When does any refrigerator or heat pump work?

The specific heat capacity of a metal at low temperature $T$ is given as $C_p = 32 \left( \frac{T}{400} \right)^3 \text{ kJ K}^{-1} \text{ kg}^{-1}$. $A$ $100 \text{ g}$ vessel of this metal is to be cooled from $20 \text{ K}$ to $4 \text{ K}$ by a special refrigerator operating at room temperature $27^\circ \text{C}$. The amount of work required to cool the vessel is

$A$ household refrigerator with a coefficient of performance $1.2$ removes heat from the refrigerated space at the rate of $60 \ kJ/min$. What would be the cost of running this fridge for one month $(30 \ days)$, assuming it is used for $4 \ hours$ each day and the cost of one electrical unit is $6 \ Rs.$?

$A$ Carnot's engine works as a refrigerator between $250\, K$ and $300\, K$. It receives $500\, cal$ heat from the reservoir at the lower temperature. The amount of work done in each cycle to operate the refrigerator is ..... $J$

Is the coefficient of performance of a refrigerator constant?

The equation $\beta = \frac{Q_2}{Q_1 - Q_2}$ is true or false?