Q38
ArithmeticAlgebra, Equations & Operations
$\left(\frac{2 + \sqrt{3}}{2 - \sqrt{3}} + \frac{2 - \sqrt{3}}{2 + \sqrt{3}} + \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\right)$ simplifies to
- a.$16 - \sqrt{3}$
- b.$4 - \sqrt{3}$
- c.$2 - \sqrt{3}$
- d.$2 + \sqrt{3}$
Answer: (A) $16 - \sqrt{3}$
Rationalising, $\frac{2 + \sqrt{3}}{2 - \sqrt{3}} = 7 + 4\sqrt{3}$ and $\frac{2 - \sqrt{3}}{2 + \sqrt{3}} = 7 - 4\sqrt{3}$, which add to 14. Also $\frac{\sqrt{3} - 1}{\sqrt{3} + 1} = \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3}$. The total is $16 - \sqrt{3}$; option C is only the last term.