Dimensional Analysis of Constants and Units

This problem requires matching fundamental units from List-I with expressions from List-II based on dimensional analysis. We first establish the dimensions of the fundamental constants provided:

• Planck's constant (h): $[M L^2 T^{-1}]$
• Gravitational constant (G): $[M^{-1} L^3 T^{-2}]$
• Speed of light (c): $[L T^{-1}]$

Expression Dimensional Analysis

We calculate the dimensions for each expression in List-II:

Expression I: $\sqrt{\frac{hc}{G}}$

Dimension calculation:

$\sqrt{\frac{[M L^2 T^{-1}][L T^{-1}]}{[M^{-1} L^3 T^{-2}]}} = \sqrt{\frac{[M L^3 T^{-2}]}{[M^{-1} L^3 T^{-2}]}} = \sqrt{[M^2]} = [M]$

The dimension is Mass, which corresponds to Kilogram (C).

Expression II: $\sqrt{\frac{Gh}{c^{5}}}$

Dimension calculation:

$\sqrt{\frac{[M^{-1} L^3 T^{-2}][M L^2 T^{-1}]}{[L T^{-1}]^5}} = \sqrt{\frac{[L^5 T^{-3}]}{[L^5 T^{-5}]}} = \sqrt{[T^2]} = [T]$

The dimension is Time, which corresponds to Second (B).

Expression III: $\sqrt{\frac{K^{2}L^{2}c^{3}}{Gh}}$

Assuming 'K' represents the dimension of Kelvin ($[\Theta]$) and 'L' represents the dimension of Meter ($[L]$) within this expression:

Dimension calculation:

$\sqrt{\frac{[\Theta]^2 [L]^2 [L T^{-1}]^3}{[M^{-1} L^3 T^{-2}][M L^2 T^{-1}]}} = \sqrt{\frac{[\Theta]^2 [L]^2 [L^3 T^{-3}]}{[L^5 T^{-3}]}} = \sqrt{\frac{[\Theta]^2 [L^5 T^{-3}]}{[L^5 T^{-3}]}} = \sqrt{[\Theta]^2} = [\Theta]$

The dimension is Temperature, which corresponds to Kelvin (D).

Expression IV: $\sqrt{\frac{Gh}{c^{3}}}$

Dimension calculation:

$\sqrt{\frac{[M^{-1} L^3 T^{-2}][M L^2 T^{-1}]}{[L T^{-1}]^3}} = \sqrt{\frac{[L^5 T^{-3}]}{[L^3 T^{-3}]}} = \sqrt{[L^2]} = [L]$

The dimension is Length, which corresponds to Meter (A).

Conclusion: Matching Units

Based on the dimensional analysis:

• A. Meter ($[L]$) matches IV.
• B. Second ($[T]$) matches II.
• C. Kilogram ($[M]$) matches I.
• D. Kelvin ($[\Theta]$) matches III.

Therefore, the correct matching is A-IV, B-II, C-I, D-III.