Class 11 Mathematics Binomial Theorem MCQs Set 06

Question. The two successive terms in the expansion of \( (1+x)^{24} \) whose coefficients are in the ratio 4 : 1 are
(a) \( t_{18}, t_{19} \)
(b) \( t_{19}, t_{20} \)
(c) \( t_{20}, t_{21} \)
(d) \( t_{21}, t_{22} \)
Answer: (c) \( t_{20}, t_{21} \)

Question. When \( x = \frac{5}{2} \), numerically greatest term in the expansion of \( (3+2x)^{15} \) is
(a) 6th
(b) 8th
(c) 10th
(d) 12th
Answer: (c) 10th

Question. The coefficients of \( x^{16} \) and \( x^{17} \) in \( (a+bx)^{101} \) are equal. If a and b are natural numbers, then the least possible value of 'a' is
(a) 1
(b) 3
(c) 5
(d) 7
Answer: (c) 5

Question. The coefficient of \( x^k \) (\( 0 \le k \le n \)) in the expansion of \( 1+(1+x)+(1+x)^2+..........+(1+x)^n \) is
(a) \( {}^{n+1}C_k \)
(b) \( {}^{n}C_k \)
(c) \( {}^{n}C_{k+1} \)
(d) \( {}^{n+1}C_{k+1} \)
Answer: (d) \( {}^{n+1}C_{k+1} \)

Question. The 21st and 22nd terms in the expansion of \( (1+x)^{44} \) are equal. Then x=
(a) 8/7
(b) 7/8
(c) 7
(d) 8
Answer: (b) 7/8

Question. Sum of the coefficients of \( (1-x)^{25} \) is
(a) -1
(b) 1
(c) 0
(d) \( 2^{25} \)
Answer: (c) 0

Question. Coefficient of \( x^{10} \) in \( (1+2x^4)(1-x)^8 \) is
(a) -56
(b) 56
(c) 112
(d) -112
Answer: (b) 56

Question. The total number of terms in the expansion of \( (x+a)^{100}+(x-a)^{100} \) is
(a) 202
(b) 51
(c) 101
(d) 50
Answer: (b) 51

Question. The number of rational terms in the expansion of \( \left( \sqrt{3} + \sqrt[4]{5} \right)^{124} \) is
(a) 31
(b) 32
(c) 33
(d) 34
Answer: (b) 32

Question. If n is odd, \( {}^{n}C_1 + {}^{n}C_3 + {}^{n}C_5 + ......... + {}^{n}C_{2[n/2]-1} = \)
where [ ] denotes greatest integer
(a) \( 2^{n-1} \)
(b) \( 2^{n-1}-1 \)
(c) \( 2^n \)
(d) \( 2^n-1 \)
Answer: (b) \( 2^{n-1}-1 \)

Question. If \( n \ge 2 \) then \( 3.C_1 - 4.C_2 + 5.C_3 - ...... + (-1)^{n-1}(n+2).C_n = \)
(a) -1
(b) 2
(c) -2
(d) 1
Answer: (b) 2

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

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

Question. \( \frac{C_0}{1.2} + \frac{C_1}{2.3} + \frac{C_2}{3.4} + ......... + \frac{C_n}{(n+1)(n+2)} = \)
(a) \( \frac{2^{n+1}-n}{(n+1)(n+2)} \)
(b) \( \frac{2^{n+2}-n-3}{(n+1)(n+2)} \)
(c) \( \frac{2^n-1}{n+1} \)
(d) \( \frac{2^{n+1}+1}{(n+1)(n+2)} \)
Answer: (b) \( \frac{2^{n+2}-n-3}{(n+1)(n+2)} \)

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

Question. If \( (x-2)^{100} = \sum_{r=0}^{100} a_r.x^r \), then \( a_1 + 2a_2 + ..... + 100a_{100} = \)
(a) 100
(b) -100
(c) 1
(d) 101
Answer: (b) -100

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

Question. If the sum of the coefficients in the expansion of \( (x+y)^n \) is 4096, then the greatest coefficient is
(a) \( {}^{11}C_5 \)
(b) \( {}^{12}C_5 \)
(c) \( {}^{12}C_6 \)
(d) \( {}^{14}C_7 \)
Answer: (c) \( {}^{12}C_6 \)

Question. In the expansion of \( (x+a)^n \), sum of the odd terms is P and the sum of the even terms is Q, then 4PQ=
(a) \( (x^2-a^2)^n \)
(b) \( (x+a)^{2n}+(x-a)^{2n} \)
(c) \( (x+a)^{2n}-(x-a)^{2n} \)
(d) \( (x-a)^{2n}-(x+a)^{2n} \)
Answer: (c) \( (x+a)^{2n}-(x-a)^{2n} \)

Question. Expansion of \( (4 - 7x)^{-\frac{2}{5}} \) is valid if
(a) \( x < \frac{4}{7} \)
(b) \( x > \frac{4}{7} \)
(c) \( -\frac{4}{7} < x < \frac{4}{7} \)
(d) \( -\frac{4}{7} \le x \le \frac{4}{7} \)
Answer: (c) \( -\frac{4}{7} < x < \frac{4}{7} \)

Question. In the expansion of \( (2-3x)^{-1} \), the 3rd term is
(a) \( \frac{9x^2}{4} \)
(b) \( \frac{9x^2}{8} \)
(c) \( \frac{27x^3}{8} \)
(d) \( \frac{27x^3}{16} \)
Answer: (b) \( \frac{9x^2}{8} \)

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

Question. Coefficient of \( x^4 \) in the expansion of \( \frac{1}{(x+1)(x+2)} \) is
(a) \( \frac{1}{32} \)
(b) \( \frac{11}{32} \)
(c) \( \frac{21}{32} \)
(d) \( \frac{31}{32} \)
Answer: (d) \( \frac{31}{32} \)

Question. Coefficient of \( x^n \) in the expansion of \( (1-2x+3x^2-4x^3+.............\infty)^{-2} \) is
(a) \( \frac{n(n+1)(n+2)}{3!} \)
(b) \( (-1)^n \frac{n(n+1)(n+2)}{3!} \)
(c) \( \frac{(n+1)(n+2)(n+3)}{3!} \)
(d) \( (-1)^n \frac{(n+1)(n+2)(n+3)}{3!} \)
Answer: (d) \( (-1)^n \frac{(n+1)(n+2)(n+3)}{3!} \)

Question. If x is small, then \( \sqrt{x^2+16} - \sqrt{x^2+9} = \)
(a) \( 1 + \frac{x^2}{24} \)
(b) \( 1 - \frac{x^2}{24} \)
(c) \( 1 + \frac{x^2}{48} \)
(d) \( 1 - \frac{x^2}{48} \)
Answer: (b) \( 1 - \frac{x^2}{24} \)

Question. \( \frac{7}{5} \left( 1 + \frac{1}{10^2} + \frac{1.3}{1.2} \frac{1}{10^4} + \frac{1.3.5}{1.2.3} \frac{1}{10^6} + ...... \infty \right) = \)
(a) \( \sqrt{2} \)
(b) \( 2\sqrt{2} \)
(c) \( 2^{1/3} \)
(d) \( \sqrt{\frac{2}{3}} \)
Answer: (a) \( \sqrt{2} \)

Question. If \( z = \frac{1}{3} + \frac{1.3}{3.6} + \frac{1.3.5}{3.6.9} + ... \) then
(a) \( z^2 - 2z + 2 = 0 \)
(b) \( z^2 - 2z - 2 = 0 \)
(c) \( z^2 + 2z - 2 = 0 \)
(d) \( z^2 + 2z + 2 = 0 \)
Answer: (c) \( z^2 + 2z - 2 = 0 \)

Question. If \( |x| < 1 \) then \( 1 + n \left( \frac{2x}{1+x} \right) + \frac{n(n+1)}{2!} \left( \frac{2x}{1+x} \right)^2 + ... \infty \)
(a) \( \left( \frac{1+x}{1-x} \right)^n \)
(b) \( \left( \frac{1-x}{1+x} \right)^n \)
(c) \( \left( \frac{1+x}{2x} \right)^n \)
(d) \( \left( \frac{2x}{1+x} \right)^n \)
Answer: (a) \( \left( \frac{1+x}{1-x} \right)^n \)

Question. \( 1 - \frac{1}{5} + \frac{1.4}{5.10} - \frac{1.4.7}{5.10.15} + ...... \infty = \)
(a) \( \sqrt[3]{5} \)
(b) \( \frac{1}{2} \sqrt[3]{5} \)
(c) \( \left( \frac{5}{2} \right)^{\frac{1}{3}} \)
(d) \( \frac{1}{2} \left( \frac{5}{2} \right)^{\frac{1}{3}} \)
Answer: (b) \( \frac{1}{2} \sqrt[3]{5} \)

Question. The sum of the series \( 1 + \frac{k}{3} + \frac{k(k+1)}{3.6} + \frac{k(k+1)(k+2)}{3.6.9} + ... \) is
(a) \( \left( \frac{2}{3} \right)^k \)
(b) \( \left( \frac{3}{2} \right)^k \)
(c) \( \frac{2}{3} \)
(d) \( \frac{3}{2} \)
Answer: (b) \( \left( \frac{3}{2} \right)^k \)

Question. which is larger \( (1.01)^{1000000} \) or 10000
(a) \( (1.01)^{1000000} \)
(b) 10000
(c) can't say
(d) both are equal
Answer: (a) \( (1.01)^{1000000} \)

Question. Sum of the coefficients of all integral power of x in \( \left( 1 + 2\sqrt{x} \right)^{40} \) is
(a) \( \frac{3^{40}-1}{2} \)
(b) \( \frac{3^{40}+1}{2} \)
(c) \( \frac{3^{38}-1}{2} \)
(d) \( \frac{3^{38}+1}{2} \)
Answer: (b) \( \frac{3^{40}+1}{2} \)

Question. If c is small in comparision with \( l \) then \( \left( \frac{l}{l+c} \right)^{\frac{1}{2}} + \left( \frac{l}{l-c} \right)^{\frac{1}{2}} = \)
(a) \( 2 + \frac{3c}{4l} \)
(b) \( 2 + \frac{3c^2}{4l^2} \)
(c) \( l + \frac{3c^2}{4l^2} \)
(d) \( l + \frac{3c}{4l} \)
Answer: (b) \( 2 + \frac{3c^2}{4l^2} \)

Question. If the number of terms in the expansion of \( (x+y+z)^n \) are 36, then the value of n
(a) 7
(b) 9
(c) 4
(d) 2
Answer: (a) 7

Question. Coefficient of \( x^7 \) in the expansion of \( (1+3x-2x^3)^{10} \).
(a) 62640
(b) 64620
(c) 65640
(d) 62330
Answer: (a) 62640

Question. The coefficient of \( x^{30} \) in \( \left( 3x^2 + \frac{2}{3x^2} \right)^{15} \) is
(a) \( ^{15}C_{2} . 3^8 . 2^7 \)
(b) \( ^{15}C_{1} . 3^7 . 2^8 \)
(c) \( 3^{15} \)
(d) \( 3^{14} . 2^2 \)
Answer: (c) \( 3^{15} \)

Question. The coefficient of y in \( \left( y^2 + \frac{c}{y} \right)^5 \) is
(a) \( 20c \)
(b) \( 10c \)
(c) \( 10c^3 \)
(d) \( 20c^2 \)
Answer: (c) \( 10c^3 \)

Question. The greatest binomial coefficient in the expansion of \( \left( \frac{x^{\frac{3}{2}} \cdot y}{2} + \frac{2}{x \cdot y^{\frac{3}{2}}} \right)^{12} \) is
(a) \( ^{12}C_{5} \)
(b) \( ^{12}C_{6} \)
(c) \( ^{12}C_{5} . 2^2 \)
(d) \( ^{12}C_{6} . 2^3 \)
Answer: (b) \( ^{12}C_{6} \)

Question. If the coefficients of (2r+4)th term and (r-2)th term in the expansion of (1+x)¹⁸ are equal, then r =
(a) 4
(b) 5
(c) 6
(d) 7
Answer: (c) 6

Question. Coefficient of \( x^5 \) in the expansion of \( (1+x)^{10} \left( 1 + \frac{1}{x} \right)^{20} \) is
(a) \( ^{30}C_{5} \)
(b) \( ^{10}C_{5} \)
(c) \( ^{20}C_{5} \)
(d) \( ^{30}C_{20} \)
Answer: (a) \( ^{30}C_{5} \)