A) 512
B) \[-512\]
C) 521
D) 251
E) 522
Correct Answer: B
Solution :
Let the coefficient of\[{{x}^{-9}}\]is in the\[(r+1)th\] term in the expansion of then \[{{T}_{r+1}}{{=}^{9}}{{C}_{r}}{{\left( \frac{{{x}^{2}}}{2} \right)}^{9-r}}{{\left( -\frac{2}{x} \right)}^{r}}\] \[{{=}^{9}}{{C}_{r}}\frac{{{x}^{18-2r}}}{{{2}^{9-r}}}.\frac{{{(-1)}^{r}}{{.2}^{r}}}{{{x}^{r}}}\] \[{{=}^{9}}{{C}_{r}}\frac{{{x}^{18-3r}}}{{{2}^{9-2r}}}{{(-1)}^{r}}\] \[\therefore \] \[{{x}^{18-3r}}={{x}^{-9}}\] \[\Rightarrow \] \[18-3r=-9\] \[\Rightarrow \] \[27-3r\Rightarrow r=9\] \[\therefore \]Coefficient of\[{{x}^{-9}}{{=}^{9}}{{C}_{9}}\frac{1}{{{2}^{-9}}}{{(-1)}^{9}}\] \[=-{{2}^{9}}=-512\]
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