Question

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

• A. 70
• B. 112
• C. 105
• D. 210

Answer 1

The correct option is D 210

Given expression $=\frac{(x^{\frac{1}{3}}{)}^3+(1{)}^3}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{(x-1)}{x^{\frac{1}{2}}(x^{\frac{1}{2}}-1)}$
$=\frac{(x^{\frac{1}{3}}+1)(x^{\frac{2}{3}}-x^{\frac{1}{3}}+1)}{(x^{\frac{2}{3}}-x^{\frac{1}{3}}+1)}-\frac{(x^{\frac{1}{2}}+1)(x^{\frac{1}{2}}-1)}{x^{\frac{1}{2}}(x^{\frac{1}{2}}-1)}$
$=(x^{\frac{1}{3}}+1)-(1+x^{\frac{-1}{2}})=x^{\frac{1}{3}}-x^{\frac{-1}{2}}$
$\Rightarrow {(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}})}^{10}$
$=(x^{\frac{1}{3}}-x^{\frac{-1}{2}}{)}^{10}$
$T_{r+1}in(x^{\frac{1}{3}}-x^{\frac{-1}{2}}{)}^{10}$ is
${}^{10}{C}_r(x^{\frac{1}{3}}{)}^{10-r}.(-1{)}^r.(x^{\frac{-1}{2}}{)}^r$
$=(-1{)}^r{}^{10}{C}_r{x}^{(\frac{10-r}{3}-\frac{r}{2})}$ Which is independent of x
If $(\frac{10-r}{3}-\frac{r}{2})=0\Rightarrow r=4$
Hence required coefficient ${=}^{10}{C}_4(-1{)}^4=210$