JEE Mathematics Binomial Theorem for Positive Integral Index MCQs Set 01

Question. The number of terms in the expansion of \( (1 + 3x + 3x^2 + x^3)^6 \) is
(a) 18
(b) 9
(c) 19
(d) 24
Answer: (c) 19

Question. The number of distinct terms in the expansion of \( (x + y - z)^{16} \) is
(a) 136
(b) 153
(c) 16
(d) 17
Answer: (b) 153

Question. The number of irrational terms in the expansion of \( (\sqrt[8]{5} + \sqrt[6]{2})^{100} \) is
(a) 97
(b) 98
(c) 96
(d) 99
Answer: (a) 97

Question. The number of terms whose values depend on x in the expansion of \( \left( x^2 - 2 + \frac{1}{x^2} \right)^n \) is
(a) 2n + 1
(b) 2n
(c) n
(d) None of the options
Answer: (b) 2n

Question. The number of real negative terms in the binomial expansion of \( (1 + ix)^{4n-2} \), \( n \in N \), \( x > 0 \), is
(a) n
(b) n + 1
(c) n – 1
(d) 2n
Answer: (a) n

Question. In the expansion of \( (x + \sqrt{x^2 - 1})^6 + (x - \sqrt{x^2 - 1})^6 \), the number of terms is
(a) 7
(b) 14
(c) 6
(d) 4
Answer: (d) 4

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

Question. The number of rational terms in the expansion of \( (1 + \sqrt{2} + \sqrt[3]{3})^6 \) is
(a) 6
(b) 7
(c) 5
(d) 8
Answer: (b) 7

Question. The number of terms with integral coefficients in the expansion of \( (7^{1/3} + 5^{1/2}x)^{600} \) is
(a) 100
(b) 50
(c) 101
(d) None of the options
Answer: (c) 101

Question. The sum of the rational terms in the expansion of \( (2 + \sqrt[5]{3})^{10} \) is
(a) 32
(b) 50
(c) 41
(d) None of the options
Answer: (c) 41

Question. The last term in the binomial expansion of \( \left( \sqrt[3]{2} - \frac{1}{\sqrt{2}} \right)^n \) is \( \left( \frac{1}{3 \cdot \sqrt[3]{9}} \right)^{\log_3 8} \). Then the 5th term from the beginning is
(a) \(^{10}C_6\)
(b) \(2 \cdot ^{10}C_4\)
(c) \( \frac{1}{2} \cdot ^{10}C_4 \)
(d) None of the options
Answer: (a) \(^{10}C_6\)

Question. If the 4th term in the expansion of \( (px + x^{-1})^m \) is 2.5 for all \( x \in R \) then
(a) \( p = \frac{5}{2}, m = 3 \)
(b) \( p = \frac{1}{2}, m = 6 \)
(c) \( p = -\frac{1}{2}, m = 6 \)
(d) None of the options
Answer: (b) \( p = \frac{1}{2}, m = 6 \)

Question. In the expansion of \( (1 + ax)^n \), \( n \in N \), the coefficient of x and \( x^2 \) are 8 and 24 respectively. Then
(a) a = 2, n = 4
(b) a = 4, n = 2
(c) a = 2, n = 6
(d) a = -2, n = 4
Answer: (a) a = 2, n = 4

Question. In the expansion of \( \left( x^3 - \frac{1}{x^2} \right)^n \), \( n \in N \), if the sum of the coefficients of \( x^5 \) and \( x^{10} \) is 0 then n is
(a) 25
(b) 20
(c) 15
(d) None of the options
Answer: (c) 15

Question. The coefficient of \( x^{20} \) in the expansion of \( (1 + x^2)^{40} \cdot (x^2 + 2 + \frac{1}{x^2})^{-5} \) is
(a) \(^{30}C_{10}\)
(b) \(^{30}C_{25}\)
(c) 1
(d) None of the options
Answer: (b) \(^{30}C_{25}\)

Question. The coefficient of \( a^8b^{10} \) in the expansion of \( (a + b)^{18} \) is
(a) \(^{18}C_8\)
(b) \(^{18}P_{10}\)
(c) \( 2^{18} \)
(d) None of the options
Answer: (a) \(^{18}C_8\)

Question. If the coefficient of the (m + 1)th term and the (m + 3)th term in the expansion of \( (1 + x)^{20} \) are equal then the value of m is
(a) 10
(b) 8
(c) 9
(d) None of the options
Answer: (c) 9

Question. The coefficient of \( x^3 \) in the expansion of \( (1 - x + x^2)^5 \) is
(a) 10
(b) -20
(c) -50
(d) -30
Answer: (d) -30

Question. If the coefficients of the 2nd, 3rd and 4th terms in the expansion of \( (1 + x)^n \), \( n \in N \), are in AP then n is
(a) 7
(b) 14
(c) 2
(d) None of the options
Answer: (a) 7

Question. The coefficient of \( x^6 \) in \( \{ (1 + x)^6 + (1 + x)^7 + \dots + (1 + x)^{15} \} \) is
(a) \(^{16}C_9\)
(b) \(^{16}C_5 - ^{6}C_5\)
(c) \(^{16}C_6 - 1\)
(d) None of the options
Answer: (a) \(^{16}C_9\)

Question. The coefficient of \( x^3 y^4 z \) in the expansion of \( (1 + x + y - z)^9 \) is
(a) \( 2 \cdot ^9C_7 \cdot ^7C_4 \)
(b) \( -2 \cdot ^9C_2 \cdot ^7C_3 \)
(c) \( ^9C_7 \cdot ^7C_4 \)
(d) None of the options
Answer: (b) \( -2 \cdot ^9C_2 \cdot ^7C_3 \)

Question. The coefficient of \( x^{13} \) in expansion of \( (1 - x)^5 (1 + x + x^2 + x^3)^4 \) is
(a) 4
(b) -4
(c) 0
(d) None of the options
Answer: (a) 4

Question. The coefficient of \( x^6 \cdot y^{-2} \) in the expansion of \( \left( \frac{x^2}{y} - \frac{y}{x} \right)^{12} \) is
(a) \(^{12}C_6\)
(b) \( -^{12}C_5 \)
(c) 0
(d) None of the options
Answer: (c) 0

Question. The greatest value of the term independent of x in the expansion of \( (x \sin \alpha + x^{-1} \cos \alpha)^{10} \), \( \alpha \in R \), is
(a) \( 2^5 \)
(b) \( \frac{10!}{(5!)^2} \)
(c) \( \frac{1}{2^5} \cdot \frac{10!}{(5!)^2} \)
(d) None of the options
Answer: (c) \( \frac{1}{2^5} \cdot \frac{10!}{(5!)^2} \)

Question. In the expansion of \( \left( x^3 - \frac{1}{x^2} \right)^{15} \), the constant term is
(a) \(^{15}C_6\)
(b) 0
(c) \( -^{15}C_6 \)
(d) 1
Answer: (c) \( -^{15}C_6 \)

Question. The constant term in the expansion of \( (1 + x)^{10} \cdot (1 + \frac{1}{x})^{12} \) is
(a) \(^{22}C_{10}\)
(b) 0
(c) \(^{22}C_{11}\)
(d) None of the options
Answer: (a) \(^{22}C_{10}\)

Question. The term independent of x in the expansion of \( (1 - x)^2 \cdot (x + \frac{1}{x})^{10} \) is
(a) \(^{11}C_5\)
(b) \(^{10}C_5\)
(c) \(^{10}C_4\)
(d) None of the options
Answer: (a) \(^{11}C_5\)

Choose the correct options. One or more options may be correct.

Question. Let \( f(x) = (\sqrt{x^2+1} + \sqrt{x^2-1})^6 + \left( \frac{2}{\sqrt{x^2+1} + \sqrt{x^2-1}} \right)^6 \). Then
(a) f(x) is a polynomial of the sixth degree in x
(b) f(x) has exactly two terms
(c) f(x) is not a polynomial in x
(d) coefficient of \( x^6 \) is 64
Answer: (a) f(x) is a polynomial of the sixth degree in x, (b) f(x) has exactly two terms, (d) coefficient of \( x^6 \) is 64

Question. The coefficient of \( a^8b^6c^4 \) in the expansion of \( (a + b + c)^{18} \) is
(a) \( ^{18}C_{14} \cdot ^{14}C_8 \)
(b) \( ^{18}C_{10} \cdot ^{10}C_6 \)
(c) \( ^{18}C_6 \cdot ^{12}C_8 \)
(d) \( ^{18}C_4 \cdot ^{14}C_6 \)
Answer: (a) \( ^{18}C_{14} \cdot ^{14}C_8 \), (b) \( ^{18}C_{10} \cdot ^{10}C_6 \), (c) \( ^{18}C_6 \cdot ^{12}C_8 \), (d) \( ^{18}C_4 \cdot ^{14}C_6 \)

Question. The term independent of x in the expansion of \( (1 + x)^n \cdot (1 - \frac{1}{x})^n \) is
(a) 0, if n is odd
(b) \( (-1)^{\frac{n-1}{2}} \cdot ^nC_{\frac{n-1}{2}} \), if n is odd
(c) \( (-1)^{n/2} \cdot ^nC_{n/2} \), if n is even
(d) None of the options
Answer: (a) 0, if n is odd, (c) \( (-1)^{n/2} \cdot ^nC_{n/2} \), if n is even

Question. The coefficient of the \( (r + 1) \)th term of \( (x + \frac{1}{x})^{20} \) when expanded in the descending powers of x is equal to the coefficient of the 6th term of \( (x^2 + 2 + \frac{1}{x^2})^{10} \) when expanded in ascending powers of x. The value of r is
(a) 5
(b) 6
(c) 14
(d) 15
Answer: (a) 5, (d) 15

Question. If \( (1 + x)^{2n} = a_0 + a_1x + a_2x^2 + \dots + a_{2n}x^{2n} \) then
(a) \( a_0 + a_2 + a_4 + \dots = \frac{1}{2} (a_0 + a_1 + a_2 + a_3 + \dots) \)
(b) \( a_{n+1} < a_n \)
(c) \( a_{n-3} = a_{n+3} \)
(d) None of the options
Answer: (a) \( a_0 + a_2 + a_4 + \dots = \frac{1}{2} (a_0 + a_1 + a_2 + a_3 + \dots) \), (b) \( a_{n+1} < a_n \), (c) \( a_{n-3} = a_{n+3} \)