Question

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

• A. $105$
• B. $210$
• C. $310$
• D. $180$

Answer 1

Answer: (B)

$210$

Let $x=t^6$

Hence,

${(\frac{t^6+1}{t^4-t^2+1}-\frac{t^6-1}{t^6-t^3})}^{10}$

$={(\frac{(t^2+1)(t^4-t^2+1)}{t^4-t^2+1}-\frac{(t^3-1)(t^3+1)}{t^3(t^3-1)})}^{10}$

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

$=(t^2-\frac{1}{t^3}{)}^{10}$

$T_{r+1}=(-1{)}^r{}^{10}{C}_r{t}^{20-2r}{t}^{-3r}$

Hence term independent of t will be given by

$20-5r=0$

$r=4$

Hence the coefficient is ${}^{10}{C}_4$ $=210.$