The coefficient of the middle term in the expansion of $(1 + x)^{10}$ is

If the coefficient of $x^9$ in $(\alpha x^3 + \frac{1}{\beta x})^{11}$ and the coefficient of $x^{-9}$ in $(\alpha x - \frac{1}{\beta x^3})^{11}$ are equal,then $(\alpha \beta)^2$ is equal to $.............$.

In the expansion of $\left( \frac{x + 1}{x^{2/3} - x^{1/3} + 1} - \frac{x - 1}{x - x^{1/2}} \right)^{10}$,the term which does not contain $x$ is:

If the $2^{\text{nd}}$,$3^{\text{rd}}$,and $4^{\text{th}}$ terms in the expansion of $(x+a)^{n}$ are $96, 216$,and $216$ respectively,and $n$ is a positive integer,then $a+x=$

Find the middle term of the expansion of $ \left(\frac{10}{x}+\frac{x}{10}\right)^{10} $.

If the constant term in the binomial expansion of $\left(\frac{x^{5/2}}{2} - \frac{4}{x^{\ell}}\right)^9$ is $-84$ and the coefficient of $x^{-3\ell}$ is $2^{\alpha}\beta$,where $\beta < 0$ is an odd number,then $|\alpha\ell - \beta|$ is equal to