Question

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

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

Answer 1

The correct option is A $210$

Let $x=a^6$.

Substituting we get,

${[\frac{a^6+1}{a^4-a^2+1}-\frac{a^6-1}{a^6-a^3}]}^{10}$

$={[\frac{(a^2{)}^3+1}{a^4-a^2+1}-\frac{(a^3{)}^2-1}{a^6-a^3}]}^{10}$

$={[\frac{(a^2+1)(a^4-a^2+1)}{a^4-a^2+1}-\frac{(a^3-1)\left(\right(a^3)+1)}{a^3(a^3-1)}]}^{10}$

$={\left[\right(a^2+1)-\frac{a^3+1}{a^3}]}^{10}$

$={\left[\frac{a^5+a^3-a^3-1}{a^3}\right]}^{10}$

$={[a^2-\frac{1}{a^3}]}^{10}$

$T_{r+1}={}^{10}{C}_r{a}^{\frac{20-5r}{3}}$

Therefore, for the term independent of $x$

$\frac{20-5r}{3}=0$

$\Rightarrow 20=5r$

$\Rightarrow r=4$

Coefficient of $T_5{=}^{10}{C}_4=210$