Question

The focal length of a concave mirror in air is f. When it is completely immersed in water will there be any change in its focal length or not? Why?

Answer 1

Focal length of  a convex mirror or  a concave mirror does not change when it is immersed in any medium.  Basically the light rays emanate from the same medium and are reflected back in to the same medium.  

The angle of reflection is same as angle of incidence.  The radius R of curvature of the mirror decides the focal length  f = R/2.  Focal length depends purely on the shape of mirror.

Only in case of refraction, the focal length depends on medium.