Mathematics - Binomial Theorem Question with Solution | TestHub

$T_{r+1}{=}^9{C}_r{\left(\frac{3x^2}{2}\right)}^{9-r}{\left(-\frac{1}{3x}\right)}^r$

${=}^9{C}_r{\left(\frac{3}{2}\right)}^{9-r}{\left(-\frac{1}{3}\right)}^r{x}^{18-3r}$

For the term independent of $x$, put $r=6$

$\Rightarrow T_7{=}^9{C}_6{\left(\frac{3}{2}\right)}^3{\left(-\frac{1}{3}\right)}^6{=}^9{C}_3{\left(\frac{1}{6}\right)}^3=\frac{9\times 8\times 7}{3\times 2\times 1}{\left(\frac{1}{6}\right)}^3=\left(\frac{7}{18}\right)$

$\Rightarrow k=T_7\Rightarrow k=\frac{7}{18}\Rightarrow 18k=7$