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 \neq 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. $=10\, C_{10}(1)(-1)^{10}=1$