Question

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?
\[\begin{align}
  & \text{A}\text{. }x\text{ and }B \\
 & \text{B}\text{. }C\text{ and }{{z}^{-1}} \\
 & \text{C}\text{. }y\text{ and }\dfrac{B}{A} \\
 & \text{D}\text{. }x\text{ and }A \\
\end{align}\]

Answer 1

Hint: We will use dimensional analysis for the given expression and by applying law of homogeneity of dimension we can find the solution for the answer. We can first apply the law on the expression and break it into part to solve further. There can also be dimensionless physical quantities.

Complete step by step answer:
According to the law of homogeneity of dimension, both sides of the equation should have the same dimension and the physical quantities having the same dimensions can only be added and subtracted.
From the above law the right hand side of the expression and the left hand side should have the same dimension. Therefore, x and Ay+Btan(Cz) should have the same dimension.
Now only quantities which have the same dimension can be added. Therefore, Ay and Btan(Cz) will have the same dimensions.
Now x, Ay and Btan(Cz) have the same dimensions. Just to show the same dimensions we can write
\[x=Ay\text{ and }x=B\tan (Cz)\]
Where x and Ay have the same dimensions and x and Btan(Cz) also have the same dimensions.
As tan(Cz) is dimensionless quantity as it is a trigonometric function. Therefore, B and x will have the same dimension and hence option A is incorrect.
Cz is dimensionless because it is radian or a degree which is dimensionless. Hence we can write
\[\begin{align}
  & Cz=1 \\
 & C=\dfrac{1}{z} \\
 & C={{z}^{-1}} \\
\end{align}\]
Which shows that C and \[{{z}^{-1}}\] have the same dimension. Hence option B is also incorrect.
As x have same dimension as Ay and B, therefore B and Ay will also have same dimension and we write it as
\[\begin{align}
  & Ay=B \\
 & y=\dfrac{B}{A} \\
\end{align}\]
Which shows that y and \[\dfrac{B}{A}\] have the same dimension and so the option C is also incorrect.
As y is not dimensionless therefore x and A cannot have the same dimensions. Therefore the correct option is D.

Note:
Here the equal sign is used to show that the light hand side quantity has the same dimension as the right hand side quantity. It doesn’t mean that they have the same numerical value or that they are numerically equal. Same is for Cz, It doesn’t have a value of one, it's just assigned to show that the quantity is dimensionless.