33
If $25^{15}$ is divided by 13 , then the remainder is .............. .
Explanation
Let
$$\begin{aligned} 25^{15} & =(26-1)^{15} \\ & ={ }^{15} C_0 26^{15}-{ }^{15} C_1 26^{14}+\ldots-{ }^{15} C_{15} \\ & ={ }^{15} C_0 26^{15}-{ }^{15} C_1 26^{14}+\ldots-1-13+13 \\ & ={ }^{15} C_0 26^{15}-{ }^{15} C_1 26^{14}+\ldots-13+12 \end{aligned}$$
It is clear that, when $25^{15}$ is divided by 13 , then remainder will be 12 .
34
The sum of the series $\sum_\limits{r=0}^{10}{ }^{20} C_r$ is $2^{19}+\frac{{ }^{20} C_{10}}{2}$.
A
TRUE
B
FALSE
35
The expression $7^9+9^7$ is divisible by 64 .
A
TRUE
B
FALSE
36
The number of terms in the expansion of $\left[\left(2 x+y^3\right)^4\right]^7$ is 8 .
A
TRUE
B
FALSE
37
The sum of coefficients of the two middle terms in the expansion of $(1+x)^{2 n-1}$ is equal to ${ }^{2 n-1} C_n$.
A
TRUE
B
FALSE