Question

The coefficient of the term independent of $x$ in the expansion of ${(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}})}^{10}$

• A. $210$
• B. $105$
• C. $70$
• D. $110$

Answer 1

The correct option is A $210$

$\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}$

$=\frac{{\left(x^{1/3}\right)}^3+1^3}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x^{1/2}(x^{1/2}-1)}$

$=\frac{(x^{1/3}+1)(x^{2/3}-x^{1/3}+1)}{(x^{2/3}-x^{1/3}+1)}-\frac{x^{1/2}+1}{x^{1/2}}$

$=x^{1/3}-x^{-1/2}$

$\therefore {(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}})}^{10}={(x^{1/3}-x^{-1/2})}^{10}$

Let $T_{r+1}$ be the general term in ${(x^{1/3}-x^{-1/2})}^{10}$.
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
$\frac{10-r}{3}-\frac{r}{2}=0\Rightarrow 20-2r-3r=0$
or, $r=4$
So, the required coefficient is ${}^{10}{C}_4(-1{)}^4=210$