Even if a physical quantity depends upon three quantities, out of which two ae dimensionally same, then the formula cannot be derived by the method of dimensions. This statement

Options

A. May be true
B. May be false
C. Must be true
D. Must be false

Answer

To analyze the statement "Even if a physical quantity depends upon three quantities, out of which two are dimensionally same, then the formula cannot be derived by the method of dimensions," we can break it down step by step.

### Step-by-Step Solution:

1. **Understanding the Principle of Dimensional Homogeneity**:
The principle of dimensional homogeneity states that in any physical equation, the dimensions on both sides must be the same. This principle is fundamental in dimensional analysis.

2. **Identifying the Physical Quantities**:
Let's denote the three quantities as A, B, and C. According to the statement, two of these quantities (let's say A and B) are dimensionally the same, meaning they have the same dimensions (for example, both could be in terms of length [L]).

3. **Setting Up the Dimensional Equation**:
If we want to express a physical quantity (let's call it Q) in terms of A, B, and C, we can write:
\[ Q = k \cdot A^m \cdot B^n \cdot C^p \]
where k is a dimensionless constant, and m, n, and p are the powers to which A, B, and C are raised.

4. **Substituting Dimensions**:
Since A and B are dimensionally the same, we can replace B with A in the equation:
\[ Q = k \cdot A^m \cdot A^n \cdot C^p = k \cdot A^{m+n} \cdot C^p \]
This shows that we can still express Q in terms of A and C, despite A and B being dimensionally identical.

5. **Applying the Method of Dimensions**:
Now, we can analyze the dimensions of Q. If Q depends on the dimensions of A and C, we can set up an equation based on their dimensions. This allows us to derive relationships between the quantities involved.

6. **Conclusion**:
Therefore, even if two quantities are dimensionally the same, we can still derive a formula for the physical quantity Q using dimensional analysis, provided we have enough independent dimensions to work with. Hence, the statement is **false**.

### Final Answer:
The statement is **false**.