UGC NET Questions (Paper – 1)

The volume of a cube is proportional to the cube of its edge, so if V₁:V₂ = 27:64, then (a₁³):(a₂³) = 27:64. Taking cube roots gives a₁:a₂ = 3:4. Surface area is proportional to the square of the edge, so the surface area ratio is a₁²:a₂² = 3²:4² = 9:16. Thus, the required ratio of surface areas is 9:16.

Option A:
Option A, 3:4, is the ratio of the edges, not of the surface areas. While it correctly comes from the cube root of the volume ratio, it does not account for the fact that surface area scales with the square of the linear dimension rather than the first power.

Option B:
Option B, 4:3, reverses the correct edge ratio and therefore also reverses the implied surface area ratio. It would suggest that the cube with smaller volume has larger surface area, which contradicts the geometric relationships given by the volume ratio.

Option C:
Option C, 27:64, simply repeats the volume ratio and ignores the need to convert from a three-dimensional measure (volume) to a two-dimensional measure (surface area). Since these scale differently, the same ratio cannot apply to both.

Option D:
Option D is correct because it follows the sequence of taking the cube root to find the edge ratio and then squaring that to find the surface area ratio. This two-step transformation, from volume to side to area, is exactly what the geometry of similar cubes requires.