## Binomial Theorem Questions and Answers Part-4

1. The greatest integer less than or equal to$\left(\sqrt{2}+1\right)^{6}$   is
a) 196
b) 197
c) 198
d) 199

Explanation:

2. Sum to (n + 1) terms of the series $\frac{C_{0}}{2}-\frac{C_{1}}{3}+\frac{C_{2}}{4}-\frac{C_{3}}{5}+....$
is
a) $\frac{1}{n+1}$
b) $\frac{1}{n+2}$
c) $\frac{1}{n\left(n+1\right)}$
d) $\frac{1}{\left(n+1\right)\left(n+2\right)}$

Explanation:

3. Value of the expression $\frac{C_{1}}{2}+\frac{C_{3}}{4}+\frac{C_{5}}{6}+....$     is
a) $\frac{2^{n}-1}{n+1}$
b) $\frac{2^{n}}{n+2}$
c) $\frac{2^{n-1}}{n}$
d) $\frac{2^{n}}{n+1}$

Explanation:

4. If $\left(5+2\sqrt{6}\right)^{n}=m+f$     , where n and m are positive integers and $0\leq f< 1$   , then $\frac{1}{1-f}-f$    is equal to
a) $\frac{1}{m}$
b) m
c) $m+\frac{1}{m}$
d) $m-\frac{1}{m}$

Explanation:

5. The number of distinct terms in the expansion of $\left(x_{1}+x_{2}+....+x_{n}\right)^{3}$     is
a) $^{n+1}C_{3}$
b) $^{n+2}C_{3}$
c) $^{n+3}C_{3}$
d) $^{n}C_{3}$

Explanation:

6. cofficient of $x^{10}$ in the expansion of$\left(1+x^{2}-x^{3}\right)^{8}$    is
a) 476
b) 496
c) 506
d) 528

Explanation: We rewrite the given expression as

7. The remainder when $2^{2003}$  is divided by 17 is
a) 2
b) 4
c) 8
d) 16

Explanation:

8. The interval in which x (> 0) must lie so that the greatest term in the expansion of $\left(1+x\right)^{2n}$    has the greatest coefficient is
a) $\left(\frac{n-1}{n},\frac{n}{n-1}\right)$
b) $\left(\frac{n}{n+1},\frac{n+1}{n}\right)$
c) $\left(\frac{n}{n+2},\frac{n+2}{n}\right)$
d) none of these

Explanation: Greatest Coefficient in the expansion of

9. The largest term in the expansion of $\left(3+2x\right)^{51}$  , where x = 1/5, is
a) 5th
b) 6th
c) 8th
d) 9th

10. Let $a_{n}=\left(\frac{3+\sqrt{5}}{2}\right)^{n}+\left(\frac{2}{3+\sqrt{5}}\right)^{n}\forall n\epsilon N$
a) $a_{1},a_{2}$ are primes
b) If $a_{k},a_{k+1}$  are integers then $a_{k+2}$  is an integer
c) $a_{n}$ is an integer for each $n\epsilon N$