## Binomial Theorem Questions and Answers Part-2

1. Value of the expression $C_0^2+C_1^2+C_2^2+....+C_n^2$       is
a) $2^{2n-1}$
b) $2n\left(^{2n}C_{n}\right)$
c) $^{2n}C_{n}$
d) none of these

Explanation:

2. Value of the expression $C_0^2+2C_1^2+....+\left(n+1\right)C_n^2$       is
a) $\left(2n+1\right)\left(^{2n}C_{n}\right)$
b) $\left(2n-1\right)\left(^{2n}C_{n}\right)$
c) $\left(\frac{n}{2}+1\right)\left(^{2n}C_{n}\right)$
d) $\left(\frac{n}{2}+1\right)\left(^{2n-1}C_{n}\right)$

Explanation:

3. If $n\epsilon N$, then value of $S=\sum_{r=0}^{n}\left(-1\right)^{r}\frac{\left(^{n}C_{r}\right)}{\left(^{r+2}C_{r}\right)}$       is
a) $\frac{1}{n+2}$
b) $\frac{2}{n+2}$
c) n+2
d) n+1

Explanation:

4. For $n\epsilon N$, let $S(n)=\sum_{r=0}^{n}\left(-1\right)^{r}\frac{1}{^{n}C_{r}}$
value of $S=\sum_{r=0}^{n}\left(-1\right)^{r}\frac{^{r+2}C_{r}}{^{n}C_{r}}$       is
a) $S\left(n+2\right)-\frac{1}{2}\left(n+1\right)^{2}$
b) $S\left(n+2\right)+\frac{1}{2}\left(n+1\right)^{2}$
c) 0
d) none of these

Explanation: From Q3,

5. If $\sum_{r=0}^{2n}a_{r}\left(x-100\right)^{r}=\sum_{r=0}^{2n}b_{r}\left(x-101\right)^{r}$
and $a_{k}=\frac{2^{k}}{^{k}C_{n}}\forall k\geq n$     , then $b_{n}$ equals
a) $2^{n}\left(2^{n+1}-1\right)$
b) $2^{n}\left(2^{n}+1\right)$
c) $2^{n}\left(2^{n}-1\right)$
d) $2^{n+1}\left(2^{n}-1\right)$

Explanation: Put x – 101 = t, so that

6. If n is even, then value of the expression $C_0^2-\frac{1}{2}C_1^2+\frac{1}{3}C_2^2- ....+\frac{\left(-1\right)^{n}}{n+1}C_n^2$
where $C_r= ^{n}C_{r}$  is
a) $\frac{\left(-1\right)^{n}n!}{\left(n+1\right)\left(n/2\right)!^{2}}$
b) $\frac{\left(-1\right)^{n-1}n!}{\left(n+1\right)\left(n/2\right)!^{2}}$
c) $\frac{-1}{\left(n+1\right)\left(n/2\right)!^{2}}$
d) $\frac{-1^{n/2}n!}{\left(n+1\right)\left(n/2\right)!^{2}}$

Explanation:

7. Sum of the coefficients of the terms of degree m in the expansion of$\left(1+x\right)^{n}\left(1+y\right)^{n}\left(1+z\right)^{n}$     is
a) $\left(^{n}C_{m}\right)^{3}$
b) $3\left(^{n}C_{m}\right)$
c) $^{n}C_{3m}$
d) $^{3n}C_{m}$

Explanation:

8. If A and B are coefficients of $x^{n}$ in the expansions of $\left(1+x\right)^{2n}$  and $\left(1+x\right)^{2n-1}$   respectively, then A/B is equal to
a) 1
b) 2
c) 1/2
d) 1/n

Explanation: We know that coefficient of xr in the

9. The coefficient of $x^{k}$ in the expansion of $E=1+\left(1+x\right)+\left(1+x\right)^{2}+...+\left(1+x\right)^{n}$        is
a) $^{n}C_{k}$
b) $^{n+1}C_{k}$
c) $^{n+1}C_{k+1}$
d) $^{n}C_{k+1}$

10. The number of irrational terms in the expansion of $\left(5^{1/6}+2^{1/8}\right)^{100}$   is