If the $k^{\text{th}}$ term in the expansion of $\left(\frac{3}{2} x^2 - \frac{1}{3x}\right)^6$ is independent of $x$,then the numerically greatest term in the expansion of $\left(\frac{3}{2} x^2 - \frac{1}{3x}\right)^k$ when $x = \frac{2}{3}$ is:

In the expansion of $\left(\frac{x}{\cos \theta}+\frac{1}{x \sin \theta}\right)^{16},$ if $\ell_{1}$ is the least value of the term independent of $x$ when $\frac{\pi}{8} \leq \theta \leq \frac{\pi}{4}$ and $\ell_{2}$ is the least value of the term independent of $x$ when $\frac{\pi}{16} \leq \theta \leq \frac{\pi}{8},$ then the ratio $\ell_{2} : \ell_{1}$ is equal to

If $21^{\text{st}}$ and $22^{\text{nd}}$ terms in the expansion of $(1+x)^{44}$ are equal,then $x$ is equal to

The coefficient of $x^6$ in the binomial expansion of ${\left( \frac{4x^2}{3} - \frac{3}{2x} \right)^9}$ is

If the coefficients of the middle terms in the binomial expansions of $(1+\alpha x)^{26}$ and $(1-\alpha x)^{28}$, where $\alpha \neq 0$, are equal, then the value of $\alpha$ is:

If the coefficient of the $3^{\text{rd}}$ term from the beginning in the expansion of $\left(ax^2 - \frac{8}{bx}\right)^9$ is equal to the coefficient of the $3^{\text{rd}}$ term from the end in the expansion of $\left(ax - \frac{2}{bx^2}\right)^9$,then the relation between $a$ and $b$ is: