Question

The term independent of $x$ in ${[\sqrt{\frac{x}{3}}+\sqrt{\frac{3}{2x^2}}]}^{10}$ is

• A. ${}^{10}{C}_1$
• B. $5/12$
• C. $1$
• D. None of these

Answer 1

The correct option is D None of these

General term is the expansion of $x$ in ${[\sqrt{\frac{x}{3}}+\sqrt{\frac{3}{2x^2}}]}^{10}$ is

$T_{r+1}={}^{10}{C}_r{\left(\sqrt{\frac{x}{3}}\right)}^{10-r}{\left(\sqrt{\frac{3}{2x^2}}\right)}^r$
$={}^{10}{C}_r{\left(\frac{1}{\sqrt{3}}\right)}^{10-r}{\left(\sqrt{\frac{3}{2}}\right)}^r{x}^{(5-\frac{r}{2}-r)}$
Let $T_{r+1}$ be the term independent of $x$.
$\therefore 5-\frac{r}{2}-r=0\Rightarrow \frac{3r}{2}=5\Rightarrow r=\frac{10}{3}$
$\therefore r$ is not a whole number.
$\therefore$ Given expression does not have any term independent of $x$.