Assertion: When a solid sphere rolls down a sufficiently rough inclined surface then its mechanical energy remains constant.
Reason: Work done by the friction in pure rolling is zero.

Options

A. If both assertion and reason are correct and reason is the correct explanation of the assertion.
B. If both assertion and reason are correct but reason is not the correct explanation of assertion
C. If assertion is correct but reason is incorrect.
D. If assertion is incorrect but reason is correct

Explanation

To analyze the assertion and reason provided in the question, we will break down the concepts involved in the rolling motion of a solid sphere on a rough inclined surface.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
The assertion states that when a solid sphere rolls down a sufficiently rough inclined surface, its mechanical energy remains constant. This implies that the total energy (potential + kinetic) is conserved during the motion.

2. **Understanding the Reason**:
The reason given is that the work done by friction in pure rolling is zero. In pure rolling motion, the point of contact between the sphere and the surface does not slide. Therefore, friction does not do any work on the sphere.

3. **Analyzing Pure Rolling Condition**:
- For a solid sphere rolling without slipping, the condition for pure rolling is that the linear acceleration \( a \) of the center of mass and the angular acceleration \( \alpha \) are related by the equation:
\[ a = r \alpha \]
where \( r \) is the radius of the sphere.

4. **Friction in Pure Rolling**:
- In pure rolling, the friction force acts at the contact point but does not cause any displacement at that point. Hence, the work done by friction is:
\[ \text{Work} = \text{Force} \times \text{Displacement} = f \times 0 = 0 \]
- Since the work done by friction is zero, it does not contribute to energy loss.

5. **Mechanical Energy Conservation**:
- As the sphere rolls down the incline, it converts potential energy into kinetic energy. The potential energy at the height \( h \) is given by \( PE = mgh \), and the kinetic energy when it reaches the bottom is given by \( KE = \frac{1}{2} mv^2 + \frac{1}{2} I \omega^2 \).
- Since there is no work done by friction, the total mechanical energy remains constant:
\[ mgh = \frac{1}{2} mv^2 + \frac{1}{2} I \omega^2 \]
- For a solid sphere, \( I = \frac{2}{5}mr^2 \) and \( \omega = \frac{v}{r} \), which confirms that the mechanical energy is conserved.

6. **Conclusion**:
- Both the assertion and the reason are correct. The mechanical energy remains constant because the work done by friction in pure rolling is zero.

### Final Answer:
Both the assertion and reason are correct. The assertion is true because the mechanical energy remains constant when a solid sphere rolls down a sufficiently rough inclined surface, and the reason is correct as the work done by friction in pure rolling is indeed zero.