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 = $

Let the coefficients of third, fourth and fifth terms in the expansion of $\left(x+\frac{a}{x^{2}}\right)^{n}, x \neq 0,$ be in the ratio $12: 8: 3 .$ Then the term independent of $x$ in the expansion, is equal to …… .

The term independent of $' x '$ in the expansion of ${\left( {9\,x\,\, – \,\,\frac{1}{{3\,\sqrt x }}} \right)^{18}}, x > 0$ , is $\alpha$ times the corresponding binomial co-efficient . Then $' \alpha '$ is :

The term independent of $x$ in the expansion of ${\left( {\frac{1}{2}{x^{1/3}} + {x^{ – 1/5}}} \right)^8}$ will be

If the ${(r + 1)^{th}}$ term in the expansion of ${\left( {\sqrt[3]{{\frac{a}{{\sqrt b }}}} + \sqrt {\frac{b}{{\sqrt[3]{a}}}} } \right)^{21}}$ has the same power of $a$ and $b$, then the value of $r$ is