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Q.
The coefficient of $x^{9}$ in the expansion of $\left(x^{3} + \frac{1}{2^{l o g_{\sqrt{2}} \left(x^{\frac{3}{2}}\right)}}\right)^{11}$ is equal to
NTA AbhyasNTA Abhyas 2022
Solution:
$log_{\sqrt{2}}x^{\frac{3}{2}}=log_{2^{\frac{1}{2}}}x^{\frac{3}{2}}=\frac{\frac{3}{2}}{\frac{1}{2}}log_{2}x=log_{2}x^{3}$
$\Rightarrow 2^{l o g_{\sqrt{2}} x^{\frac{3}{2}}}=2^{l o g_{2} x^{3}}=x^{3}$
We consider the expansion of $\left(x^{3} + \frac{1}{x^{3}}\right)^{11}.$
$t_{r + 1}=^{11}C_{r}\left(x^{3}\right)^{11 - r}\left(\frac{1}{x^{3}}\right)^{r}=^{11}C_{r}x^{33 - 3 r - 3 r}$
$=^{11}C_{r}x^{33 - 6 r}$
For the coefficient of $x^{9},$ we get $33-6r=9$
$\Rightarrow 6r=24\Rightarrow r=4.$
Thus, the coefficient of $x^{9}$ is $^{11}C_{4}=\frac{11 \times 10 \times \cancel{9} 3 \times \cancel{8}}{\cancel{4} \times \cancel{3} \times \cancel{2}}=330.$