An ideal gas is taken from state $1$ to state $2$ through optional paths $A, B, C$ and $D$ as shown in the $P-V$ diagram. Let $Q, W$ and $\Delta U$ represent the heat supplied,work done,and change in internal energy of the gas,respectively. Then:

An ideal gas expands from volume $V_1$ to $V_2$. This may be achieved by either of the three processes: isobaric,isothermal,and adiabatic. Let $\Delta U$ be the change in internal energy of the gas,$Q$ be the quantity of heat added to the system,and $W$ be the work done by the system. Identify which of the following statements is false for $\Delta U$?

Two identical vessels $A$ and $B$ contain equal amounts of an ideal monoatomic gas. The piston of $A$ is fixed,while the piston of $B$ is free to move. The same amount of heat $\Delta Q$ is absorbed by both $A$ and $B$. If the internal energy of $B$ increases by $100 \, J$,what is the change in the internal energy of $A$ (in $, J$)?

One mole of a monatomic ideal gas undergoes the cyclic process $J \rightarrow K \rightarrow L \rightarrow M \rightarrow J$,as shown in the $P - T$ diagram. Match the quantities mentioned in $List-I$ with their values in $List-II$ and choose the correct option. [$R$ is the gas constant]

$List-I$	$List-II$
$(P)$ Work done in the complete cyclic process	$(1)$ $R T_0 - 4 R T_0 \ln 2$
$(Q)$ Change in the internal energy of the gas in the process $JK$	$(2)$ $0$
$(R)$ Heat given to the gas in the process $KL$	$(3)$ $3 R T_0$
$(S)$ Change in the internal energy of the gas in the process $MJ$	$(4)$ $-2 R T_0 \ln 2$
	$(5)$ $-3 R T_0 \ln 2$

In changing the state of a gas adiabatically from an equilibrium state $A$ to another equilibrium state $B$,an amount of work equal to $22.3 \; J$ is done on the system. If the gas is taken from state $A$ to $B$ via a process in which the net heat absorbed by the system is $9.35 \; cal$,how much is the net work done (in $J$) by the system in the latter case? (Take $1 \; cal = 4.19 \; J$)

$A$ heating element of resistance $r$ is fitted inside an adiabatic cylinder which carries a frictionless piston of mass $m$ and cross-sectional area $A$. The cylinder contains one mole of a diatomic gas. The temperature of the gas varies with time $t$ as $T = \alpha t + \frac{1}{2} \beta t^2$ (where $\alpha$ and $\beta$ are constants),while the pressure remains constant. The atmospheric pressure above the piston is $P_0$. Then: