$A$ cylinder made of perfectly non-conducting material,closed at both ends,is divided into two equal parts by a heat-proof piston. Both parts of the cylinder contain the same mass of a gas at a temperature $t_0 = 27^{\circ}C$ and pressure $P_0 = 1 \text{ atm}$. If the gas in one of the parts is slowly heated to $t = 57^{\circ}C$ while the temperature of the first part is maintained at $t_0$,the distance moved by the piston from the middle of the cylinder will be $x \text{ cm}$. Find $x$ (total length of the cylinder $= 84 \text{ cm}$).

$A$ monoatomic gas at a pressure $P$,having a volume $V$ expands isothermally to a volume $4V$ and then adiabatically to volume $16V$. The final pressure of the gas is (Take $\gamma = \frac{5}{3}$)

$A$ gas follows $VT^2 = \text{constant}$. The coefficient of volume expansion of the gas is

$A$ monoatomic gas at pressure $P$ having volume $V$ expands isothermally to a volume $2V$ and then adiabatically to a volume $16V$. The final pressure of the gas is $\left(\gamma = \frac{5}{3}\right)$.

$A$ thermodynamic cycle of an ideal gas is shown in the figure. Choose the correct option which represents the same cycle.

Answer the following by appropriately matching the lists based on the information given in the paragraph.
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 the 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 the gas constant,$V$ is the volume of the 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.

List-$I$	List-$II$
$(I)$ Work done by the system in process $1 \rightarrow 2 \rightarrow 3$	$(P)$ $\frac{1}{3} R T_0 \ln 2$
$(II)$ Change in internal energy in process $1 \rightarrow 2 \rightarrow 3$	$(Q)$ $\frac{1}{3} R T_0$
$(III)$ Heat absorbed by the system in process $1 \rightarrow 2 \rightarrow 3$	$(R)$ $R T_0$
$(IV)$ Heat absorbed by the system in process $1 \rightarrow 2$	$(S)$ $\frac{4}{3} R T_0$
	$(T)$ $\frac{1}{3} R T_0 (3 + \ln 2)$
	$(U)$ $\frac{5}{6} R T_0$

If the process carried out on one mole of monatomic ideal gas is as shown in the figure in the $PV$-diagram with $P_0 V_0 = \frac{1}{3} R T_0$,the correct match is:
$(1)$ $I \rightarrow Q, II \rightarrow R, III \rightarrow P, IV \rightarrow U$
$(2)$ $I \rightarrow S, II \rightarrow R, III \rightarrow Q, IV \rightarrow T$
$(3)$ $I \rightarrow Q, II \rightarrow R, III \rightarrow S, IV \rightarrow U$
$(4)$ $I \rightarrow Q, II \rightarrow S, III \rightarrow R, IV \rightarrow U$
If the process on one mole of monatomic ideal gas is as shown in the $TV$-diagram with $P_0 V_0 = \frac{1}{3} R T_0$,the correct match is:
$(1)$ $I \rightarrow S, II \rightarrow T, III \rightarrow Q, IV \rightarrow U$
$(2)$ $I \rightarrow P, II \rightarrow R, III \rightarrow T, IV \rightarrow S$
$(3)$ $I \rightarrow P, II \rightarrow R, III \rightarrow Q, IV \rightarrow T$
$(4)$ $I \rightarrow P, II \rightarrow R, III \rightarrow T, IV \rightarrow P$
Give the answer for question $(1)$ and $(2)$.