CAPF AC 2023 Q44: General Mental Ability – Quantitative Aptitude
- (a) East only
- (b) West only
- (c) East or West but not any other direction
- (d) Any fixed direction
Answer: (d) Any fixed direction
Explanation
Travelling round a spherical Earth
Given: Earth is a smooth sphere of radius x km. You travel east a distance of πx km from point P.
How to think: the distance from a point to its opposite point (antipode) along the surface is exactly half a great circle, which is πx.
So, starting at P on the equator (the natural reading of 'along the East direction'), you now stand at the antipode of P.
| Fact | Value |
|---|---|
| Circumference of a great circle | 2πx |
| Half of it | πx, the distance to the antipode |
| Shortest path from the antipode back to P | πx, along any great circle |
Every great circle through the antipode of P also passes through P, so all directions lead back to P over the same shortest distance.
No direction is better than another; the minimum distance is πx in each.
Answer: (d) Any fixed direction.
Why each option is right or wrong
- (a) Incorrect. East is only one of the directions that return to P with distance πx.
- (b) Incorrect. West is only one of the directions that return to P with distance πx.
- (c) Incorrect. Directions other than east and west also reach P in distance πx.
- (d) CORRECT. From the antipode, every direction along a great circle returns to P in the minimum distance πx.
Key point
Antipodal points: distance on the surface = πR for any great circle through both. A sphere has infinitely many great circles through two antipodal points, so all directions give the same shortest route.
Remember
Antipode: all great-circle routes home are equally short.
Common mistakes
- Choosing east or west only because the journey started eastward.
- Forgetting that πx is the distance to the antipode, not a full circle.
- Mixing up a great circle with a latitude circle; only the equator is a great circle among latitudes.
Exam tip
Educated guess: think of the North Pole and the South Pole. From either pole, every direction is south (or north) and every path of length πx reaches the other pole. The same symmetry shows up here.
Sources
- NCERT Class VI Social Science — The Earth: Our Habitat, Chapter 2 (Globe: Latitudes and Longitudes)
- NCERT Class IX Mathematics — Surface Areas and Volumes, Chapter 13 (the sphere)