The numerically greatest term in the expansion of $(2x - 3y)^{13}$ when $x = \frac{7}{2}$ and $y = \frac{3}{7}$ is:

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}})^{n}$,in the increasing powers of $\frac{1}{\sqrt[4]{3}}$ be $\sqrt[4]{6}: 1$. If the sixth term from the beginning is $\frac{\alpha}{\sqrt[4]{3}}$,then $\alpha$ is equal to $.......$

The number of integral terms in the expansion of $(7^{1/3} + 11^{1/9})^{6561}$ is:

Let $\alpha > 0, \beta > 0$ be such that $\alpha^{3} + \beta^{2} = 4$. If the maximum value of the term independent of $x$ in the binomial expansion of $(\alpha x^{\frac{1}{9}} + \beta x^{-\frac{1}{6}})^{10}$ is $10k$,then $k$ is equal to

In the expansion of ${\left( \frac{a}{x} + bx \right)^{12}}$,the coefficient of $x^{-10}$ is:

The positive integer $k$ for which $\frac{(101)^{k/2}}{k!}$ is a maximum is