Question

If sum of the coefficient of the first, the second and the third term of the expansion of ${(x^2+\frac{1}{x})}^m$ is $46$, then the coefficient of the term that does not contain $x$ is

• A. $84$
• B. $92$
• C. $98$
• D. $106$

Answer 1

The correct option is A $84$

Since, sum of the coefficient of the first, the second and the third term of the expansion of ${(x^2+\frac{1}{x})}^m$ is $46$.
$\therefore {}^m{C}_0{+}^m{C}_1{+}^m{C}_2=46\Rightarrow m^2+m-90=0\Rightarrow m=9\text{as}m>0\text{Now,}{T}_{r+1}{=}^m{C}_r(x^2{)}^{m-r}{\left(\frac{1}{x}\right)}^r$
For the term which is independent of x, $2m-3r=0\Rightarrow r=6.$ So, coefficient of term independent of $x$ is ${}^9{C}_6=84.$