UPSC NDA 2 2019 Mathematics Solved Paper

Concept:Use trigonometry and the Pythagorean theorem to relate the ladder length, flagstaff height, and angle of elevation.Explanation:Let the height of the flagstaff be $h$ metres.The ladder, $9$ m long, reaches a point $9$ m below the top, so the vertical height from ground to that point is $h-9$ m.From the foot of the ladder, the angle of elevation to the top of the flagstaff is $60^{\circ}$.Let the horizontal distance from the foot to the base of the flagstaff be $d$ m.Using the ladder as hypotenuse: $d^2 + (h-9)^2 = 9^2 = 81$.From the angle of elevation: $\tan 60^{\circ} = \frac{h}{d} \Rightarrow d = \frac{h}{\sqrt{3}}$.Substitute $d$ into the first equation: $\left(\frac{h}{\sqrt{3}}\right)^2 + (h-9)^2 = 81$.Simplify: $\frac{h^2}{3} + (h^2 - 18h + 81) = 81$.Multiply by $3$: $h^2 + 3h^2 - 54h + 243 = 243$.This gives $4h^2 - 54h = 0 \Rightarrow 2h(2h - 27) = 0$.Ignoring $h=0$, we get $h = \frac{27}{2} = 13.5$ m.Answer:$13.5$ m (Option C).