A

Step 1: Understanding the Problem
We have a body that moves from point A to point B along two different curved paths with friction acting on it. The coefficient of friction is constant for both paths, and the radii of the paths are equal. We need to determine which path allows the body to reach point B with a greater velocity.

Step 2: Analyzing Forces and Motion
As the body moves along the curved surfaces, it experiences gravitational force, normal force, and frictional force. The gravitational potential energy (PE) lost while descending translates into kinetic energy (KE), but some energy is also converted into heat due to friction.

Step 3: Energy Considerations
Let's denote:
- Mass of the body = m
- Height of point A = h
- Frictional force = f = μN (where μ is the coefficient of friction and N is the normal force)

The gravitational potential energy lost when the body descends to point B is given by:
$$ PE = mgh $$
This is converted partly into kinetic energy (KE) and partly lost to friction:
$$ KE = PE - ext{Friction Energy} $$

Step 4: Path Comparison
For path 1, the body might maintain more consistent acceleration due to a smoother gradient at different heights, while path 2 may involve more abrupt changes in trajectory which can create greater normal forces and, subsequently, friction. Hence, it can be inferred that energy losses due to friction will be higher in the path with a more complicated curve.

Step 5: Conclusion
Thus, the body will have greater velocity at point B when taking the more gently sloped route (path 1). Therefore, the answer is option A.