In which of the following cases, the centre of mass of a metallic rod is definitely not at its centre?
I. The density of the metallic rod continuously decreases from left to right.
II. The density of the metallic rod continually increases from left to right.
III. The density of the metallic rod decreases from left to right up to the centre and then increases.
IV. The density of the metallic rod increases from left to right up to the centre and then decreases.

The center of mass (CoM) of a rod is located at its geometric center if the mass distribution is symmetric about that center. For a rod of length L, if the linear density λ(x) satisfies λ(x) = λ(L - x), the CoM will be at L/2.

• Statements I and II: In these cases, the density is strictly monotonic (always decreasing or always increasing). This creates an asymmetric mass distribution where one half of the rod is heavier than the other. Therefore, the CoM definitely cannot be at the center.
• Statements III and IV: These describe density profiles that change direction at the center (e.g., a V-shape or an inverted V-shape). If these profiles are symmetric about the midpoint, the CoM will be at the center. Since they can be at the center, they are not "definitely not" at the center.

Thus, only I and II satisfy the condition.