Question

The coefficient of the term independent of $x$ in the expansion of $\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)$ is

• A. 210
• B. 105
• C. 70
• D. 112

Answer 1

Answer: (A)

We have,
$\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}$
$=\frac{{\left(x^{1/3}\right)}^3+1^3}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x^{1/2}\left(x^{1/2}-1\right)}$
$=\frac{\left(x^{1/3}+1\right)\left(x^{2/3}-x^{1/3}+1\right)}{x^{2/3}-x^{1/3}+1}-\frac{x^{1/2}+1}{x^{1/2}}$
$=x^{1/3}+1-1-x^{-1/2}=x^{1/3}-x^{-1/2}$
${\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)}^{10}={\left(x^{1/3}-x^{-1/2}\right)}^{10}$
Let $T_{r+1}$ be the general term in ${\left(x^{1/3}-x^{-1/2}\right)}^{10}$.
Then, $T_{r+1}={}^{10}{C}_r{\left(x^{1/3}\right)}^{10-r}(-1{)}^r{\left(x^{-1/2}\right)}^r$
For this term to be independent of $x$, we must have $\frac{10-r}{3}-\frac{r}{2}=0$
or $20-2r-3r=0$
or $r=4$
So, the required coefficient is ${}^{10}{C}_4(-1{)}^4=210$.