$A$ student focused the image of a candle flame on a white screen using a convex lens. He noted down the positions of the candle,screen,and lens as follows:
Position of candle $= 12.0 \, cm$
Position of convex lens $= 50.0 \, cm$
Position of the screen $= 88.0 \, cm$
$(i)$ What is the focal length of the convex lens?
$(ii)$ Where will the image be formed if he shifts the candle towards the lens to a position of $31.0 \, cm$?

Where is the object placed to observe it through a compound microscope?

Where should an object be placed in front of a concave mirror to obtain its image at the same position?

The figure below shows an object $AB$ placed on the principal axis of a lens $L$. The two foci of the lens are $F_1$ and $F_2$. The image formed by the lens is erect,virtual and diminished. Copy the diagram and answer the following questions: $(i)$ Draw the outline of the lens $(L)$ used. $(ii)$ Draw a ray from $B$ passing through $O$. Show the ray after refraction by the lens. $(iii)$ Draw a ray of light starting from $B$,which after passing parallel to the principal axis,is incident on the lens and emerges after refraction from it. $(iv)$ Locate the final image formed.

The velocity of light in air is $3 \times 10^{8} \, m \, s^{-1}$. What is the velocity of light in water? (The absolute refractive index of water is $\frac{4}{3}$.)

Write down four important characteristics of the image formed by a plane mirror.