Question

Which term in the expansion of ${(\sqrt{x}-\frac{3}{x^2})}^{10}$ is independent of $x$? Find its value.

Answer 1

${(\sqrt{x}-\frac{3}{x^2})}^{10}$

General formula of $n^{th}$ term of $(a+b{)}^m{=}^m{C}_{n-1}{a}^{m+1-n}{b}^{n-1}$

Here, $n^{th}$ term ${=}^{10}{C}_{n-1}(\sqrt{x}{)}^{11-n}{\left(\frac{-3}{x^2}\right)}^{n-1}-----(1)$

As it is independent of $n$,

$\frac{11-n}{2}=2(n-1)$

$11-n=4n-4$

$15=5n\Rightarrow n=3$.

Putting $n=3$ os formula $\left(1\right)$

value ${=}^{10}{c}_2(\sqrt{x}{)}^8\frac{(-3{)}^2}{(x^2{)}^2}$

$=\frac{10\times 9}{2}\times 9$

$=5\times 81$

$=405$