Binomial Theorem Questions and Answers Part-3

1. Sum of the last 30 coefficients in the expansion of $\left(1+x\right)^{59}$  , when expanded in ascending powers of x, is
a) $2^{29}$
b) $2^{28}$
c) $^{60}C_{30}2^{19}$
d) $2^{58}$

Explanation: There are 60 terms in the expansion of

2. The sum of the rational terms in the expansion of $\left(2^{1/5}+\sqrt{3}\right)^{20}$   is
a) 71
b) 85
c) 97
d) none of these

Explanation:

3. If sum of the coefficients of the first , second and third terms of the expansion of $\left(x^{2}+\frac{1}{x}\right)^{m}$    is 46,then the coefficients of the term that does not contain x is
a) 84
b) 92
c) 98
d) 106

Explanation:

4. Value of $\sum_{k=1}^{n}\left(C_{k}\right)\left(C_{k-1}\right)$    is
a) $^{2n}C_{n}$
b) $\frac{1}{2}\left(^{2n+2}C_{n+1}\right)-^{2n}C_{n}$
c) $^{2n}C_{n+2}$
d) $^{2n}C_{n+1}$

Explanation:

5. Value of $S = 1 \times 2 \times 3 \times 4 + 2 \times 3 \times 4 \times 5 + .... + n (n + 1) (n + 2) (n + 3)$
is
a) $\frac{1}{5}n\left(n+1\right)\left(n+2\right)\left(n+3\right)\left(n+4\right)$
b) $\frac{1}{5!}\left(^{n+3}C_{5}\right)$
c) $\frac{1}{5}\left(^{n+4}C_{4}\right)$
d) $\frac{1}{5}\left(^{n}C_{4}\right)$

Explanation:

6. If in the expansion of $\left(x^{3}-\frac{1}{x^{2}}\right)^{n}$    , $n\epsilon N$ , sum of the coefficient of $x^{5}$ and $x^{10}$ is 0, then value of n is
a) 5
b) 10
c) 15
d) 20

Explanation: (r + 1)th term in the expansion of

7. If the last term in the binomial expansion of $\left(2^{1/3}-\frac{1}{\sqrt{2}}\right)^{n}$    is $\left(\frac{1}{3^{5/3}}\right)^{\log_{3}8}$     , then the 5th term from the beginning is
a) 210
b) 420
c) 105
d) 35

Explanation:

8. If coefficients of $x^{20}$ in $\left(1-x+x^{2}\right)^{20}$    and in $\left(1+x-x^{2}\right)^{20}$    are respectively a and b, then
a) a = b
b) a > b
c) a < b
d) a+b=0

Explanation:

9. if coefficients of 2nd ,3rd and the 4th terms in the expansion of $\left(1+x\right)^{n}$  are in A.P., then value of n is
a) 5
b) 7
c) 11
d) 14

Explanation:

10. The coefficient of the term independent of x in the expansion of $\left(1+x+2x^{3}\right)\left(\frac{3}{2}x^{2}-\frac{1}{3x}\right)^{9}$
is
a) 1/3
b) 19/54
c) 17/54
d) 1/4