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$(A)$ The dimensions are matched as follows:$(A)$ Entropy $(S)$: $S = \frac{\Delta Q}{T}$. The unit is $J/K$, which is the same as the unit of the Bol

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$A-II, B-III, C-IV, D-I$

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$(A)$ The dimensions are matched as follows:$(A)$ Entropy $(S)$: $S = \frac{\Delta Q}{T}$. The unit is $J/K$, which is the same as the unit of the Boltzmann constant $(k_B)$. Thus, $(A) ightarrow (II)$.$(B)$ Young's modulus $(Y)$: $Y = \frac{	ext{Stress}}{	ext{Strain}} = \frac{	ext{Force}}{	ext{Area}}$. The unit is $N/m^2$ or $J/m^3$, which is the same as the unit of energy density $(U/V)$. Thus, $(B) ightarrow (III)$.$(C)$ Angular momentum $(L)$: $L = mvr$. The unit is $kg \cdot m^2/s$, which is the same as the unit of Planck's constant $(h)$. Thus, $(C) ightarrow (IV)$.$(D)$ Decay constant $(\lambda)$: $\lambda = \frac{1}{	ext{time}}$. The unit is $s^{-1}$, which is the same as the unit of angular velocity $(\omega)$. Thus, $(D) ightarrow (I)$.Therefore, the correct matching is $(A-II, B-III, C-IV, D-I)$.

$A$. Entropy	$I$. Angular velocity
$B$. Young's modulus of elasticity	$II$. Boltzmann constant
$C$. Angular momentum	$III$. Energy density
$D$. Decay constant	$IV$. Planck's constant

$A$. $Pa \cdot s$	$(i)$. $[L^2 T^{-2} K^{-1}]$
$B$. $N \cdot m \cdot K^{-1}$	$(ii)$. $[MLT^{-3} K^{-1}]$
$C$. $J \cdot kg^{-1} \cdot K^{-1}$	$(iii)$. $[ML^{-1} T^{-1}]$
$D$. $W \cdot m^{-1} \cdot K^{-1}$	$(iv)$. $[ML^2 T^{-2} K^{-1}]$

$(A)$ Thermal conductivity	$(i)$ $MLT^{-3}K^{-1}$
$(B)$ Boltzmann constant	(ii) $M^0L^2T^{-2}K^{-1}$
$(C)$ Latent heat	(iii) $ML^2T^{-2}K^{-1}$
$(D)$ Specific heat	(iv) $M^0L^2T^{-2}$

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Match the following physical quantities with their respective dimensional formulas:
$(A)$ Thermal conductivity $(i)$ $MLT^{-3}K^{-1}$
$(B)$ Boltzmann constant (ii) $M^0L^2T^{-2}K^{-1}$
$(C)$ Latent heat (iii) $ML^2T^{-2}K^{-1}$
$(D)$ Specific heat (iv) $M^0L^2T^{-2}$

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Match the physical quantities in Column-$I$ with those in Column-$II$ having the same dimensions. The correct matching is:$A$. Entropy$I$. Angular velocity$B$. Young's modulus of elasticity$II$. Boltzmann constant$C$. Angular momentum$III$. Energy density$D$. Decay constant$IV$. Planck's constant