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

• A
• B
• C
• D