UPSC NDA I 2024 Mathematics Paper

Concept:The area $A_1$ consists of a right triangle and three-quarters of a unit circle.Explanation:The given equation $x^2 + y^2 - 2x = 0$ is rewritten as $(x-1)^2 + y^2 = 1$.This represents a circle with centre $(1,0)$ and radius $1$.The area $A_1$ is the sum of triangle $\triangle ABF$ and region $AFBDEA$.Triangle $\triangle ABF$ has base and height both equal to $1$, so its area is $\frac{1}{2} \times 1 \times 1 = \frac{1}{2}$.Region $AFBDEA$ is three‑quarters of the full circle: $\frac{3}{4} \pi (1)^2 = \frac{3\pi}{4}$.Adding them: $A_1 = \frac{1}{2} + \frac{3\pi}{4} = \frac{2 + 3\pi}{4} = \frac{3\pi + 2}{4}$.Answer:$\frac{3\pi+2}{4}$ (Option D).