Question

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

• A. $4$
• B. $120$
• C. $210$
• D. $310$

Answer 1

The correct option is C $210$

${(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}})}^{10}$
$={(\frac{(x^{1/3}+1)(x^{2/3}-x^{1/3}+1)}{(x^{2/3}-x^{1/3}+1)}-\frac{(x^{1/2}+1)(x^{1/2}-1)}{x^{1/2}(x^{1/2}-1)})}^{10}$
$={\left(\right(x^{1/3}+1)-\frac{x^{1/2}-1}{x^{1/2}})}^{10}$
$=(x^{1/3}+1-1-x^{-1/2}{)}^{10}$
$=(x^{1/3}-x^{-1/2}{)}^{10}$
The general term :
$T_r={}^{10}{C}_r(-1{)}^r(x^{1/3}{)}^r(x^{-1/2}{)}^{10-r}$
$={}^{10}{C}_r(-1{)}^r{x}^{r/3}{x}^{-5+r/2}$
$={}^{10}{C}_r(-1{)}^r{x}^{\frac{5r}{6}-5}$
$\therefore \frac{5r}{6}-5=0$
$\Rightarrow r=6$
$T_7={}^{10}{C}_6=\frac{10!}{4!6!}=210$