 + 2} + \frac{1 ^{2} + 2 ^{2} + 3 ^{2}}{1 + 2 + 3} + is$ (1) $\frac{n ^{2} - 2 n}{3}$ (2) $\frac{2 n ^{2} + n}{3}$ (3) $\frac{n ( n + 2 )}{3}$ (4) $\frac{2 n ^{2} - n}{3}$ 73. $(\frac{1}{3} + \frac{4}{7}) + (\frac{1}{3 ^{2}} + \frac{1}{3} \times \frac{4}{7} + \frac{4 ^{2}}{7 ^{2}}) + (\frac{1}{3 ^{3}} + \frac{1}{3 ^{2}} \times \frac{4}{7} + \frac{1}{3} \times \frac{4 ^{2}}{7 ^{2}} + \frac{4 ^{3}}{7 ^{3}}) + \dots upto infinite terms, is equal to$ (1) $\frac{4}{3}$ (2) $\frac{6}{5}$ (3) $\frac{5}{2}$ (4) $\frac{7}{4}$ 74. $If the sum of first n terms of an A.P. is cn2, then the sum of squares of these n terms is$ (1) $\frac{n ( 4 n ^{2} - 1 ) c ^{2}}{6}$ (2) $\frac{n ( 4 n ^{2} + 1 ) c ^{2}}{3}$ (3) $\frac{n ( 4 n ^{2} - 1 ) c ^{2}}{3}$ (4) $\frac{n ( 4 n ^{2} + 1 ) c ^{2}}{6}$ 75. $If ω is a complex cube root of unity, then ω^{(\frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \dots \infty)} + ω^{(\frac{1}{2} + \frac{3}{8} + \frac{9}{32} + \dots \infty)} is equal to$ (1) $1$ (2) $-1$ (3) $ω$ (4) $i$ 76. $If a_{1}, a_{2}, a_{3}, \dots \dots are in A . P . such that a_{1} + a_{7} + a_{16} = 40, then the sum of the first 15 terms of this A . P . is:$ (1) $200$ (2) $280$ (3) $120$ (4) $150$ 77. $The expression for a_{n} which satisfies a_{0} = 0, a_{1} = 1 and a_{n} = a_{n - 1} + a_{n - 2}, \forall n \in N - 0, 1 from the following is$ (1) $\frac{1}{5} (\frac{1 + 5}{2})^{n} - \frac{1}{5} (\frac{1 - 5}{2})^{n}$ (2) $\frac{1}{7} (\frac{1 + 7}{2})^{n} - \frac{1}{7} (\frac{1 - 7}{2})^{n}$ (3) $\frac{1}{2} (\frac{1 + 2}{2})^{n} - \frac{1}{2} (\frac{1 - 2}{2})^{n}$ (4) $\frac{1}{3} (\frac{1 + 3}{2})^{n} - \frac{1}{3} (\frac{1 - 3}{2})^{n}$ 78. $Let S_{a} denote the sum of first n terms an arithmetic progression. If S_{20} = 790 and S_{10} = 145, then S_{15} - S_{5} is :$ (1) $395$ (2) $390$ (3) $405$ (4) $410$ 79. $If the roots of the equation x^{3} - 7 x^{2} + 14 x - 8 = 0 are in geometric progression, then the difference between the largest and the smallest roots is$ (1) $4$ (2) $2$ (3) $\frac{1}{2}$ (4) $3$ 80. $Let a_{1}, \frac{a _{2}}{2}, \frac{a _{3}}{2 ^{2}}, \dots, \frac{a _{10}}{2 ^{9}} be a G.P. of common ratio \frac{1}{2} . If a_{1} + a_{2} + \dots + a_{10} = 62, then a_{1} is equal to :$ (1) $2 - 2$ (2) $2 (2 - 2)$ (3) $2 - 1$ (4) $2 (2 - 1)$ 81. $If the sum of the first 20 terms of the series log_{(7^{1/2})} x + log_{(7^{1/3})} x + log_{(7^{1/4})} x + \dots is 460 then x is equal to$ (1) $7^{2}$ (2) $e^{2}$ (3) $7^{1} /2$ (4) $7^{46} /21$ 82. $If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:$ (1) $\frac{4}{3}$ (2) $1$ (3) $\frac{7}{4}$ (4) $\frac{8}{5}$ 83. $If gcd(m, n)= 1 and 1^{2} - 2^{2} + 3^{2} - 4^{2} +.... + (2021)^{2} - (2022)^{2} + (2023)^{2} = 1012 m^{2} n then m^{2} - n^{2} is equal to$ (1) $240$ (2) $200$ (3) $220$ (4) $180$ 84. $Let a_{1} = 8, a_{2}, a_{3}, \dots a_{n} be an A.P. If the sum of its first four terms is 50 the sum of its last four terms is 170, then the product of its middle two terms is ..........$ 85. $Let 3, 7, 11, 15, \dots, 403 and 2, 5, 8, 11, \dots, 404 be two arithmetic progressions. Then the sum, of the common terms in them, is equal to$ 86. $Let 3, 6, 9, 12,... upto 78 terms and 5, 9, 13, 17,... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to___.$ 87. $A pack contains n card numbered from 1 to n. Two consecutive numbered card are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then k - 20 =$ 88. $The sum of the common terms of the following three arithmetic progressions. a. 3, 7, 11, 15, ......................, 399. b. 2, 5, 8, 11, ......................... 359 and c. 2, 7, 12, 17, .........................197, is equal to_____$ 89. $Let a,b,c be positive integers such that \frac{b}{a} is an integer. If a,b,c are in geometric progression and the arithmetic mean of a,b,c is b + 2, then the value of \frac{a ^{2} + a -14}{a + 1} is$ 90. $The roots of the quadratic equation 3 x^{2} - p x + q = 0 are 10^{th} and 11^{th} terms of an arithmetic progression with common difference \frac{3}{2} . If the sum of the first 11 terms of this arithmetic progression is 88, then q - 2 p is equal to$