Question:
In the expansion of $\left(x-\frac{1}{3 x^{2}}\right)^{9}$, the term independent of $x$ is
(a) T3
(b) T4
(c) T5
(d) none of these
Solution:
(b) $T_{4}$
Suppose $T_{\mathrm{r}+1}$ is the term in the given expression that is independent of $x$.
Thus, we have :
$T_{r+1}={ }^{9} C_{r} x^{9-r}\left(\frac{-1}{3 x^{2}}\right)^{r}$
$=(-1)^{r}{ }^{9} C_{r} \frac{1}{3^{r}} x^{9-r-2 r}$
For this term to be independent of $x$, we must have
$9-3 r=0$
$\Rightarrow r=3$
Hence, the required term is the 4 th term $i . e . T_{4}$