Sequence & Series PYQs Set 4 (PDF)

Set 4 of 10 · Questions 91-120 of 284

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sequence & Series · Set 4 (Q91-120)
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91. $Let S_{n} = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \dots upto n terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is 2026 S_{2025}, then the absolute difference betwen 20^{th} and 15^{th} terms of the A.P. is$ (1) $20$ (2) $90$ (3) $45$ (4) $25$ 92. $If θ = \frac{π}{6}, then the 10^{th} term of the series 1 + (cos θ + i sin θ)^{1} + (cos θ + i sin θ)^{2} + \dots is$ (1) $-1$ (2) $-i$ (3) $\frac{1}{2} + \frac{3 i}{2}$ (4) $1$ 93. $If the roots α, β, γ of the equation x^{3} - 6 x^{2} + px + 10 = 0 are in arithmetic progressions, then α^{3} + β^{3} + γ^{3} =$ (1) $132$ (2) $134$ (3) $629$ (4) $645$ 94. $Let S_{1} be the sum of first 2 n terms of an arithmetic progression. Let S_{2} be the sum of first 4 n terms of thesame arithmetic progression. If (S_{2} - S_{1}) is 1000, then the sum of the first 6 n terms of the arithmeticprogression is equal to:$ (1) $1000$ (2) $7000$ (3) $5000$ (4) $3000$ 95. $If tan B = \frac{2 sin A sin C}{sin ( A + C )}, then tan A, tan B and tan C are in$ (1) $Arithmetic progression$ (2) $Harmonic progression$ (3) $Geometric progression$ (4) $Arithmetico - geometric progression$ 96. $The sum of the series, \frac{1}{2 \cdot 3} \cdot 2 + \frac{2}{3 \cdot 4} \cdot 2^{2} + \frac{3}{4 \cdot 5} \cdot 2^{3} + to n terms is$ (1) $\frac{2 ^{n + 1}}{n + 2} + 1$ (2) $\frac{2 ^{n + 1}}{n + 2} - 1$ (3) $\frac{2 ^{n + 1}}{n + 2} + 2$ (4) $\frac{2 ^{n + 1}}{n + 2} - 2$ 97. $Let the arithmetic mean of \frac{1}{a} and \frac{1}{b} be \frac{5}{16}, a > 2 . If α is such that a, 4, α, b are n A.P., then the equation α x^{2} - ax + 2 (α - 2 b) = 0 has:$ (1) $one root in (0, 2) and another in (- 4, - 2)$ (2) $one root in (1, 4) and another in (-2,0)$ (3) $both roots in the interval (- 2, 0)$ (4) $complex roots of magnitude less than 2$ 98. $If log_{e} a, log_{e} b, log_{e} c are in an A.P. and log_{e} a - log_{e} 2 b, log_{e} 2 b - log_{e} 3 c, log_{e} 3 c - log_{e} a are also in an A.P, then a : b : c is equal to$ (1) $9 : 6 : 4$ (2) $16 : 4 : 1$ (3) $25 : 10 : 4$ (4) $6 : 3 : 2$ 99. $If a, b, c are three consecutive terms of an A . P . and x, y, z are three consecutive terms of a G.P, then the value of x^{b - c} \cdot y^{c - a} \cdot z^{a - b} is$ (1) $0$ (2) $xyz$ (3) $-1$ (4) $1$ 100. $Five numbers are in A . P ., whose sum is 25 and product is 2520 . If one of these five numbers is - \frac{1}{2}, then the greatest number amongst them is :$ (1) $\frac{21}{2}$ (2) $16$ (3) $5$ (4) $7$ 101. $\frac{1}{2 \cdot 5} + \frac{1}{5 \cdot 8} + \frac{1}{8 \cdot 11} + \dots \dots . + \frac{1}{( 3 n - 1 ) ( 3 n + 2 )} =$ (1) $\frac{n}{6 n - 4}$ (2) $\frac{n}{6 n + 3}$ (3) $\frac{n}{6 n + 4}$ (4) $\frac{n + 1}{6 n + 4}$ 102. $If the set R = (a, b) : a+5b=42, a, b \in N has m elements and \sum_{n = 1}^{m} (1 - i^{n!}) = x + iy, where i = - 1, then the value of m + x + y is$ (1) $12$ (2) $4$ (3) $8$ (4) $5$ 103. $For x \in R, let [x] denote the greatest integer \leq x, then the sum of the series (- \frac{1}{3}) + (- \frac{1}{3} - \frac{1}{100}) + (- \frac{1}{3} - \frac{2}{100}) + \dots + (- \frac{1}{3} - \frac{99}{100}) is$ (1) $- 153$ (2) $- 133$ (3) $- 131$ (4) $- 135$ 104. $If the sum of n terms of an A.P. is given by S_{n} = n^{2} + n, then the common difference of the A.P. is$ (1) $1$ (2) $2$ (3) $6$ (4) $4$ 105. $\frac{1}{3 ^{2} - 1} + \frac{1}{5 ^{2} - 1} + \frac{1}{7 ^{2} - 1} + \dots + \frac{1}{( 201 ) ^{2} - 1} is equal to$ (1) $\frac{101}{404}$ (2) $\frac{25}{101}$ (3) $\frac{101}{408}$ (4) $\frac{99}{400}$ 106. $The sum (\frac{3 \times 1 ^{3}}{1 ^{2}} + \frac{5 \times ( 1 ^{3} + 2 ^{3} )}{1 ^{2} + 2 ^{2}} + \frac{7 \times ( 1 ^{3} + 2 ^{3} + 3 ^{3} )}{1 ^{2} + 2 ^{2} + 3 ^{2}}) + \dots \dots upto 10^{th} term, is :$ (1) $680$ (2) $600$ (3) $660$ (4) $620$ 107. $If x = \frac{3}{10} + \frac{3.7}{10.15} + \frac{3.7 . 9}{10.15 . 20} + \dots . , then 5 x + 8 =$ (1) $\frac{5 5}{3 3}$ (2) $\frac{5 5}{3}$ (3) $\frac{3 3}{5}$ (4) $\frac{25 5}{3 3}$ 108. $If \| x \| < 1, \| y \| < 1 and x \neq = y, then the sum to infinity of the following series (x + y) + (x^{2} + xy + y^{2}) + (x^{3} + x^{2} y + x y^{2} + y^{3}) + \dots \dots is$ (1) $\frac{x + y + xy}{( 1 + x ) ( 1 + y )}$ (2) $\frac{x + y - xy}{( 1 - x ) ( 1 - y )}$ (3) $\frac{x + y - xy}{( 1 + x ) ( 1 + y )}$ (4) $\frac{x + y + xy}{( 1 - x ) ( 1 - y )}$ 109. $The common difference of the A . P . b_{1}, b_{2}, \dots, b_{m} is 2 more than the common difference of A . P . a_{1}, a_{2}, \dots, a_{n} if a_{40} = - 159, a_{100} = - 399 and b_{100} = a_{70} then b_{1} is equal to:$ (1) $- 127$ (2) $- 81$ (3) $127$ (4) $81$ 110. $The 20^{th} term from the end of the progression 20, 19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \dots, - 129 \frac{1}{4} is :-$ (1) $-118$ (2) $-110$ (3) $-115$ (4) $-100$ 111. $Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11^{th} term is:$ (1) $90$ (2) $84$ (3) $122$ (4) $108$ 112. $If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to$ (1) $7$ (2) $4$ (3) $5$ (4) $6$ 113. $The minimum value of 2^{sin x} + 2^{cos x} is:$ (1) $2^{- 1 + 2}$ (2) $2^{1 - 2}$ (3) $2^{1 - \frac{1}{2}}$ (4) $2^{- 1 + \frac{1}{2}}$ 114. $If the roots of x^{3} + a x^{2} + bx + c = 0 are in arithmetic progression with common difference 1 , then$ (1) $9 c = a (b - 2)$ (2) $9 c = a (2 - b)$ (3) $9 c - a^{2} (b - 2) = 0$ (4) $9 c - a^{2} (2 - b) = 0$ 115. $If 1 + (1 - 2^{2} \cdot 1) + (1 - 4^{2} \cdot 3) + (1 - 6^{2} \cdot 5) + \dots \dots + (1 - 20^{2} \cdot 19) = α - 220 β then an ordered pair (α, β) is equal to :$ (1) $(10, 103)$ (2) $(10, 97)$ (3) $(11, 97)$ (4) $(11, 103)$ 116. $The number of terms common the two A . P . 3, 7, 11, \dots, 407 and 2, 9, 16, \dots, 709 is$ 117. $If the arithmetic mean and geometric mean of the p^{th} and q^{th} terms of the sequence - 16, 8, - 4, 2, \dots \dots \dots satisfy the equation 4 x^{2} - 9 x + 5 = 0, then p + q is equal to$ 118. $The sum of first four terms of a geometric progression (G . P .) is \frac{65}{12} and the sum of their respective reciprocals is \frac{65}{18} . If the product of first three terms of the G . P . is 1, and the third term is α, then 2 α is$ 119. $The sum \sum_{k = 1}^{20} (1 + 2 + 3 + \dots + k) is$ 120. $The sum 1 \cdot 1^{2} - 2 \cdot 3^{2} + 3 \cdot 5^{2} - 4 \cdot 7^{2} + 5 \cdot 9^{2} - \dots + 15 \cdot 2 9^{2} is ________$