CAPF AC 2025 Q60: General Mental Ability – Quantitative Aptitude
- (a) (√2 + 1) m
- (b) (√2 − 1) m
- (c) (π√2 − 1) m
- (d) (π√2 + 1) m
Answer: (b) (√2 − 1) m
Explanation
Pipe between four drums
Given: four drums, each of radius 1 m, are placed so that each touches two others (a square arrangement).
A pipe is to fit in the gap in the middle.
How to think: join the centres of the drums; they form a square of side 2 m (two radii).
The middle gap is centred on the square's centre.
| Step | Working | Result |
|---|---|---|
| Diagonal of the square | 2 × √2 | 2√2 m |
| Centre of square to a drum centre | half the diagonal | √2 m |
| Pipe touches a drum | √2 = 1 + r (drum radius + pipe radius) | r = √2 − 1 |
The pipe radius is about 0.41 m.
Bank key corrected: the stored key was (c) (π√2 − 1) m; the right answer is (b) (√2 − 1) m.
Answer: (b) (√2 − 1) m.
Why each option is right or wrong
- (a) Incorrect. (√2 + 1) m is larger than a drum and cannot fit in the gap.
- (b) CORRECT. The pipe radius is √2 − 1 metres.
- (c) Incorrect. Pi has no role here, and the value is bigger than a drum.
- (d) Incorrect. This is larger than the gap.
Key point
Four unit circles at the corners of a square of side 2: distance from the centre to a corner is √2, so the inner circle has radius √2 − 1 ≈ 0.414.
Remember
Half-diagonal of the square minus the drum radius: √2 − 1.
Common mistakes
- Using the side (2) as the distance to the centre instead of half the diagonal.
- Adding the radii instead of subtracting: √2 + 1.
- Thinking pi appears because circles are involved.
Exam tip
Educated guess: the pipe must be smaller than a drum (radius 1). Only (√2 − 1) is less than 1; the others exceed 1.
Sources
- NCERT Class X Mathematics, Chapter 6 (Triangles: Pythagoras theorem) and Chapter 10 (Circles: touching circles)
- NCERT Class IX Mathematics, Chapter 10 (Circles)