Question

If the $5^{th}$ term in the expansion of ${(3\sqrt{x}+\frac{1}{x})}^n$ is independent of $x$, then $n$ $=$

• A. $8$
• B. $12$
• C. $16$
• D. $20$

Answer 1

The correct option is B $12$

We know, $T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r$
Applying to the above question, we get
$T_{r+1}={}^n{C}_r{.3}^{n-r}.x^{\frac{n-r}{2}}.x^{-r}$
$={}^n{C}_r{.3}^{n-r}.x^{\frac{n-3r}{2}}$ ...(i)
It is given that the $5^{th}$ term is independent of $x$
$T_{4+1}={}^n{C}_4{.3}^{n-4}.x^{\frac{n-12}{2}}$
Therefore $\frac{n-12}{2}=0$
$\Rightarrow n=12$