Three moles of an ideal gas $\left( {{C_P} = \frac{7}{2}R} \right)$ at pressure ${P_A}$ and temperature ${T_A}$ is isothermally expanded to twice its initial volume. It is then compressed at constant pressure to its original volume. Finally, the gas is compressed at constant volume to its original pressure ${P_A}.$ The correct $P-V$ and $P-T$ diagrams indicating the process are

The graph for $y = 1 - e^{-x}$ is represented by (for $x > 0$):

The specific heat at constant pressure of a real gas obeying $PV^2=RT$ equation 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:

The efficiency of a Carnot engine operating with a hot reservoir kept at a temperature of $1000 K$ is $0.4$. It extracts $150 J$ of heat per cycle from the hot reservoir. The work extracted from this engine is being fully used to run a heat pump which has a coefficient of performance $10$. The hot reservoir of the heat pump is at a temperature of $300 K$. Which of the following statements is/are correct:
$(A)$ Work extracted from the Carnot engine in one cycle is $60 J$.
$(B)$ Temperature of the cold reservoir of the Carnot engine is $600 K$.
$(C)$ Temperature of the cold reservoir of the heat pump is $270 K$.
$(D)$ Heat supplied to the hot reservoir of the heat pump in one cycle is $540 J$.

$A$ monoatomic gas performs a work of $\frac{Q}{4}$,where $Q$ is the heat supplied to it. The molar heat capacity of the gas during this transformation will be $xR$,where $R$ is the gas constant. Find the value of $x$.