$A$ Carnot engine has the same efficiency between $800 \ K$ and $500 \ K$ and between $x \ K$ and $600 \ K$. The value of '$x$' is: (in $K$)

$A$ Carnot engine whose low temperature reservoir is at $7\,^{\circ}C$ has an efficiency of $50\%$. It is desired to increase the efficiency to $70\%$. By how many degrees should the temperature of the high temperature reservoir be increased? (in $K$)

$A$ Carnot engine,whose efficiency is $40 \%$,takes heat from a source maintained at a temperature of $600 \ K$. If it is desired to have an efficiency of $60 \%$,then the intake temperature for the same exhaust (sink) temperature should be: (in $K$)

$A$ steam engine delivers $5.4 \times 10^{8} \;J$ of work per minute and receives $3.6 \times 10^{9} \;J$ of heat per minute from its boiler. What is the efficiency of the engine? How much heat is wasted per minute?

Explain why the efficiency of a Carnot engine cannot be $100\%$?

Consider a two-stage Carnot engine. In the first stage, heat $Q_1$ is absorbed at temperature $T$ and heat $Q_2$ is expelled at temperature $\alpha T$ (where $\alpha < 1$). In the second stage, heat $Q_2$ is absorbed at temperature $\alpha T$ and heat $Q_3$ is expelled at temperature $\beta T$ (where $\beta < \alpha$). The efficiency of the Carnot engine will be