If $1 + {x^4} + {x^5} = \sum\limits_{i = 0}^5 {{a_i}\,(1 + {x})^i,} $ for all $x$ in $R,$ then $a_2$ is

If the coefficients of $x^{7}$ in $\left(x^{2}+\frac{1}{b x}\right)^{11}$ and $x^{-7}$ in $\left(x-\frac{1}{b x^{2}}\right)^{11}, b \neq 0$, are equal, then the value of $b$ is equal to:

If coefficient of ${(2r + 3)^{th}}$ and ${(r – 1)^{th}}$ terms in the expansion of ${(1 + x)^{15}}$ are equal, then value of r is

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 = $

Coefficient of ${x^2}$ in the expansion of ${\left( {x – \frac{1}{{2x}}} \right)^8}$ is