A, B, C and D are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation `AD= C 1n(BD)` holds true. Then which of the combination is not a meaningful quantity :-

Options

A. `A^(2)-B^(2)C^(2)`
B. `C/(BD)-(AD^(2))/C`
C. `((A-C))/D`
D. `A/B-C`

Answer

To determine which combination of the physical quantities A, B, C, and D is not meaningful based on the equation \( AD = C \ln(BD) \), we will analyze the dimensions of each term step by step.

### Step-by-Step Solution:

1. **Understand the Equation**:
We start with the equation \( AD = C \ln(BD) \). Here, \( A \), \( B \), \( C \), and \( D \) are physical quantities with different dimensions.

2. **Natural Logarithm**:
The argument of the natural logarithm must be dimensionless. Therefore, \( BD \) must have dimensions that cancel out to give a dimensionless quantity. This implies:
\[ [B][D] = 1 \quad \text{(dimensionless)} \]
Thus, we can express the dimension of \( B \) in terms of \( D \):
\[ [B] = \frac{1}{[D]} \]

3. **Dimensions of the Left Side**:
The left side of the equation is \( AD \). Therefore, the dimension of \( AD \) is:
\[ [A][D] \]

4. **Dimensions of the Right Side**:
The right side is \( C \ln(BD) \). Since \( \ln(BD) \) is dimensionless, the dimension of the right side is simply the dimension of \( C \):
\[ [C] \]

5. **Equating Dimensions**:
From the equation \( AD = C \ln(BD) \), we can equate the dimensions:
\[ [A][D] = [C] \]

6. **Analyzing Options**:
Now we will analyze each option to see if they yield meaningful dimensions.

**Option 1**: \( A^2 - B^2C^2 \)
- Dimension of \( A^2 \): \( [A]^2 \)
- Dimension of \( B^2C^2 \): \( \left(\frac{1}{[D]}\right)^2 [C]^2 \)
- Since \( [C] = [A][D] \), substituting gives \( [C]^2 = [A]^2[D]^2 \).
- Thus, \( B^2C^2 \) has dimension \( \frac{[A]^2[D]^2}{[D]^2} = [A]^2 \).
- Therefore, \( A^2 - B^2C^2 \) is meaningful.

**Option 2**: \( \frac{C}{BD} - \frac{AD^2}{C} \)
- Dimension of \( \frac{C}{BD} = \frac{[C]}{[B][D]} = \frac{[C]}{\frac{1}{[D]}[D]} = [C] \).
- Dimension of \( \frac{AD^2}{C} = \frac{[A][D^2]}{[C]} \).
- Since \( [C] = [A][D] \), substituting gives \( \frac{[A][D^2]}{[A][D]} = [D] \).
- The dimensions do not match, thus this option is not meaningful.

**Option 3**: \( \frac{A}{C} - \frac{A}{C} \)
- Both terms are \( \frac{[A]}{[C]} \), hence they are the same.
- This option is meaningful.

**Option 4**: \( \frac{A}{B} - C \)
- Dimension of \( \frac{A}{B} = [A][D] \) (since \( [B] = \frac{1}{[D]} \)).
- Dimension of \( C \) is \( [C] = [A][D] \).
- Both terms are equal, hence this option is meaningful.

### Conclusion:
The combination that is not meaningful is **Option 2**: \( \frac{C}{BD} - \frac{AD^2}{C} \).