Q57 / 170 · Manipal (MET) 2018 · Mathematics
The coefficient of $x^{-9}$ in the expansion of $\left(\frac{x^2}{2}-\frac{2}{x}\right)^9$ is
- A512
- Correct answerB-512
- C521
- D251
Solution
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
$\begin{aligned}
T_{r+1} & ={ }^9 C_r\left(\frac{x^2}{2}\right)^{9-r}\left(-\frac{2}{x}\right)^r \\
& ={ }^9 C_r \frac{x^{18-2 r}}{2^{9-r}} \cdot \frac{(-1)^r \cdot 2^r}{x^r} \\
& ={ }^9 C_r \frac{x^{18-3 r}}{2^{9-2 r}}(-1)^r
\end{aligned}$
For coefficient of $x^{-9}$, put $\begin{aligned} & & x^{18-3 r} & =x^{-9} \\ \Rightarrow & & 18-3 r & =-9 \\ \Rightarrow & & & r \end{aligned}$ $\therefore$ Coefficient of $x^{-9}={ }^9 C_9 \frac{1}{\gamma^{-9}}(-1)^9=-2^9=-512$
For coefficient of $x^{-9}$, put $\begin{aligned} & & x^{18-3 r} & =x^{-9} \\ \Rightarrow & & 18-3 r & =-9 \\ \Rightarrow & & & r \end{aligned}$ $\therefore$ Coefficient of $x^{-9}={ }^9 C_9 \frac{1}{\gamma^{-9}}(-1)^9=-2^9=-512$
Hence, the correct answer is option: 'B'.