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)

$(A)$ $q_{A C}=\Delta U_{B C}$ and $W_{A B}=P_2\left(V_2-V_1\right)$ $(B)$ $W _{ BC }= P _2\left( V _2- V _1\right)$ and $q _{ BC }= H _{ AC }$ $(C)$ $\Delta H _{ CA }<\Delta U _{ CA }$ and $q _{ AC }=\Delta U _{ BC }$ $(D)$ $q_{B C}=\Delta H_{A C}$ and $\Delta H_{C A}>\Delta U_{C A}$

A vertical cylinder with heat-conducting walls is closed at the bottom and is fitted with a smooth light piston. It contains one mole of an ideal gas. The temperature of the gas is always equal to the surrounding’s temperature, $T_0$. The piston is moved up slowly to increase the volume of the gas to $\eta$ times. Which of the following is incorrect?

A closed container is fully insulated from outside. One half of it is filled with an ideal gas $X$ separated by a plate Pfrom the other half $Y$ which contains a vacuum as shown in figure. When $P$ is removed, $X$ moves into $Y$. Which of the following statements is correct?

The internal energy of an ideal gas increases during an isothermal process when the gas is 

A gas is taken through the cycle $A\to B\to C\to A$ as shown. What is the net work done by the gas …… $J$ $?$