A) \[{{\,}^{15}}{{C}_{9}}\]
B) zero
C) \[-{{\,}^{15}}{{C}_{9}}\]
D) 1
Correct Answer: C
Solution :
Let \[(r+1)\]the term be the constant term in the expansion of \[{{\left( {{x}^{3}}-\frac{1}{{{x}^{2}}} \right)}^{15}}\] Then, \[{{T}_{r+1}}={{\,}^{15}}{{C}_{r}}{{({{x}^{3}})}^{15-r}}{{\left( -\frac{1}{{{x}^{2}}} \right)}^{r}}\]is independent of \[x\]or \[{{T}_{r+1}}={{\,}^{15}}{{C}_{r}}\]\[{{x}^{45-5r}}{{(-1)}^{r}}\]is independent of \[x\] \[\Rightarrow 45-5r=0\Rightarrow r=9.\] Thus, tenth term is independent of \[x\] and is given by \[{{T}_{10}}={{\,}^{15}}{{C}_{9}}{{(-1)}^{9}}=-{{\,}^{15}}{{C}_{9}}\]
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