Find the term independent of x in the expansion of the expression:
(v) \[\left( \frac{\sqrt{x}}{3} + \frac{3}{2 x^2} \right)^{10}\]
Solution
(v) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
\[\left( \sqrt{\frac{x}{3}} + \frac{3}{2 x^2} \right)^{10} \]
\[ T_{r + 1} =^{10}{}{C}_r \left( \sqrt{\frac{x}{3}} \right)^{10 - r} \left( \frac{3}{2 x^2} \right)^r \]
\[ = ^{10}{}{C}_r . \frac{3^{r - \frac{10 - r}{2}}}{2^r} x^\frac{10 - r}{2} - 2r \]
\[\text{ For this term to be independent of x, we must have} \]
\[\frac{10 - r}{2} - 2r = 0\]
\[ \Rightarrow 10 - 5r = 0\]
\[ \Rightarrow r = 2\]
\[\text{ Hence, the required term is the 3rd term } . \]
\[\text{ Now, we have } \]
\[^{10}{}{C}_2 \times \frac{3^{2 - \frac{10 - 2}{2}}}{2^2}\]
\[ = \frac{10 \times 9}{2 \times 4 \times 9}\]
\[ = \frac{5}{4}\]
