Browse Questions

# The position of the term independent of $x$ in the expansion of $\big(\sqrt{\large\frac{x}{3}}+\large\frac{3}{2x^2}\big)^{10}$ is

$\begin{array}{1 1}(A)\;10C_1\\(B)\;\large\frac{5}{12}\\(C)\;1\\(D)\;\text{Third term}\end{array}$

Toolbox:
• $T_{r+1}=nC_r a^{n-r} b^r$
$T_{r+1}=10C_r \big(\sqrt{\large\frac{x}{3}}\big)^{10-r}.(\large\frac{3}{2x^2})^r$
$\Rightarrow 10C_r \big((\large\frac{x}{3})^{1/2}\big)^{10-r}.\big(\large\frac{3}{2x^2}\big)^r$
$\Rightarrow 10C_r \big[\large\frac{x}{3}^{\Large\frac{10-r}{2}}\big(\large\frac{3}{2^r x^{2r}}\big)$
$\Rightarrow 10C_r (\large\frac{1}{\sqrt 3})^{10-r}(\large\frac{3}{2})^r.$$x^{5-\Large\frac{r}{2}-\normalsize 2r} Let T_{r+1} be the term independent of x 5-\large\frac{r}{2}$$-2r=0$