$A$ sample of an ideal gas is taken through the cyclic process $ABCA$ as shown in the figure. It absorbs $40 \ J$ of heat during the part $AB$,no heat during $BC$,and rejects $60 \ J$ of heat during $CA$. $A$ work of $50 \ J$ is done on the gas during the part $BC$. The internal energy of the gas at $A$ is $1560 \ J$. The work done by the gas during the part $CA$ is.............$J$.

Match List-$I$ with List-$II$:

List-$I$	List-$II$
$(a)$ Isothermal	$(i)$ Pressure constant
$(b)$ Isochoric	$(ii)$ Temperature constant
$(c)$ Adiabatic	$(iii)$ Volume constant
$(d)$ Isobaric	$(iv)$ Heat content is constant

Choose the correct answer from the options given below:

In the given $P-V$ diagram,a monoatomic gas $\left(\gamma = \frac{5}{3}\right)$ is first compressed adiabatically from state $A$ to state $B$. Then it expands isothermally from state $B$ to state $C$. [Given: $\left(\frac{1}{3}\right)^{0.6} \simeq 0.5, \ln 2 \simeq 0.7$].
Which of the following statement$(s)$ is(are) correct?
$(A)$ The magnitude of the total work done in the process $A \rightarrow B \rightarrow C$ is $144 \text{ kJ}$.
$(B)$ The magnitude of the work done in the process $B \rightarrow C$ is $84 \text{ kJ}$.
$(C)$ The magnitude of the work done in the process $A \rightarrow B$ is $60 \text{ kJ}$.
$(D)$ The magnitude of the work done in the process $C \rightarrow A$ is zero.

For a monoatomic ideal gas following the cyclic process $ABCA$ shown in the $U$ vs $\rho$ plot,identify the incorrect option:

An office room contains about $2000$ moles of air. The change in the internal energy of this much air when it is cooled from $34^{\circ} C$ to $24^{\circ} C$ at a constant pressure of $1.0 \text{ atm}$ is (Use $\gamma_{\text{air}} = 1.4$ and universal gas constant $R = 8.314 \text{ J/mol-K}$)

When a system is taken from state $i$ to state $f$ along the path $iaf$,$Q = 50 \ J$ and $W = 20 \ J$. Along the path $ibf$,$Q = 35 \ J$ and $W = -13 \ J$. For the curved path $fi$,$Q = \dots \ J$.