UPSC CDS 2 2021 Math Paper

Concept:Use right-triangle trigonometry (tan and Pythagoras) to relate ladder length, flagstaff height, and distance.Explanation:Let the height of the flagstaff be $h$ metres.The ladder reaches a point $10$ m below the top, so the height of that point is $h - 10$ m.Let the horizontal distance from the foot of the ladder to the base of the flagstaff be $x$ metres.From the foot of the ladder, the angle of elevation to the top of the flagstaff is $60^{\circ}$.In the right triangle formed: $\tan 60^{\circ} = \frac{h}{x} \Rightarrow \sqrt{3} = \frac{h}{x} \Rightarrow x = \frac{h}{\sqrt{3}}$.The ladder of length $10$ m touches the flagstaff at height $h-10$. So, by Pythagoras:$(h-10)^2 + x^2 = 10^2$.Substitute $x$: $(h-10)^2 + \left(\frac{h}{\sqrt{3}}\right)^2 = 100$.Simplify: $h^2 - 20h + 100 + \frac{h^2}{3} = 100$.Cancel $100$ on both sides: $\frac{4h^2}{3} - 20h = 0$.Multiply by $3$: $4h^2 - 60h = 0$ $\Rightarrow$ $4h(h - 15) = 0$.Since $h > 0$, we get $h = 15$.Answer:$15$ m (Option B).