Question

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

• A. 120
• B. 210
• C. 310
• D. 4

Answer 1

Answer: (B)

$\left(\frac{x+1}{x^{2/3} -x^{1/3} +1}-\frac{x-1}{x-x^{1/2}}\right)^{10}$
$ =\left\{\frac{\left(x^{1/3} +1\right)\left(x^{2/3} -x^{1/3}+1\right)}{x^{2/3} -x^{1/3}+1} -\frac{\left(\sqrt{x} +1\right)\left(\sqrt{x} -1\right)}{\sqrt{x}\left(\sqrt{x} -1\right)}\right\}^{10} $
$=\left(x^{13} -x^{-12}\right)^{10} $
$\therefore \:\: T_{r+1} = (-1)^{r} \,{^{10}C_{r}} x^{\frac{20 - 5r}{6} } $
Thus $\frac{20-5r}{6}=0 \Rightarrow r=4 $
$\therefore $ Term = $^{10}C_{4} =210$.