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.

To determine which pair of quantities do not share the same dimensions, we can apply the principle of homogeneity of dimensions. According to this principle, the terms on both sides of a physical equation, as well as terms added or subtracted, must have identical dimensional formulas.

Let us analyze the given equation:
$$x=Ay+Btan\left(Cz\right)$$

1. Dimensions of C and z^-1:
Trigonometric arguments and functions are always dimensionless. Since $tan\left(Cz\right)$ is a trigonometric function, its argument $Cz$ must be dimensionless:
$$\left[Cz\right]=\left[M^0{L}^0{T}^0\right]=1$$
This allows us to find the relationship between the dimensions of $C$ and $z$:
$$\left[C\right]=\left[\frac{1}{z}\right]=\left[z^{-1}\right]$$
Thus, C and z^-1 have the same dimensions.

2. Comparing x and B:
Since $tan\left(Cz\right)$ is a dimensionless quantity, the dimensions of the term $Btan\left(Cz\right)$ are determined solely by the dimensions of $B$:
$$[Btan(Cz\left)\right]=\left[B\right]$$
Applying the principle of homogeneity, the dimensions of $x$ must match the dimensions of each individual term in the sum:
$$\left[x\right]=\left[B\right]$$
Thus, x and B have the same dimensions.

3. Comparing y and B / A:
From the principle of homogeneity, the dimensions of the two terms on the right-hand side of the equation must be equal:
$$\left[Ay\right]=\left[B\right]$$
Dividing by the dimension of $A$ on both sides:
$$\left[y\right]=\left[\frac{B}{A}\right]$$
Thus, y and B / A have the same dimensions.

4. Comparing x and A:
We established that:
$$\left[x\right]=\left[Ay\right]$$
This means the dimensions of $A$ are:
$$\left[A\right]=\left[\frac{x}{y}\right]$$
Because the dimension of $A$ differs from the dimension of $x$ by a factor of $\left[y\right]$, x and A do not have the same dimensions.