Class 11 Mathematics Binomial Theorem MCQs Set 07

Question. The coefficient of \( x^{-17} \) in \( \left( x^4 - \frac{1}{x^3} \right)^{15} \) is
(a) \( -^{15}C_{11} \)
(b) \( ^{15}C_{11} \)
(c) \( ^{15}C_{5} \)
(d) \( -^{15}C_{5} \)
Answer: (a) \( -^{15}C_{11} \)

Question. If the coefficients of (3r)th and (r+2)th terms in the expansion of \( (1+x)^{2n} \) are equal (r>1, n>2), then
(a) n=2r
(b) n=3r
(c) n=2r+1
(d) n=4r
Answer: (a) n=2r

Question. The term independent of x in \( \left( 2x - \frac{1}{2x^2} \right)^{12} \) is
(a) \( -^{12}C_{3} . 2^6 \)
(b) \( -^{12}C_{5} . 2^2 \)
(c) \( ^{12}C_{6} \)
(d) \( ^{12}C_{4} . 2^4 \)
Answer: (d) \( ^{12}C_{4} . 2^4 \)

Question. The term independent of x in \( \left( 2\sqrt{x} - \frac{3}{x^{\frac{1}{3}}} \right)^{20} \) is
(a) \( ^{20}C_{8} . 2^8 . 3^{12} \)
(b) \( -^{20}C_{8} . 2^8 . 3^{12} \)
(c) \( ^{20}C_{7} . 2^7 . 3^{13} \)
(d) \( -^{20}C_{7} . 2^7 . 3^{13} \)
Answer: (a) \( ^{20}C_{8} . 2^8 . 3^{12} \)

Question. The term independent of x in \( \left( x^2 - \frac{1}{x} \right)^9 \) is
(a) 1
(b) -1
(c) 48
(d) 84
Answer: (d) 84

Question. The term independent of x in the expansion of \( \left( \frac{\sqrt{x}}{2} - \frac{2}{x^2} \right)^{10} \) is ...
(a) 45/64
(b) 64/45
(c) 6/5
(d) 4/5
Answer: (a) 45/64

Question. In \( \left( \sqrt{x} - \frac{2}{x} \right)^{18} \), the term independent of x is
(a) \( ^{18}C_{6} \)
(b) \( 2 . ^{18}C_{6} \)
(c) \( ^{18}C_{6} . 2^6 \)
(d) \( 16 . ^{18}C_{6} \)
Answer: (c) \( ^{18}C_{6} . 2^6 \)

Question. The middle term in the expansion of \( (1+x)^{2n} \) is
(a) \( ^{2n}C_{n} \)
(b) \( ^{2n}C_{n-1} . x^{n+1} \)
(c) \( ^{2n}C_{n-1} . x^{n-1} \)
(d) \( ^{2n}C_{n} . x^{n} \)
Answer: (d) \( ^{2n}C_{n} . x^{n} \)

Question. The ratio of the coefficient of \( x^{10} \) in \( (1-x^2)^{10} \) and the term independent of x in \( \left( x - \frac{2}{x} \right)^{10} \) is
(a) 32 : 1
(b) -32 : 1
(c) -1 : 32
(d) 1 : 32
Answer: (d) 1 : 32

Question. The ratio of the rth term and the (r+1)th term in the expansion of \( (1+x)^n \) is
(a) \( \frac{r}{(n-r+1)x} \)
(b) \( \frac{1}{(n-r+1)x} \)
(c) \( \frac{r}{n-r+1} \)
(d) \( \frac{(n-r+1)x}{r} \)
Answer: (a) \( \frac{r}{(n-r+1)x} \)

Question. When x=1, y=\( \frac{3}{2} \), numerically greatest term in the expansion of \( (2x-3y)^{12} \) is
(a) 6th
(b) 8th
(c) 10th
(d) 12th
Answer: (c) 10th

Question. If the first three terms in the expansion of \( (1+ax)^n \) are 1, 8x, \( 24x^2 \) respectively, then a=
(a) 1
(b) 2
(c) 4
(d) 8
Answer: (b) 2

Question. The coefficients of 2nd, 3rd and 4th terms in the expansion of \( (1+x)^n \) are in A.P. then n=
(a) 7
(b) 8
(c) 9
(d) 14
Answer: (a) 7

Question. In the binomial expansion of \( \left( \sqrt{y} + \frac{1}{2\sqrt[4]{y}} \right)^8 \), the terms in the expansion in which the power of y is a natural number are
(a) \( T_1, T_5 \)
(b) \( T_1, T_2, T_3, T_4 \)
(c) \( T_2, T_4 \)
(d) \( T_1, T_3, T_5 \)
Answer: (a) \( T_1, T_5 \)

Question. In the expansion of \( (1+x)^n \), the binomial coefficients of 3 consecutive terms are respectively 220, 495 and 792, then n=
(a) 4
(b) 8
(c) 12
(d) 16
Answer: (c) 12

Question. The coefficients of 9th, 10th and 11th terms in the expansion of \( (1+x)^n \) are in A.P. then n =
(a) 7
(b) 7 or 14
(c) 14
(d) 21
Answer: (c) 14

Question. In the expansion of \( (x+y)^n \), if the binomial coefficient of the third term is greater by 9 then that of the second term, then the sum of the binomial coefficients of the terms occupying the odd places is
(a) \( 2^9 \)
(b) \( 2^6 \)
(c) \( 2^5 \)
(d) \( 2^8 \)
Answer: (c) \( 2^5 \)

Question. The value of \( (1.02)^4 + (0.98)^4 \) correct to three decimal places is
(a) 2.004
(b) 2.005
(c) 2.006
(d) 1.995
Answer: (b) 2.005

Question. Coefficient of \( x^4 \) in \( (1+x-2x^2)^6 \) is
(a) -60
(b) -45
(c) 45
(d) 15
Answer: (b) -45

Question. Sum of the third from the begining and the third from the end of the binomial coefficients of the expansion of \( \left( \sqrt[4]{3} + \sqrt[3]{4} \right)^n \) is equal to 9900. Number of rational terms contained in the expansion is
(a) 8
(b) 9
(c) 10
(d) 11
Answer: (b) 9

Question. The sum of the coefficients in the expansion of \( (1+x-3x^2)^{2143} \) is
(a) 0
(b) 1
(c) -1
(d) \( (-2)^{2143} \)
Answer: (c) -1

Question. Sum of the coefficients in the expansion of \( (5x-4y)^n \) where n is a positive integer is
(a) 1
(b) \( 9^n \)
(c) \( (-1)^n \)
(d) \( 5^n \)
Answer: (a) 1

Question. If \( (1+x-2x^2)^6 = \sum_{r=0}^{12} a_r x^r \), then \( a_2 + a_4 + ....... + a_{12} = \)
(a) 64
(b) 63
(c) 32
(d) 31
Answer: (d) 31

Question. \( ^{2n}C_2 + ^{2n}C_4 + ...... + ^{2n}C_{2n} = \)
(a) \( 2^{2n} \)
(b) \( 2^{2n}-1 \)
(c) \( 2^{2n-1} \)
(d) \( 2^{2n-1}-1 \)
Answer: (d) \( 2^{2n-1}-1 \)

Question. \( C_0 + 2.C_1 + 3.C_2 + ............ + (n+1).C_n = \)
(a) \( (n+2)2^n \)
(b) \( n.2^n \)
(c) \( n.2^{n-1} \)
(d) \( (n+2)2^{n-1} \)
Answer: (d) \( (n+2)2^{n-1} \)

Question. \( \frac{C_1}{C_0} + 2\frac{C_2}{C_1} + 3\frac{C_3}{C_2} + ........... + n\frac{C_n}{C_{n-1}} = \)
(a) \( \frac{(n+1)(n+2)}{2} \)
(b) \( \frac{n(n+1)}{2} \)
(c) \( \frac{n(n-1)}{2} \)
(d) \( \frac{n(n+2)}{2} \)
Answer: (b) \( \frac{n(n+1)}{2} \)

Question. The sum to (n+1) terms of the series \( \frac{C_0}{2} - \frac{C_1}{3} + \frac{C_2}{4} - \frac{C_3}{5} + ....... = \)
(a) \( \frac{1}{n+1} \)
(b) \( \frac{1}{n+2} \)
(c) \( \frac{1}{n(n+1)} \)
(d) \( \frac{1}{(n+1)(n+2)} \)
Answer: (d) \( \frac{1}{(n+1)(n+2)} \)

Question. \( C_0 + 3.C_1 + 5.C_2 + .......... + (2n+1).C_n = \)
(a) \( (n+1)2^n \)
(b) \( (2n+1)2^{n-1} \)
(c) \( (2n+1)2^n \)
(d) \( (n+1)2^{n-1} \)
Answer: (a) \( (n+1)2^n \)

Question. \( 5.C_0 + 8.C_1 + 11.C_2 + ........ \text{ to } (n+1) \text{ terms} = \)
(a) \( (3n+5)2^n \)
(b) \( (3n+5)2^{n-1} \)
(c) \( (3n+10)2^n \)
(d) \( (3n+10)2^{n-1} \)
Answer: (d) \( (3n+10)2^{n-1} \)

Question. \( ^5C_0 + 2.^5C_1 + 2^2.^5C_2 + 2^3.^5C_3 + 2^4.^5C_4 + 2^5.^5C_5 = \)
(a) 32
(b) 243
(c) 64
(d) 729
Answer: (b) 243

Question. In the expansion of \( \frac{1}{1-3x} \), coefficient of \( x^4 \) is
(a) 81
(b) 27
(c) -27
(d) 28
Answer: (a) 81

Question. In the expansion of \( (1 + x + x^2 + x^3 + x^4)^{-2} \), coefficient of \( x^{10} \) is
(a) 3
(b) 5
(c) 7
(d) 9
Answer: (a) 3

Question. In the expansion of \( \frac{3-x}{(1-x)^2} \) coefficient of \( x^r \) is 9 then r =
(a) 3
(b) 4
(c) 3.5
(d) 4.5
Answer: (a) 3

Question. If n is a positive integer, then the coefficient of \( x^n \) in the expansion of \( \frac{(1+2x)^n}{1-x} \) is
(a) \( n.3^n \)
(b) \( (n-1)3^n \)
(c) \( (n+1)3^n \)
(d) \( 3^n \)
Answer: (d) \( 3^n \)

Question. \( 1 + n \left( \frac{2n}{1+n} \right) + \frac{n(n+1)}{2!} \left( \frac{2n}{1+n} \right)^2 + ....... \infty = \)
(a) \( \left( \frac{1-n}{1+n} \right)^n \)
(b) \( \left( \frac{1+n}{1-n} \right)^n \)
(c) \( \left( \frac{1+3n}{1+n} \right)^n \)
(d) \( \left( \frac{1+n}{1+3n} \right)^n \)
Answer: (b) \( \left( \frac{1+n}{1-n} \right)^n \)

Question. Neglecting \( x^n \) for \( n \ge 2 \), value of \( (1-7x)^{\frac{1}{3}} (1+2x)^{-\frac{3}{4}} \) is
(a) \( 1 - \frac{5x}{6} \)
(b) \( 1 + \frac{5x}{6} \)
(c) \( 1 + \frac{23x}{6} \)
(d) \( 1 - \frac{23x}{6} \)
Answer: (d) \( 1 - \frac{23x}{6} \)

Question. \( 1 + \frac{1}{4} + \frac{1.4}{4.8} + \frac{1.4.7}{4.8.12} + ........ \infty = \)
(a) \( \left( \frac{1}{4} \right)^{\frac{1}{3}} \)
(b) \( \left( \frac{1}{2} \right)^{\frac{1}{3}} \)
(c) \( (2)^{\frac{1}{3}} \)
(d) \( (4)^{\frac{1}{3}} \)
Answer: (d) \( (4)^{\frac{1}{3}} \)

Question. \( \frac{3}{6} + \frac{3.5}{6.9} + \frac{3.5.7}{6.9.12} + ....... \infty = \)
(a) \( 3\sqrt{3} \)
(b) \( 3\sqrt{3} - \frac{4}{3} \)
(c) \( 3\sqrt{3} - 4 \)
(d) \( 2\sqrt{3} \)
Answer: (c) \( 3\sqrt{3} - 4 \)

Question. If \( 1 + \frac{2}{x} + \frac{3}{x^2} + \frac{4}{x^3} + ....... \infty = \frac{1}{2} \), then x=
(a) \( -1+\sqrt{2} \)
(b) \( -1-\sqrt{2} \)
(c) \( 1+\sqrt{2} \)
(d) \( 1-\sqrt{2} \)
Answer: (b) \( -1-\sqrt{2} \)

Question. \( 1 + \frac{1}{2} \cdot \frac{3}{5} + \frac{1.3}{2.4} \left( \frac{3}{5} \right)^2 + \frac{1.3.5}{2.4.6} \left( \frac{3}{5} \right)^3 + ....... \infty = \)
(a) \( \sqrt{\frac{5}{2}} \)
(b) \( \sqrt{\frac{2}{5}} \)
(c) \( \frac{1}{2}\sqrt{\frac{5}{2}} \)
(d) \( \sqrt{\frac{5}{3}} \)
Answer: (a) \( \sqrt{\frac{5}{2}} \)