Question

The coefficient of $x^{-9}$ in the expansion of $\left(\frac{x^{2}}{2}-\frac{2}{x}\right)^{9}$ is

• A. 512
• B. -512
• C. 521
• D. 251

Answer 1

Answer: (B)

Let the coefficient of $x^{-9}$ is in the $(r+1)$ th term in the expansion of $\left(\frac{x^{2}}{2}-\frac{2}{x}\right)^{9}$, then
$T_{r+1}={ }^{9} C_{r}\left(\frac{x^{2}}{2}\right)^{9-r}\left(-\frac{2}{x}\right)^{r}$
$={ }^{9} C_{r} \frac{x^{18-2 r}}{2^{9-r}} \cdot \frac{(-1)^{r} \cdot 2^{r}}{x^{r}}$
$={ }^{9} C_{r} \frac{x^{18-3 r}}{2^{9-2 r}}(-1)^{r}$
For coefficient of $x^{-9}$, put $x^{18-3 r}=x^{-9}$
$\Rightarrow 18-3 r=-9$
$ \Rightarrow 27=3 r$
$\Rightarrow r=9$
$\therefore $ Coefficient of $x^{-9}={ }^{9} C_{9} \cdot \frac{1}{2^{-9}}(-1)^{9}$
$=-2^{9}=-512$