Match List-I with List-II. List-I List-II (A) Planck Length (I) c^...

First, write the dimensional formulas for the fundamental constants:
Speed of light, $[c] = [L T^{-1}]$
Universal gravitational constant, $[G] = [M^{-1} L^3 T^{-2}]$
Planck's constant, $[h] = [M L^2 T^{-1}]$

Let us evaluate the common term $G^{1/2} h^{1/2}$:
$[G h]^{1/2} = (M^{-1} L^3 T^{-2} \times M L^2 T^{-1})^{1/2} = (L^5 T^{-3})^{1/2} = [L^{5/2} T^{-3/2}]$

To obtain Planck Length $[L^1]$, we need to multiply by a power of $c$ that yields $L^1 T^0$.
Multiplying by $c^{-3/2} = (L T^{-1})^{-3/2} = L^{-3/2} T^{3/2}$ gives:
$L^{5/2} T^{-3/2} \times L^{-3/2} T^{3/2} = [L^1]$.
Thus, Planck Length $\propto c^{-3/2} G^{1/2} h^{1/2}$. (A $\rightarrow$ II)

To obtain Planck Time $[T^1]$, multiply by $c^{-5/2} = L^{-5/2} T^{5/2}$:
$L^{5/2} T^{-3/2} \times L^{-5/2} T^{5/2} = [T^1]$.
Thus, Planck Time $\propto c^{-5/2} G^{1/2} h^{1/2}$. (C $\rightarrow$ IV)

Now evaluate the term $G^{-1/2} h^{1/2}$:
$[G^{-1} h]^{1/2} = (M L^{-3} T^2 \times M L^2 T^{-1})^{1/2} = (M^2 L^{-1} T^1)^{1/2} = [M^1 L^{-1/2} T^{1/2}]$

To obtain Planck Mass $[M^1]$, multiply by $c^{1/2} = L^{1/2} T^{-1/2}$:
$M^1 L^{-1/2} T^{1/2} \times L^{1/2} T^{-1/2} = [M^1]$.
Thus, Planck Mass $\propto c^{1/2} G^{-1/2} h^{1/2}$. (B $\rightarrow$ I)

To obtain Planck Energy $[M L^2 T^{-2}]$, multiply by $c^{5/2} = L^{5/2} T^{-5/2}$:
$M^1 L^{-1/2} T^{1/2} \times L^{5/2} T^{-5/2} = [M L^2 T^{-2}]$.
Thus, Planck Energy $\propto c^{5/2} G^{-1/2} h^{1/2}$. (D $\rightarrow$ III)

The correct matching is (A)-(II), (B)-(I), (C)-(IV), (D)-(III).

Answer: (A)-(II), (B)-(I), (C)-(IV), (D)-(III)