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. `PQ//R`
D. `(R+Q)//P`

Explanation

To solve the problem of determining which combinations of the physical quantities P, Q, and R can never be a meaningful quantity, we need to analyze the dimensions of these quantities. Since P, Q, and R have different dimensions, we will evaluate each option provided.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given three physical quantities P, Q, and R, which have different dimensions. We need to identify which combinations of these quantities cannot yield a meaningful physical quantity.

2. **Option A: P/Q**:
- When dividing two physical quantities, the dimensions must be compatible.
- Since P and Q have different dimensions, the division P/Q will result in a quantity with dimensions that do not correspond to any meaningful physical quantity.
- **Conclusion**: P/Q cannot be a meaningful quantity.

3. **Option B: PQ - R**:
- Here, we are multiplying P and Q, which will yield a new quantity with its own dimension.
- However, to subtract R from PQ, the dimensions of PQ must equal the dimensions of R.
- Since P and Q have different dimensions, we cannot definitively say that PQ will have the same dimension as R.
- **Conclusion**: This combination may or may not be meaningful, depending on the specific dimensions.

4. **Option C: P + Q**:
- Similar to option A, when adding two physical quantities, their dimensions must be the same.
- Since P and Q have different dimensions, P + Q cannot yield a meaningful quantity.
- **Conclusion**: P + Q cannot be a meaningful quantity.

5. **Option D: PR - Q²/R**:
- In this case, we are multiplying P and R, and then we have Q squared divided by R.
- For this to be meaningful, the dimensions of PR must equal the dimensions of Q²/R.
- This can happen if the dimensions align correctly, but we cannot definitively say it will not be meaningful.
- **Conclusion**: This combination may or may not be meaningful, depending on the specific dimensions.

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