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GATE 2021 CE (CE2) – Question 9

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

In square PQRS of side r, the shaded lens is the overlap of quarter-circles of radius r centred at S and Q. Find the probability that a uniformly random point in the square lies in the lens.

the source diagram and labels are reproduced in the attached image.
  1. $4-\pi/2$
  2. $1/2$
  3. $\pi/2-1$
  4. $\pi/4$

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Correct answer: (C) $\pi/2-1$

Explanation

The two quarter-discs together cover the square. Their areas sum to $2(\pi r^2/4)$, so their overlap is $\pi r^2/2-r^2$. Divide by square area r² to obtain **π/2−1 (C)**. This lies between zero and one, as a probability must.