$A$ candle flame of $3\, cm$ is placed at $3\, m$ from a wall. $A$ concave mirror is kept at a distance $x$ from the wall in such a way that the image of the flame on the wall is $9\, cm$. Then $x$ is......$cm$.

Suppose that the lower half of the concave mirror's reflecting surface in the figure is covered with an opaque (non-reflective) material. What effect will this have on the image of an object placed in front of the mirror?

An object is placed at a distance of $40 \ cm$ in front of a concave mirror of focal length $20 \ cm$. The image produced is

An object placed in front of a concave mirror at a distance of $x \ cm$ from the pole gives a $3$ times magnified real image. If it is moved to a distance of $(x+5) \ cm$, the magnification of the image becomes $2$. The focal length of the mirror is: (in $cm$)

$A$ concave mirror of focal length $10$ cm forms an image which is double the size of the object when the object is placed at two different positions. The distance between the two positions of the object is . . . . . . cm.

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