Question

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

• A. 210
• B. -252
• C. None
• D. -210

Answer 1

The correct option is A

210

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

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

$\therefore$ $T_{r+1}$ in $(x^{\frac{1}{3}}-x^{\frac{1}{2}}{)}^{10}$ = ${}^{10}{C}_r\left(x^{\frac{1}{3}}{)}^{10r}\right(-1{)}^r.\left(x^{\frac{1}{2}}\right)r$

= $(-1{)}^r{}^{10}{C}_r{x}^{\frac{(20-5r)}{6}}$

For coefficient of $x^0$,5r = 20 $\Rightarrow$ r = 4

Hence, term independent of x = $(-1{)}^4.{}^{10}{C}_4=210$.