# If the term free from $x$ in the expansion of $(\sqrt x-\large\frac{k}{x^2})^{10}$ is $405$,find the value of $k$.

$\begin{array}{1 1}(A)\;\pm 1\\(B)\;\pm 2\\(C)\;\pm 3\\(D)\;\pm 4\end{array}$

Toolbox:
• $T_{r+1}=nC_ra^{n-r}b^r$
Given :
$(\sqrt x-\large\frac{k}{x^2})^{10}$
$\Rightarrow 10C_r(\sqrt x)^{10-r}(\large\frac{k}{x^2})^r$
$\Rightarrow 10C_r(x)^{\Large\frac{10-r}{2}}.\large\frac{k^r}{x^{2r}}$
$\Rightarrow 10C_r k^r x^{\Large\frac{10-r}{2}-2r}$
Since the term is independent of x we have
$\large\frac{10-r}{2}$$-2r=0 10-r-4r=0 10-5r=0 r=2 Hence 3^{rd} term is independent of x T_3=10C_2k^2=405 \large\frac{10!}{2!8!}$$k^2=405$