Match the following:

List-$I$	List-$II$
$i)$ Isothermal process	$a)$ $0$
$ii)$ Isobaric process	$b)$ $\frac{1}{\gamma-1}[P_2 V_2 - P_1 V_1]$
$iii)$ Isochoric process	$c)$ $\mu RT \ln(\frac{V_2}{V_1})$
$iv)$ Adiabatic process	$d)$ $P(V_2 - V_1)$

The correct answer is:

If $dQ$, $dU$, and $dW$ are heat energy absorbed, change in internal energy, and external work done respectively by a diatomic gas at constant pressure, then the ratio $dW: dU: dQ$ is:

Obtain the relation between the coefficient of performance and the efficiency of a heat engine.

One mole of a monoatomic ideal gas goes through a thermodynamic cycle,as shown in the volume versus temperature $(V-T)$ diagram. The correct statement$(s)$ is/are :
[$R$ is the gas constant]
$(1)$ Work done in this thermodynamic cycle $(1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 1)$ is $|W| = \frac{1}{2} RT_0$
$(2)$ The ratio of heat transfer during processes $1 \rightarrow 2$ and $2 \rightarrow 3$ is $\left|\frac{Q_{1 \rightarrow 2}}{Q_{2 \rightarrow 3}}\right| = \frac{5}{3}$
$(3)$ The above thermodynamic cycle exhibits only isochoric and adiabatic processes.
$(4)$ The ratio of heat transfer during processes $1 \rightarrow 2$ and $3 \rightarrow 4$ is $\left|\frac{Q_{1 \rightarrow 2}}{Q_{3 \rightarrow 4}}\right| = \frac{1}{2}$

In Column-$I$ devices and in Column-$II$ efficiency are given. Match them appropriately:

Column-$I$	Column-$II$
$(a)$ Heat engine	$(i)$ $\eta = \frac{Q_2}{Q_1 - Q_2}$
$(b)$ Heat pump	$(ii)$ $\eta = \frac{Q_1 - Q_2}{Q_1}$
	$(iii)$ $\eta = \frac{T_1 - T_2}{T_1}$

Two cylinders $A$ and $B$ fitted with pistons contain equal number of moles of an ideal monoatomic gas at $400 \,K$. The piston of $A$ is free to move while that of $B$ is held fixed. The same amount of heat energy is given to the gas in each cylinder. If the rise in temperature of the gas in $A$ is $42 \,K$, what is the rise in temperature of the gas in $B$ (in $\,K$)? (Given $\gamma = 5/3$)