A) 10
B) 15
C) 16
D) none of these
Correct Answer: B
Solution :
Key Idea: The general term in the expansion of\[{{(x+a)}^{n}}\]is\[^{n}{{C}_{r}}{{x}^{n-r}}{{a}^{r}}={{T}_{r+1}}\]. Given, \[{{\left( x+\frac{1}{{{x}^{2}}} \right)}^{6}}\] \[\therefore \]General term is \[{{T}_{r+1}}{{=}^{6}}{{C}_{r}}{{x}^{6-r}}{{\left( \frac{1}{{{x}^{2}}} \right)}^{r}}\] \[\Rightarrow \] \[{{T}_{r+1}}{{=}^{6}}{{C}_{r}}{{x}^{6-r}}{{x}^{-2r}}\] \[\Rightarrow \] \[{{T}_{r+1}}{{=}^{6}}{{C}_{r}}{{x}^{6-3r}}\] Now, for term independent of\[x\] \[\Rightarrow \] \[3r=6\Rightarrow r=2\] \[\therefore \] \[{{T}_{3}}{{=}^{6}}{{C}_{2}}{{x}^{o}}{{=}^{6}}{{C}_{2}}=\frac{6\times 5}{2!}=15\] \[\Rightarrow \] \[{{T}_{3}}=15\]
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