## Binomial Theorem Questions and Answers Part-9

1. If $\left(1+x+x^{2}\right)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+....+a_{2n}x^{2n}$         , then value of $N=a_{0}a_{1}-a_{1}a_{2}+a_{2}a_{3}-a_{3}a_{4}+....$         is equal to
a) 0
b) n
c) -n
d) 1

Explanation:

2. If three succesive coffiecients in the exapansion of $\left(1+x\right)^{n}$  are in A.P., then (n + 2) is
a) at least 19
b) at most 19
c) a prefect square
d) a prefect cube

Explanation:

3. Sum of the first 20 terms of the series $\frac{1}{\left(2\right)\left(4\right)}+\frac{\left(1\right)\left(3\right)}{\left(2\right)\left(4\right)\left(6\right)}+\frac{\left(1\right)\left(3\right)\left(5\right)}{\left(2\right)\left(4\right)\left(6\right)\left(8\right)}+....$
is
a) $\frac{1}{2}-\frac{1}{2^{40}}\left(^{40}C_{20}\right)$
b) $\frac{1}{2}-\frac{1}{2^{41}}\left(^{42}C_{21}\right)$
c) $\frac{1}{2}-\frac{1}{2^{42}}\left(^{42}C_{21}\right)$
d) $\frac{1}{2}-\frac{1}{2^{43}}\left(^{40}C_{20}\right)$

Explanation:

4. Coefficients of $x^{n}$ in the expression $\left(1-2x+3x^{2}-4x^{3}+....\right)^{-n}$
equals
a) $^{2n}C_{n}$
b) $^{2n}C_{n+1}$
c) $^{3n}C_{n}$
d) 0

Explanation:

5. If $a=99^{100},b=100^{99}$    , then
a) a > b
b) a=b
c) a < b
d) $a+2b=101^{100}$

Explanation:

6. If $\alpha\epsilon\left[-1,1\right]$   and $I\left(\alpha\right)$  denotes the term independent of x in the expansion of
$\left(x\sin ^{-1}\alpha+\frac{1}{x}\cos^{-1}\alpha\right)^{10}$
then $\mid I \left(\alpha\right)\mid$  cannot exceed
a) $\frac{1}{2^{10}}$
b) $^{10}C_{5}\frac{\pi^{5}}{2^{10}}$
c) $^{10}C_{5}\frac{\pi^{10}}{2^{20}}$
d) $^{10}C_{5}\left(\frac{1}{2}\right)^{10}$

Explanation:

7. For $n\epsilon N$ , $x\epsilon R$ , the sum of the series $C_{1}x\left(1-x\right)^{n-1}+2C_{2}x^{2}\left(1-x\right)^{n-2}+3C_{3}x^{3}\left(1-x\right)^{n-3}+...+nC_{n}x^{n}$
is
a) nx
b) 0
c) $2^{n}x$
d) $n2^{n-1}x$

Explanation:

8. Let $a_{n}=\sum_{k=0}^{n}\left(1+2^{k}\right)C_{k}$     then which of the following an equals $a_{n+1}$
a) $5a_{n}-6a_{n-1}$
b) $5a_{n}+6a_{n}$
c) $2a_{n}+3a_{n-1}$
d) $a_{n}+a_{n-1}$

Explanation:

9. If $\log x=\log _{10}x$    , the values of x for which 4th term in the expansion of
$\left[\sqrt{x^{2/\left(1+\log x\right)}}+x^{1/12}\right]^{6}$
is 200 are
a) 0.001, 0.1
b) 0.0001, 10
c) 0.01, 100
d) 0.1, 10000

10. If $\sum_{r=1}^{n}r\left(r+1\right)\frac{C_{r}}{C_{r-1}}=77$     , then n equals