Carnot Engine:

Hot reservoir temperature ($T_H$) = 1000 K

Efficiency ($\eta$) = 0.4

Heat extracted from the hot reservoir ($Q_H$) = 150 J per cycle

Heat Pump:

Coefficient of Performance (COP) = 10

Work input ($W_{pump}$) = Work output from the engine ($W_{engine}$)

Hot reservoir temperature ($T_{H,hp}$) = 300 K

Let's analyze each statement:

Statement A: Work extracted from the Carnot engine in one cycle is 60 J.

The efficiency of an engine is defined as the ratio of the work output to the heat input from the hot reservoir.

$$ \eta = \frac{W_{engine}}{Q_H} $$

We are given $\eta = 0.4$ and $Q_H = 150$ J. We can find the work done by the engine:

$$ W_{engine} = \eta \times Q_H = 0.4 \times 150 \text{ J} = 60 \text{ J} $$

This statement is correct.

Statement B: Temperature of the cold reservoir of the Carnot engine is 600 K.

For a Carnot engine, the efficiency can also be expressed in terms of the temperatures of the hot and cold reservoirs:

$$ \eta = 1 - \frac{T_C}{T_H} $$

where $T_C$ is the temperature of the cold reservoir and $T_H$ is the temperature of the hot reservoir (both in Kelvin).

We know $\eta = 0.4$ and $T_H = 1000$ K. We can solve for $T_C$:

$$ 0.4 = 1 - \frac{T_C}{1000} $$

$$ \frac{T_C}{1000} = 1 - 0.4 = 0.6 $$

$$ T_C = 0.6 \times 1000 \text{ K} = 600 \text{ K} $$

This statement is also correct.

Statement C: Temperature of the cold reservoir of the heat pump is 270 K.

The coefficient of performance (COP) for a heat pump is defined as the ratio of the heat delivered to the hot reservoir ($Q_{H,hp}$) to the work input ($W_{pump}$). For an ideal (Carnot) heat pump operating between temperatures $T_{C,hp}$ and $T_{H,hp}$, the COP is also given by:

$$ \text{COP} = \frac{T_{H,hp}}{T_{H,hp} - T_{C,hp}} $$

We are given the COP of the heat pump is 10 and its hot reservoir temperature $T_{H,hp} = 300$ K. Let's assume this heat pump is operating ideally with this COP. We can find $T_{C,hp}$:

$$ 10 = \frac{300 \text{ K}}{300 \text{ K} - T_{C,hp}} $$

Let's solve for $T_{C,hp}$:

$$ 10 \times (300 - T_{C,hp}) = 300 $$

$$ 3000 - 10 T_{C,hp} = 300 $$

$$ 10 T_{C,hp} = 3000 - 300 = 2700 $$

$$ T_{C,hp} = \frac{2700}{10} \text{ K} = 270 \text{ K} $$

This statement is correct, assuming the heat pump is ideal.

Statement D: Heat supplied to the hot reservoir of the heat pump in one cycle is 540 J.

The COP of a heat pump is also defined as the ratio of the heat delivered to the hot reservoir ($Q_{H,hp}$) to the work input ($W_{pump}$).

$$ \text{COP} = \frac{Q_{H,hp}}{W_{pump}} $$

We know the COP is 10 and the work input to the heat pump is the work output from the engine, which we found to be $W_{pump} = W_{engine} = 60$ J.

So, the heat supplied to the hot reservoir of the heat pump is:

$$ Q_{H,hp} = \text{COP} \times W_{pump} = 10 \times 60 \text{ J} = 600 \text{ J} $$

This statement says the heat supplied is 540 J, which does not match our calculation of 600 J.

This statement is incorrect.

Based on our analysis, statements A, B, and C are correct.