## Binomial Theorem Questions and Answers Part-5

1. Suppose $\alpha,\beta>0$   and
$x_{n}=^{n}C_{0}\alpha^{n}+^{n}C_{2}\alpha^{n-2}\beta^{2}+....$
$Y_{n}=^{n}C_{1}\alpha^{n-1}\beta+^{n}C_{3}\alpha^{n-3}\beta^{3}+....$
Then
a) $2x_{n}=\left(\alpha+\beta\right)^{n}+\left(\alpha-\beta\right)^{n}$
b) $2y_{n}=\left(\alpha+\beta\right)^{n}-\left(\alpha-\beta\right)^{n}$
c) $\lim_{n \rightarrow \infty}\frac{x_{n}}{y_{n}}=1$
d) All of the above

Explanation:

2. Let $a= \sum_{k\geq0}{\left(\begin{array}{c}n\\ 2k\end{array}\right)}2^{n-2k}3^{k} n \epsilon N , n\geq2$
then
a) 2a is an integer
b) 2a – 1, 2a, 2a + 1 are sides of a triangle
c) inradius of triangle with sides 2a – 1, 2a,2a+1 is an integer
d) All of the above

Explanation: 2a is clearly an integer

3. Values of x for which the sixth term of the expansion of $E= \left(3^{\log_{3}\sqrt{9^{\mid x-2\mid}}}+7^{\left(1/5\right)\log_{7}\left[\left(4\right).3^{\mid x-2\mid}-9\right]}\right)$         is 567 ,are
a) 1
b) 2
c) 3
d) Both a and c

Explanation:

4. If $C_{r}=\left(\begin{array}{c}n\\ r\end{array}\right)$    , then sum of the series $S=C_0^2+\frac{1}{2}C_1^2+\frac{1}{3}C_2^2+....$      upto (n + 1) terms is:
a) $\frac{1}{n+1}\left(\begin{array}{c}2n+1\\ n+1\end{array}\right)$
b) $\frac{1}{2\left(n+1\right)}\left(\begin{array}{c}2n+2\\ n+1\end{array}\right)$
c) $\frac{1}{n+1}\left(\begin{array}{c}2n+1\\ n\end{array}\right)$
d) All of the above

Explanation:

5. Let $a_{n}=\left(1+\frac{1}{n}\right)^{n}$     . Then for each $n\epsilon N$
a) $a_{n}\geq 2$
b) $a_{n}< 3$
c) $a_{n}< 4$
d) All of the above

Explanation: We have a1 = 2 and for n > 2,

6. If in the expansion of $\left(\frac{1}{x}+x\tan x\right)^{5}$     the ratio of 4th term to the 2nd term is $\frac{2}{27}\pi^{4}$   , then value of x can be
a) $\frac{-\pi}{6}$
b) $\frac{-\pi}{3}$
c) $\frac{\pi}{3}$
d) Both b and c

Explanation:

7. The value of the expression $C_0^2-C_1^2+C_2^2-....+\left(-1\right)^{n}C_n^2$
is
a) 0 if n is odd
b) $\left(-1\right)^{n}$    if n is odd
c) $\left(-1\right)^{n/2}.^{n}C_{n/2}$     if n is even
d) Both a and c

Explanation: When n is odd, taken n = 2m + 1, so that

8. The value of the expression $C_0^2-2C_1^2+3C_2^2-....+\left(-1\right)^{n}\left(n+1\right)C_n^2$
is
a) 0 if n is odd
b) $\left(n+1\right)\left(-1\right)^{n}$    if n is odd
c) $\left(-1\right)^{n/2}\left(\frac{n}{2}+1\right)\frac{n!}{\left(\frac{n}{2}\right)!\left(\frac{n}{2}\right)!}$       if n is even
d) Both a and c

Explanation: When n is odd, we take n = 2m – 1, so that

9. If m is a positive integer, then $\left[\left(\sqrt{3}+1\right)^{2m}\right]+1$    , where [x] denotes greatest integer $\leq n$ , is divisible by
a) $2^{m}$
b) $2^{m+1}$
c) $2^{m+2}$
d) Both a and b

10. If $x= \left(7+4\sqrt{3}\right)^{2n}=\left[x\right]+f$      then $x \left(1-f\right)$   is equal to