Final answer:
The assertion that the radius of curvature is double the focal length is correct, but the reason given, which claims that the focal length changes when the medium changes, is false. Hence, the correct answer is option C.
Step-by-step explanation:
The radius of curvature of a spherical mirror is indeed double the focal length of the mirror in the case of a spherical mirror under the small-angle approximation.
According to the law of reflection and under the small-angle condition, for a concave spherical mirror, the focal length (f) is half of its radius of curvature (R), evident from the relationship R = 2f.
However, the reason provided that a concave mirror of focal length f in air used in a medium of refractive index 2 changes its focal length to 2f is incorrect, as the focal length of a mirror is not affected by the medium in which it is used; instead, it is a property dependent on the mirror's curvature.
Therefore, the answer to the student's question is option C: Assertion is true but reason is false.