UPSC CDS II 2025 Mathematics Paper

Concept:The largest square inside a circle has its diagonal equal to the circle's diameter. The largest circle inside a square has its diameter equal to the square's side.Explanation:For the square inscribed in a circle of unit radius:- Circle radius = 1, so diameter = 2.- Square diagonal = diameter = 2.- Side of square = diagonal / √2 = 2 / √2 = √2.- Area of square, x = (side)² = (√2)² = 2.For the circle inscribed in a square of unit side:- Square side = 1, so circle diameter = 1.- Circle radius = diameter / 2 = 1/2.- Area of circle, y = π × (radius)² = π × (1/2)² = π / 4.Now find relation between x and y:- x = 2, y = π/4.- Multiply both sides: πx = π × 2 = 2π.- Also, 4y = 4 × (π/4) = π.- But that gives πx = 2π and 4y = π, not equal. Wait, check: Actually 4y = π, and πx = 2π? That would be πx = 2π => x=2, which is true. So we need to express πx in terms of y. πx = 2π. And 4y = π, so 2π = 8y? Let's re-evaluate: y = π/4 => 4y = π. Then πx = π * 2 = 2π = 2*(4y) = 8y? That would be πx = 8y, but that's option D. But we know from the original solution that πx = 4y. Let's recalc: x=2, y=π/4. Then πx = 2π. And 4y = π. So πx = 2 * (4y) = 8y. That gives πx = 8y. But original says πx = 4y. Mist? Let's verify: If πx = 4y, then LHS = 2π, RHS = 4*(π/4)=π, not equal. So original must have been πx = 8y? Wait, the original solution ends with "πx = 4y". Let's double-check the geometry: Largest square in a circle of radius 1: side = √2, area = 2. Largest circle in a square of side 1: radius = 0.5, area = π/4 ≈ 0.785. So x=2, y≈0.785. Now check options: A. πx = 2y => 2π = 2*(π/4)=π/2? No. B. 2πx = y => 4π = π/4? No. C. πx = 4y => 2π = π? No. D. πx = 8y => 2π = 8*(π/4)=2π? Yes. So correct relation is πx = 8y. The original solution had a math error: they wrote πx = 4y but then correctly computed πx = 2π and 4y = π, which are not equal. Actually, they wrote "$⇒ πx = 4y$" but then "∴ The correct answer is πx = 4y." That is inconsistent. So we must correct it. The correct answer is option D: πx = 8y. Verify: πx = π*2 = 2π, 8y = 8*(π/4)=2π. So yes.Thus the rewritten solution should reflect the correct relation. I'll compute again: x=2, y=π/4. Then πx = 2π, and 8y = 8*(π/4)=2π, so πx = 8y. Alternatively, we can derive: y = π/4 => π = 4y, so πx = (4y)x = 4yx = 4*2*y = 8y. So indeed πx = 8y.Therefore answer is D. I'll write the solution accordingly, with correct final answer.Now format: Use Concept:, Explanation:, Answer:. Keep math inline with $...$ for simplicity. Use for line breaks. Ensure sentences short. No extra text. Output only the rewritten solution.Concept:For a square inscribed in a circle, the diagonal equals the circle’s diameter. For a circle inscribed in a square, the diameter equals the square’s side.Explanation:First, find $x$, the area of the largest square in a circle of radius $1$.- Circle diameter = $2 \times 1 = 2$.- So square diagonal = $2$.- Side of square = diagonal / √2 = $2 / √2 = √2$.- Area $x = (√2)^2 = 2$.Next, find $y$, the area of the largest circle in a square of side $1$.- Square side = $1$, so circle diameter = $1$.- Circle radius = $1 / 2 = 0.5$.- Area $y = π \times (0.5)^2 = π \times 0.25 = π/4$.Now relate $x$ and $y$:- $x = 2$, $y = π/4$.- Multiply both sides: $πx = π \times 2 = 2π$.- Also, $8y = 8 \times (π/4) = 2π$.- Hence $πx = 8y$.Answer:$πx = 8y$ (Option D).