An object is placed $20 \ cm$ in front of a $4 \ cm$ thick plane mirror. The image of the object is finally formed at $45 \ cm$ from the object itself. The refractive index of the material of the unpolished side of the mirror is (considering near normal incidence):

$A$ fish and a bird are moving as shown in the figure. Find the velocity of the bird as observed by the fish.

$A$ coin is placed at the bottom of an empty beaker and is focused by a microscope. When water is poured into the beaker to a height of $10 \, cm$,how much must the microscope be moved to focus it again?

$A$ glass tumbler having an inner depth of $17.5 \, cm$ is kept on a table. $A$ student starts pouring water $(\mu = 4/3)$ into it while looking at the surface of water from above. When he feels that the tumbler is half-filled,he stops pouring water. Up to what height,the tumbler is actually filled? (In $cm$)

$A$ light ray is incident on a glass slab of thickness $4 \sqrt{3} \text{ cm}$ and refractive index $\sqrt{2}$. The angle of incidence is equal to the critical angle for the glass slab with air. The lateral displacement of the ray after passing through the glass slab is . . . . . . $\text{cm}$. (Given $\sin 15^{\circ} = 0.25$)

$A$ bird is flying $3\, m$ above the surface of water. If the bird is diving vertically down with speed $= 6\, m/s$,its apparent velocity as seen by a stationary fish underwater is $m/s$.