In a thermodynamics process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by $T \Delta X$, where $T$ is temperature of the system and $\Delta X$ is the infinitesimal change in a thermodynamic quantity $X$ of the system. For a mole of monatomic ideal gas

$X=\frac{3}{2} R \ln \left(\frac{T}{T_A}\right)+R \ln \left(\frac{V}{V_A}\right)$. Here, $R$ is gas constant, $V$ is volume of gas, $T_A$ and $V_A$ are constants.

The $List-I$ below gives some quantities involved in a process and $List-II$ gives some possible values of these quantities.

If the process carried out on one mole of monatomic ideal gas is as shown in figure in the PV-diagram with $P _0 V _0=\frac{1}{3} RT _0$, the correct match is,

$(1)$$I \rightarrow Q, II \rightarrow R , III \rightarrow P , IV \rightarrow U$

$(2)$ $I \rightarrow S , II \rightarrow R , III \rightarrow Q , IV \rightarrow T$

$(3)$ $I \rightarrow Q , II \rightarrow R , III \rightarrow S , IV \rightarrow U$

$(4)$ $I \rightarrow Q , II \rightarrow S , III \rightarrow R , IV \rightarrow U$

($2$) If the process on one mole of monatomic ideal gas is an shown is as shown in the $TV$-diagram with $P _0 V _0=\frac{1}{3} RT _0$, the correct match is

$(1)$ $I \rightarrow S, II \rightarrow T, III \rightarrow Q , IV \rightarrow U$

$(2)$ $I \rightarrow P , II \rightarrow R, III \rightarrow T , IV \rightarrow S$

$(3)$ $I \rightarrow P, II \rightarrow, III \rightarrow Q, IV \rightarrow T$

$(4)$ $I \rightarrow P, II \rightarrow R, III \rightarrow T, IV \rightarrow P$

Give the answer or quetion $(1)$ and $(2)$

Choose the correct answer from the options given below :