## Binomial Theorem Questions and Answers Part-8

1. If the cofficients of rth ,(r+1)th and (r+2)th terms in the expansion of (1 + x)n are in A.P., then n is a root of the equation
a) $x^{2}+x\left(4r+1\right)+4r^{2}-2=0$
b) $x^{2}-x\left(4r+1\right)+4r^{2}-2=0$
c) $x^{2}-x\left(4r+1\right)+\left(4r^{2}+2\right)=0$
d) $x^{2}-x\left(4r+1\right)-4r^{2}-2=0$

Explanation:

2. If $\frac{1}{\sqrt{2x+1}}\left\{\left(1+\sqrt{2x+1}\right)^{n}-\left(1-\sqrt{2x+1}\right)^{n}\right\}= a_{0}+a_{1}x+a_{2}x^{2}+....+a_{10}x^{10}$
then n must be equal to
a) 20, 21
b) 21, 22
c) 22, 23
d) 23, 24

Explanation:

3. If in the expansion of $\left(2^{1/3}+3^{-1/3}\right)^{n}$    , the ratio of the 7th term from the beginning to the 7th term from the end is 1 : 6, then n
a) 6
b) 7
c) 8
d) 9

Explanation:

4. if sum of the coefficients of $x^{7}$ and $x^{4}$ in the expansion of $\left(\frac{x^{2}}{a}-\frac{b}{x}\right)^{11}$    is zero ,then
a) ab=1
b) a=b
c) ab=-1
d) a+b=0

Explanation:

5. If n > 2 and $\alpha,\beta,\gamma$   are three real numbers, then value of$S=\alpha C_{0}-\left(\alpha+\beta\right)C_{1}+\left(\alpha+2\beta+2^{n}\gamma\right)C_{2}-\left(\alpha+3\beta+3^{2}\gamma\right)C_{3}-....$              upto (n + 1) terms is
a) 0
b) $2^{n-2}\gamma$
c) $n^{2}2^{n-2}\gamma$
d) $n\gamma$

Explanation:

6. Value of$\sum_{k=0}^{n}.^{n}C_{k}\sin\left(kx\right)\cos\left(n-k\right)x$       is
a) $2^{n-1}\sin\left(nx\right)$
b) $2^{n}\sin\left(nx\right)$
c) $2^{n}\cos\left(nx\right)$
d) none of these

Explanation:

7. If $S_{1}=\sum_{k=0}^{n}.^{n}C_{k}\cos\left(kx\right)\cos\left(n-k\right)x$
and $S_{2}=\sum_{k=1}^{n-1}.^{n}C_{k}\sin\left(kx\right)\sin\left(n-k\right)x$
then $S_{1}-S_{2}$   equals
a) $2^{n-1}\cos\left(nx\right)$
b) $2^{n}\cos\left(nx\right)$
c) 0
d) $2+2^{n}\cos nx$

Explanation:

8. Sum of cofficients of all the powers of x in the expansion of $\left(1-2x\right)^{9}+\left(1+x\right)^{10}+\left(1-x+x^{2}\right)^{5}$
is
a) $2^{10}$
b) $2^{10}-1$
c) $2^{10}-2$
d) $2^{10}+1$

Explanation: Put x = 1

9. Cofficients of $x^{9}$ in the expansion of
$\left(x^{3}+\frac{1}{2^{\log_{\sqrt{2}}{\left(x^{3/2}\right)}}}\right)^{11}$
is
a) -5
b) 330
c) 520
d) $5+\log_{\sqrt{2}}$

10. If n ≥ r – 1 and $^{n}C_{r-1}+^{n+1}C_{r-1}+^{n+2}C_{r-1}+....+^{2n}C_{r-1}=^{2n+1}C_{r^{2}-182}-^{n}C_{r}$