The relation between the efficiency $\eta$ of a heat engine and the coefficient of performance $\alpha$ of a refrigerator is:

The $VT$ curve for a thermodynamic process is given. Identify the corresponding $PV$ curve.

$A$ monoatomic gas performs a work of $\frac{Q}{4}$,where $Q$ is the heat supplied to it. The molar heat capacity of the gas during this transformation will be $xR$,where $R$ is the gas constant. Find the value of $x$.

$A$ cyclic process on an ideal monatomic gas is shown in the figure. The correct statement is:

$A$ small particle of mass $m$ moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in the figure. When the distance of the piston from the closed end is $L = L_0$, the particle speed is $v = v_0$. The piston is moved inward at a very low speed $V$ such that $V \ll \frac{dL}{L} v_0$, where $dL$ is the infinitely small displacement of the piston. Which of the following statement(s) is/are correct?
$(1)$ The rate at which the particle strikes the piston is $v / (2L)$
$(2)$ After each collision with the piston, the particle speed increases by $2V$
$(3)$ The particle's kinetic energy increases by a factor of $4$ when the piston is moved inward from $L_0$ to $L_0 / 2$
$(4)$ If the piston moves inward by $dL$, the particle speed increases by $v \frac{dL}{L}$

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 $MJ$	$(4)$ $-2 R T_0 \ln 2$
	$(5)$ $-3 R T_0 \ln 2$