If the coefficients of $x^2$ and $x^3$ in the expansion of $(3 + ax)^9$ are the same,then the value of $a$ is

If $ab \neq 0$ and the sum of the coefficients of $x^7$ and $x^4$ in the expansion of $\left(\frac{x^2}{a}-\frac{b}{x}\right)^{11}$ is $0$,then

If the coefficient of $x$ in the expansion of $\left(x^2+\frac{k}{x}\right)^5$ is $270$,then $k$ is equal to

Let the coefficients of $x^{-1}$ and $x^{-3}$ in the expansion of $(2x^{1/5} - x^{-1/5})^{15}$,$x > 0$,be $m$ and $n$ respectively. If $r$ is a positive integer such that $mn^2 = {}^{15}C_r \cdot 2^r$,then the value of $r$ is equal to

The $13^{th}$ term in the expansion of $\left(x^{2}+\frac{2}{x}\right)^{n}$ is independent of $x$. Then the sum of the divisors of $n$ is:

If $p$ and $q$ are respectively the coefficients of $x^{-3}$ and $x^{-5}$ in the expansion of $\left(x^{1/3} + \frac{1}{2x^{1/3}}\right)^{21}, x > 0$,then $\frac{5p}{4q} = $