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 = ${}^{10}{C}_4=210$.