Two identical beakers with negligible thermal expansion are filled with water to the same level at `4^(@)C`. If one says `A` is heated while the other says `B` is colled, then:

Options

A. water level in `A` must rise
B. water level in `B` must rise
C. water level in `A` must fall
D. water level in `B` must fall

Answer

To solve the problem, we need to analyze the behavior of water in two identical beakers when one is heated and the other is cooled.

### Step-by-Step Solution:

1. **Understanding Initial Conditions**:
- We have two identical beakers, A and B, filled with water at the same level at a temperature of \(4^\circ C\).

2. **Heating Beaker A**:
- When beaker A is heated, the temperature of the water in A increases. As the temperature of water increases, its density decreases.

3. **Effect of Heating on Volume**:
- The relationship between density (\(\rho\)), mass (m), and volume (V) is given by the formula:
\[ \rho = \frac{m}{V} \]
- Since the mass of water remains constant, a decrease in density implies an increase in volume. Therefore, when beaker A is heated, the volume of water in A increases.

4. **Cooling Beaker B**:
- Simultaneously, when beaker B is cooled, the temperature of the water in B decreases. As the temperature drops below \(4^\circ C\), the density of water also decreases.

5. **Effect of Cooling on Volume**:
- Similar to the heating scenario, the decrease in density due to cooling means that the volume of water in beaker B also increases.

6. **Conclusion on Water Levels**:
- Since both beakers A and B experience an increase in volume due to the changes in temperature, the water level in both beakers will rise.

### Final Answer:
- The water level in beaker A must rise.
- The water level in beaker B must also rise.