$A$ Carnot heat engine absorbs $600 \ J$ of heat from a source at a temperature of $127^{\circ} C$ and rejects $400 \ J$ of heat to a sink in each cycle. The temperature of the sink is (in $K$)

$A$ Carnot engine whose sink is at $300\, K$ has an efficiency of $40\%.$ By how much should the temperature of the source be increased so as to increase its efficiency by $50\%$ of its original efficiency? (in $K$)

If a Carnot engine functions at a source temperature of $127^{\circ}C$ and a sink temperature of $87^{\circ}C$,what is its efficiency in $\%$?

Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$.
Assertion $A$: Efficiency of a reversible heat engine will be highest at $-273^{\circ} C$ temperature of cold reservoir.
Reason $R$: The efficiency of Carnot's engine depends not only on the temperature of the cold reservoir but it depends on the temperature of the hot reservoir too and is given as $\eta = (1 - \frac{T_2}{T_1})$.
In the light of the above statements,choose the correct answer from the options given below:

One mole of an ideal monoatomic gas undergoes the following four reversible processes:
Step $1$: It is first compressed adiabatically from volume $V_{1}$ to $1 \; m^{3}$.
Step $2$: Then expanded isothermally to volume $10 \; m^{3}$.
Step $3$: Then expanded adiabatically to volume $V_{3}$.
Step $4$: Then compressed isothermally to volume $V_{1}$.
If the efficiency of the above cycle is $3/4$, then $V_{1}$ is ............ $m^{3}$.

$A$ heat engine receives $200 \, cal$ of heat and rejects $150 \, cal$ of heat. If the temperature of the source is $400 \, K$,what is the temperature of the sink in $K$ (in $, K$)?