Question

Assertion :The coefficient of term independent of 'x' in the expansion of ${(x+1/x+2)}^{25}$ is ${}^{50}{C}_{25}$. Reason: Middle term in the binomial expansion is independent of 'x'.

• 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 C Assertion is true but Reason is false,

Given expansion is
${(x+1/x+2)}^{25}$
$={\left(\frac{(x+1{)}^2}{x}\right)}^{25}$
$=\frac{(x+1{)}^{5}0}{x^{2}5}$
Let $T_{r+1}$ term be general term of $(x+1{)}^{50}$
$T_{r+1}{=}^{50}{C}_r{x}^{50-r}$
Now, the term independent of x in the given expansion
$\frac{{}^{50}{C}_r{x}^{50-r}}{x^{2}5}$
${=}^{50}{C}_r{x}^{25-r}$
So, this term will be independent of x if
$25-r=0$
$r=25$
So the coefficient of independent term in the given expansion is ${}^{50}{C}_{25}$
Hence, assertion is true.
Reason is incorrect as middle term cannot be independent of x in an expansion.