Question

If a independent of x in the expansion of $(x+\frac{1}{x^2}{)}^n$ exists, then one of the possible value of n is

• A. 2010
• B. 2011
• C. 2012
• D. 2009

Answer 1

The correct option is A 2010

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

${=}^n{c}_r(x{)}^{n-r}{\left(\frac{1}{x^2}\right)}^n$

${=}^n{c}_r\left(x^{n-r-2r}\right)$

$n-2r-r=0$

(If term is independent of $x$ )

$n=3r$

$n$ is a multiple of 3

$\therefore$ Ans $=2010$

Option $\left(a\right)$