Question

The term is independent of $x$ in the expansion of ${(x+\frac{1}{x^2})}^6$ is :

• A. $20$
• B. $15$
• C. $6$
• D. $1$
• E. $0$

Answer 1

The correct option is B $15$

The general term of ${(x+\frac{1}{x^2})}^6$ is
$T_{r+1}{=}^6{C}_r{x}^r{\left(\frac{1}{x^2}\right)}^{6-r}$
${=}^6{C}_r{x}^{r-12+2r}$
For independent value of $x,$
$r-12+2r=0$
$\Rightarrow 3r=12$
$\Rightarrow r=4$
Therefore, required term is ${}^6{C}_4=\frac{6\times 5}{2\times 1}=15$.