UPSC NDA I 2023 Mathematics Solved Paper

Concept:Use trigonometric ratios and angle sum formula to relate distances and heights in the geometry.Explanation:Let the pillar height be $10$ m and flagstaff height $20$ m above the pillar, so total height $30$ m.From $\Delta PAB$: $\tan\theta = \frac{10}{x}$.From $\Delta PAC$: $\tan(\alpha+\theta) = \frac{30}{x}$, where $\alpha = \tan^{-1}(0.5)$ so $\tan\alpha = \frac{1}{2}$.Using the formula $\tan(\alpha+\theta) = \frac{\tan\alpha + \tan\theta}{1 - \tan\alpha \tan\theta}$, we get $\frac{\frac{1}{2} + \frac{10}{x}}{1 - \frac{1}{2}\cdot\frac{10}{x}} = \frac{30}{x}$.Simplify: $\frac{\frac{x+20}{2x}}{\frac{2x-10}{2x}} = \frac{30}{x}$ leads to $\frac{x+20}{2x-10} = \frac{30}{x}$.Cross-multiply: $x(x+20) = 30(2x-10)$ → $x^2 + 20x = 60x - 300$ → $x^2 - 40x + 300 = 0$.Factor: $(x-10)(x-30)=0$ → $x = 10$ m or $x = 30$ m.Statement 1: The two values are in ratio $10:30 = 1:3$. Correct.Statement 2: Height of flagstaff is $20$ m, but neither $x=10$ nor $x=30$ equals $20$. Incorrect.Answer:Only statement 1 is correct. Option A.