If the $r^{th}$ term in the expansion of $(\large\frac{x}{3}-\frac{2}{x^2})^{10}$ contains $x^4$ then $r$ is equal to

Question

$\begin{array}{1 1}(A)\;4\\(B)\;3\\(C)\;2\\(D)\;\text{none of these}\end{array} $

Answer 1

Toolbox:

• $T_{r+1}=nC_ra^{n-r} b^r$

$T_r=T_{(r-1)}+1=10C_{r-1}(\large\frac{x}{3})^{10-(r-1)}(-\large\frac{2}{x^2})^{r-1}$

$\Rightarrow 10C_{r-1}(\large\frac{1}{3})^{n-r} $$(-2)^{r-1} x^{13-3r}$

Let $T_{r+1}$ contains $x^4$

$\therefore 13-3r=4$

(i.e) $r=3$

Hence (B) is the correct answer.