Describe a spherical mirror.

$(a)$ 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.

Give the mathematical expression for the power of a lens.

For what position of the object does a concave mirror form a real and diminished image?

How is the refractive index of a medium related to the speed of light? Obtain an expression for the refractive index of a medium with respect to another in terms of the speed of light in these two media.

$A$ concave lens made of a material of refractive index $n_{1}$ is kept in a medium of refractive index $n_{2}$. $A$ parallel beam of light is incident on the lens. Complete the path of rays of light emerging from the concave lens if
$(i)$ $n_{1} > n_{2}$ $(ii)$ $n_{1}=n_{2}$ $(iii)$ $n_{1} < n_{2}$