The minimum distance between the object and its real image for a concave mirror is

An object is placed at a distance of $40 \,cm$ from a concave mirror of focal length $20 \,cm$. The image formed is . . . . . . .

$A$ concave mirror of focal length $60 \ cm$ forms a real image of size $5$ times the size of a real object. Find the distance between the mirror and the object. (in $cm$)

When an object is placed at a distance of $25 \ cm$ from a concave mirror,the magnification is $m_1$. The object is moved $15 \ cm$ farther away with respect to the earlier position and the magnification becomes $m_2$. If $m_1/m_2 = 4$,the focal length of the mirror is $....... \ cm$ (Assume image is real and $m_1, m_2$ are numerical values).

Derive the relationship between the focal length $(f)$ and the radius of curvature $(R)$ for a spherical mirror.

The focal length $f$ is related to the radius of curvature $R$ of a spherical convex mirror by