Question

Sol. [Key Concept: In $T_{r + 1}$ set the power of $x$ to $0$ (gives $r = 12$); the resulting coefficient equals $221k$.]

General term in above expression is

$${T_{r + 1} = \ ^{18}C_{r}(9x)^{18 - r}\left( - \frac{1}{3\sqrt{x}} \right)^{r} }{= \left( - \frac{1}{3} \right)^{r}\ ^{18}C_{r}9^{18 - r} \cdot x^{18 - \frac{3r}{2}} }$$For term independent of $x,18 - \frac{3r}{2} = 0 \Rightarrow r = 12$

∴ Coefficient of term independent of x

$${= \left( - \frac{1}{3} \right)^{12}\ ^{18}C_{12}9^{18 - 12} }{= 18564 }$$$= 221k$ (given)

$$\Rightarrow k = 84 $$