If the term independent of $x$ in the expansion of $\left(\sqrt{\mathrm{ax}}{ }^2+\frac{1}{2 \mathrm{x}^3}\right)^{10}$ is 105 , then $\mathrm{a}^2$ is equal to :

The value of $x$ in the expression ${[x + {x^{{{\log }_{10}}}}^{(x)}]^5}$, if the third term in the expansion is $10,00,000$

Find the $4^{\text {th }}$ term in the expansion of $(x-2 y)^{12}$

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

The ratio of coefficient of $x^2$ to coefficient of $x^{10}$ in the expansion of ${\left( {{x^5} + {{4.3}^{ - {{\log }_{\sqrt 3 }}\sqrt {{x^3}} }}} \right)^{10}}$ is

Find the cocfficient of $x^{5}$ in $(x+3)^{8}$