UPSC CDS 2 2019 Math Paper

Concept:Recognize the expression under the square root as a perfect square of a binomial, then simplify the sum $a + 1/a$ by rationalization.Explanation:We are given $a = \sqrt{7 + 4\sqrt{3}}$.Observe that $7 + 4\sqrt{3} = 4 + 3 + 2 \cdot 2 \cdot \sqrt{3}$.This can be written as $(2)^2 + (\sqrt{3})^2 + 2 \cdot 2 \cdot \sqrt{3}$.It matches the identity $(x + y)^2 = x^2 + y^2 + 2xy$ with $x = 2$ and $y = \sqrt{3}$.Thus, $7 + 4\sqrt{3} = (2 + \sqrt{3})^2$.Since $a$ is the positive square root, $a = 2 + \sqrt{3}$.Now compute $1/a = \frac{1}{2 + \sqrt{3}}$.Rationalize by multiplying numerator and denominator by $2 - \sqrt{3}$: $\frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}$.Therefore, $a + \frac{1}{a} = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4$.Answer:$4$ (Option C).