One mole of an ideal gas undergoes a cyclic process,consisting of two isochores and two isobars. Temperatures at points $1$ and $3$ are $T_1$ and $T_3$ respectively. Find the work done by the gas over the cycle,if points $2$ and $4$ lie on the same isotherm.

An ideal monoatomic gas is carried along the cycle $ABCDA$ as shown in the figure. The total heat absorbed during this process is (in $p_0 V_0$)

Which of the following statements is correct regarding the $p-V$ graph?
$1$. The slope of the $p-V$ graph in an isothermal process is $-\frac{p}{V}$.
$2$. The slope of the $p-V$ graph in an adiabatic process is $-\frac{p}{V}$.
$3$. The slope of the $p-V$ graph in an isochoric process is $-\frac{\gamma p}{V}$.
$4$. The slope of the $p-V$ graph in an isobaric process is zero.

$1$ mole of rigid diatomic gas performs a work of $Q/5$ when heat $Q$ is supplied to it. The molar heat capacity of the gas during this transformation is $\frac{xR}{8}.$ The value of $x$ is $\ldots \ldots \ldots .$ $[R =$ universal gas constant $]$

$A$ Carnot cycle for a gas is represented in the pressure-volume $(P-V)$ diagram shown below. Consider the following statements:
$I.$ The area of $ABCD$ is the work done by the gas.
$II.$ The area of $ABCD$ is the total heat absorbed.
$III.$ The change in internal energy of the cycle is zero.
Which of the following is correct?

When $Q_1$ amount of heat is supplied to a monatomic gas, the work done by the gas is $W$. When $Q_2$ amount of heat is supplied to a diatomic gas, the work done by the gas is $2W$. Then $Q_1: Q_2=$