Class 11 Mathematics Binomial Theorem MCQs Set 10

Question. If \( \{x\} \) denotes fractional part of x then \( \left\{ \frac{2^{2003}}{17} \right\} = \)
(a) \( \frac{2}{17} \)
(b) \( \frac{4}{17} \)
(c) \( \frac{6}{17} \)
(d) \( \frac{8}{17} \)
Answer: (d) \( \frac{8}{17} \)

Question. If \( \left(3 + x^{2008} + x^{2009}\right)^{2010} = a_0 + a_1x + a_2x^2 + \dots + a_nx^n \) then \( a_0 - \frac{a_1}{2} - \frac{a_2}{2} + a_3 - \frac{a_4}{2} - \frac{a_5}{2} + a_6 \dots \)
(a) \( 2^{2010} \)
(b) \( 2^{2011} \)
(c) \( 2^{2012} \)
(d) \( 2^{2013} \)
Answer: (a) \( 2^{2010} \)

STATEMENTS TYPE

Question. (i) The no. of distinct terms in the expansion of \( (x_1+x_2+\dots+x_n)^3 \) is \( ^{n+2}c_3 \)
(ii) The no. of irrational terms in the expansion \( (2^{1/5} + 3^{1/10})^{55} \) is 55
(a) (i) is true (ii) is false
(b) both (i) & (ii) are true
(c) both (i) & (ii) are false
(d) (i) is false (ii) is true
Answer: (a) (i) is true (ii) is false

Question. i. Three consecutive binomial coefficients can not be in G.P.
ii. Three consecutive binomial coefficients can not be in A.P.
Which of the above statement is correct?
(a) both i and ii
(b) neither i nor ii
(c) Only i
(d) Only ii
Answer: (c) Only i

Question. S1 : If the coefficients of \( x^6 \) and \( x^7 \) in the expansion of \( \left( \frac{x}{4} + 3 \right)^n \) are equal, then the number of divisors of n is 12.
S2 : If the expansion of \( \left( x^2 + \frac{2}{x} \right)^n \) for positive integer n has 13 th term independent of x. Then the sum of divisors of n is 39.
(a) Only S1 is true
(b) Only S2 is true
(c) Both S1 and S2 are true
(d) Neither S1 nor S2 is true
Answer: (c) Both S1 and S2 are true

Question. S1 : The fourth term in the expansion of \( \left( 2x + \frac{1}{x^2} \right)^9 \) is equal to the second term in the expansion of \( (1+x^2)^{84} \) then the positive value of x is \( \frac{1}{2\sqrt{3}} \)
S2 : In the expansion of \( \left( x^2 + \frac{a}{x^3} \right)^{10} \), the co-efficients of \( x^5 \) and \( x^{15} \) are equal, then the positive value of a is 8
(a) Only S1 is true
(b) Only S2 is true
(c) Both S1 and S2 are true
(d) Neither S1 nor S2 is true
Answer: (d) Neither S1 nor S2 is true

Question. i. The sum of the binomial coefficients of the expansion \( \left( x + \frac{1}{x} \right)^n \) is \( 2^n \)
ii. The term independent of x in the expansion of \( \left( x + \frac{1}{x} \right)^n \) is 0 when is even.
Which of the above statements is correct?
(a) Only i
(b) Only ii
(c) both i and ii
(d) neither i nor ii
Answer: (a) Only i

Question. A. \( ^{2n}c_n = C_0^2 + C_1^2 + C_2^2 + C_3^2 + \dots + C_n^2 \)
B. \( ^{2n}c_n = \) term independent of x in \( (1+x)^n \left( 1 + \frac{1}{x} \right)^n \)
C. \( ^{2n}c_n = \frac{1 \cdot 3 \cdot 5 \cdot 7 \dots (2n-1)}{n!} \) then
(a) A, B are false, C is true
(b) A is false, B and C are true
(c) A and B are true; C is false
(d) A, B, C are true
Answer: (c) A and B are true; C is false

INCREASING AND DECREASING ORDER

Question. The arrangement of the following binomial expansions in the ascending order of their independent terms
A. \( \left( \sqrt{x} - \frac{3}{x^2} \right)^{10} \)
B. \( \left( x + \frac{1}{x} \right)^6 \)
C. \( (1+x)^{32} \)
D. \( \left( \frac{3}{2}x^2 - \frac{1}{3x} \right)^9 \)
(a) C,A,B,D
(b) B,C,A,D
(c) C,A,D,B
(d) D,C,B,A
Answer: (a) C,A,B,D

Question. A : If the term independent of x in the expansion of \( \left( \sqrt{x} - \frac{n}{x^2} \right)^{10} \) is 405, then n =
B: If the third term in the expansion of \( \left( \frac{1}{n} + n^{\log_n 10} \right)^5 \) is 1000, then n = (here n<10)
C: If in the binomial expansion of \( (1+x)^n \), the coefficients of 5th , 6th and 7th terms are in A.P then n =
[Arranging the values of n in ascending order]
(a) A,B,C
(b) B,A,C
(c) A,C,B
(d) C,A,B
Answer: (b) B,A,C

Question. The arrangement of the following with respect to coefficient of \( x^r \) in ascending order where \( |x| < 1 \)
A) \( x^5 \) in \( (1-x)^{-3} \) where \( |x| < 1 \)
B) \( x^7 \) in \( (1 + 2x + 3x^2 + \dots \infty) \) where \( |x| < 1 \)
C) \( x^{10} \) in \( (1+x)^{-1} \) where \( |x| < 1 \)
D) \( x^3 \) in \( (1+x)^4 \)
(a) B,A,C,D
(b) C,D,B,A
(c) A,B,D,C
(d) C,D,A,B
Answer: (b) C,D,B,A

Question. If A= \( (300)^{600} \), B=600!, C= \( (200)^{600} \) then
(a) A<B<C
(b) A>B>C
(c) A>C and C=B
(d) A=B and B>C
Answer: (b) A>B>C

ASSERTION AND REASON TYPE

Question. Assertion (A): Number of the disimilar terms in the sum of expansion \( (x+a)^{102} + (x-a)^{102} \) is 206
Reason (R): Number of terms in the expansion of \( (x+b)^n \) is n+1
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (d) A is false but R is true

Question. Assertion (A): \( 2^{4n} - 2^n(7n+1) \) is divisible by the square of 14 where n is a natural number
Reason (R): \( (1+x)^n = 1 + ^nC_1x + \dots + ^nC_nx^n \forall n \in N \)
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (a) Both A and R are true and R is correct explanation of A

Question. Assertion (A): The sum of the last ten coefficients in the expansion of \( (1+x)^{19} \), when expanded in ascending powers of x is \( 2^{18} \)
Reason (R): \( ^nC_r = ^nC_{n-r} \left( r > \frac{n}{2} \right) \forall n \in N \) and \( r \in \) whole number
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (a) Both A and R are true and R is correct explanation of A

Question. Assertion (A): The third term in the expansion of \( \left( 2x + \frac{1}{x^2} \right)^m \) does not contain x. The value of x for which that term equal to the second term in the expansion of \( (1+x^3)^{30} \) is 4
Reason (R): \( (a+x)^n = \sum_{r=0}^{n} {}^nC_r a^{n-r} x^r \)
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (d) A is false but R is true

Question. Assertion (A): Number of terms in the expansion of \( (x+y+z)^5 \) is 21.
Reason (R) : The number of terms in the expansion of \( (x+y+z)^n \) is \( ^{(n+2)}C_2 \)
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (a) Both A and R are true and R is correct explanation of A

Question. Assertion (A): In the expansion of \( (x+x^{-2})^n \) the coefficient of eighth term and nineteenth term are equal then n=25
Reason (R): Middle term in the expansion of \( (x+a)^n \) has greatest binomial coefficient where n is an even integer
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (b) Both A and R are true but R is not correct explanation of A

Question. Assertion (A): The expansion of \( (1+x)^n = C_0 + C_1x + C_2x^2 + \dots + C_nx^n \)
Reason (R): If \( x = -1 \), then the above expansion is zero
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (b) Both A and R are true but R is not correct explanation of A

Question. Assertion (A): The coefficient of \( x^7 \) in \( \left( \frac{x^2}{2} - \frac{2}{x} \right)^9 \) is zero
Reason (R) : r in \( t_{r+1} \) that contain coefficient of \( x^7 \) is not positive integer
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (a) Both A and R are true and R is correct explanation of A

Question. Assertion (A) : The coefficient of \( x^{-2} \) in the expansion of \( \left( x^2 + \frac{1}{x} \right)^5 \) is equal to \( ^5C_4 \)
Reason (R) : The value of r for the above ( r in \( t_{r+1} \) ) expansion is 3.
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (c) A is true but R is false

Question. Assertion (A): The sum of the coefficients of the middle terms in the expansion of \( (1+x)^{2n-1} \) is equal to \( ^{2n}C_n \)
Reason (R): To find the sum of the coefficients of the two middle terms in the expansion \( (1+x)^{2n-1} \) the Value of n is any natural number.
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (b) Both A and R are true but R is not correct explanation of A

Question. Assertion (A): In \( (1+x)^n \) sum of coefficients of even powers of x is not equal to the sum of coefficients off odd powers of x.
Reason (R): The value \( (1+x)^n \) for x = -1 is zero
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (d) A is false but R is true

Question. Assertion (A) : Greatest binomial coefficient in the expansion of \( (1+5x)^8 \) is \( ^8C_4 5^4 \)
Reason ( R) : Greatest coefficient in the expansion of \( (1+x)^{2n} \) is the middle term.
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (d) A is false but R is true