there is a rod AB of uniform cross section and its centre is C. In which of the following case(s) centre of mass of the rod may lie at C.

Options

A. Density of the material of rod increases continuously from A to B.
B. Density of the material of rod increases continuously from B to A.
C. Density of the material of rod increases from A to C and then decreases from C to B
D. Density of the material of rod decreases from A to C and then increases from C to B.

Explanation

To determine in which cases the center of mass of a rod lies at its geometric center (point C), we need to analyze the mass distribution around point C for each of the given scenarios. The center of mass will lie at point C if the mass distribution is symmetrical on both sides of this point.

### Step-by-Step Solution:

1. **Understanding the Rod and Center of Mass**:
- We have a rod AB with a uniform cross-section and a center point C. The center of mass will be at point C if the mass distribution is symmetrical around this point.

2. **Analyzing Option A**:
- **Statement**: Density of the material of the rod increases continuously from A to B.
- **Analysis**: The density is lowest at point A and highest at point B. This means the mass distribution is not symmetrical around point C. Therefore, the center of mass will not lie at C.
- **Conclusion**: **Not valid**.

3. **Analyzing Option B**:
- **Statement**: Density of the material of the rod increases continuously from B to A.
- **Analysis**: Here, the density is lowest at point B and highest at point A. Again, the mass distribution is not symmetrical around point C. Thus, the center of mass will not lie at C.
- **Conclusion**: **Not valid**.

4. **Analyzing Option C**:
- **Statement**: Density of material of the rod increases from A to C and then decreases from C to B.
- **Analysis**: The density increases as we move from A to C, reaching a maximum at C, and then decreases as we move from C to B. This creates a symmetrical mass distribution around point C.
- **Conclusion**: **Valid**.

5. **Analyzing Option D**:
- **Statement**: Density of the material of the rod decreases from A to C, then increases from C to B.
- **Analysis**: The density is highest at point A, decreases to a minimum at C, and then increases again towards point B. This also results in a symmetrical mass distribution around point C.
- **Conclusion**: **Valid**.

### Final Answer:
The center of mass of the rod may lie at point C in the following cases:
- **Option C**: Density increases from A to C and decreases from C to B.
- **Option D**: Density decreases from A to C and increases from C to B.