An ideal gas expands isothermally from volume $V_1$ to volume $V_2$. It is then compressed to the original volume $V_1$ adiabatically. If $p_1$ and $p_2$ represent the initial pressure and final pressure respectively, and $W$ represents the net work done by the gas during the entire process, then:

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?
$[A] (III) (iv) (R)$ $[B] (I) (ii) (Q)$ $[C] (I) (iv) (Q)$ $[D] (I) (iv) (R)$

An ideal gas at pressure $P$ and volume $V$ is expanded to volume $2V$. Column $I$ represents the thermodynamic processes used during expansion. Column $II$ represents the work done during these processes in random order:

Column $I$	Column $II$
$(p)$ isobaric	$(x)$ $\frac{PV(1 - 2^{1 - \gamma})}{\gamma - 1}$
$(q)$ isothermal	$(y)$ $PV$
$(r)$ adiabatic	$(z)$ $PV \ln 2$

The correct matching of column $I$ and column $II$ is given by:

An ideal gas is subjected to a cyclic process $ABCD$ as depicted in the $p-V$ diagram given below. Which of the following curves represents the equivalent cyclic process?

For an ideal gas, a cyclic process $ABCA$ as shown in the $P-T$ diagram, when presented in a $P-V$ plot, would be:

An ideal gas undergoes four different processes from the same initial state as shown in the figure below. Those processes are adiabatic,isothermal,isobaric and isochoric. The curve which represents the adiabatic process among $1, 2, 3$ and $4$ is