The greatest value of the term independent of $x$ in the expansion of ${\left( {x\sin \theta + \frac{{\cos \theta }}{x}} \right)^{10}}$ is

The index of the power of $x$ occurring in the $5^{\text{th}}$ term from the end in the expansion of $\left(\frac{x^3}{2} - \frac{2}{x^2}\right)^{12}$ is

Sum of the coefficients of $x^4$ and $x^6$ in the expansion of $(1+x-x^2)^6$ is

Find the term independent of $x$ in the expansion of $\left(\sqrt[3]{x}+\frac{1}{2 \sqrt[3]{x}}\right)^{18}, x > 0$.

If $A$ and $B$ are coefficients of $x^{n}$ in the expansions of $(1+x)^{2n}$ and $(1+x)^{2n-1}$ respectively, then $A / B$ is equal to

Let the coefficients of the middle terms in the expansion of $\left(\frac{1}{\sqrt{6}}+\beta x\right)^{4}$,$(1-3 \beta x)^{2}$ and $\left(1-\frac{\beta}{2} x\right)^{6}$,where $\beta > 0$,respectively form the first three terms of an $A.P.$ If $d$ is the common difference of this $A.P.$,then $50-\frac{2 d}{\beta^{2}}$ is equal to.