Class 11 Mathematics Binomial Theorem MCQs Set 09

Question. The number of non zero terms in \( (x+a)^{75} + (x-a)^{75} \)
(a) 38
(b) 76
(c) 34
(d) 32
Answer: (a) 38

Question. \( (\sqrt{2}+1)^6 + (\sqrt{2}-1)^6 = \)
(a) 99
(b) 98
(c) 196
(d) 198
Answer: (d) 198

Question. Let \( R = (5\sqrt{5} + 11)^{2n+1}, f = R - [R] \), then \( Rf = \)
(a) 1
(b) \( 2^n \)
(c) \( 2^{2n} \)
(d) \( 4^{2n+1} \)
Answer: (d) \( 4^{2n+1} \)

Question. If \( (6 + \sqrt{35})^n = I + F \) when I is odd and \( 0 \lt F \lt 1 \), then (I+F)(1-F)=
(a) 1
(b) 1/2
(c) 2
(d) 4
Answer: (a) 1

Question. \( (x-1)^4 + 4(x-1)^3 + 6(x-1)^2 + 4(x-1) + 1 = \)
(a) \( x^4 \)
(b) \( x^3 \)
(c) \( x^2 \)
(d) 1
Answer: (a) \( x^4 \)

Question. Let \( (1+\sqrt{2})^n = x_n + y_n\sqrt{2} \) where \( x_n, y_n \) are integers, then
(a) \( x_n^2 - 2y_n^2 = (-1)^n \)
(b) \( x_n + 2y_n - x_{n+1} = 3 \)
(c) \( x_n^2 - 2y_n^2 = 1 \)
(d) \( x_{n+1} - x_n - 2y_n = 1 \)
Answer: (a) \( x_n^2 - 2y_n^2 = (-1)^n \)

Question. The expansion \( \left[x + (x^3 - 1)^{\frac{1}{2}}\right]^5 + \left[x - (x^3 - 1)^{\frac{1}{2}}\right]^5 \) is a polynomial of degree
(a) 8
(b) 7
(c) 6
(d) 5
Answer: (b) 7

Question. If \( x = (99)^{50} + (100)^{50} \) and \( y = (101)^{50} \), then
(a) x > y
(b) x < y
(c) x = y
(d) xy = 1
Answer: (b) x < y

Question. If the coefficients of \( x^{39} \) and \( x^{40} \) are equal in the expansion of \( (p+qx)^{49} \). then the possible values of p and q are
(a) 1, 5
(b) 1, 4
(c) 1, 3
(d) 2, 7
Answer: (b) 1, 4

Question. If \( x+y=1 \), then \( \sum_{r=0}^{n} r \cdot {}^nC_r x^r \cdot y^{n-r} = \)
(a) 1
(b) n
(c) nx
(d) ny
Answer: (c) nx

Question. If \( x \) is nearly equal to 1, then \( \frac{mx^m - nx^n}{m - n} = \)
(a) \( x^{m+n} \)
(b) \( x^{m-n} \)
(c) \( x^m \)
(d) \( x^n \)
Answer: (a) \( x^{m+n} \)

Question. If p and q are the coefficients of \( x^n \) in \( (1+x)^{2n} \) and \( (1-4x)^{-1/2} \), \( |x| < \frac{1}{4} \) then
(a) p = q
(b) p = 2q
(c) q = 2p
(d) p+q = \( {}^{2n}C_n \)
Answer: (a) p = q

Question. If \( (1+x+x^2)^n = a_0 + a_1x + a_2x^2 + \dots + a_{2n}x^{2n} \) then \( a_0 + a_3 + a_6 + \dots = \)
(a) \( 3^n \)
(b) \( 3^{n-1} \)
(c) \( 3^{n-2} \)
(d) 1
Answer: (b) \( 3^{n-1} \)

Question. If m and n are +ve integers and \( m > n \) and if \( (1+x)^{m+n}(1-x)^{m-n} \) is expanded as a polynomial in x, then the coefficient of \( x^2 \) is
(a) \( 2m^2 - n \)
(b) \( 2n^2 - m \)
(c) \( 2m^2 + n \)
(d) \( 2n^2 + m \)
Answer: (b) \( 2n^2 - m \)

Question. \( 9^{11} + 11^9 \) is divisible by
(a) 7
(b) 8
(c) 9
(d) 10
Answer: (d) 10

Question. For \( n \in \mathbb{N} \); \( (1+x)^n - nx - 1 \) is divisible by
(a) 2
(b) x
(c) \( x^2 \)
(d) \( x^3 \)
Answer: (c) \( x^2 \)

Question. If \( t_0, t_1, t_2, \dots, t_n \) are the consecutive terms in the expansion \( (x+a)^n \) then \( (t_0 - t_2 + t_4 - t_6 + \dots)^2 + (t_1 - t_3 + t_5 - \dots)^2 = \)
(a) \( x^2 + a^2 \)
(b) \( (x^2 + a^2)^n \)
(c) \( x^2 - a^2 \)
(d) \( (x^2 - a^2)^n \)
Answer: (b) \( (x^2 + a^2)^n \)

Question. Coefficient of \( x^{50} \) in \( (1+x)^{1000} + 2x(1+x)^{999} + 3x^2(1+x)^{998} + \dots \) is
(a) \( {}^{1000}C_{50} \)
(b) \( {}^{1001}C_{50} \)
(c) \( {}^{1002}C_{50} \)
(d) \( {}^{1002}C_{49} \)
Answer: (c) \( {}^{1002}C_{50} \)

Question. A positive integer which is just greater than \( (1 + 0.0001)^{10000} \) is
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (a) 3

Question. If n is an integer lying between 0 and 21, then the least value of \( n!(21-n)! \) is
(a) 1!.20!
(b) 11!.10!
(c) 9!.12!
(d) 2!.19!
Answer: (b) 11!.10!

Question. Coefficient of \( x^5 \) in the expansion of \( \frac{3x}{(x-1)^2(x+2)} \) is
(a) \( \frac{171}{32} \)
(b) \( \frac{171}{64} \)
(c) \( \frac{57}{32} \)
(d) \( \frac{57}{16} \)
Answer: (a) \( \frac{171}{32} \)

Question. Coefficient of \( x^n \) in \( \frac{(1+x)(1+2x)(1+3x)}{(1-x)(1-2x)(1-3x)} \) is
(a) \( 12 - 30 \cdot 2^n - 20 \cdot 3^n \)
(b) \( 12 - 30 \cdot 2^n + 20 \cdot 3^n \)
(c) \( 12 + 30 \cdot 2^n + 20 \cdot 3^n \)
(d) \( 12 + 30 \cdot 2^n - 20 \cdot 3^n \)
Answer: (b) \( 12 - 30 \cdot 2^n + 20 \cdot 3^n \)

Question. The coefficient of \( x^9 \) in \( (x+2)(x+4)(x+8)\dots(x+1024) \) is
(a) 2046
(b) 1023
(c) 55
(d) 0
Answer: (a) 2046

Question. If \( s_n = \sum_{r=0}^n \frac{1}{{}^nC_r} \) and \( t_n = \sum_{r=0}^n \frac{r}{{}^nC_r} \), then \( \frac{t_n}{s_n} = \)
(a) \( \frac{1}{2}n \)
(b) \( \frac{2n-1}{2} \)
(c) \( n-1 \)
(d) \( \frac{1}{2}n - 1 \)
Answer: (a) \( \frac{1}{2}n \)

Question. If \( |x| < \frac{1}{2} \), then the coefficient of \( x^r \) in the expansion of \( \frac{1+2x}{(1-2x)^2} \) is
(a) \( r \cdot 2^r \)
(b) \( (2r-1)2^r \)
(c) \( r \cdot 2^{2r+1} \)
(d) \( (2r+1)2^r \)
Answer: (d) \( (2r+1)2^r \)

Question. For natural numbers m, n if \( (1-y)^m(1+y)^n = 1 + a_1y + a_2y^2 + \dots \), and \( a_1 = a_2 = 10 \), then \( (m, n) \) is
(a) (45, 35)
(b) (35, 45)
(c) (20, 45)
(d) (35, 20)
Answer: (b) (35, 45)

Question. \( \sum_{r=0}^n (-1)^r \cdot {}^nC_r \frac{1+r\log_e 10}{(1+\log_e 10^n)^r} = \)
(a) 1
(b) -1
(c) n
(d) 0
Answer: (d) 0

Question. In the expansion of \( (1+x)^n \cdot (1+y)^n \cdot (1+z)^n \), the sum of the coefficients of the terms of degree r is
(a) \( ({}^nC_r)^3 \)
(b) \( {}^{3n}C_r \)
(c) \( 3 \cdot {}^nC_r \)
(d) \( {}^nC_{3r} \)
Answer: (b) \( {}^{3n}C_r \)

Question. The coefficient of \( x^n \) in the polynomial \( (x + {}^nC_0)(x + 3 \cdot {}^nC_1)(x + 5 \cdot {}^nC_2) \dots [x + (2n+1) \cdot {}^nC_n] \) is
(a) \( n \cdot 2^n \)
(b) \( n \cdot 2^{n+1} \)
(c) \( (n+1)2^n \)
(d) \( n \cdot 2^n + 1 \)
Answer: (c) \( (n+1)2^n \)

Question. Coefficient of \( x^{2^{m+1}} \) in the expansion of \( \frac{1}{(1+x)(1+x^2)(1+x^4)(1+x^8)\dots(1+x^{2^m})} \), (\(|x| < 1\)) is
(a) 3
(b) 2
(c) 1
(d) 0
Answer: (c) 1

Question. If \( n > 2 \) and \( C_r = {}^nC_r \), then \( 1^2 \cdot C_0 - 2^2 \cdot C_1 + 3^2 \cdot C_2 - \dots = \)
(a) 0
(b) \( (-1)^n \)
(c) n
(d) -n
Answer: (a) 0

Question. If \( x = (2+\sqrt{3})^n, n \in \mathbb{N} \) and \( f = x - [x] \), then \( \frac{f^2}{1-f} \) is
(a) an irrational number
(b) a non integer rational number
(c) an odd number
(d) an even number
Answer: (d) an even number

Question. The last four digits of the natual number \( 3^{100} \) are
(a) 7231
(b) 1231
(c) 3451
(d) 2001
Answer: (d) 2001

Question. If \( C_r = {}^{30}C_r \), then \( C_0 + C_4 + C_8 + \dots + C_{28} = \)
(a) \( 2^{28} \)
(b) \( 2^{29} \)
(c) \( 2^{30} \)
(d) \( 2^{15} \)
Answer: (a) \( 2^{28} \)

Question. If \( C_r = {}^{32}C_r \), then \( \sum_{r=0}^5 C_{6r} = \)
(a) \( \frac{1}{6} \left(2^{32} - 3^{16} - 1\right) \)
(b) \( \frac{1}{6} \left(2^{32} + 3^{16} - 1\right) \)
(c) \( \frac{1}{6} \left(2^{32} - 3^{16} + 1\right) \)
(d) \( \frac{1}{6} \left(2^{32} + 3^{16}\right) \)
Answer: (a) \( \frac{1}{6} \left(2^{32} - 3^{16} - 1\right) \)

Question. The coefficient of \( x^8 \) in the expasnsion of \( \left(1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \frac{x^8}{8!}\right)^2 \) is
(a) \( \frac{1}{315} \)
(b) \( \frac{2}{315} \)
(c) \( \frac{1}{105} \)
(d) \( \frac{1}{210} \)
Answer: (a) \( \frac{1}{315} \)

Question. The remainder when \( 23^{23} \) is divided by 53 is
(a) 17
(b) 21
(c) 30
(d) 43
Answer: (c) 30

Question. If \( 2^{2006} - 2006 \) divided by 7, the remainder is
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (a) 0

Question. The number of rational terms in the expansion of \( \left(1 + \sqrt{2} + \sqrt[3]{3}\right)^6 \) is
(a) 6
(b) 7
(c) 3
(d) 8
Answer: (b) 7

Question. \( {}^{404}C_4 - {}^{303}C_4 \cdot {}^4C_1 + {}^{202}C_4 \cdot {}^4C_2 - {}^{101}C_4 \cdot {}^4C_3 = \)
(a) \( (101)^3 \)
(b) \( (101)^4 \)
(c) \( (202)^3 \)
(d) \( (202)^4 \)
Answer: (b) \( (101)^4 \)