Question

The coefficient of $x^{-5}$ in the binomial expansion of ${\left(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}}\right)}^{10}$ where $x\ne 0,1,$ is :

• A. 1
• B. 4
• C. -4
• D. -1

Answer 1

Answer: (A)

${\left[\frac{\left(x^{1/3}+1\right)\left(x^{2/3}-x^{1/3}+1\right)}{\left(x^{2/3}-x^{1/3}+1\right)}-\frac{(\sqrt{x}-1)(\sqrt{x+1})}{\sqrt{x}(\sqrt{x}-1)}\right]}^{10}$
$={\left(x^{1/3}+1-1-1/x^{1/2}\right)}^{10}$
$=\left(x^{1/3}-1/x^{1/2}\right)$
$r=1/3,$
$b=1/2$
$r=\frac{\frac{10}{3}-(-5)}{1/3+\frac{1}{2}}$
$r=\frac{25/3}{\left(5\frac{1}{2}\right)}=10$
cos. $=10C_{10}(1)(-1{)}^{10}=1$