A physical quantity `x` depends on quantities `y and z` as follows : ` x = Ay + B tan ( C z)`, where `A , B and C` are constants. Which of the followings do not have the same dimensions?

Options

A. ` x and B`
B. `C and z^(-1)`
C. `y and B//A`
D. ` x and A`

Explanation

To solve the problem, we need to analyze the given equation and the dimensions of the quantities involved. The equation is:

\[ x = Ay + B \tan(Cz) \]

Where \( A \), \( B \), and \( C \) are constants, and \( y \) and \( z \) are physical quantities.

### Step 1: Identify the dimensions of the quantities involved

1. **Identify the dimensions of \( x \)**:
- The left side of the equation is \( x \), which has its own dimension, say \([X]\).

2. **Identify the dimensions of \( y \)**:
- Let the dimension of \( y \) be \([Y]\).

3. **Identify the dimensions of \( z \)**:
- Let the dimension of \( z \) be \([Z]\).

### Step 2: Analyze the equation

1. **For the term \( Ay \)**:
- The term \( Ay \) must have the same dimension as \( x \).
- Therefore, the dimension of \( A \) must be such that:
\[ [A][Y] = [X] \implies [A] = \frac{[X]}{[Y]} \]

2. **For the term \( B \tan(Cz) \)**:
- The function \( \tan \) is dimensionless, which means \( Cz \) must also be dimensionless.
- Therefore, \( C \) must have dimensions that cancel out the dimensions of \( z \):
\[ [C][Z] = 1 \implies [C] = \frac{1}{[Z]} \]
- Since \( B \tan(Cz) \) must also have the same dimension as \( x \):
\[ [B] = [X] \]

### Step 3: Compare dimensions

Now we have the following dimensions:
- \( [x] = [X] \)
- \( [A] = \frac{[X]}{[Y]} \)
- \( [B] = [X] \)
- \( [C] = \frac{1}{[Z]} \)

### Step 4: Determine which quantities do not have the same dimensions

1. **Comparing \( x \) and \( A \)**:
- \( x \) has dimension \([X]\) while \( A \) has dimension \(\frac{[X]}{[Y]}\). They do not have the same dimensions.

2. **Comparing \( x \) and \( B \)**:
- Both \( x \) and \( B \) have the same dimension \([X]\).

3. **Comparing \( A \) and \( B \)**:
- \( A \) has dimension \(\frac{[X]}{[Y]}\) while \( B \) has dimension \([X]\). They do not have the same dimensions.

4. **Comparing \( A \) and \( C \)**:
- \( A \) has dimension \(\frac{[X]}{[Y]}\) while \( C \) has dimension \(\frac{1}{[Z]}\). They do not have the same dimensions.

### Conclusion

The quantities that do not have the same dimensions are \( A \), \( B \), and \( C \). However, since the question asks for which of the following do not have the same dimensions, the answer is:

**The correct option is \( B \)**.