$A$ heating element of resistance $r$ is fitted inside an adiabatic cylinder which carries a frictionless piston of mass $m$ and cross-sectional area $A$. The cylinder contains one mole of a diatomic gas. The temperature of the gas varies with time $t$ as $T = \alpha t + \frac{1}{2} \beta t^2$ (where $\alpha$ and $\beta$ are constants), while the pressure remains constant. The atmospheric pressure above the piston is $P_0$. Then:

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 $(2)$.

The slopes of isothermal and adiabatic curves are related as

The efficiency of a heat engine is $\eta$ and the coefficient of performance of a refrigerator is $\beta$. Then:

Two samples of gas $A$ and $B$ are initially at the same pressure and temperature. They are compressed from volume $V$ to $V/2$. If $A$ is compressed isothermally and $B$ is compressed adiabatically,then the final pressure of $A$ is:

The internal energy of the air in a room of volume $V$ at temperature $T$, with outside pressure $P$ increasing linearly with time, varies as