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GATE 2021 PH – Question 9

General Aptitude · Quantitative Aptitude: Data Interpretation, Arithmetic and Number Computation · 2 marks · Multiple choice

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

Diagram for GATE 2021 PH question 9
  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-circle sectors together cover the square, with the shaded lens counted twice. Thus the lens area is $2(\pi r^2/4)-r^2=(\pi/2-1)r^2$. Uniform selection makes the required probability equal to this area divided by $r^2$, giving $\pi/2-1$.

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