Question

The coefficient of term independent of $x$ in the expansion of${(\sqrt{\frac{x}{3}}-\frac{\sqrt{3}}{x^2})}^{10},$ is

• A. $\frac{5}{3}$
• B. $\frac{4}{5}$
• C. $6$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $\frac{5}{3}$

The (r+1)th term in the expansion of ${(\sqrt{\frac{x}{3}}-\frac{\sqrt{3}}{x^2})}^{10}$ is given by

$T_{r+1}=(-1{)}^r{.}^{10}{C}_r{\left(\sqrt{\frac{x}{3}}\right)}^{10-r}.{\left(\frac{\sqrt{3}}{x^2}\right)}^r$

$T_{r+1}=(-1{)}^r{}^{10}{C}_r{x}^{5-\frac{5r}{2}}{3}^{r-5}$

For term to be independent of x,
$5-\frac{5r}{2}=0$

$\Rightarrow r=2$

Substituting we get
$T_3=(-1{)}^2{}^{10}{C}_2{3}^{-3}$

$=\frac{10\left(9\right)}{2\left(27\right)}$

$=\frac{5}{3}$