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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?

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

Statement $-I$ : What $\mu$ amount of an ideal gas undergoes adiabatic change from state $\left( P _{1}, V _{1}, T _{1}\right)$ to state $\left( P _{2}, V _{2}, T _{2}\right)$, the work done is $W =\frac{1 R \left( T _{2}- T _{1}\right)}{1-\gamma}$, where $\gamma=\frac{ C _{ P }}{ C _{ V }}$ and $R =$ universal gas constant,

Statement $-II$ : In the above case. when work is done on the gas. the temperature of the gas would rise.

Choose the correct answer from the options given below