Check whether the following statements are true or false:
$1.$ The change in internal energy $\Delta U = 0$ in a cyclic process.
$2.$ In an adiabatic process,temperature remains constant.
$3.$ The internal energy of a system during an isothermal process decreases.

In Column $-I$ processes and in Column $-II$ formulas of work are given. Match them appropriately:

Column $-I$	Column $-II$
$(a)$ Isothermal process	$(i)$ $W = \frac{\mu R(T_1 - T_2)}{\gamma - 1}$
$(b)$ Adiabatic process	$(ii)$ $W = P\Delta V$
	$(iii)$ $W = 2.303\mu RT \log_{10} \left( \frac{V_2}{V_1} \right)$

$A$ quantity of heat $Q$ is supplied to a monoatomic ideal gas which expands at constant pressure. What is the fraction of heat converted into work? Given $\gamma = \frac{C_p}{C_v} = \frac{5}{3}$.

Answer the following questions based on the $P-T$ phase diagram of $CO_{2}$:
$(a)$ $CO_{2}$ at $1 \; atm$ pressure and temperature $-60^{\circ} C$ is compressed isothermally. Does it go through a liquid phase?
$(b)$ What happens when $CO_{2}$ at $4 \; atm$ pressure is cooled from room temperature at constant pressure?
$(c)$ Describe qualitatively the changes in a given mass of solid $CO_{2}$ at $10 \; atm$ pressure and temperature $-65^{\circ} C$ as it is heated up to room temperature at constant pressure.
$(d)$ $CO_{2}$ is heated to a temperature $70^{\circ} C$ and compressed isothermally. What changes in its properties do you expect to observe?

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

$P-V$ diagram of a cyclic process $ABCA$ is as shown in figure. Choose the correct statement.