A : Coefficient of performanceof a refreigerator is always greater than 1.
R : Efficiencyof heat engine is greater than 1.

Options

A. If both Assertion & Reason are true and the reason is the correct explantion of the assertion, then mark(1)
B. If both Assertion & Reason are true but the reason is not the correct explanation of the assertion, then mark (2).
C. If Assertion is true statement but Reason is false, then mark (3).
D. If both Assertion and Reason are false statements, then mark (4)

Answer

To solve the assertion-reason question, we will analyze both the assertion (A) and the reason (R) separately.

### Step 1: Analyze the Assertion (A)
The assertion states: "The coefficient of performance (COP) of a refrigerator is always greater than 1."

1. **Understanding COP**: The Coefficient of Performance (COP) for a refrigerator is defined as:
\[ \text{COP} = \frac{Q_L}{W} \]
where \( Q_L \) is the heat removed from the cold reservoir and \( W \) is the work input.

2. **Relation to Temperatures**: The COP can also be expressed in terms of the temperatures of the hot and cold reservoirs:
\[ \text{COP} = \frac{T_L}{T_H - T_L} \]
where \( T_L \) is the temperature of the cold reservoir and \( T_H \) is the temperature of the hot reservoir.

3. **Condition for COP > 1**: For the COP to be greater than 1:
\[ T_L > T_H - T_L \]
Rearranging gives:
\[ 2T_L > T_H \quad \Rightarrow \quad T_L > \frac{T_H}{2} \]
This condition does not always hold true; thus, the assertion is not universally valid.

### Conclusion for Assertion:
The assertion is **false** because the COP of a refrigerator is not always greater than 1.

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### Step 2: Analyze the Reason (R)
The reason states: "The efficiency of a heat engine is greater than 1."

1. **Understanding Efficiency**: The efficiency (\( \eta \)) of a heat engine is defined as:
\[ \eta = \frac{W}{Q_H} \]
where \( W \) is the work output and \( Q_H \) is the heat absorbed from the hot reservoir.

2. **Relation to Temperatures**: The efficiency can also be expressed in terms of the temperatures of the hot and cold reservoirs:
\[ \eta = 1 - \frac{T_L}{T_H} \]
where \( T_L \) is the temperature of the cold reservoir and \( T_H \) is the temperature of the hot reservoir.

3. **Condition for Efficiency**: Since \( T_L < T_H \), it follows that:
\[ \frac{T_L}{T_H} < 1 \quad \Rightarrow \quad 1 - \frac{T_L}{T_H} < 1 \]
Therefore, the efficiency of a heat engine is always less than 1.

### Conclusion for Reason:
The reason is **false** because the efficiency of a heat engine is never greater than 1.

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### Final Conclusion:
Both the assertion (A) and the reason (R) are false. Therefore, the correct answer is option D: "Both A and R are false."

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