UPSC CDS II 2025 Mathematics Paper

Concept:Use the identity sin²θ + cos²θ = 1. Substitute cosθ from the given equation to form a quadratic in sinθ. Solve for sinθ and then find cosecθ.Explanation:Given: 8 sinθ − cosθ = 4. Rearrange to get cosθ = 8 sinθ − 4.Square both sides after substituting into sin²θ + cos²θ = 1:$sin²θ + (8 sinθ − 4)² = 1$Expand: $sin²θ + 64 sin²θ − 64 sinθ + 16 = 1$Simplify: $65 sin²θ − 64 sinθ + 15 = 0$Solve using quadratic formula: discriminant = $64² − 4 × 65 × 15 = 196$, √196 = 14.Roots: $sinθ = (64 ± 14)/(130)$ −> $78/130 = 3/5$ and $50/130 = 5/13$.Check both. For θ in (0, π/2), cosθ must be positive. For sinθ = 3/5, cosθ = 8(3/5) − 4 = 24/5 − 20/5 = 4/5 > 0. For sinθ = 5/13, cosθ = 8(5/13) − 4 = 40/13 − 52/13 = −12/13 < 0, invalid.Thus sinθ = 3/5. Then cosecθ = 1 / sinθ = 5/3.Answer:cosecθ = $\frac{5}{3}$, which corresponds to option C.