$(a)$ Draw a ray diagram in each of the following cases to show the position and nature of the image formed when an object is placed:
$(i)$ Between the focus and the centre of curvature of a concave mirror.
$(ii)$ Between the focus and the pole of a concave mirror.
$(iii)$ Between the centre of curvature and infinity for a concave mirror.
$(b)$ Give mathematical formulae for determining the magnification produced by a spherical mirror. What does $m = -1$ signify? Identify the mirror that can produce it.

(N/A) The ray diagrams are as shown in the provided image.
$(b)$ Magnification $(m)$ is defined as the ratio of the height of the image $(h')$ to the height of the object $(h)$,which is also equal to $-\frac{v}{u}$,where $v$ is the image distance and $u$ is the object distance.
$m = \frac{h'}{h} = -\frac{v}{u}$
$m = -1$ signifies:
$(i)$ The size of the image is equal to the size of the object.
$(ii)$ The negative sign indicates that the image is real and inverted.
This magnification is produced by a concave mirror when the object is placed at the centre of curvature $(C)$.