UPSC CDS II 2025 Mathematics Paper

Concept:Pythagoras theorem and perimeter conditions are used to find the sides of a right triangle.Explanation:Let AB = x, BC = y, AC = z.Perimeter: x + y + z = 60.Given: x + y exceeds z by 10, so x + y = z + 10.Substitute z = x + y – 10 into perimeter: x + y + (x + y – 10) = 60.This gives 2(x + y) – 10 = 60, so 2(x + y) = 70 → x + y = 35.Then z = 35 – 10 = 25.Apply Pythagoras: $z^2 = x^2 + y^2$ → $625 = x^2 + y^2$.From x + y = 35, write y = 35 – x.Substitute: $625 = x^2 + (35 – x)^2 = x^2 + 1225 – 70x + x^2 = 2x^2 – 70x + 1225$.Simplify: $2x^2 – 70x + 600 = 0$ → divide by 2: $x^2 – 35x + 300 = 0$.Factor: $(x – 20)(x – 15) = 0$ → x = 20 or x = 15.Thus sides are 20 and 15 (AB = 20, BC = 15 or vice versa).Area = $\frac{1}{2} \times AB \times BC = \frac{1}{2} \times 20 \times 15 = 150$ square units.Answer:150 square units (Option D)