GATE 2021 AE – Question 9
PQRS is a square of side r. The shaded area is the intersection of sectors of circles of radius r centered at S and Q. The probability that a uniformly selected point in the square falls in the shaded area is
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Correct answer: (C) $\pi/2-1$
Explanation
The square has side $r$, so its area is $r^2$.
**The shaded region** is the overlap of two quarter-circle sectors of radius $r$, one centred at S and the other at Q (opposite corners of the square). Each sector has area $\dfrac{\pi r^2}{4}$ and lies inside the square.
**Area of the overlap.** The two sectors together cover the whole square, and the region where they overlap is counted twice. By inclusion-exclusion,
$$A_{lens}=\frac{\pi r^2}{4}+\frac{\pi r^2}{4}-r^2=\left(\frac\pi2-1\right)r^2 .$$
**Probability** = shaded area / area of the square:
$$P=\frac{(\pi/2-1)r^2}{r^2}=\frac\pi2-1\quad(\text{option C}).$$