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

Given expression can be written as
${\left(\right(x^{1/3}+1)-\left(\frac{\sqrt{x}+1}{\sqrt{x}}\right))}^{10}={(x^{1/3}+1-1-\frac{1}{\sqrt{x}})}^{10}=(x^{1/3}-x^{-1/2}{)}^{10}$
$\text{general term}=T_{r+1}{=}^{10}{C}_r\left(x^{1/3}{)}^{10-r}\right(-x^{-1/2}{)}^r={}^{10}{C}_r{x}^{\frac{10-r}{3}}\cdot (-1{)}^r\cdot x^{-\frac{r}{2}}={}^{10}{C}_r(-1{)}^r.x^{\frac{10-r}{3}-\frac{r}{2}}$
Term will be independent of $x$ when $\frac{10-r}{3}-\frac{r}{2}=0$
$\Rightarrow r=4$
So, required term $=T_5{=}^{10}{C}_4=210$