## Binomial Theorem Questions and Answers Part-1

1. Suppose m and n are positive integers and let
$S=\sum_{k=0}^{n}\left(-1\right)^{k}\frac{1}{k+m+1}\left(^{n}C_{k}\right)$
and $T=\sum_{k=0}^{n}\left(-1\right)^{k}\frac{1}{k+n+1}\left(^{m}C_{k}\right)$
then S-T is equal to
a) 0
b) $n^{m}-m^{n}$
c) $\left(n+1\right)^{m}-\left(m+1\right)^{n}$
d) $\left(1-n\right)^{m}-\left(1-m\right)^{n}$

Explanation:

2. Let $S=\sum_{r=0}^{n}\frac{^{n}C_{r}}{\sum_{s=0}^{r}\frac{^{s}C_{r}}{\sum_{t=0}^{s}.^{s}C_{t}\left(2\sin \frac{\pi}{10}\right)^{t}}}$
Then $S^{1/n}$ is equal to
a) $2\cos\left(\pi/5\right)$
b) $2\cos\left(\pi/10\right)$
c) $2\sin\left(\pi/5\right)$
d) $2\sin\left(\pi/10\right)$

Explanation:

3. Let P(x) be a polynomial with real coefficients such that $\int_{0}^{1}x^{m}P \left(1-x\right)dx=0 \forall m \epsilon N\cup\left\{0\right\}$
,then
a) $P \left(x\right)=x^{n}\left(1-x\right)^{n}$     for some $n\epsilon N$
b) $P \left(x\right)=\left(1-x\right)^{2n}$     for some $n\epsilon N$
c) $P \left(x\right)=1-x^{m}\left(1-x\right)^{n}$     for some m, $n\epsilon N$
d) P(x) $\equiv$ 0

Explanation:

4. Suppose [x] denote the greatest integer $\leq x$ , and $n\epsilon N$ , then
$\lim_{n \rightarrow \infty}\frac{\left[^{n}C_{0}x^{2}\right]+\left[^{n}C_{1}x^{2}\right] +....+\left[^{n}C_{n}x^{2}\right]}{2^{n}-2}$
a) $\frac{1}{2}x^{2}$
b) $x^{2}$
c) $2x^{2}$
d) $4x^{2}$

Explanation:

5. If in the expansion of $\left(a-2b\right)^{n}$  , the sum of 5th and 6th term is 0, then value of $\frac{a}{b}$ is
a) $\frac{n-4}{5}$
b) $\frac{2(n-4)}{5}$
c) $\frac{5}{n-4}$
d) $\frac{5}{2(n-4)}$

Explanation:

6. The coefficient of $t^{50}$ in
$\left(1+t^{2}\right)^{25}\left(1+t^{25}\right)\left(1+t^{40}\right)\left(1+t^{45}\right)\left(1+t^{47}\right)$               (1)
is
a) $1+^{25}C_{5}$
b) $1+^{25}C_{5}+^{25}C_{7}$
c) $1+^{25}C_{7}$
d) 1

Explanation: As we are interested in coefficient of t50, we shall ignore all the term with exponent more than 50

7. The expression $P\left(x\right)=\left(\sqrt{x^{5}-1}+x\right)^{7}-\left(\sqrt{x^{5}-1}-x\right)^{7}$
is a polynomial of degree
a) 16
b) 18
c) 20
d) 27

Explanation:

8. The expression $\left(\sqrt{2x^{2}+1}+\sqrt{2x^{2}-1}\right)^{6}+\left(\frac{2}{\left(\sqrt{2x^{2}+1}\right)+\left(\sqrt{2x^{2}-1}\right)}\right)^{6}$
is a polynomial of degree
a) 6
b) 8
c) 10
d) 12

Explanation:

9. The expression $C_{0}+2C_{1}+3C_{2}+....+\left(n+1\right)C_{n}$       is equal to
a) $2^{n-1}$
b) $n\left( 2^{n-1}\right)$
c) $n\left( 2^{n-1}\right)+2^{n}$
d) $\left( n+1\right)2^{n}$

10. If n > 1 then value of the expression is $C_{0}-2C_{1}+3C_{2}-4C_{3}+....+\left(-1\right)^{n}\left(n+1\right)C_{n}$