Question

Assertion :The term independent of 'x' in the expansion of ${(x+\frac{1}{x}+2)}^{2m}$ is $\frac{\left(4m\right)!}{\left(2m\right)!\left(2m\right)!}$. Reason: The coefficient of $x^a$ in the expansion of ${(1+x)}^n$ is ${}^n{C}_a$.

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,
• B. Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion,
• C. Assertion is true but Reason is false,
• D. Assertion is false but Reason is true.

Answer 1

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,

$(x+\frac{1}{x}+2{)}^{2m}$

$=\left(\right(\sqrt{x}{)}^2+\frac{1}{(\sqrt{x}{)}^2}+2{)}^{2m}$

$=\left(\right(\sqrt{x}+\frac{1}{\sqrt{x}}{)}^2{)}^{2m}$

$=(\sqrt{x}+\frac{1}{\sqrt{x}}{)}^{4m}$

The general term will be

$T_{r+1}={}^{4m}{C}_r{x}^{2m-\frac{r}{2}}.x^{\frac{-r}{2}}$

$T_{r+1}={}^{4m}{C}_r{x}^{2m-r}$

For the term independent of x,

$2m-r=0$
$r=2m$

Hence

$T_{2m+1}={}^{4m}{C}_{2m}$

$=\frac{4m!}{2m!.2m!}$.