Question

The coefficient of $x^9$ in the expansion of ${(x^3+\frac{1}{2^t})}^{11}$, where $t={\log }_{\sqrt{2}}{(x}^{\frac{3}{2}})$,

• A. $-5$
• B. $330$
• C. $520$
• D. $5+{\log }_{\sqrt{2}}(3)$

Answer 1

Answer: (B)

$330$

$2^t=2^{{\log }_{\sqrt{2}}{x}^{\frac{3}{2}}}=2^{{\log }_2{x}^3}=x^3$
The general term in the expansion of ${(x^3+\frac{1}{x^3})}^{11}$ is
$T_{r+1}{=}^{11}{C}_r{\left(x^3\right)}^{11-r}\frac{1}{x^{3r}}{=}^{11}{C}_r{x}^{33-6r}$
To find the coefficient of $x^9$
Hence, $33-6r=9$
$\Rightarrow r=4$
Hence, the coefficient of $x^9$ is ${}^{11}{C}_4=330$.