If P, Q, R are physical quantities, having different dimensions, which of the following combinations can never be a meaningful quantity ?

Options

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

Explanation

To solve the problem of determining which combinations of the physical quantities P, Q, and R can never be meaningful, we need to analyze the combinations based on the rules of dimensional analysis. Here’s a step-by-step breakdown:

### Step 1: Understand the Problem
We have three physical quantities P, Q, and R, each with different dimensions. We need to evaluate the combinations given to see which ones cannot yield a meaningful result.

### Step 2: Analyze Each Combination
1. **Combination A: P - Q**
- Since P and Q have different dimensions, we cannot subtract them. The operation of subtraction requires that both quantities have the same dimensions.
- **Conclusion**: This combination is not meaningful.

2. **Combination B: P * Q - R**
- Here, we are multiplying P and Q. The result of multiplying two quantities can have a different dimension than R. We cannot definitively say this combination is not meaningful without knowing the specific dimensions.
- **Conclusion**: This combination can potentially be meaningful.

3. **Combination C: P + Q**
- Similar to Combination A, since P and Q have different dimensions, we cannot add them. The addition operation also requires that both quantities have the same dimensions.
- **Conclusion**: This combination is not meaningful.

4. **Combination D: P * R - Q^2 / R**
- In this case, we are looking at a more complex expression. The multiplication and division of quantities can yield a meaningful result if the dimensions are compatible. We cannot conclude definitively without specific dimensions.
- **Conclusion**: This combination can potentially be meaningful.

### Step 3: Final Evaluation
From our analysis:
- Combinations A (P - Q) and C (P + Q) are not meaningful because they involve operations (subtraction and addition) between quantities of different dimensions.
- Combinations B (P * Q - R) and D (P * R - Q^2 / R) could potentially be meaningful depending on the specific dimensions.

### Final Answer
The combinations that can never be a meaningful quantity are:
- **A (P - Q)**
- **C (P + Q)**