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(D) Given the equation: $x = Ay + B 	an(Cz)$.According to the principle of dimensional homogeneity,the dimensions of each term in an equation must be

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$x$ and $A$

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(D) Given the equation: $x = Ay + B 	an(Cz)$.According to the principle of dimensional homogeneity,the dimensions of each term in an equation must be the same.$1$. For the term $B 	an(Cz)$,the argument of the trigonometric function $(Cz)$ must be dimensionless. Therefore,$[Cz] = [M^0 L^0 T^0]$,which implies $[C] = [z^{-1}]$. Thus,$C$ and $z^{-1}$ have the same dimensions.$2$. The dimensions of $x$ must be equal to the dimensions of $Ay$ and $B$. Therefore,$[x] = [Ay] = [B]$. This means $x$ and $B$ have the same dimensions.$3$. From $[x] = [Ay]$,we get $[y] = [x/A]$. Also,from $[x] = [B]$,we get $[A] = [B/x]$. Combining these,$[y] = [x/A] = [B/A]$. Thus,$y$ and $B/A$ have the same dimensions.$4$. Comparing $x$ and $A$: Since $[x] = [Ay]$,the dimension of $A$ is $[x/y]$. Therefore,$x$ and $A$ do not have the same dimensions.Hence,the correct option is $(d)$.

$A$ 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?

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