In the figure shown, $O$ is a point object placed in the air on the axis of a spherical surface $APB$ of a glass block. $OA$ is an incident ray. The direction of the refracted ray $AD$ in the glass block is parallel to the principal axis $PC$. $CP=CA=R$ is the radius of the spherical surface. Use the formula $\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$ to show that the object distance $PO=u=\raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$(n-1)$}\right.$; where $n_2=n$ and $n_1=1$.

A convex lens $L$ and plane mirror $M$ are arranged as shown in figure. Position of the object pin $O$ is adjusted in such a way that the inverted image $I$ formed by the lens mirror combinations, coincides with the object pin $O$. Explain how and when this happens.

An equiconvex lens of focal length $f$ and power $P$ is cut into two halves in thickness. What are the focal length and power of each half?

Sunglasses (goggles) have curved surfaces, yet their power is zero, why?

A lens of focal length $f$ forms the image of a luminous object on a screen $m$ times large. Show that the distance of the screen from the lens is $f(1+m)$.