Three samples $X, Y$,and $Z$ of the same gas have equal volumes and temperatures. The volume of each sample is doubled. The process is isothermal for $X$,adiabatic for $Y$,and isobaric for $Z$. If the final pressures are equal for the three samples,find the ratio of the initial pressures. (Take adiabatic exponent $\gamma = 3/2$)

The $P-V$ diagram of a certain process (Carnot cycle) is as shown. The process is also represented as:

The relation between efficiency $(\eta)$ of a Carnot engine and the coefficient of performance $(\beta)$ of a refrigerator working between the same temperatures is:

When an ideal gas is heated at constant pressure,what percentage of the heat supplied is used to perform external work? $(\gamma = 5/3)$

$A$ thermodynamic cycle of an ideal gas is shown in the figure. Choose the correct option which represents the same cycle.

$A$ heating element of resistance $r$ is fitted inside an adiabatic cylinder which carries a frictionless piston of mass $m$ and cross-sectional area $A$. The cylinder contains one mole of a diatomic gas. The temperature of the gas varies with time $t$ as $T = \alpha t + \frac{1}{2} \beta t^2$ (where $\alpha$ and $\beta$ are constants), while the pressure remains constant. The atmospheric pressure above the piston is $P_0$. Then: