JEE Mathematics Binomial Theorem for Positive Integral Index MCQs Set 02

Question. The middle term in the expansion of \( \left( \frac{2x}{3} - \frac{3}{2x^2} \right)^{2n} \) is
(a) \(^{2n}C_n\)
(b) \( (-1)^n \frac{(2n)!}{(n!)^2} \cdot x^{-n} \)
(c) \( ^{2n}C_n \cdot \frac{1}{x^n} \)
(d) None of the options
Answer: (b) \( (-1)^n \frac{(2n)!}{(n!)^2} \cdot x^{-n} \)

Question. The middle term in the expansion of \( \left( 1 - \frac{1}{x} \right)^n \cdot (1 - x)^n \) is
(a) \(^{2n}C_n\)
(b) \( -^{2n}C_n \)
(c) \( -^{2n}C_{n-1} \)
(d) None of the options
Answer: (a) \(^{2n}C_n\)

Question. If the rth term is the middle term in the expansion of \( \left( x^2 - \frac{1}{2x} \right)^{20} \) then the (r + 3)th term is
(a) \( ^{20}C_{14} \cdot \frac{1}{2^{14}} \cdot x \)
(b) \( ^{20}C_{12} \cdot \frac{1}{2^{12}} \cdot x^2 \)
(c) \( -\frac{1}{2^{13}} \cdot ^{20}C_7 \cdot x \)
(d) None of the options
Answer: (c) \( -\frac{1}{2^{13}} \cdot ^{20}C_7 \cdot x \)

Question. Let \( n \in N \) and \( n < (\sqrt{2} + 1)^6 \). Then the greatest value of n is
(a) 199
(b) 198
(c) 197
(d) 196
Answer: (c) 197

Question. If the coefficient of the 5th term be the numerically greatest coefficient in the expansion of \( (1 - x)^n \) then the positive integral value of n is
(a) 9
(b) 8
(c) 7
(d) 10
Answer: (b) 8

Question. The greatest coefficient in the expansion of \( (1 + x)^{2n} \) is
(a) \( \frac{1.3.5. \dots .(2n - 1)}{n!} \cdot 2^n \)
(b) \(^{2n}C_{n-1}\)
(c) \(^{2n}C_{n+1}\)
(d) None of the options
Answer: (a) \( \frac{1.3.5. \dots .(2n - 1)}{n!} \cdot 2^n \)

Question. Let n be an odd natural number greater than 1. Then the number of zeros at the end of the sum \( 99^n + 1 \) is
(a) 3
(b) 4
(c) 2
(d) None of the options
Answer: (c) 2

Question. Let \( f(n) = 10^n + 3 \cdot 4^{n+2} + 5 \), \( n \in N \). The greatest value of the integer which divides f(n) for all n is
(a) 27
(b) 9
(c) 3
(d) None of the options
Answer: (b) 9

Question. \( 2^{60} \) when divided by 7 leaves the remainder
(a) 1
(b) 6
(c) 5
(d) 2
Answer: (a) 1

Question. If {x} denotes the fractional part of x then \( \{ \frac{3^{2n}}{9} \} \), \( n \in N \), is
(a) 3/8
(b) 7/8
(c) 1/8
(d) None of the options
Answer: (c) 1/8

Question. The sum of the coefficients in the binomial expansion of \( \left( \frac{1}{x} + 2x \right)^n \) is equal to 6561. The constant term in the expansion is
(a) \(^8C_4\)
(b) \( 16 \cdot ^8C_4 \)
(c) \( ^6C_4 \cdot 2^4 \)
(d) None of the options
Answer: (b) \( 16 \cdot ^8C_4 \)

Question. The sum of the numerical coefficients in the expansion of \( \left( 1 + \frac{x}{3} + \frac{2y}{3} \right)^{12} \) is
(a) 1
(b) 2
(c) \( 2^{12} \)
(d) None of the options
Answer: (c) \( 2^{12} \)

Question. The sum of the last ten coefficients in the expansion of \( (1 + x)^{19} \) when
(a) \( 2^{18} \)
(b) \( 2^{19} \)
(c) \( 2^{18} - ^{19}C_{10} \)
(d) None of the options
Answer: (a) \( 2^{18} \)

Question. The sum of the coefficients of \( x^{2r} \), r = 1, 2, 3,...., in the expansion of \( (1 + x)^n \) is
(a) \( 2^n \)
(b) \( 2^{n-1} - 1 \)
(c) \( 2^n - 1 \)
(d) \( 2^{n-1} + 1 \)
Answer: (b) \( 2^{n-1} - 1 \)

Question. The sum of the coefficients in the polynomial expansion of \( (1 + x - 3x^2)^{2163} \) is
(a) 1
(b) -1
(c) 0
(d) None of the options
Answer: (b) -1

Question. The sum of the coefficients of all the integral powers of x in the expansion of \( (1 + 2\sqrt{x})^{40} \) is
(a) \( 3^{40} + 1 \)
(b) \( 3^{40} - 1 \)
(c) \( \frac{1}{2}(3^{40} - 1) \)
(d) \( \frac{1}{2}(3^{40} + 1) \)
Answer: (d) \( \frac{1}{2}(3^{40} + 1) \)

Question. If \( (1 + x - 2x^2)^8 = a_0 + a_1x + a_2x^2 + \dots + a_{16}x^{16} \) then the sum \( a_1 + a_3 + a_5 + \dots + a_{15} \) is equal to
(a) \( -2^7 \)
(b) \( 2^7 \)
(c) \( 2^8 \)
(d) None of the options
Answer: (a) \( -2^7 \)

Question. The sum \( ^{20}C_0 + ^{20}C_1 + ^{20}C_2 + \dots + ^{20}C_{10} \) is equal to
(a) \( 2^{20} + \frac{20!}{(10!)^2} \)
(b) \( 2^{19} - \frac{1}{2} \cdot \frac{20!}{(10!)^2} \)
(c) \( 2^{19} + ^{20}C_{10} \)
(d) None of the options
Answer: (d) None of the options

Question. The sum \( ^{10}C_3 + ^{11}C_3 + ^{12}C_3 + \dots + ^{20}C_3 \) is equal to
(a) \(^{21}C_4\)
(b) \(^{21}C_4 + ^{10}C_4\)
(c) \(^{21}C_{17} - ^{10}C_6\)
(d) None of the options
Answer: (c) \(^{21}C_{17} - ^{10}C_6\)

Question. If \( (1 + x)^{10} = a_0 + a_1x + a_2x^2 + \dots + a_{10}x^{10} \) then \( (a_0 - a_2 + a_4 - a_6 + a_8 - a_{10})^2 + (a_1 - a_3 + a_5 - a_7 + a_9)^2 \) is equal to
(a) \( 3^{10} \)
(b) \( 2^{10} \)
(c) \( 2^9 \)
(d) None of the options
Answer: (b) \( 2^{10} \)

Question. The sum \( \frac{1}{2} \cdot ^{10}C_0 - ^{10}C_1 + 2 \cdot ^{10}C_2 - 2^2 \cdot ^{10}C_3 + \dots + 2^9 \cdot ^{10}C_{10} \) is equal to
(a) 1/2
(b) 0
(c) \( \frac{1}{2} \cdot 3^{10} \)
(d) None of the options
Answer: (a) 1/2

Question. \( 1 \cdot ^nC_1 + 2 \cdot ^nC_2 + 3 \cdot ^nC_3 + \dots + n \cdot ^nC_n \) is equal to
(a) \( \frac{n(n+1)}{4} \cdot 2^n \)
(b) \( n \cdot 2^{n+1} - 3 \)
(c) \( n \cdot 2^{n-1} \)
(d) None of the options
Answer: (c) \( n \cdot 2^{n-1} \)

Question. If \( a_n = \sum_{r=0}^{n} \frac{1}{^nC_r} \) then \( \sum_{r=0}^{n} \frac{r}{^nC_r} \) equals
(a) \( (n-1)a_n \)
(b) \( n \cdot a_n \)
(c) \( \frac{1}{2} n a_n \)
(d) None of the options
Answer: (c) \( \frac{1}{2} n a_n \)

Question. The sum of the series \( \sum_{r=1}^{n} (-1)^{r-1} \cdot ^nC_r \cdot (a - r) \) is equal to
(a) \( n \cdot 2^{n-1} + a \)
(b) 0
(c) a
(d) None of the options
Answer: (c) a

Question. Let \( (1 + x)^n = \sum_{r=0}^{n} a_r x^r \). Then \( \left( 1 + \frac{a_1}{a_0} \right) \left( 1 + \frac{a_2}{a_1} \right) \dots \left( 1 + \frac{a_n}{a_{n-1}} \right) \) is equal to
(a) \( \frac{(n+1)^{n+1}}{n!} \)
(b) \( \frac{(n+1)^n}{n!} \)
(c) \( \frac{n^{n-1}}{(n-1)!} \)
(d) \( \frac{(n+1)^{n-1}}{(n-1)!} \)
Answer: (b) \( \frac{(n+1)^n}{n!} \)

Question. The value of \( \sum_{r=1}^{10} r \cdot \frac{^nC_r}{^nC_{r-1}} \) is equal to
(a) 5(2n – 9)
(b) 10n
(c) 9(n – 4)
(d) None of the options
Answer: (a) 5(2n – 9)

Question. The sum \( \sum_{r=1}^{n} r \cdot ^{2n}C_r \) is equal to
(a) \( n \cdot 2^{2n-1} \)
(b) \( 2^{2n-1} \)
(c) \( 2^{n-1} + 1 \)
(d) None of the options
Answer: (a) \( n \cdot 2^{2n-1} \)

Question. The sum \( 1 \cdot ^{20}C_1 - 2 \cdot ^{20}C_2 + 3 \cdot ^{20}C_3 - \dots - 20 \cdot ^{20}C_{20} \) is equal to
(a) \( 2^{19} \)
(b) 0
(c) \( 2^{20-1} \)
(d) None of the options
Answer: (b) 0

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

Question. In the expansion of \( \left( \sqrt[3]{4} + \frac{1}{\sqrt[4]{6}} \right)^{20} \),
(a) the number of rational terms = 4
(b) the number of irrational terms = 18
(c) the middle term is irrational
(d) the number of irrational terms = 17
Answer: (b) the number of irrational terms = 18, (c) the middle term is irrational

Question. Let \( n \in N \). If \( (1 + x)^n = a_0 + a_1x + a_2x^2 + \dots + a_nx^n \), and \( a_{n-3}, a_{n-2}, a_{n-1} \) are in AP then
(a) \( a_1, a_2, a_3 \) are in AP
(b) \( a_1, a_2, a_3 \) are in HP
(c) n = 7
(d) n = 14
Answer: (a) \( a_1, a_2, a_3 \) are in AP, (c) n = 7

Question. Let \( R = (8 + 3\sqrt{7})^{20} \) and [R] = the greatest integer less than or equal to R.
(a) [R] is even
(b) [R] is odd
(c) \( R - [R] = 1 - \frac{1}{(8 + 3\sqrt{7})^{20}} \)
(d) None of the options
Answer: (b) [R] is odd, (c) \( R - [R] = 1 - \frac{1}{(8 + 3\sqrt{7})^{20}} \)

Question. \( \frac{1}{1!.(n-1)!} + \frac{1}{3!.(n-3)!} + \frac{1}{5!(n-5)!} + \dots \) is equal to
(a) \( \frac{2^{n-1}}{n!} \) for even values of n only
(b) \( \frac{2^{n-1} + 1}{n!} - 1 \) for odd values of n only
(c) \( \frac{2^{n-1}}{n!} \) for all \( n \in N \)
(d) None of the options
Answer: (c) \( \frac{2^{n-1}}{n!} \) for all \( n \in N \)

Question. In the expansion of \( (x + y + z)^{25} \)
(a) every term is of the form \( ^{25}C_r \cdot ^rC_k \cdot x^{25-r} \cdot y^{r-k} \cdot z^k \)
(b) the coefficient of \( x^8 y^9 z^9 \) is 0
(c) the number of terms is 325
(d) None of the options
Answer: (a) every term is of the form \( ^{25}C_r \cdot ^rC_k \cdot x^{25-r} \cdot y^{r-k} \cdot z^k \), (b) the coefficient of \( x^8 y^9 z^9 \) is 0