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 question step by step, we need to analyze the given equation and the dependencies of the physical quantities involved.

### Step 1: Analyze the given equation
The equation provided is:
\[ x = Ay + B \tan(Cz) \]
Here, \( A \), \( B \), and \( C \) are constants, and \( y \) and \( z \) are physical quantities.

### Step 2: Identify the dimensions of each term
1. **Dimension of \( x \)**: Since \( x \) is a physical quantity, it has its own dimension, which we will denote as \([x]\).
2. **Dimension of \( y \)**: Let the dimension of \( y \) be denoted as \([y]\).
3. **Dimension of \( z \)**: Let the dimension of \( z \) be denoted as \([z]\).

### Step 3: Analyze the term \( B \tan(Cz) \)
- The term \( \tan(Cz) \) is a trigonometric function, and it is dimensionless. Therefore, the argument \( Cz \) must also be dimensionless.
- This implies that the dimensions of \( C \) and \( z \) must satisfy:
\[ [C][z] = 1 \]
Thus, we can write:
\[ [C] = [z]^{-1} \]

### Step 4: Analyze the term \( Ay \)
- The term \( Ay \) must also have the same dimension as \( x \):
\[ [x] = [A][y] \]
From this, we can express the dimension of \( A \):
\[ [A] = \frac{[x]}{[y]} \]

### Step 5: Analyze the dimension of \( B \)
- Since \( B \tan(Cz) \) must also have the same dimension as \( x \), we can write:
\[ [x] = [B][\tan(Cz)] \]
Since \( \tan(Cz) \) is dimensionless, we have:
\[ [B] = [x] \]

### Step 6: Compare the dimensions
Now we have:
- \( [A] = \frac{[x]}{[y]} \)
- \( [B] = [x] \)
- \( [C] = [z]^{-1} \)

### Step 7: Determine which dimensions do not match
From the above analysis:
- The dimensions of \( x \) and \( B \) are the same: \([x]\).
- The dimensions of \( A \) and \( x \) are different because \( [A] \) depends on \( [y] \).
- The dimensions of \( C \) and \( z \) are also different since \( [C] \) is the inverse of \( [z] \).

Thus, the quantities that do not have the same dimensions are \( A \), \( x \), and \( B \).

### Conclusion
The correct answer to the question is that the dimensions of \( A \) do not match with those of \( x \) and \( B \).