Binomial Theorem PYQs Set 3 (PDF)

Set 3 of 14 · Questions 61-90 of 391

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Binomial Theorem · Set 3 (Q61-90)
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61. $If the term independent of x in the expansion of (x - \frac{k}{x ^{2}})^{10} is 405 , then k =$ (1) $\pm 1$ (2) $0$ (3) $\pm 3$ (4) $\pm 5$ 62. $If x is so small that x^{2} and higher powers of x may be neglected, then the approximate value of \frac{( 1 + \frac{2}{3} x ) ^{- 3} ( 1 - 15 x ) ^{- 1/5}}{( 2 - 3 x ) ^{4}}$ (1) $\frac{1}{8} (1 + 7 x)$ (2) $116(1 - 7 x)$ (3) $1 - 7 x$ (4) $\frac{1}{16} (1 + 7 x)$ 63. $If the constant term in the expansion of (\frac{5 3}{x} + \frac{2 x}{3 5})^{12}, x \neq = 0, is α \times 2^{8} \times 53, then 25 α is equal to :$ (1) $724$ (2) $742$ (3) $639$ (4) $693$ 64. $The corner points of the feasible region of an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5), then the minimum value of z = 4 x + 6 y occurs at$ (1) $finite number of points$ (2) $only one point$ (3) $infinite number of points$ (4) $only two points$ 65. $The coefficient of x^{7} in the expression (1 + x)^{10} + x (1 + x)^{9} + x^{2} (1 + x)^{8} + \dots + x^{10} is:$ (1) $330$ (2) $210$ (3) $420$ (4) $260$ 66. $The number of rational terms in the binomial expansion of (45 + 54)^{100} is$ (1) $50$ (2) $5$ (3) $6$ (4) $51$ 67. $The term independent of x in the expansion of (x - \frac{2}{x})^{21} is$ (1) $^{21} C_{15} (- 2)^{15}$ (2) $^{21} C_{14} 2^{14}$ (3) $-^{21} C_{7} (2)^{7}$ (4) $-^{21} C_{7} 2^{14}$ 68. $Corner points of the feasible region determined by the system of linear constraints are (0, 3), (1, 1) and (3, 0) . Let z = px + qy, where p, q > 0 . Condition on p and q so that the minimum of z occurs at (3, 0) and (1, 1) is$ (1) $p = \frac{q}{2}$ (2) $p = 3 q$ (3) $p = q$ (4) $p = 2 q$ 69. $If the expansion in powers of x of the function \frac{1}{( 1 - ax ) ( 1 - bx )} is a0 + a1x + a2x2 + a3x3 + ......., then an is :$ (1) $\frac{a ^{n} - b ^{n}}{b - a}$ (2) $\frac{a ^{n + 1} - b ^{n + 1}}{b - a}$ (3) $\frac{b ^{n + 1} - a ^{n + 1}}{b - a}$ (4) $\frac{b ^{n} - a ^{n}}{b - a}$ 70. $In the expansion of (1 + x)^{n}, the coefficients of p th and (p + 1) th terms are respectively p and q then p + q =$ (1) $n + 3$ (2) $n + 2$ (3) $n$ (4) $n + 1$ 71. $If n is even and the middle term in the expansion of (x^{2} + \frac{1}{x})^{n} is 924 x^{6}, then n is equal to$ (1) $12$ (2) $10$ (3) $8$ (4) $14$ 72. $For some n \neq = 10, let the coefficients of the 5 th, 6 th and 7 th terms in the binomial expansion of (1 + x)^{n + 4} be in A.P. Then the largest coefficient in the expansion of (1 + x)^{n + 4} is:$ (1) $20$ (2) $10$ (3) $35$ (4) $70$ 73. $If the coefficients of rth, (r + 1)th and (r +2)th terms in the binomial expansion of (1 +y)m are in AP, then m and r satisfy the equation :$ (1) $m2 - m(4r - 1) + 4r2 + 2 = 0.$ (2) $m2 - m(4r +1) + 4r2 - 2 = 0.$ (3) $m2 - m(4r+1) + 4r2 + 2 = 0.$ (4) $m2 - m(4r -1) + 4r2 - 2 = 0.$ 74. $Given below are two statements : Statement I: 25^{13} + 20^{13} + 8^{13} + 3^{13} is divisible by 7 . Statement II: The integral part of (7 + 43)^{25} is an odd number. In the light of the above statements, choose the correct answer from the options given below:$ (1) $Statement I is true but Statement II is false$ (2) $Both Statement I and Statement II are true$ (3) $Statement I is false but Statement II is true$ (4) $Both Statement I and Statement II are false$ 75. $If n is a positive integer and f (n) is the coefficient of x^{n} in the expansion of (1 + x) (1 - x)^{n}, then f (2023) =$ (1) $-2021$ (2) $2022$ (3) $2023$ (4) $-2023$ 76. $The value of -^{15} C_{1} + 2 \cdot^{15} C_{2} - 3 \cdot^{15} C_{3} + \dots . - 15 \cdot^{15} C_{15} +^{14} C_{1} +^{14} C_{3} +^{14} C_{5} + \dots, +^{14} C_{11} is:$ (1) $2^{14}$ (2) $2^{13} - 13$ (3) $2^{16} - 1$ (4) $2^{13} - 14$ 77. $If x = 1 + \frac{3}{1 !} \times \frac{1}{6} + \frac{3 \times 7}{2 !} (\frac{1}{6})^{2} + \frac{3 \times 7 \times 11}{3 !} (\frac{1}{6})^{3} + \dots, then x^{4} equals$ (1) $81$ (2) $54$ (3) $27$ (4) $8$ 78. $If the sum of the coefficients of even powers of x in the expansion of (1 - x + x^{2})^{2 n} is 3281 , then n =$ (1) $4$ (2) $5$ (3) $6$ (4) $3$ 79. $Numerically greatest term in the expansion of (2 x - 3 y)^{11} when x = \frac{1}{3} and y = \frac{1}{2} is$ (1) $^{11} C_{8} (\frac{2}{3})^{5}$ (2) $^{11} C_{3} (\frac{3}{2})^{5}$ (3) $^{11} C_{2} (\frac{3}{2})^{7}$ (4) $^{11} C_{2} (\frac{2}{3})^{7}$ 80. $If the constant term in the binomial expansion of (x - \frac{k}{x ^{2}})^{10} is 405 then \| k \| equals:$ (1) $2$ (2) $3$ (3) $9$ (4) $1$ 81. $The coefficient of x^{3} in the expansion of (1 - 2 x)^{\frac{1}{2}} (1 + 3 x)^{- \frac{1}{3}} is$ (1) $- \frac{20}{3}$ (2) $\frac{20}{3}$ (3) $\frac{17}{3}$ (4) $- \frac{17}{3}$ 82. $If the constant term in the binomial expansion of (2 x^{r} + \frac{1}{x ^{2}})^{10} is 180, then r is equal to ____.$ 83. $The remainder when 19^{200} + 23^{200} is divided by 49, is _________$ 84. $If the coefficients of x and x^{2} in the expansion of (1 + x)^{p} (1- x)^{q}, p, q \leq 15, are -3 and-5 respectively, then the coefficient of x^{3} is equal to______________.$ 85. $50th root of a number x is 12 and 50th root of another number y is 18. Then the remainder obtained on dividing (x + y) by 25 is _______$ 86. $Let the coefficients of three consecutive terms in the binomial expansion of (1 + 2 x)^{n} be in the ratio 2 : 5 : 8. Then the coefficient of the term, which is in the middle of these three terms, is$ 87. $Find the coefficient of x^{4} in (1 + x + x^{2})^{10}$ 88. $If the sum of the coefficients of all even powers of x in the product (1 + x + x^{2} + \dots + x^{2 n}) (1 - x + x^{2} - x^{3} + \dots + x^{2 n}) is 61, then n is equal to$ 89. $If the coefficient of a^{7} b^{8} in the expansion of (a + 2 b + 4 ab)^{10}, is K. 2^{16} then K is equal to .$ 90. $If second, third and fourth terms in the expansion of (x + a)n are 240, 720 and 1080 respectively, then n is equal to$