Question

The position of the term independent of x in the expansion of ${\left(\sqrt{\frac{x}{3}}+\frac{3}{2x^2}\right)}^{10}$ is ___________.

Answer 1

$$\begin{gathered} \mathrm{for}{\left(\sqrt{\frac{x}{3}}+\frac{3}{2x^2}\right)}^{10} \\ {T}_{r+1}{=}^{10}{C}_r{\left(\sqrt{\frac{x}{3}}\right)}^{10-r}{\left(\frac{3}{2x^2}\right)}^r \\ ={}^{10}{C}_r\frac{1}{{\left(\sqrt{3}\right)}^{10-r}}{x}^{\frac{10-r}{2}}{x}^{-2r}{\left(\frac{3}{2}\right)}^r \\ {T}_{r+1}=\frac{{}^{10}{C}_r}{{\left(\sqrt{3}\right)}^{10-r}}{\left(\frac{3}{2}\right)}^r{x}^{\frac{10-5r}{2}} \\ \mathrm{for}\mathrm{constant}\mathrm{term},\frac{10-5r}{2}=0\mathrm{i}.\mathrm{e}.r=2 \end{gathered}$$
Hence, third term is independent of x.