In the $p-V$ diagram below,the dashed curved line is an adiabat. For a process that is described by a straight line joining two points $X$ and $Y$ on the adiabat (solid line in the diagram),heat is (Hint: consider the variation in temperature from $X$ to $Y$ along the straight line)

An engineer claims to have made an engine delivering $10 \, kW$ power with fuel consumption of $1 \, g \, s^{-1}$. The calorific value of the fuel is $2 \, kcal \, g^{-1}$. His claim:

$A$ reversible cyclic process for an ideal gas is shown below. Here,$P, V$,and $T$ are pressure,volume,and temperature,respectively. The thermodynamic parameters $q, w, H$,and $U$ are heat,work,enthalpy,and internal energy,respectively.
The correct option$(s)$ is (are):
$(A)$ $q_{AC} = \Delta U_{BC}$ and $W_{AB} = P_2(V_2 - V_1)$
$(B)$ $W_{BC} = P_2(V_2 - V_1)$ and $q_{BC} = H_{AC}$
$(C)$ $\Delta H_{CA} < \Delta U_{CA}$ and $q_{AC} = \Delta U_{BC}$
$(D)$ $q_{BC} = \Delta H_{AC}$ and $\Delta H_{CA} > \Delta U_{CA}$

Starting at temperature $300 \; K,$ one mole of an ideal diatomic gas $(\gamma=1.4)$ is first compressed adiabatically from volume $V_{1}$ to $V_{2}=\frac{V_{1}}{16}.$ It is then allowed to expand isobarically to volume $2V_{2}.$ If all the processes are quasi-static,then the final temperature of the gas (in $K$) is (to the nearest integer):

One mole of an ideal gas is taken through a cyclic process with alternating isothermal and adiabatic curves. In the $P-V$ diagram, $AB, CD, EF$ are isothermal curves at absolute temperatures $T_1, T_2,$ and $T_3$ respectively, and $BC, DE,$ and $FA$ are adiabatic curves. If $\frac{V_B}{V_A} = 2$ and $\frac{V_D}{V_C} = 2$, then for the cycle shown in the figure, four statements are made below. (Figure is not drawn to scale)
Statement $1$: Ratio of volumes $\frac{V_E}{V_F} = 4$
Statement $2$: Magnitude of work done in isothermal compression $EF$ is $2RT_3 \ln(2)$
Statement $3$: Ratio of heat supplied to the gas in process $AB$ to heat rejected by the gas in process $EF$ is $\frac{T_1}{T_3}$
Statement $4$: Net work done by the gas in the cycle $ABCDEFA$ is $(T_1 + T_2 - 2T_3) R \ln(2)$
Find the number of correct statements given for the cyclic process followed by the gas.

$A$ thermodynamic cycle takes in heat energy at a high temperature and rejects energy at a lower temperature. If the amount of energy rejected at the low temperature is $3$ times the amount of work done by the cycle,the efficiency of the cycle is