CAPF AC 2023 Q46: General Mental Ability – Quantitative Aptitude
- (a) It is equal to the surface area of the large cube.
- (b) It is less than the surface area of the large cube.
- (c) It is more than the surface area of the large cube.
- (d) Insufficient data
Answer: (c) It is more than the surface area of the large cube.
Explanation
Surface area after removing four cubes
Given: 64 small cubes of side 2 cm make a large cube of side 8 cm (4 × 4 × 4).
Four small cubes along the diagonal of one face are removed.
How to think: removing a cube takes away the faces it showed and uncovers the faces of its neighbours that touched it.
Count the change for each removed cube; one small face has area 2 × 2 = 4 cm².
| Removed cube | Faces lost | Faces uncovered | Net change |
|---|---|---|---|
| Corner of the face (2 cubes) | 3 outside faces | 3 inner faces | 0 |
| Middle of the face (2 cubes) | 1 outside face | 5 inner faces (4 sides, 1 below) | +4 faces = +16 cm² each |
The four diagonal cubes only touch at their corners, so no two share a face and each can be counted alone.
Total change: 0 + 0 + 16 + 16 = +32 cm².
The original surface area is 6 × 64 = 384 cm², so the new solid has 384 + 32 = 416 cm², which is more.
Answer: (c) It is more than the surface area of the large cube.
Why each option is right or wrong
- (a) Incorrect. The two middle cubes each add 16 cm², so the area is not equal.
- (b) Incorrect. Digging a hole in a surface exposes new walls; the area cannot fall.
- (c) CORRECT. The area rises from 384 cm² to 416 cm².
- (d) Incorrect. The data given is enough to work out the change.
Key point
Large cube side 8 cm, surface area 6 × 8² = 384 cm². Removing an inner-face cube: area changes by +4 faces; a corner cube: 0 change. Total +32 cm², new area 416 cm².
Remember
Cut a hole in the face: net area goes up, except at a corner.
Common mistakes
- Assuming removed material always reduces the surface area.
- Treating the two corner cubes like the middle ones; their change is zero.
- Forgetting the bottom face of the hole, which is also uncovered.
Exam tip
Educated guess: a dent in a surface exposes inner walls. A notch cut into a cube, like a bite from a cheese block, adds new faces, so the area cannot drop; with the corners adding nothing, 'more' is the only option that holds.
Sources
- NCERT Class IX Mathematics — Surface Areas and Volumes, Chapter 13 (surface area of a cube)
- NCERT Class VIII Mathematics — Mensuration, Chapter 11 (surface area of cuboid and cube)