## Binomial Theorem Questions and Answers Part-11

1. If the third term in the expansion of $\left[\left(1/x\right)+x^{\log}10^{x}\right]^{5}$    is 1000, then x is equal to
a) 100
b) 10
c) $1/\sqrt{10}$
d) Both a and c

Explanation:

2. If the third term in the expansion of $\left(x+x^{\log10x}\right)^{5}$    is $10^{6}$ , then x can be
a) $10^{-1/3}$
b) 10
c) $10^{-5/2}$
d) Both b and c

Explanation:

3. Positive integer (s) which is (are) greater than $\left(1+0.0001\right)^{10000}$     is (are)
a) 3
b) 4
c) 5
d) All of the above

Explanation: Use the fact that

4. If $\left(1+2x+3x^{2}\right)^{10}=a_{0}+a_{1}x+a_{2}x^{2}+....+a_{20}x^{20}$
then
a) $a_{1}=20$
b) $a_{2}=210$
c) $a_{4}=8085$
d) All of the above

Explanation:

5. The number $101^{100}-1$   is divisible by
a) 100
b) 1000
c) 10000
d) All of the above

Explanation:

6. If the second, third and fourth terms in the expansion of $\left(a+b\right)^{n}$  are 135, 30 and 10/3 respectively, then
a) a=3
b) b=1/3
c) n=5
d) All of the above

Explanation:

7. The coefficient of the middle term in the expansion of $\left(1+x\right)^{2n}$   is
a) $^{2n}C_{n}$
b) $\frac{1.3.5....\left(2n-1\right)}{n!}{2^{n}}$
c) $2 \times 6....\left(4n-2\right)$
d) Both a and b

Explanation:

8. If n > 1, then $\left(1+x\right)^{n}-nx-1$     is divisible by
a) x
b) $x^{2}$
c) $x^{3}$
d) Both a and b

Explanation:

9. If the middle term of $\left(x+\frac{1}{x}\sin^{-1}x\right)^{8}$     is equal to 630/16 , then value of x is (are)
a) $\pi/3$
b) $\pi/6$
c) $-\pi/3$
d) Both a and c

10. Let $S_{n}=\sum_{r=0}^{n}\left(-2\right)^{r}\left(\frac{^{n}C_{r}}{^{r+2}C_{r}}\right)$     then
a) $S_{n}=\frac{1}{n+1}$   if n is odd
b) $S_{n}=\frac{1}{n+2}$   if n is odd
c) $S_{n}=\frac{1}{n+1}$   if n is even