Which of the following combinations of three different physical quantities P, Q, R can never be a meaningful quantity?

Options

A. `PQ - R`
B. `(PQ)/(R )`
C. `(P -Q)/(r)`
D. `(PR - Q^2)/(QR)`

Answer

To determine which combination of three different physical quantities \( P, Q, R \) can never be a meaningful quantity, we need to analyze the dimensional consistency of the combinations provided.

### Step-by-Step Solution:

1. **Understanding Physical Quantities**:
- Physical quantities have dimensions associated with them. For example, mass (M), length (L), time (T), volume (L³), and density (M/L³) all have different dimensions.

2. **Identifying the Combinations**:
- Let's consider the combinations provided in the question. For instance, if we take:
- \( P \) as mass (M)
- \( Q \) as volume (L³)
- \( R \) as density (M/L³)

3. **Checking Dimensional Consistency**:
- To check if a combination is meaningful, we need to see if we can perform operations like addition or subtraction.
- The dimensions of mass (M), volume (L³), and density (M/L³) are different.

4. **Performing Operations**:
- If we try to subtract or add these quantities:
- \( P - Q \) (mass - volume) is not meaningful because they have different dimensions.
- \( Q - R \) (volume - density) is also not meaningful.
- \( R - P \) (density - mass) is again not meaningful.

5. **Conclusion**:
- Since all combinations of operations between \( P, Q, R \) result in quantities with different dimensions, we conclude that this combination cannot yield a meaningful quantity.

### Final Answer:
The combination of \( P \) (mass), \( Q \) (volume), and \( R \) (density) can never be a meaningful quantity due to the dimensional inconsistency.

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