$A$ convex lens forms a real and inverted image of a needle at a distance of $50\, cm$ from the lens. Where should the needle be placed in front of the convex lens so that this image is of the same size as the object? Also, find the power of the lens.

$(a)$ For the same angle of incidence $45^{\circ}$,the angle of refraction in two transparent media $I$ and $II$ is $20^{\circ}$ and $30^{\circ}$ respectively. Out of $I$ and $II$,which medium is optically denser and why?
$(b)$ Light enters from air to diamond,which has a refractive index of $2.42$. Calculate the speed of light in diamond,if the speed of light in air is $3.00 \times 10^{8} \text{ m s}^{-1}$.

Draw a ray diagram showing the path of rays of light when it enters with oblique incidence: $(i)$ from air into water; $(ii)$ from water into air.

$(a)$ $A$ thin converging lens forms a:
$(i)$ Real magnified image.
$(ii)$ Virtual magnified image of an object placed in front of it.
Write the positions of the objects in each case.
$(b)$ Draw labelled ray diagrams to show the image formation in each case.
$(c)$ How will the following be affected on cutting this lens into two halves along the principal axis?
$(i)$ Focal length,
$(ii)$ Intensity of the image formed by half lens.

Which of the lenses with focal lengths $10 \, cm$,$20 \, cm$,$25 \, cm$,and $50 \, cm$ has the maximum power (in $, cm$)?

In the case of a convex mirror,if the object is moved away from the surface of the mirror,how do the position and size of the image change?