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(D) The pressure of the gas is $P = P_1 + \frac{kx}{A}$. Since $V = V_1 + Ax$,we have $x = \frac{V-V_1}{A}$. Thus,$P = P_1 + \frac{k(V-V_1)}{A^2}$.Wor

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$(A), (B), (C)$

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(D) The pressure of the gas is $P = P_1 + \frac{kx}{A}$. Since $V = V_1 + Ax$,we have $x = \frac{V-V_1}{A}$. Thus,$P = P_1 + \frac{k(V-V_1)}{A^2}$.Work done by the gas $W = \int_{V_1}^{V_2} P dV = P_1(V_2-V_1) + \frac{k(V_2-V_1)^2}{2A^2}$.Since $P_2 = P_1 + \frac{k(V_2-V_1)}{A^2}$,we have $\frac{k(V_2-V_1)}{A^2} = P_2 - P_1$. Substituting this,$W = P_1(V_2-V_1) + \frac{1}{2}(P_2-P_1)(V_2-V_1) = \frac{1}{2}(P_1+P_2)(V_2-V_1)$.Change in internal energy $\Delta U = nC_V\Delta T = \frac{3}{2}(P_2V_2 - P_1V_1)$.Energy stored in spring $U_s = \frac{1}{2}kx^2 = \frac{1}{2}k(\frac{V_2-V_1}{A})^2 = \frac{1}{2}(P_2-P_1)(V_2-V_1)$.Case $I$: $V_2=2V_1, T_2=3T_1$. From $PV=nRT$,$P_2(2V_1) = nR(3T_1) = 3P_1V_1 \implies P_2 = 1.5P_1$.$U_s = \frac{1}{2}(1.5P_1-P_1)(2V_1-V_1) = \frac{1}{2}(0.5P_1)(V_1) = 0.25P_1V_1 = \frac{1}{4}P_1V_1$. ($A$ is correct)$\Delta U = \frac{3}{2}(1.5P_1 \cdot 2V_1 - P_1V_1) = \frac{3}{2}(3P_1V_1 - P_1V_1) = 3P_1V_1$. ($B$ is correct)Case $II$: $V_2=3V_1, T_2=4T_1$. $P_2(3V_1) = nR(4T_1) = 4P_1V_1 \implies P_2 = \frac{4}{3}P_1$.$W = \frac{1}{2}(P_1 + \frac{4}{3}P_1)(3V_1-V_1) = \frac{1}{2}(\frac{7}{3}P_1)(2V_1) = \frac{7}{3}P_1V_1$. ($C$ is correct)$Q = W + \Delta U = \frac{7}{3}P_1V_1 + \frac{3}{2}(\frac{4}{3}P_1 \cdot 3V_1 - P_1V_1) = \frac{7}{3}P_1V_1 + \frac{3}{2}(3P_1V_1) = \frac{7}{3}P_1V_1 + 4.5P_1V_1 = \frac{41}{6}P_1V_1$. ($D$ is correct)

$List-I$	$List-II$
$(I)$ $10^{-3} \, kg$ of water at $100^{\circ} C$ is converted to steam at the same temperature, at a pressure of $10^5 \, Pa$. The volume of the system changes from $10^{-6} \, m^3$ to $10^{-3} \, m^3$. Latent heat of water $= 2250 \, kJ/kg$.	$(P)$ $2 \, kJ$
$(II)$ $0.2 \, moles$ of a rigid diatomic ideal gas with volume $V$ at temperature $500 \, K$ undergoes an isobaric expansion to volume $3V$. Assume $R = 8.0 \, J \, mol^{-1} \, K^{-1}$.	$(Q)$ $7 \, kJ$
$(III)$ One mole of a monatomic ideal gas is compressed adiabatically from volume $V = 1/3 \, m^3$ and pressure $2 \, kPa$ to volume $V/8$.	$(R)$ $4 \, kJ$
$(IV)$ Three moles of a diatomic ideal gas whose molecules can vibrate, is given $9 \, kJ$ of heat and undergoes isobaric expansion.	$(S)$ $5 \, kJ$
	$(T)$ $3 \, kJ$

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