The given problem involves a thin plano-convex lens with a silvered plane surface. We need to determine the focal length of the lens when it behaves like a concave mirror.

1. Understanding the Given Data:
- Radius of curvature of the curved surface of the lens = 10 cm.
- Refractive index of the lens material = 1.5.

2. Behavior of a Plano-Convex Lens with a Silvered Plane Surface:
When the plane surface of a lens is silvered, it acts as a mirror. The behavior of the lens changes from being a converging lens to that of a concave mirror. The focal length of the lens as a concave mirror can be determined using the lens-maker's formula.

3. Lens-Maker's Formula:
The lens-maker's formula relates the focal length of a lens with its refractive index and the radii of curvature of its surfaces. For a thin lens, the formula can be expressed as:

\(\frac{1}{f} = (n - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)\)

where:
- f is the focal length of the lens.
- n is the refractive index of the lens material.
- R1 is the radius of curvature of the first surface.
- R2 is the radius of curvature of the second surface.

4. Applying the Lens-Maker's Formula:
In the given problem, the lens has a plano-convex shape with one curved surface and one plane surface. The radius of curvature of the curved surface is given as 10 cm. The radius of curvature of the plane surface is considered infinite (∞) since it is silvered and behaves as a mirror.

We can substitute the values into the lens-maker's formula:

\(\frac{1}{f} = (1.5 - 1)\left(\frac{1}{10} - \frac{1}{\infty}\right)\)

Since \(\frac{1}{\infty}\) is approximately zero, the equation simplifies to:

\(\frac{1}{f} = 0.5 \times \frac{1}{10}\)

Simplifying further, we get:

\(\frac{1}{f} = \frac{0.5}{10}\)

\(\frac{1}{f} = \frac{1}{20}\)

Comparing the equation with the lens-maker's formula, we can see that the focal length of the lens as a concave mirror is 20 cm.

5. Answer:
Therefore, the correct option is (c) 20 cm.