If a parallel beam of light is incident on spherical mirrors, then after reflection, they pass through the focal point. If $F$ is the focal length and $R$ is the radius of curvature of the mirror, then:

$A$ convex lens (of focal length $20\, cm$) and a concave mirror,having their principal axes along the same lines,are kept $80\, cm$ apart from each other. The concave mirror is to the right of the convex lens. When an object is kept at a distance of $30\, cm$ to the left of the convex lens,its image remains at the same position even if the concave mirror is removed. The maximum distance of the object for which this concave mirror,by itself,would produce a virtual image would be.....$cm$.

$A$ person sees his virtual image by holding a mirror very close to the face. When he moves the mirror away from his face,the image becomes inverted. What type of mirror is he using?

$A$ small linear object is placed on the optical axis of a concave mirror. If the distance of the nearest end of the object from the mirror is greater than the radius of curvature,then:

Use the mirror equation to deduce that:
$(a)$ an object placed between $f$ and $2f$ of a concave mirror produces a real image beyond $2f$.
$(b)$ a convex mirror always produces a virtual image independent of the location of the object.
$(c)$ the virtual image produced by a convex mirror is always diminished in size and is located between the focus and the pole.
$(d)$ an object placed between the pole and focus of a concave mirror produces a virtual and enlarged image.

Write the relationship between the radius of curvature and the focal length for a spherical mirror.