Introduction:
To determine the minimum value of the coefficient of friction required for a disc to roll, we need to consider the forces acting on the disc and the conditions for rolling without slipping.

Conditions for Rolling Without Slipping:
For a disc to roll without slipping, the following conditions must be met:
1. The net force acting on the disc must be zero.
2. The torque about the center of mass of the disc must be zero.
3. The frictional force between the disc and the surface must provide sufficient torque to prevent slipping.

Analysis:
Let's consider the forces acting on the disc when a force F is applied at the center:
1. Gravitational force (mg) acting vertically downward.
2. Normal force (N) exerted by the surface in the upward direction.
3. Applied force (F) acting horizontally at the center.
4. Frictional force (f) acting in the opposite direction of motion.

Net Force:
Since the net force acting on the disc must be zero for rolling without slipping, we have:
F - f = 0
=> f = F

Torque about the Center of Mass:
To ensure the torque about the center of mass is zero, the frictional force must provide sufficient torque. The torque due to the applied force is zero since it acts through the center of mass.

Minimum Value of Coefficient of Friction:
To find the minimum value of the coefficient of friction, we need to determine the maximum possible frictional force. This occurs when the disc is on the verge of slipping.

The maximum frictional force (f_max) can be calculated as:
f_max = μN

Since the normal force (N) is equal to the weight of the disc (mg), we have:
f_max = μmg

Substituting the value of f from the net force equation, we get:
F = μmg

Solving for the coefficient of friction (μ), we have:
μ = F/mg

Answer:
The minimum value of the coefficient of friction required for rolling is given by:
μ_min = F/mg

Comparing the answer choices:
(a) F/2Mg
(b) F/3Mg
(c) 2F/5Mg
(d) 2F/7Mg

We can see that the correct answer is (a) F/2Mg.

Summary:
When a force F is applied at the center of a disc of mass M, the minimum value of the coefficient of friction required for rolling is F/2Mg. This ensures that the net force is zero and the torque about the center of mass is also zero, allowing the disc to roll without slipping.