Class 11 Mathematics Binomial Theorem MCQs Set 08

Question. \( 2 + \frac{5}{2!.3} + \frac{5.7}{3!.3^2} + ....... \infty = \)
(a) \( 3\sqrt{3} - 1 \)
(b) \( \sqrt{3} \)
(c) \( 3\sqrt{3} + 1 \)
(d) \( 3\sqrt{3} \)
Answer: (d) \( 3\sqrt{3} \)

Question. If \( 17^{th} \) and \( 18^{th} \) terms of the expansion of \( (2+a)^{50} \) are equal then value of a is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. \( c_0^2 + c_1^2 + c_2^2 + ... + c_{25}^2 = \) (where \( n c_r = ^{25}C_r \))
(a) \( ^{49}C_{50} \)
(b) \( ^{49}C_{25} \)
(c) \( ^{50}C_{25} \)
(d) \( ^{39}C_{40} \)
Answer: (c) \( ^{50}C_{25} \)

Question. let \( p_n \) denotes product of binomial coefficients in \( (1+x)^n \) then \( \frac{p_{n+1}}{p_n} = \)
(a) \( \frac{(n+1)^n}{n!} \)
(b) \( \frac{(n+1)^{2n}}{n!} \)
(c) \( \frac{(n+1)^{2n}}{(n!)^2} \)
(d) \( \frac{(n+1)^2}{(n!)^2} \)
Answer: (a) \( \frac{(n+1)^n}{n!} \)

Question. If \( 9^7 + 7^9 \) is divisible by \( 2^n \) then the greatest value of n, where \( n \in N \)
(a) 6
(b) 31
(c) 5
(d) 7
Answer: (a) 6

Question. The sum of \( \sum_{r=0}^n {^{n+r}C_r} \)
(a) \( ^{2n}C_n \)
(b) \( ^{2n+1}C_n \)
(c) \( ^{2n+1}C_{n+1} \)
(d) \( 3 ^{n+1}C_{n+1} \)
Answer: (c) \( ^{2n+1}C_{n+1} \)

Question. Coefficient of \( x \) in the expansion of \( (1 - 2x^3 + 3x^5) \left(1 + \frac{1}{x}\right)^8 \) is
(a) 154
(b) 164
(c) 146
(d) 156
Answer: (a) 154

Question. The middle term in the expansion of \( (1 - 3x + 3x^2 - x^3)^{2n} \) is
(a) \( {}^{6n}C_{3n} (-x)^{3n} \)
(b) \( {}^{6n}C_{2n} (-x)^{2n+1} \)
(c) \( {}^{4n}C_{3n} (-x)^{3n} \)
(d) \( {}^{6n}C_{3n-1} (-x)^{3n-1} \)
Answer: (a) \( {}^{6n}C_{3n} (-x)^{3n} \)

Question. In the expansion of \( (1+x)^n \), the 5th term is 4 times the 4th term and the 4th term is 6 times the 3rd term. then n=
(a) 9
(b) 10
(c) 11
(d) 12
Answer: (c) 11

Question. In the expansion \( (1+x)^m (1-x)^n \), the coefficients of \( x \) and \( x^2 \) are 3 and -6 respectively, then \( m \) is
(a) 6
(b) 9
(c) 12
(d) 24
Answer: (c) 12

Question. If \( (1+kx)^{10} = a_0 + a_1 x + a_2 x^2 + \dots + a_{10} x^{10} \) and \( a_2 + \frac{7}{5} a_1 + 1 = 0 \) then \( k = \)
(a) \( -1/5, -1/9 \)
(b) \( 1/5, 1/9 \)
(c) \( -1/5, -1/7 \)
(d) \( 1/5, -1/9 \)
Answer: (a) \( -1/5, -1/9 \)

Question. \( C_1 + 2.5C_2 + 3.5^2C_3 + \dots = \)
(a) \( n.6^{n-1} \)
(b) \( 6^{n-1} \)
(c) \( 6^n \)
(d) \( n.6^n \)
Answer: (a) \( n.6^{n-1} \)

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

Question. \( \sum_{r=2}^n (5r-3) C_r = \)
(a) \( (5n+6)2^{n-1} - 2n + 2 \)
(b) \( (5n+6)2^{n-1} - 2n + 3 \)
(c) \( (5n-6)2^{n-1} - 2n + 2 \)
(d) \( (5n-6)2^{n-1} - 2n + 3 \)
Answer: (d) \( (5n-6)2^{n-1} - 2n + 3 \)

Question. The sum of the coefficients of even powers of x in the expansion \( (1 - x + x^2 - x^3)^5 \) is
(a) 512
(b) 0
(c) -512
(d) 510
Answer: (a) 512

Question. The sum of the coefficients of middle terms in the expansion of \( (1+x)^{2n-1} \)
(a) \( (2n)! \)
(b) \( \frac{(2n)!}{n!} \)
(c) \( \frac{(2n)!}{(n!)^2} \)
(d) \( \frac{(2n-1)!}{n!} \)
Answer: (c) \( \frac{(2n)!}{(n!)^2} \)

Question. The sum of the coefficients of the first 10 terms in the expansion of \( (1-x)^{-3} \) is
(a) 220
(b) 286
(c) 120
(d) 150
Answer: (a) 220

Question. If \( (1 - x - x^2)^{20} = \sum_{r=0}^{40} a_r x^r \), then \( a_1 + 3a_3 + 5a_5 + \dots + 39a_{39} = \)
(a) 40
(b) -40
(c) 80
(d) -80
Answer: (a) 40

Question. The coefficient of \( x^{24} \) in the expansion of \( (1 + 3x + 6x^2 + 10x^3 + \dots + \infty)^{2/3} = \)
(a) 300
(b) 250
(c) 25
(d) 205
Answer: (c) 25

Question. If \( S_n \) denotes the sum of first 'n' natural numbers, then
\( S_1 + S_2 x + S_3 x^2 + \dots + S_n x^{n-1} + \dots \infty = \)
(a) \( (1-x)^{-1} \)
(b) \( (1-x)^{-2} \)
(c) \( (1-x)^{-3} \)
(d) \( (1-x)^{-4} \)
Answer: (c) \( (1-x)^{-3} \)

Question. If \( \frac{1}{1 - 2x + x^2} = 1 + b_1 x + b_2 x^2 + b_3 x^3 + \dots \infty \)
\( |x| < 1 \), then the value of \( b_1 \) is
(a) 3
(b) 2
(c) 4
(d) 1
Answer: (b) 2

Question. If \( y = 2x + 3x^2 + 4x^3 + \dots \), then
\( \frac{y}{2} - \frac{1 \cdot 3}{2!} \left( \frac{y}{2} \right)^2 + \frac{1 \cdot 3 \cdot 5}{3!} \left( \frac{y}{2} \right)^3 - \dots \infty = \)
(a) \( x+2 \)
(b) \( x \)
(c) \( x-2 \)
(d) \( x+4 \)
Answer: (b) \( x \)

Question. \( 1 + \frac{2 \cdot 1}{3 \cdot 2} + \frac{2 \cdot 5 \cdot 1}{3 \cdot 6 \cdot 2^2} + \frac{2 \cdot 5 \cdot 8}{3 \cdot 6 \cdot 9} \frac{1}{2^3} + \dots \infty = \)
(a) 0.4 (nearly)
(b) 0.3
(c) \( \sqrt{3} \)
(d) \( \sqrt{3} - 1 \)
Answer: (a) 0.4 (nearly)

Question. The greatest integer less than or equal to \( (2 + \sqrt{3})^6 \) is
(a) 2702
(b) 2701
(c) 2700
(d) 2699
Answer: (b) 2701

Question. \( (2 - \sqrt{5})^6 + (2 + \sqrt{5})^6 = \)
(a) 1264
(b) 1964
(c) 2889
(d) 5778
Answer: (d) 5778

Question. The number of terms in the expansion of \( \left[ (a + 4b)^3 + (a - 4b)^3 \right]^2 \) are
(a) 6
(b) 8
(c) 7
(d) 3
Answer: (d) 3

Question. The coefficient of \( x^{17} \) in the expansion of \( (x - 1)(x - 2)(x - 3) \dots (x - 18) \) is
(a) 164
(b) -171
(c) 194
(d) 221
Answer: (b) -171

Question. If \( (1 + x + x^2)^n = \sum_{r=0}^{2n} a_r x^r \) then
\( a_1 - 2a_2 + 3a_3 - \dots - 2n a_{2n} = \dots \)
(a) 0
(b) 1
(c) \( n \)
(d) \( -n \)
Answer: (d) \( -n \)

Question. The term independent of x in the expansion \( \left(1+2x+\frac{2}{x}\right)^3 \) is
(a) 12
(b) 18
(c) 36
(d) 25
Answer: (d) 25

Question. If the \( 5^{\text{th}} \) term of \( \left(\frac{q}{2x}-px\right)^8 \) is 1120 and \( p+q=5, p \gt q \) then p=
(a) 3
(b) 6
(c) 4
(d) 7
Answer: (c) 4

Question. The coefficient of \( x^{53} \) in the expansion of \( \sum_{m=0}^{100} {}^{100}C_m (x-3)^{100-m} \cdot 2^m \) is
(a) \( C(100,53) \)
(b) \( -C(100,43) \)
(c) \( -C(100,53) \)
(d) \( {}^{100}C_{50} \)
Answer: (c) \( -C(100,53) \)

Question. If \( C_0 + C_1 + C_2 + \dots + C_n = 128 \) then \( C_0 - \frac{C_1}{2} + \frac{C_2}{3} - \frac{C_3}{4} + \dots = \)
(a) 0
(b) 8
(c) 1/8
(d) 7/8
Answer: (c) 1/8

Question. \( {}^{(n+1)}C_1 + {}^{(n+1)}C_2 + {}^{(n+1)}C_3 + \dots + {}^{(n+1)}C_n = \)
(a) \( 2(2^n + 1) \)
(b) \( 2(2^n - 1) \)
(c) \( 2^{n+1} \)
(d) \( (2^{n+1} - 1) \)
Answer: (b) \( 2(2^n - 1) \)

Question. \( \int_0^1 (1-x^3)^n dx = \)
(a) \( C_0 - \frac{C_1}{4} + \frac{C_2}{7} - \frac{C_3}{10} + \dots + (-1)^n \frac{C_n}{3n+1} \)
(b) \( C_0 + \frac{C_1}{4} + \frac{C_2}{7} + \frac{C_3}{10} + \dots + \frac{C_n}{3n+1} \)
(c) \( C_0 + \frac{C_1}{4} + \frac{C_2}{7} + \frac{C_3}{10} + \dots + \frac{C_{n-1}}{3n-2} \)
(d) \( C_0C_1 + C_1C_2 + C_2C_3 + \dots + C_{n-1}C_n \)
Answer: (a) \( C_0 - \frac{C_1}{4} + \frac{C_2}{7} - \frac{C_3}{10} + \dots + (-1)^n \frac{C_n}{3n+1} \)

Question. The sum of the coefficients of odd powers of x in the expansion \( (1+x+x^2+x^3)^5 \) is
(a) 520
(b) 525
(c) 576
(d) 512
Answer: (d) 512

Question. If the sum of the coefficients in the expansion of \( (a^2x^2 - 2ax + 1)^{51} \) vanishes, then the value of a is
(a) 2
(b) -1
(c) 1
(d) -2
Answer: (c) 1

Question. The sum of the binomial coefficients of the \( 3^{\text{rd}}, 4^{\text{th}} \) terms from the beginning and from the end of \( (a+x)^n \) is 440 then n=
(a) 10
(b) 11
(c) 12
(d) 13
Answer: (b) 11

Question. Using the binomial expansion upto 3 terms the approximate value of \( (8.8)^{1/3} \) can be shown to be
(a) \( 2 + \frac{67}{900} \)
(b) \( 2 + \frac{107}{900} \)
(c) \( 2 + \frac{58}{900} \)
(d) \( 2 + \frac{47}{900} \)
Answer: (c) \( 2 + \frac{58}{900} \)

Question. If cubes and higher powers of x are negligible then one can write \( \sqrt{\frac{1-x}{1+x}} = 1 + Ax + Bx^2 \), then A=
(a) -1
(b) 1
(c) 0
(d) 2
Answer: (a) -1

Question. The coefficient of \( x^4 \) in the expansion of \( \frac{3x-8}{4-4x+x^2} \) is
(a) \( -1/4 \)
(b) -4
(c) 4
(d) 1/4
Answer: (a) \( -1/4 \)