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GATE 2020 CS – Question 8

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

The figure below shows an annular ring with outer and inner radii as $b$ and $a$, respectively. The annular space has been painted in the form of blue colour circles touching the outer and inner periphery of annular space. If maximum $n$ number of circles can be painted, then the unpainted area available in annular space is ______.

annular ring with radii $a$ (inner) and $b$ (outer) and $n$ blue circles packed between them
  1. $\pi\left[(b^2-a^2)-\frac{n}{4}(b-a)^2\right]$
  2. $\pi[(b^2-a^2)-n(b-a)^2]$
  3. $\pi\left[(b^2-a^2)+\frac{n}{4}(b-a)^2\right]$
  4. $\pi[(b^2-a^2)+n(b-a)^2]$

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Correct answer: (A) $\pi\left[(b^2-a^2)-\frac{n}{4}(b-a)^2\right]$

Explanation

Each circle has diameter $b-a$, so its area is $\pi(b-a)^2/4$. Unpainted area = ring area minus $n$ circle areas = $\pi[(b^2-a^2)-\frac{n}{4}(b-a)^2]$.