The efficiency of the cycle shown in the figure (consisting of one isobar,one adiabat,and one isotherm) is $50 \%$. The ratio $x$ between the highest and lowest temperatures attained in this cycle obeys (the working substance is an ideal gas):

Two identical adiabatic vessels are filled with oxygen at pressure $P_1$ and $P_2$ $(P_1 > P_2)$. The vessels are interconnected with each other by a non-conducting pipe. If $U_{01}$ and $U_{02}$ denote the initial internal energy of oxygen in the first and second vessel respectively,and $U_{f1}$ and $U_{f2}$ denote the final internal energy values,then:

An ideal gas is made to undergo a cycle depicted by the $p-V$ diagram given below. The curved line from $A$ to $B$ is an adiabat. Then,

One mole of an ideal gas is taken through a cyclic process with alternating isothermal and adiabatic curves. In the $P-V$ diagram, $AB, CD, EF$ are isothermal curves at absolute temperatures $T_1, T_2,$ and $T_3$ respectively, and $BC, DE,$ and $FA$ are adiabatic curves. If $\frac{V_B}{V_A} = 2$ and $\frac{V_D}{V_C} = 2$, then for the cycle shown in the figure, four statements are made below. (Figure is not drawn to scale)
Statement $1$: Ratio of volumes $\frac{V_E}{V_F} = 4$
Statement $2$: Magnitude of work done in isothermal compression $EF$ is $2RT_3 \ln(2)$
Statement $3$: Ratio of heat supplied to the gas in process $AB$ to heat rejected by the gas in process $EF$ is $\frac{T_1}{T_3}$
Statement $4$: Net work done by the gas in the cycle $ABCDEFA$ is $(T_1 + T_2 - 2T_3) R \ln(2)$
Find the number of correct statements given for the cyclic process followed by the gas.

The volume $V$ of a given mass of monoatomic gas changes with temperature $T$ according to the relation $V = KT^{2/3}$. The work done when temperature changes by $90\,K$ will be $xR$. The value of $x$ is $[R = \text{universal gas constant}]$

$A$ gas is compressed from a volume of $2 \,m^3$ to a volume of $1 \,m^3$ at a constant pressure of $100 \,N m^{-2}$. Then it is heated at constant volume by supplying $150 \,J$ of energy. As a result,the internal energy of the gas