Which of the following quantities are dimensionally independent? I. Dielectric constant    II. Momentum   III. Efficiency   IV. Refractive index

The wording "dimensionally independent" is awkward here. The question most likely means "dimensionless" (independent of dimensions, with no units). I'll work with that meaning.

Step 1: Analyze the question We need to find which quantities have no dimensions, meaning they are pure numbers or ratios of like quantities.

Step 2: Check each quantity

• I. Dielectric constant (relative permittivity): ε/ε₀ is a ratio of two permittivities, so it is dimensionless.
• II. Momentum: p = mv, with dimensions MLT⁻¹. It has dimensions, so it is not dimensionless.
• III. Efficiency: (useful output energy)/(input energy) is a ratio of like quantities, so it is dimensionless.
• IV. Refractive index: n = c/v (or sin i / sin r) is a ratio of like quantities, so it is dimensionless.

Step 3: Choose the option The dimensionless quantities are I, III and IV. Momentum (II) must be excluded, which rules out A, B and C.

Answer: D (I, III and IV)

Exam tip: Whenever you see "dimensionless" (or this odd "dimensionally independent" wording), look for ratios of like quantities: refractive index, efficiency, relative density, dielectric constant, strain, and angles. Anything with mass, length or time left over, such as momentum, has dimensions.