UPSC NDA I 2023 Mathematics Solved Paper

Concept:For a quadratic equation $ax^2 + bx + c = 0$, the harmonic mean of its roots is $HM = \frac{2x_1x_2}{x_1+x_2} = \frac{2P}{S}$, where $S = -\frac{b}{a}$ (sum of roots) and $P = \frac{c}{a}$ (product of roots).Explanation:Step 1: Identify coefficients from the given equation: $a = 3 + 2\sqrt{2}$, $b = -(4 + 2\sqrt{3})$, $c = 8 + 4\sqrt{3}$.Step 2: Compute the sum of roots: $S = -\frac{b}{a} = \frac{4 + 2\sqrt{3}}{3 + 2\sqrt{2}}$.Step 3: Compute the product of roots: $P = \frac{c}{a} = \frac{8 + 4\sqrt{3}}{3 + 2\sqrt{2}}$.Step 4: Substitute into the harmonic mean formula: $HM = \frac{2P}{S} = \frac{2 \times \frac{8 + 4\sqrt{3}}{3 + 2\sqrt{2}}}{\frac{4 + 2\sqrt{3}}{3 + 2\sqrt{2}}}$.Step 5: Simplify by canceling the common denominator $(3+2\sqrt{2})$: $HM = \frac{2(8+4\sqrt{3})}{4+2\sqrt{3}} = \frac{2 \times 2(4+2\sqrt{3})}{4+2\sqrt{3}} = \frac{4(4+2\sqrt{3})}{4+2\sqrt{3}} = 4$.Answer:$4$