An ideal gas is undergoing 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 $n$.

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($1$) Which of the following options is the only correct representation of a process in which $\Delta U=\Delta Q-P \Delta V$ ?

$[A] (II) (iv) (R)$    $[B] (II) (iii) (P)$    $[C] (II) (iii) (S)$   $[D] (III) (iii) (P)$

($2$)  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?

$[A] (III) (iv) (R)$  $[B] (I) (ii)$ $(\mathrm{Q})$   $[C] (IV) (ii) (R)$    $[D] (I) (iv) (Q)$