A transparent glass cube of length $$24 \mathrm{~cm}$$ has a small air bubble trapped inside. When seen normally through one surface from air outside, its apparent distance is $$10 \mathrm{~cm}$$ from the surface. When seen normally from opposite surface, its apparent distance is $$6 \mathrm{~cm}$$. The distance of the air bubble from first surface is

The angle of prism is $$A$$ and one of its refracting surface is silvered. Light rays falling at an angle of incidence '$$2 \mathrm{A}$$' on the first surface return back through the same path after suffering reflection at the silvered surface. The refractive index of the material of the prism is

The speed of light in two media $$M_1$$ and $$M_2$$ are $$1.5 \times 10^8$$ $$\mathrm{m} / \mathrm{s}$$ and $$2 \times 10^8 \mathrm{~m} / \mathrm{s}$$ respectively. If the light undergoes total internal reflection, the critical angle between the two media is

The minimum distance between an object and its real image formed by a convex lens of focal length 'f' is