If the $6^{th}$ term in the expansion of the binomial ${\left[ {\sqrt {{2^{\log (10 – {3^x})}}} + \sqrt[5]{{{2^{(x – 2)\log 3}}}}} \right]^m}$ is equal to $21$ and it is known that the binomial coefficients of the $2^{nd}$, $3^{rd}$ and $4^{th}$ terms in the expansion represent respectively the first, third and fifth terms of an $A.P$. (the symbol log stands for logarithm to the base $10$), then $x = $

If the coefficents of ${x^3}$ and ${x^4}$ in the expansion of  $\left( {1 + ax + b{x^2}} \right){\left( {1 – 2x} \right)^{18}}$ in powers of $x$ are both zero, then $ (a,b) $ is equal to 

Let $\alpha$ be the constant term in the binomial expansion of $\left(\sqrt{ x }-\frac{6}{ x ^{\frac{3}{2}}}\right)^{ n }, n \leq 15$. If the sum of the coefficients of the remaining terms in the expansion is $649$ and the coefficient of $x^{-n}$ is $\lambda \alpha$, then $\lambda$ is equal to $……….$.

The coefficient of $x^5$ in the expansion of $\left(2 x^3-\frac{1}{3 x^2}\right)^5$ is

Find the value of $\left(a^{2}+\sqrt{a^{2}-1}\right)^{4}+\left(a^{2}-\sqrt{a^{2}-1}\right)^{4}$