Explanation

(a)
$In\ the\ expansion\ of\ \left( {ax}^{2} + \frac{1}{bx} \right)^{11},$
$\ \ T_{r + 1} = \ ^{11}C_{r}{(ax}^{2})^{11 - r}\left( \frac{1}{bx} \right)^{r}$
$= \ ^{11}C_{r}\frac{a^{11 - r}}{b^{r}} \bullet x^{22 - 3r}$
For coefficient of $x^{7}$, put $22 - 3r = 7$
$\Longrightarrow \ \ r = 5$
$\therefore\ \ \ T_{6} = \ ^{11}C_{5}\frac{a^{6}}{b^{5}} \bullet x^{7}$
$\therefore Coefficient\ of\ x^{7}\ in\ the\ expansion\ of\$
$\left( {ax}^{2} + \frac{1}{bx} \right)^{11}is$
$11_{C_{5}}\frac{a^{6}}{b^{5}}$
$Similarly,\ coefficient\ of\ x^{- 7}\ in\ the\ expansion\ of\$
$\left( ax + \frac{1}{bx} \right)^{11}is$
$\ ^{11}C_{5}\frac{a^{5}}{b^{6}}$
$Now,\ \ \ ^{11}C_{5}\frac{a^{6}}{b^{5}} = \ ^{11}C_{6}\frac{a^{5}}{b^{6}}$
$\Longrightarrow \ \ ab = 1$