## Binomial Theorem Questions and Answers Part-10

1. If n is an odd natural number, then number of zeros at the end of $99^{n}+1$  is
a) 2
b) n
c) 2n
d) none of these

Explanation:

2. Value of $S=\sum\sum_{0\leq r< s\leq n}^{}\left(C_{r}+C_{s}\right)^{2}$       is
a) $2^{2n}\left(n+1\right)\left(^{2n}C_{n}\right)$
b) $2^{2n}+\left(n+1\right)\left(^{2n}C_{n}\right)$
c) $2^{2n-1}+\left(n+1\right)\left(^{2n}C_{n}\right)$
d) none of these

Explanation:

3. Value of $S=\sum\sum_{0\leq r< s\leq n}^{}\left(r+s\right)\left(C_{r}+C_{s}\right)^{2}$
a) $n\left[\left(n-1\right)\left(^{2n}C_{n}\right)+2^{2n}\right]$
b) $n\left[\left(n+1\right)\left(^{2n}C_{n}\right)+2^{2n}\right]$
c) $n\left[2^{2n}-n\left(^{2n}C_{n}\right)\right]$
d) none of these

Explanation:

4. Value of S = $\sum_{k=1}^{\infty}\sum_{r=0}^{k}\frac{1}{3^{k}}\left(^{k}C_{r}\right)$    is
a) 2
b) $\frac{2}{3}$
c) $\frac{1}{3}$
d) none of these

Explanation:

5. Value of $S=^{n}C_{r}+3\left(^{n-1}C_{r}\right)+5\left(^{n-2}C_{r}\right)+....upto \left(n-r+1\right)$           terms is
a) $^{n+2}C_{r+2}$
b) $^{n+2}C_{r+2}+^{n+1}C_{r+2}$
c) $^{n+2}C_{r+1}$
d) $^{n+2}C_{r+2}+^{n+1}C_{r}$

Explanation:

6. If n is a multiple of 3 and $\left(1+x+x^{2}\right)^{n}=\sum_{r=0}^{2n}a_{r}x_{r}$
and $\sum_{r=0}^{n}(-1)^{r}a_{r}\left(^{n}C_{r}\right)=k\left(^{n}C_{n/3}\right)$
then value of k is
a) 0
b) 1
c) 2
d) none of these

Explanation: none of these

7. If the middle term of $\left(1+x\right)^{2n}$    is also the largest term then x lies in the interval
a) $\left(1-\frac{1}{n+1},1+\frac{1}{n}\right)$
b) $\left(1-\frac{1}{n},1+\frac{1}{n}\right)$
c) (0, 1)
d) (– 1/n, 1/n)

Explanation:

8. Sum of the series $S=\sum_{r=0}^{n}\left(-1\right)^{r}\left(^{n}C_{r}\right)\left[\frac{2^{r}}{3^{r}}+\frac{8^{r}}{3^{2r}}+\frac{26^{r}}{3^{3r}}+...upto \infty\right]$
is
a) $\frac{2}{3^{n}-1}$
b) $\frac{1}{3^{n}+1}$
c) $\frac{1}{3^{n}-1}$
d) $\frac{2}{3^{n}+1}$

Explanation:

9. The ratio of the coefficients of $x^{15}$ to the term independent of x in the expansion of $\left(x^{2}+\frac{2}{x}\right)^{15}$    is
a) 1 : 8
b) 1 : 12
c) 1 : 16
d) 1 : 32

10. If $\left(1+x\right)\left(1+x+x^{2}\right)\left(1+x+x^{2}+x^{3}\right)...\left(1+x+...+x^{n}\right)=a_{0}+a_{1}x+a_{2}x^{2}+...a_{m}x^{m},$
a) $m=\frac{1}{2}n\left(n+1\right)$
b) $a_{1}=m$
c) $a_{m}=1$