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

The correct option is x and A.

Let us analyze the dimensions of each term in the given physical equation step-by-step to understand why this option is correct.

The given equation is:
$$x=Ay+Btan\left(Cz\right)$$

According to the principle of dimensional homogeneity, physical quantities added or subtracted must have the same dimensions, and the dimensions of the quantity on the left-hand side must equal the dimensions of each individual term on the right-hand side.

Therefore, we have:
[x] = [Ay] = [B tan(Cz)]

Since trigonometric functions like tan(θ) are dimensionless, the term tan(Cz) is dimensionless. This gives:
[tan(Cz)] = 1
Consequently, the dimension of the term B tan(Cz) is simply the dimension of B:
[B tan(Cz)] = [B]

Equating the dimensions of the terms, we get:
[x] = [B] and [x] = [Ay]

Let us evaluate each of the options based on these dimensional relations:

1. x and B:
As derived above, [x] = [B]. Therefore, x and B have the same dimensions.

2. C and z^-1:
The argument of a trigonometric function must be dimensionless. Thus, the quantity Cz is dimensionless:
[Cz] = 1
This implies that:
[C] = [z^-1]
Therefore, C and z^-1 have the same dimensions.

3. y and B / A:
We know that [x] = [Ay] and [x] = [B]. This means:
[Ay] = [B]
Dividing both sides by the dimension of A, we get:
[y] = [B / A]
Therefore, y and B / A have the same dimensions.

4. x and A:
We have the relation [x] = [Ay]. This shows that the dimension of x is equal to the dimension of A multiplied by the dimension of y. Unless y is a dimensionless quantity (which is not stated), the dimensions of x and A must be different:
[x] ≠ [A]
Therefore, x and A do not have the same dimensions.