Binomial Theorem PYQs Set 4 (PDF)

Set 4 of 14 · Questions 91-120 of 391

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Binomial Theorem · Set 4 (Q91-120)
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91. $The middle term in the expansion of (4 x^{3} - \frac{15}{4 x})^{8} is$ (1) $70 (15 x)^{4}$ (2) $1820 x^{8}$ (3) $70 (15 x^{2})^{4}$ (4) $2560 x^{4}$ 92. $If an LPP admits optimal solution at two consecutive vertices of a feasible region, then$ (1) $the required optimal solution is at the midpoint of the line joining two points.$ (2) $the optimal solution occurs at every point on the line joining these two points.$ (3) $the LPP under consideration is not solvable.$ (4) $the LPP under consideration must be reconstructed$ 93. $The coefficient of x^{3} in the expansion of (1 - \frac{3}{4} x)^{\frac{1}{2}} is$ (1) $\frac{27}{1024}$ (2) $\frac{- 27}{1024}$ (3) $\frac{81}{1024}$ (4) $\frac{- 81}{1024}$ 94. $n \in N then, the statement 8 n + 16 \leq 2^{n} is true for$ (1) $n = 2$ (2) $n = 3$ (3) $n = 6$ (4) $n = 5$ 95. $The remainder of n^{4} - 2 n^{3} - n^{2} + 2 n - 26 when divided by 24 , is$ (1) $20$ (2) $21$ (3) $22$ (4) $23$ 96. $Choose the correct option regarding the following statements 1. C_{0} + C_{2} + C_{4} + \dots + C_{n} = 2^{n - 1}, if n is even 2. C_{1} + C_{3} + C_{5} + \dots + C_{n - 1} = 2^{n - 1}, if n is even$ (1) $1 is true, 2 is false$ (2) $1 is false, 2 is true$ (3) $Both 1 and 2 are false$ (4) $Both 1 and 2 are true$ 97. $The coefficient of x^{48} in (1 + x) + 2 (1 + x)^{2} + 3 (1 + x)^{3} + \dots + 100 (1 + x)^{100} is equal to$ (1) $100 \cdot^{100} C_{49} -^{100} C_{48}$ (2) $^{100} C_{50} +^{101} C_{49}$ (3) $100 \cdot^{100} C_{49} -^{101} C_{50}$ (4) $100 \cdot^{100} C_{49} -^{100} C_{50}$ 98. $The sum of all rational terms in the expansion of (2^{\frac{1}{5}} + 5^{\frac{1}{3}})^{15} is equal to :$ (1) $3133$ (2) $931$ (3) $6131$ (4) $633$ 99. $The sum of the coefficients of three consecutive terms in the binomial expansion of (1 + x)^{n + 2}, which are in the ratio 1 : 3 : 5, is equal to$ (1) $92$ (2) $63$ (3) $41$ (4) $25$ 100. $a^{n} + b^{n} is divisible by .......... if n is any odd positive integer.$ (1) $a - b$ (2) $a^{2} - b^{2}$ (3) $a^{2} + b^{2}$ (4) $a + b$ 101. $The expansion of (1 + x + x^{2})^{- 3/2} in powers of x is valid if$ (1) $\| x \| < 1$ (2) $∣ x ∣ < \frac{1}{2}$ (3) $x + \frac{1}{2} < \frac{5}{2}$ (4) $- \frac{1}{2} - \frac{5}{2} < x < 1$ 102. $The number of ordered pairs (r, k) for which 6 .^{35} C_{r} = (k^{2} - 3) \cdot^{36} C_{r + 1} where k is an integer, is:$ (1) $6$ (2) $2$ (3) $3$ (4) $4$ 103. $The cost and revenue functions of a product are given by c (x) = 20 x + 4000 and R (x) = 60 x + 2000 respectively where x is the number of items produced and sold. The value of x to earn profit is$ (1) $> 50$ (2) $> 60$ (3) $> 80$ (4) $> 40$ 104. $The 11^{th} term in the expansion of (x + \frac{1}{x})^{14}$ (1) $\frac{999}{x}$ (2) $\frac{1001}{x}$ (3) $i$ (4) $\frac{x}{1001}$ 105. $The coefficient of x^{n} in the expansion of \frac{1}{x ^{2} - 5 x + 6} for \| x \| < 1 is$ (1) $\frac{1}{2 ^{n - 1}} - \frac{1}{3 ^{n - 1}}$ (2) $\frac{1}{2 ^{n + 1}} - \frac{1}{3 ^{n + 2}}$ (3) $\frac{1}{2 ^{n + 1}} - \frac{1}{3 ^{n + 1}}$ (4) $\frac{1}{2 ^{n}} - \frac{1}{3 ^{n}}$ 106. $The ninth term of the expansion (3 x - \frac{1}{2 x})^{8} is$ (1) $\frac{1}{512 x ^{9}}$ (2) $\frac{- 1}{512 x ^{9}}$ (3) $\frac{- 1}{256 x ^{8}}$ (4) $\frac{1}{256 x ^{8}}$ 107. $Among the statements : (S_{1}) : 202 3^{2022} - 199 9^{2022} is divisible by 8. (S_{2}) : 13 (13)^{n} - 11n- 13 is divisible by 144 for infinitely many n \in N$ (1) $Only (S_{2}) is correct$ (2) $Only (S_{1}) is correct$ (3) $Both (S_{1}) and (S_{2}) are correct$ (4) $Both (S_{1}) and (S_{2}) are incorrect$ 108. $Let the coefficients of third, fourth and fifth terms in the expansion of (x + \frac{a}{x ^{2}})^{n}, x \neq = 0, be in the ratio 12 : 8 : 3 . Then the term independent of x in the expansion, is equal to$ 109. $The remainder on dividing 5^{99} by 11 is ___________$ 110. $Let m, n \in N and g cd (2, n) = 1 . If 30 (0 30) + 29 (1 30) + \dots + 2 (28 30) + 1 (29 30) = n \cdot 2^{m} then n + m is equal to$ 111. $The number of elements in the set {n ∈ {1, 2, 3,…, 100} for > + }is ____.$ 112. $Let the sixth term in the binomial expansion of (2^{l o g_{2} (10 - 3^{x})} + 5 2^{(x - 2) l o g_{2} 3})^{m}, in the increasing powers of 2^{(x - 2) l o g_{2} 3}, be 21. If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of x is __________$ 113. $Let X = + + + ....... + where 10Cr, r ∈ {1, 2, ......., 10} denote binomial coefficients. Then the value of X is _______$ 114. $Let the coefficients of the middle terms in the expansion of (\frac{1}{6} + βx)^{4}, (1 - 3 βx)^{2} and (1 - \frac{β}{2} x)^{6}, β > 0, respectively form the first three terms of an A.P. If d is the common difference of this A.P., then 50 - \frac{2 d}{β ^{2}} is equal to_____$ 115. $Let P be a plane containing the line \frac{x - 1}{3} = \frac{y + 6}{4} = \frac{z + 5}{2} and parallel to the line \frac{x - 3}{4} = \frac{y - 2}{- 3} = \frac{z + 5}{7} . If the point (1, - 1, α) lies on the plane P, then the value of \| 5 α \| is equal to$ 116. $If the sum of the coefficients of all the positive powers of x, in the binomial expansion of (x^{n} + \frac{2}{x ^{5}})^{7} is 939, then the sum of all the possible integral values of n is :$ 117. $If the remainder when x is divided by 4 is 3, then the remainder when (2020 + x)^{2022} is divided by 8 is$ 118. $The coefficient of x^{2012} in the expansion of (1 - x)^{2008} (1 + x + x^{2})^{2007} is equal to$ 119. $Let α = \sum_{k = 0}^{n} (\frac{( ^{n} C _{k} ) ^{2}}{k + 1}) and β = \sum_{k = 0}^{n - 1} (\frac{^{n} C _{k} ^{n} C _{k + 1}}{k + 2}) If 5 α = 6 β, then n equals$ 120. $If \frac{^{11} c _{1}}{2} + \frac{^{11} c _{2}}{3} + \dots + \frac{^{11} c _{9}}{10} = \frac{n}{m} with gcd (n, m) = 1, then n + m is equal to$