Question

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

• A. $210$
• B. $105$
• C. $70$
• D. $112$

Answer 1

The correct option is A $210$

Let $x=a^6$
Substituting we get
$[\frac{a^6+1}{a^4-a^2+1}-\frac{a^6-1}{a^6-a^3}{]}^{10}$
$=[\frac{(a^2{)}^3+1}{a^4-a^2+1}-\frac{(a^3{)}^2-1}{a^6-a^3}{]}^{10}$
$=[\frac{(a^2+1)(a^4-a^2+1)}{a^4-a^2+1}-\frac{(a^3-1)\left(\right(a^3)+1)}{a^3(a^3-1)}{]}^{10}$
$=\left[\right(a^2+1)-\frac{a^3+1}{a^3}{]}^{10}$
$=[\frac{a^5+a^3-a^3-1}{a^3}{]}^{10}$
$=[a^2-\frac{1}{a^3}{]}^{10}$
Term independent of a implies
$T_{r+1}={}^{10}{C}_r{a}^{\frac{20-5r}{3}}$
Therefore
$\frac{20-5r}{3}=0$
$20=5r$
$r=4$
Coefficient of $T_5$=${}^{10}{C}_4$
$=210$