A

Step 1: At the center of a thin spherical shell, we can apply the shell theorem which states that the gravitational field inside a uniformly distributed spherical shell is zero. Therefore, at the center of the shell,

• Gravitational Field (E) = 0
• Gravitational Potential (V) ≠ 0 (as it is constant throughout the shell).

Step 2: At the center of a solid sphere, by using the formula for gravitational field within the solid sphere (which increases linearly from zero at the center to a maximum at the surface),

• Gravitational Field (E) = 0 at the center.
• Gravitational Potential (V) ≠ 0 (it is maximum here).

Step 3: For a solid sphere with a non-concentric spherical cavity, the gravitational potential at any point inside the cavity (including the center) is still non-zero, but the gravitational field could be non-zero as well depending on the positions of mass. Thus,

• Gravitational Field (E) ≠ 0 because the cavity disrupts the symmetry.
• Gravitational Potential (V) ≠ 0 when considering the influence of mass outside the cavity and at least some contribution from the mass that is present.

Step 4: For the center of the line joining two point masses of equal magnitude, by symmetry, the gravitational field due to both masses cancels out:

• Gravitational Field (E) = 0 at the midpoint.
• Gravitational Potential (V) = 0 at the midpoint as the potentials from both masses cancel each other out.

Conclusion: By matching the statements based on our analysis, we have:

• (A) matches with (P): E = 0.
• (B) matches with (R): V ≠ 0
• (C) matches with (Q): E ≠ 0, V ≠ 0.
• (D) matches with (S): V = 0.

Therefore, the correct match for A is P: E = 0.