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GATE 2026 MT – Question 8

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

A string P of length $l$ is laid out as a straight-line segment. Another string K is laid out as a semicircular arc with string P as its diameter, as represented in Figure (i). When both strings are shortened by a length $x$, they can be rearranged such that the shortened string K forms a full circle with the shortened string P as its diameter, as represented in Figure (ii). The value of $x/l$ is ________.

Diagram for GATE 2026 MT question 8
  1. $\pi$
  2. $\frac{\pi-1}{2\pi}$
  3. $\frac{\pi}{2(\pi-1)}$
  4. $\frac{\pi}{\pi-1}$

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Correct answer: (C) $\frac{\pi}{2(\pi-1)}$

Explanation

The initial semicircular arc has length $\pi l/2$. After shortening, its length is $\pi l/2-x$. The shortened diameter is $l-x$, so the new full circle has circumference $\pi(l-x)$.

Equating these lengths gives $\pi l/2-x=\pi(l-x)$, hence $(\pi-1)x=\pi l/2$.

Thus $x/l=\pi/[2(\pi-1)]$, **C**.