Question

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

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

Answer 1

Answer: (D)

None of these

${\left(\sqrt{\frac{x}{3}}+\sqrt{\frac{3}{2x^2}}\right)}^{10}$

General term

$\Rightarrow {}^{10}{c}_r{\left(\sqrt{\frac{x}{3}}\right)}^{10-r}{\left(\sqrt{\frac{3}{2x^2}}\right)}^r$

$\Rightarrow {}^{10}{c}_r\frac{x^{\frac{10-r}{2}}}{3^{\frac{10-r}{2}}}\frac{3^{\frac{r}{2}}}{\sqrt{2}{x}^r}$

$\Rightarrow \text{P of}x=\frac{10-r}{2}-r$

$\Rightarrow \frac{10-r-2r}{2}$

$\Rightarrow 10-3r$

$\Rightarrow \frac{10-3r}{2}=0$

$\Rightarrow 10=3r$

there exists no 'r'

where term is independent of $x$

$\therefore$ None of these