A rod of length L is positioned along the X-axis between x = 0 and...

The linear density $\rho$ is defined as mass per unit length, so its SI unit is $\text{kg/m}$.

According to the principle of dimensional homogeneity, each term in the equation $\rho = a + bx$ must have the same unit.

Therefore, the unit of $a$ is the same as the unit of $\rho$, which is $\text{kg/m}$.

Similarly, the unit of the term $bx$ must be $\text{kg/m}$. Since $x$ represents distance, its SI unit is $\text{m}$.

Unit of $b = \dfrac{\text{Unit of } \rho}{\text{Unit of } x} = \dfrac{\text{kg/m}}{\text{m}} = \text{kg/m}^2$.

Thus, the SI units of $a$ and $b$ are $\text{kg/m}$ and $\text{kg/m}^2$ respectively.