Question

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

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

Answer 1

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

Given : ${(\sqrt{\frac{x}{3}}+\frac{\sqrt{3}}{x^2})}^{10}$
The general term of the expansion is
$T_{r+1}={}^{10}{C}_r{\left(\frac{x}{3}\right)}^{\frac{10-r}{2}}\cdot (\sqrt{3}{)}^r\cdot x^{-2r}$
For the term to be independent of $x$, we get
$\frac{10-r}{2}-2r=0\Rightarrow r=2$

Hence, the coefficient is
$T_3={}^{10}{C}_2{\left(\frac{1}{3}\right)}^4(\sqrt{3}{)}^2\therefore T_3=\frac{5}{3}$