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
$T_{r+1}={}^9{C}_r{\left(\frac{x^2}{2}\right)}^{9-r}{\left(-\frac{2}{x}\right)}^r$
$={}^9{C}_r\frac{x^{18-2r}}{2^{9-r}}\cdot \frac{(-1{)}^r\cdot 2^r}{x^r}$
$={}^9{C}_r\frac{x^{18-3r}}{2^{9-2r}}(-1{)}^r$
For coefficient of $x^{-9}$, put $x^{18-3r}=x^{-9}$
$\Rightarrow 18-3r=-9$
$\Rightarrow 27=3r$
$\Rightarrow r=9$
$\therefore$ Coefficient of $x^{-9}={}^9{C}_9\cdot \frac{1}{2^{-9}}(-1{)}^9$
$=-2^9=-512$