Class 11 Mathematics Binomial Theorem MCQs Set 05

Question. If n is a positive integer, then the number of terms in the expansion of \( (x+a)^n \) is
(a) n
(b) n+1
(c) n+2
(d) infinitely many
Answer: (b) n+1

Question. The number of terms in the expansion of \( (x+y+z)^n \) is
(a) \( \frac{n(n+1)}{2} \)
(b) \( \frac{(n+1)(n+2)}{2!} \)
(c) \( \frac{n(n+3)}{2} \)
(d) \( \frac{(n+1)(n+3)}{2} \)
Answer: (b) \( \frac{(n+1)(n+2)}{2!} \)

Question. If the coefficient of \( x^n \) in \( (1+x)^{2n} \) is 'a' and the coefficient of \( x^n \) in \( (1+x)^{2n-1} \) is b, then \( \frac{a}{b} = \)
(a) 2n + 1
(b) 4n + 1
(c) 2n
(d) n
Answer: (a) 2n + 1

Question. If the middle term of \( (1+x)^{2n} \) is the greatest term, then x lies between
(a) \( n-1 < x < n \)
(b) \( \frac{n}{n+1} < x < \frac{n+1}{n} \)
(c) \( n < x < n+1 \)
(d) \( \frac{n+1}{n} < x < \frac{n}{n+1} \)
Answer: (b) \( \frac{n}{n+1} < x < \frac{n+1}{n} \)

Question. If n is a positive integer, \( \sum_{r=0}^n \left(^nC_r\right)^2 = \)
(a) 0
(b) \( ^nC_{n/2} \)
(c) \( \frac{(n!)^2}{(2n)!} \)
(d) \( \frac{(2n)!}{(n!)^2} \)
Answer: (d) \( \frac{(2n)!}{(n!)^2} \)

Question. In \( (1+ax)^n \), sum of the coefficients is \( S_1 \). If we double 'a' and half 'n', the new sum is \( S_2 \). Then
(a) \( S_1 > S_2 \)
(b) \( S_1 < S_2 \)
(c) \( S_1 = S_2 \)
(d) Cannot be decided
Answer: (a) \( S_1 > S_2 \)

Question. In the binomial expansion of \( (a-b)^n, n \geq 5 \), the sum of 5th and 6th terms is zero then a/b equal to [ AIEEE - 2008 ]
(a) \( \frac{5}{n-4} \)
(b) \( \frac{6}{n-5} \)
(c) \( \frac{n-5}{6} \)
(d) \( \frac{n-4}{5} \)
Answer: (d) \( \frac{n-4}{5} \)

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

Question. The coefficient of \( x^p \) in the expansion of \( \left(x^2 + \frac{1}{x}\right)^{2n} \), when exists is
(a) \( ^{2n}C_{\frac{4n+p}{3}} \)
(b) \( ^{2n}C_{\frac{4n-p}{3}} \)
(c) \( ^{2n}C_{\frac{n+p}{3}} \)
(d) \( ^nC_{\frac{n+p}{3}} \)
Answer: (b) \( ^{2n}C_{\frac{4n-p}{3}} \)

Question. If a term independent of x exist in the expansion of \( \left(x + \frac{1}{x^2}\right)^n \), then n must be
(a) a multiple of 2
(b) a multiple of 3
(c) a multiple of 5
(d) a multiple of 7
Answer: (b) a multiple of 3

Question. In the expansion of \( (a+b)^n \) if two consecutive terms are equal, then \( \frac{(n+1)b}{a+b} \) and \( \frac{(n+1)a}{a+b} \) are
(a) integers
(b) complex numbers
(c) irrational numbers
(d) does not exist
Answer: (a) integers

Question. In the expansion of \( (1+x)^n \) if the coefficients of three consecutive terms are in A.P. then n+2 is
(a) \( n^2 \)
(b) a perfect square
(c) a perfect cube
(d) \( n^3 \)
Answer: (b) a perfect square

Question. If the coefficient of \( x^{2r} \) in the expansion of \( \left(x + \frac{1}{x^2}\right)^{n-3} \) is not zero, then \( \frac{n-2r}{3} \) is
(a) a rational number
(b) a positive integer
(c) a negative integer
(d) a positive rational number
Answer: (b) a positive integer

Question. \( ^mC_r + ^mC_{r-1} \cdot ^nC_1 + ^mC_{r-2} \cdot ^nC_2 + \ldots + ^nC_r = \)
(a) \( ^{m+n}C_{n+r} \)
(b) \( ^{m+n}C_{m+r} \)
(c) \( ^{m+n}C_r \)
(d) 0
Answer: (c) \( ^{m+n}C_r \)

Question. The coefficient of \( \frac{1}{x} \) in the expansion of \( (1+x)^n \left(1+\frac{1}{x}\right)^n \) is
(a) \( \frac{n!}{(n-1)!(n+1)!} \)
(b) \( \frac{2n!}{(n-1)!(n+1)!} \)
(c) \( \frac{n!}{(2n-1)!(2n+1)!} \)
(d) \( \frac{2n!}{(2n-1)!(2n+1)!} \)
Answer: (b) \( \frac{2n!}{(n-1)!(n+1)!} \)

Question. The ratio of \( (r+1) \) th and \( (r-1) \) th terms in the expansion of \( (a-b)^n \) is
(a) \( \frac{(n-r+2)(n-r+1)}{r(r-1)} \cdot \frac{b^2}{a^2} \)
(b) \( \frac{(n-r+2)(n-r+1)}{r(r-1)} \cdot \frac{a^2}{b^2} \)
(c) \( \left(\frac{n-r+2}{r}\right) \frac{b}{a} \)
(d) \( \left(\frac{n-r+1}{r-1}\right) \frac{b}{a} \)
Answer: (a) \( \frac{(n-r+2)(n-r+1)}{r(r-1)} \cdot \frac{b^2}{a^2} \)

Question. If a, b, c, d are any four consecutive coefficients in the expansion of \( (1+x)^n \), then \( \frac{a}{a+b}, \frac{b}{b+c}, \frac{c}{c+d} \) are in
(a) A.P.
(b) G.P.
(c) H.P.
(d) A.G.P
Answer: (a) A.P.

Question. If \( (1+x)^n = \sum_{i=0}^n C_i x^i \), then the sum of the products of \( C_i \)'s taken two at a time is represented by \( \sum_{0 \leq i < j \leq n} C_i C_j \)
(a) \( 2^n - \frac{(2n)!}{2(n!)^2} \)
(b) \( 2^{2n} - \frac{(2n)!}{2(n!)^2} \)
(c) \( \frac{1}{2} \left( 2^{2n} - \frac{(2n)!}{(n!)^2} \right) \)
(d) \( \frac{2^{2n}}{2(n!)^2} \)
Answer: (c) \( \frac{1}{2} \left( 2^{2n} - \frac{(2n)!}{(n!)^2} \right) \)

Question. If \( (1+x) (1+x+x^2) (1+x+x^2+x^3) \ldots (1+x+x^2+\ldots+x^n) = a_0 + a_1 x + a_2 x^2 + \ldots + a_m x^m \)
Then the value of \( a_0 - a_1 + a_2 - a_3 + \ldots + (-1)^m a_m \) is
(a) (n-2)
(b) (m-1)
(c) 0
(d) mn
Answer: (c) 0

Question. The sum of \( ^rC_r + ^{r+1}C_r + ^{r+2}C_r + \ldots + ^nC_r \quad (n \geq r) \)
(a) \( ^nC_{r+1} \)
(b) \( ^{n+1}C_{r+1} \)
(c) \( ^{n+1}C_r \)
(d) \( ^{n+1}C_{r-1} \)
Answer: (b) \( ^{n+1}C_{r+1} \)

Question. The digit in the units place of the number \( 183! + 3^{183} \) is
(a) 0
(b) 3
(c) 6
(d) 7
Answer: (d) 7

Question. The last digit in \( 7^{100} \) is
(a) 1
(b) 3
(c) 7
(d) 9
Answer: (a) 1

Question. The coefficient of \( a^4 b^3 c^2 d \) in the expansion of \( (a-b+c-d)^{10} \) is
(a) 12600
(b) 16200
(c) 21600
(d) 26100
Answer: (a) 12600

Question. If \( T_r \) denotes the rth term in the expansion of \( \left(x + \frac{1}{x}\right)^{23} \), then
(a) \( T_{12} = T_{13} \)
(b) \( T_{13} = x^2 T_{12} \)
(c) \( T_{12} = x^2 T_{13} \)
(d) \( T_{12} + T_{13} = 25 \)
Answer: (c) \( T_{12} = x^2 T_{13} \)

Question. In the expansion of \( \left(x + x^{\log_{10} x}\right)^5 \) the third term is \( 10^6 \) then x
(a) 1
(b) 2
(c) 10
(d) 100
Answer: (c) 10

Question. The value of \( (1.03)^{\frac{1}{3}} \) up to 4 decimal places is
(a) 1.0998
(b) 1.0099
(c) 1.0098
(d) 1.0989
Answer: (b) 1.0099

Question. The binomial coefficients which are in decreasing order
(a) \( ^{15}C_5, ^{15}C_6, ^{15}C_7 \)
(b) \( ^{15}C_{10}, ^{15}C_8, ^{15}C_9 \)
(c) \( ^{15}C_6, ^{15}C_7, ^{15}C_8 \)
(d) \( ^{15}C_7, ^{15}C_6, ^{15}C_5 \)
Answer: (d) \( ^{15}C_7, ^{15}C_6, ^{15}C_5 \)

Question. The number of terms in the expansion of \( (1+x)^{21} \) is
(a) 20
(b) 21
(c) 22
(d) 24
Answer: (c) 22

Question. The 4th term in the expansion of \( \left( \sqrt{x} + \frac{1}{x} \right)^{12} \) is
(a) \( 110x^{\frac{3}{2}} \)
(b) \( 220x^{\frac{3}{2}} \)
(c) \( 220x^2 \)
(d) \( 110x^2 \)
Answer: (b) \( 220x^{\frac{3}{2}} \)

Question. The (n+1)th term from the end in \( \left( x - \frac{1}{x} \right)^{3n} \) is
(a) \( {}^{3n}C_n . x^{-n} \)
(b) \( (-1)^n . {}^{3n}C_n . x^{-n} \)
(c) \( {}^{3n}C_n . x^n \)
(d) \( (-1)^n . {}^{3n}C_n . x^n \)
Answer: (a) \( {}^{3n}C_n . x^{-n} \)

Question. If the coefficient of x in \( \left( x^2 + \frac{k}{x} \right)^5 \) is 270, then k=
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (a) 3

Question. The coefficients of \( x^p \) and \( x^q \) (p and q are positive integers) in the expansion of \( (1+x)^{p+q} \) are
(a) equal
(b) equal with opposite signs
(c) reciprocal to each other
(d) unequal
Answer: (a) equal

Question. If the coefficients of 5th, 6th and 7th terms in the expansion of \( (1+x)^n \) are in A.P. then n =
(a) 7
(b) 14
(c) 7 or 14
(d) 8
Answer: (c) 7 or 14

Question. The coefficient of \( x^9 \) in \( (1+9x+27x^2+27x^3)^6 \) is
(a) \( {}^{18}C_9 . 3^9 \)
(b) \( {}^{18}C_8 . 3^9 \)
(c) \( {}^{18}C_{10} . 3^8 \)
(d) \( {}^{8}C_9 . 3^7 \)
Answer: (a) \( {}^{18}C_9 . 3^9 \)

Question. If the coefficient of x in the expansion of \( (1+ax)^8(1+3x)^4-(1+x)^3(1+2x)^4 \) is zero, then a=
(a) 1/4
(b) -1/4
(c) 1/8
(d) -1/8
Answer: (d) -1/8

Question. The term independent of \( x \) in the expansion of \( (1+x)^{10} . \left( 1 + \frac{1}{x} \right)^{10} \) is
(a) \( {}^{10}C_5 \)
(b) \( {}^{20}C_{10} \)
(c) \( {}^{20}C_5 \)
(d) \( {}^{10}C_2 \)
Answer: (b) \( {}^{20}C_{10} \)

Question. If the fourth term in the expansion of \( \left( px + \frac{1}{x} \right)^n \) is 5/2, then (n, p) =
(a) (3, 1/2)
(b) (6, 1/2)
(c) (5, 1/2)
(d) (6, 2)
Answer: (b) (6, 1/2)

Question. The term independent of x in the expansion of \( \left( \sqrt{\frac{x}{3}} + \frac{\sqrt{3}}{x^2} \right)^{10} \) is
(a) 5/9
(b) 5/3
(c) 1/3
(d) 4/3
Answer: (b) 5/3

Question. The middle term of \( \left( x - \frac{1}{x} \right)^{2n+1} \) is
(a) \( {}^{2n+1}C_n . x \)
(b) \( {}^{2n+1}C_n \)
(c) \( (-1)^n {}^{2n+1}C_n \)
(d) \( (-1)^n {}^{2n+1}C_n . x \)
Answer: (d) \( (-1)^n {}^{2n+1}C_n . x \)

Question. In the expansion of \( \left( \sqrt{a} + \frac{1}{\sqrt{3a}} \right)^n \) if the ratio of the binomial coefficient of the 4th term to the binomial coefficient of the 3rd term is \( \frac{10}{3} \), the 5th term is
(a) 55a
(b) 45a²
(c) 50a²
(d) 55a²
Answer: (d) 55a²