Explanation

(a)
$Given,\ \left\lbrack \sqrt{\frac{x}{3}} + \frac{\sqrt{3}}{x^{2}} \right\rbrack^{10}$
$General\ term,\ T_{r + 1} = \ ^{10}C_{r}\left( \frac{x}{3} \right)^{\frac{1}{2}(10 - r)}\left( \frac{\sqrt{3}}{x^{2}} \right)^{r}$
$\Longrightarrow T_{r + 1} = \ ^{10}C_{r}\left( \frac{1}{3} \right)^{\frac{10 - r}{2}}\left( \sqrt{3} \right)^{r}x^{\frac{1}{2}(10 - r) - 2r}$
For term independent of $x$ put
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{2}(10 - r) - 2r = 0$
$\Longrightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r = 2$
$\therefore T_{2 + 1} = T_{3} = \ ^{10}C_{2}\left( \frac{1}{3} \right)^{\frac{8}{2}}\left( \sqrt{3} \right)^{2}$
$\Longrightarrow = 45 \times \frac{1 \times 3}{81} = \frac{5}{3}$