UPSC NDA I 2025 Mathematics Paper

Concept:The area under a curve from $x=0$ to $x=4$ is found using integration or by recognizing the shape as a right triangle.Explanation:The curve is $y = 4x$. It passes through the origin and the point $(4,16)$. The region bounded by the curve, the $x$-axis, and the vertical line $x=4$ forms a right triangle. Base of triangle lies on $x$-axis from $x=0$ to $x=4$, so base length $= 4$ units. Height is the $y$-coordinate at $x=4$: $y = 4 \times 4 = 16$ units. Area of triangle $= \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 16 = 32$ square units.Answer:32 square units (Option C).