A) \[\frac{405}{256}\]
B) \[\frac{504}{259}\]
C) \[\frac{450}{263}\]
D) None of these
Correct Answer: A
Solution :
In the expansion of\[{{\left( \frac{x}{2}-\frac{3}{{{x}^{2}}} \right)}^{10}}\], the general term is \[{{T}_{r+1}}={{\,}^{10}}{{C}_{r}}{{\left( \frac{x}{2} \right)}^{10-r}}.\,\,{{\left( -\frac{3}{{{x}^{2}}} \right)}^{r}}\] \[{{=}^{10}}{{C}_{r}}{{(-1)}^{r}}.\frac{{{3}^{r}}}{{{2}^{10-r}}}{{x}^{10-r-2r}}\] Here, the exponent of x is \[10-3r=4\Rightarrow r=2\] \[\therefore \,\,\,\,{{T}_{2+1}}{{=}^{10}}{{C}_{2}}{{\left( \frac{x}{\text{2}} \right)}^{8}}{{\left( -\frac{3}{{{x}^{2}}} \right)}^{2}}=\frac{10.9}{1.2}.\frac{1}{{{2}^{8}}}{{.3}^{2}}.{{x}^{4}}\] = \[\frac{405}{256}{{x}^{4}}\] \[\therefore \] The required coefficient\[=\frac{405}{256}\].
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