One mole of an ideal gas goes from an initial state $A$ to a final state $B$ via two processes: It first undergoes isothermal expansion from volume $V$ to $3V$ and then its volume is reduced from $3V$ to $V$ at constant pressure. The correct $P-V$ diagram representing the two processes is:

Initial pressure and volume of a gas are $P$ and $V$ respectively. First,it is expanded isothermally to volume $4V$ and then compressed adiabatically to volume $V$. The final pressure of the gas will be

One mole of a monatomic ideal gas undergoes the cyclic process $J \rightarrow K \rightarrow L \rightarrow M \rightarrow J$,as shown in the $P - T$ diagram. Match the quantities mentioned in $List-I$ with their values in $List-II$ and choose the correct option. [$R$ is the gas constant]

$List-I$	$List-II$
$(P)$ Work done in the complete cyclic process	$(1)$ $R T_0 - 4 R T_0 \ln 2$
$(Q)$ Change in the internal energy of the gas in the process $JK$	$(2)$ $0$
$(R)$ Heat given to the gas in the process $KL$	$(3)$ $3 R T_0$
$(S)$ Change in the internal energy of the gas in the process $MJ$	$(4)$ $-2 R T_0 \ln 2$
	$(5)$ $-3 R T_0 \ln 2$

$A$ certain amount of gas is taken through a cyclic process $(A-B-C-D-A)$ that has two isobars, one isochore, and one isothermal process. The cycle can be represented on a $P-V$ indicator diagram as:

Two identical samples of a gas are allowed to expand $(i)$ isothermally and $(ii)$ adiabatically. The work done is:

An ideal gas undergoes a cyclic thermodynamic process in different ways as shown in the corresponding $P-V$ diagrams in column $3$ of the table. Consider only the path from state $1$ to $2$. $W$ denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here $\gamma$ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is $n$.

Column $I$	Column $II$	Column $III$
$(I)$ $W_{1-2} = \frac{1}{\gamma-1}(P_2V_2 - P_1V_1)$	$(i)$ Isothermal	$(P)$ [Graph $P$]
$(II)$ $W_{1-2} = -P(V_2 - V_1)$	(ii) Isochoric	$(Q)$ [Graph $Q$]
$(III)$ $W_{1-2} = 0$	(iii) Isobaric	$(R)$ [Graph $R$]
$(IV)$ $W_{1-2} = -nRT \ln(\frac{V_2}{V_1})$	(iv) Adiabatic	$(S)$ [Graph $S$]

$(1)$ Which of the following options is the only correct representation of a process in which $\Delta U = \Delta Q - P \Delta V$?
$[A] (II) (iii) (P)$ $[B] (II) (iii) (R)$ $[C] (II) (iv) (S)$ $[D] (III) (iii) (P)$
$(2)$ Which one of the following options is the correct combination?
$[A] (III) (ii) (S)$ $[B] (II) (iv) (R)$ $[C] (II) (iv) (P)$ $[D] (IV) (ii) (S)$
$(3)$ Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?
$[A] (III) (iv) (R)$ $[B] (I) (ii) (Q)$ $[C] (I) (iv) (Q)$ $[D] (I) (iv) (R)$