An object is placed at a distance of $$12 \mathrm{~cm}$$ from a convex lens on its principal axis and a virtual image of certain size is formed. If the object is moved $$4 \mathrm{~cm}$$ away from the lens, a real image of the same size as that of the virtual image is formed. The focal length of the lens in $$\mathrm{cm}$$ is

To a fish under water, viewing obliquely, a fisherman standing on the bank of a lake looks

In a container of height $$21 \mathrm{~cm}$$, certain transparent liquid is taken to a height of $$12 \mathrm{~cm}$$. When seen from above, it appears half filled. The refractive index of the liquid is

Light enters at an angle of incidence in a transparent rod of refractive index '$$n$$'. The least value of '$$n$$' for which the light once entered into it will not leave it through its lateral face whatsoever be the value of the angle of incidence is