Match the following:

Column $I$	Column $II$
$A$. Ratio of $\frac{\Delta Q}{\Delta U}$ in an isobaric process	$1$. $\frac{T_1}{T_1-T_2}$
$B$. Ratio of $\frac{\Delta Q}{\Delta W}$ in an isobaric process	$2$. $\frac{T_2}{T_1-T_2}$
$C$. Coefficient of performance of a refrigerator	$3$. $\frac{\gamma}{\gamma-1}$
$D$. Coefficient of performance of a heat pump	$4$. $\gamma$

Codes:
$A \quad B \quad C \quad D$

$A$ certain amount of gas of volume $V$ at $27^{\circ}C$ temperature and pressure $2 \times 10^{7} \; N m^{-2}$ expands isothermally until its volume gets doubled. Later it expands adiabatically until its volume gets redoubled. The final pressure of the gas will be (Use $\gamma = 1.5$)

The cycle shown in the figure represents an engine (the engine consists of one mole of gas in a cylinder with a piston). $A$ to $B$ is isochoric,$B$ to $C$ is isothermal,$C$ to $D$ is isochoric,and $D$ to $A$ is isothermal. Also,$V_C = V_D = 2V_A = 2V_B$.
$(a)$ In which part of the cycle is heat supplied to the engine from the outside?
$(b)$ In which part of the cycle can the engine give energy to its surroundings?
$(c)$ How much work is done by the engine during one cycle? Give your answer in terms of $P_A, P_B$ and $V_A$.
$(d)$ What is the efficiency of the engine? (For the gas,$\gamma = 5/3$,and for one mole,$C_V = 3/2 R$)

$n$ moles of a perfect gas undergo a cyclic process $ABCA$ (see figure) consisting of the following processes:
$A \rightarrow B :$ Isothermal expansion at temperature $T$ so that the volume is doubled from $V_{1}$ to $V_{2}=2V_{1}$ and pressure changes from $P_{1}$ to $P_{2}$.
$B \rightarrow C :$ Isobaric compression at pressure $P_{2}$ to initial volume $V_{1}$.
$C \rightarrow A :$ Isochoric change leading to a change of pressure from $P_{2}$ to $P_{1}$.
Total work done in the complete cycle $ABCA$ is

Two moles of a gas are expanded to double their volume by two different processes. One is isobaric and the other is isothermal. If $W_1$ and $W_2$ are the works done,respectively,then:

One mole of a monoatomic ideal gas $\left(c_{V} = \frac{3}{2} R\right)$ undergoes a cycle where it first goes isochorically from the state $\left(\frac{3}{2} P_{0}, V_{0}\right)$ to $\left(P_{0}, V_{0}\right)$,and then is isobarically contracted to the volume $\frac{1}{2} V_{0}$. It is then taken back to the initial state by a path which is a quarter ellipse on the $P-V$ diagram. The efficiency of this cycle is