Question

If the middle term in the binomial expansion of $\left(\frac{1}{x}+x \sin x\right)^{10}$ is $\frac{63}{8}$, then the value of $6 \sin ^2 x+\sin x-2$ is

Answer 1

Here, $n=10$, which is even.
Middle term $=\frac{10}{2}+1^{t h}$ term $=6^{t h}$ term
$T_6={ }^{10} C_5\left(\frac{1}{x}\right)^5(x \sin x)^5$
$\Rightarrow \frac{63}{8}=252 \sin x^5$
$\Rightarrow \sin x^5=\frac{1}{32} $
$\Rightarrow \sin x^5=\frac{1}{2} $
$\Rightarrow \sin x=\frac{1}{2} $
$\Rightarrow 2 \sin x-1=0$
$\Rightarrow 6 \sin ^2 x+\sin x-2=2 \sin x-13 \sin x+2=0$