UPSC Prelims 2005 Q120: Science & Technology – Quantitative Aptitude
- (a) 196
- (b) 216
- (c) 256
- (d) 296
Answer: (c) 256
Explanation
> Note for students: Both 196 and 256 are perfect squares and so both are possible values of n; the question has two correct options. The published key is 256. Choose (c) 256 or skip it to avoid a negative mark.
Cutting an equilateral triangle into identical equilateral triangles
| Number of small triangles n | Possible? |
|---|---|
| n = k^2 (perfect square) | Yes — divide each side into k equal parts |
| 196 = 14^2 | Yes — also a perfect square |
| 256 = 16^2 | Yes — also a perfect square |
| 216 and 296 | Not perfect squares, so not possible |
Dividing each side into k equal parts gives k^2 identical small triangles, so any perfect square is possible. Both 196 and 256 are perfect squares, so the question has two valid options; the commonly keyed answer is 256, and the entry flags that 196 is equally valid.
Answer: (c) 256 (and 196 also works).
Why each option is right or wrong
- (a) Incorrect. Note that 196 = 14^2 is also a valid value of n, so the question has two valid options.
- (b) Incorrect. 216 is not a perfect square.
- (c) CORRECT. 256 = 16^2 is a valid value of n.
- (d) Incorrect. 296 is not a perfect square.
Key point
Equilateral triangle divided into k x k small triangles: n = k^2.
Remember
n must be a perfect square: k^2 small triangles.
Common mistakes
- Choosing the first square number without checking the other options.
- Thinking that any even number of triangles works.
- Using a rule such as n = 6k instead of squares.
Exam tip
Both 196 and 256 are squares, so the question itself has two valid options.
Sources
- NCERT Class X Mathematics — Triangles
- Arun Sharma — How to Prepare for Quantitative Aptitude for the CAT (Geometry and Mensuration)