Number of rational terms in the expansion of $( \sqrt{2} + \sqrt[4]{3} )^{100}$ is:

If the coefficients of $x^{10}$ and $x^{11}$ in the expansion of $(1+\alpha x+\beta x^2)(1+x)^{11}$ are $396$ and $144$ respectively,then $\alpha^2+\beta^2=$

If $n$ is the degree of the polynomial,$\left[ {\frac{1}{{\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }}} \right]^8 + \left[ {\frac{1}{{\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} }}} \right]^8$ and $m$ is the coefficient of $x^{12}$ in it,then the ordered pair $(n, m)$ is equal to

If $t_r$ is the $r^{\text{th}}$ term in the expansion of $(2^x + 4^{-x})^8$ and if $t_3 = 7t_2$,then $x =$

The natural number $m$,for which the coefficient of $x$ in the binomial expansion of $\left( x^{m} + \frac{1}{x^{2}} \right)^{22}$ is $1540$,is

Find the term independent of $x$ $(x > 0, x \neq 1)$ in the expansion of $\left[\frac{(x+1)}{\left(x^{2/3}-x^{1/3}+1\right)}-\frac{(x-1)}{(x-\sqrt{x})}\right]^{10}$.