Assertion : A solid is floating in a liquid of density `rho_(1)`. When the solid melts its density becomes `rho_(2)` in liquid state. If `rho_(1) gt rho_(2)` level of liquids will increase after melting.
Reason : In liquid state volume always increases after a solid melts.

Options

A. If both Assertion and Reason are correct and Reason is the correct explanation of Assertion.
B. If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. If Assertion is true but Reason is false.
D. If Assertion is false but Reason is true.

Answer

To analyze the assertion and reason provided in the question, we will break down the problem step by step.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
- The assertion states that a solid is floating in a liquid of density \( \rho_1 \). When the solid melts, its density becomes \( \rho_2 \) in the liquid state. If \( \rho_1 > \rho_2 \), the level of the liquid will increase after melting.
- This means that the solid is less dense than the liquid when it melts, which implies that the solid will float in the liquid.

2. **Analyzing the Situation**:
- Initially, the solid is floating, which means it displaces a volume of liquid equal to the weight of the solid. According to Archimedes' principle, the weight of the liquid displaced is equal to the weight of the solid.
- When the solid melts, it turns into a liquid with density \( \rho_2 \). Since \( \rho_1 > \rho_2 \), the density of the molten solid is less than that of the liquid.

3. **Effect of Melting on Liquid Level**:
- When the solid melts, it occupies a volume that is less than or equal to the volume it displaced while solid. However, because the density of the molten solid is less than that of the liquid, it will still float.
- The volume of the molten solid will be less than the volume displaced when it was solid, leading to an increase in the liquid level.

4. **Conclusion on Assertion**:
- Therefore, the assertion is true: the level of the liquid will indeed increase after the solid melts.

5. **Understanding the Reason**:
- The reason states that in a liquid state, the volume always increases after a solid melts.
- This statement is not universally true. While many solids do expand upon melting, there are exceptions (e.g., ice to water). Therefore, the reason is false.

6. **Final Evaluation**:
- Since the assertion is true and the reason is false, the correct conclusion is that the assertion is true but the reason is false.

### Final Answer:
- The correct option is **C**: Assertion is true, Reason is false.