To find the total surface area of the remaining solid, we need to calculate the surface area of both the cylinder and the conical cavity and subtract the area of the cavity from the total area of the cylinder.

1. Surface Area of the Cylinder:
The formula to calculate the surface area of a cylinder is given by:
Surface Area = 2πr² + 2πrh
where r is the radius and h is the height of the cylinder.

Since the diameter is given as 10 cm, the radius of the cylinder is half of the diameter, i.e., 5 cm. And the height is given as 12 cm.

Substituting these values into the formula, we get:
Surface Area of the Cylinder = 2π(5)² + 2π(5)(12)
= 2π(25) + 2π(60)
= 50π + 120π
= 170π

2. Surface Area of the Conical Cavity:
The formula to calculate the surface area of a cone is given by:
Surface Area = πr(r + √(r² + h²))
where r is the radius and h is the height of the cone.

Since the diameter is given as 10 cm, the radius of the cone is half of the diameter, i.e., 5 cm. And the height is given as 12 cm.

Substituting these values into the formula, we get:
Surface Area of the Conical Cavity = π(5)(5 + √(5² + 12²))
= π(5)(5 + √(25 + 144))
= π(5)(5 + √(169))
= π(5)(5 + 13)
= 18π

3. Total Surface Area of the Remaining Solid:
To find the total surface area of the remaining solid, we subtract the area of the conical cavity from the surface area of the cylinder.
Total Surface Area = Surface Area of Cylinder - Surface Area of Conical Cavity
= 170π - 18π
= 152π

Now, to find the numerical value of the total surface area, we can approximate the value of π to be 3.14.
Total Surface Area ≈ 152 × 3.14
≈ 477.28 cm²

Therefore, the total surface area of the remaining solid is approximately 477.28 cm², which is closest to option A (660 cm²) in the given options.