Question

The rational expression $\frac{a^{\frac{1}{2}}+a^{-\frac{1}{2}}}{1-a}+\frac{1-a^{-\frac{1}{2}}}{1+\sqrt{a}},a\ne 1$, is equal to

• A. $\frac{1}{a-1}$
• B. $\frac{a}{1-a}$
• C. $\frac{2}{a-1}$
• D. $\frac{2}{1-a}$

Answer 1

The correct option is D $\frac{2}{1-a}$

$\frac{a^{\frac{1}{2}}+a^{-\frac{1}{2}}}{1-a}+\frac{1-a^{-\frac{1}{2}}}{1+\sqrt{a}}=\frac{\sqrt{a}+\frac{1}{\sqrt{a}}}{1-a}+\frac{1-\frac{1}{\sqrt{a}}}{1+\sqrt{a}}\times \frac{1-\sqrt{a}}{1-\sqrt{a}}=\frac{\sqrt{a}+\frac{1}{\sqrt{a}}}{1-a}+\frac{1-\sqrt{a}-\frac{1}{\sqrt{a}}+1}{1-a}$
$=\frac{\sqrt{a}+\frac{1}{\sqrt{a}}+2-\sqrt{a}-\frac{1}{\sqrt{a}}}{1-a}=\frac{2}{1-a}$