A) \[\frac{70}{243}\]
B) \[\frac{60}{423}\]
C) \[\frac{50}{13}\]
D) none of these
Correct Answer: A
Solution :
The general term in the expansion of \[{{\left( \frac{1}{3}{{x}^{1/2}}+{{x}^{-1/4}} \right)}^{10}}\]is \[{{T}_{r+1}}={{\,}^{10}}{{C}_{r}}{{\left( \frac{1}{3}{{x}^{1/2}} \right)}^{10-r}}{{({{x}^{-1/4}})}^{r}}\] \[={{\,}^{10}}{{C}_{r}}\frac{1}{{{30}^{10-r}}}{{x}^{5-r/2}}{{x}^{-r/4}}\] \[={{\,}^{10}}{{C}_{r}}\frac{1}{{{30}^{10-r}}}{{x}^{5-3r/4}}\] For coefficient of \[{{x}^{2}},\]we take \[5-\frac{3r}{4}=2\] \[\Rightarrow \] \[3=\frac{3r}{4}\] \[\Rightarrow \] \[r=4\] \[\therefore \] Coefficient of \[{{x}^{2}}={{\,}^{10}}{{C}_{4}}\frac{1}{{{3}^{10-4}}}\] \[=\frac{210}{729}\] \[=\frac{70}{243}\]
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