Question

The coefficient of the term independent of $x$ in $\left(\left[\sqrt{\left(\frac{x}{3}\right)} + \frac{\sqrt{3}}{x^{2}}\right]\right)^{10} \, is$

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

Answer 1

Answer: (A)

Given, $\left[\sqrt{\frac{x}{3}} + \frac{\sqrt{3}}{x^{2}}\right]^{10}$
General term, $T_{r + 1}=\left( \, \right)^{10}C_{r}\left(\frac{x}{3}\right)^{\frac{1}{2} \left(10 - r\right)}\left(\frac{\sqrt{3}}{x^{2}}\right)^{r}$
$\Longrightarrow T_{r + 1}=\left( \, \right)^{10}C_{r}\left(\frac{1}{3}\right)^{\frac{10 - r}{2}}\left(\sqrt{3}\right)^{r}x^{\frac{1}{2} \left(10 - r\right) - 2 r}$
For the term independent of $x$ , put
$ \, \, \frac{1}{2}\left(10 - r\right)-2r=0$
$\Longrightarrow \, r=2$
$\therefore T_{2 + 1}=T_{3}=\left( \, \right)^{10}C_{2}\left(\frac{1}{3}\right)^{\frac{8}{2}}\left(\sqrt{3}\right)^{2}$
$=45\times \frac{1 \times 3}{81}=\frac{5}{3}$