The angle of deviation produced by a thin prism when placed in air is '$$\delta_1$$' and that when immersed in water is '$$\delta_2$$'. The refractive index of glass and water are $$\frac{3}{2}$$ and $$\frac{4}{3}$$ respectively. The ratio $$\delta_1: \delta_2$$ is

A ray of light passes through an equilateral prism such that the angle of incidence $$(i)$$ is equal to angle of emergence $$(e)$$. The angle of emergence is equal to $$\left(\frac{3}{4}\right)$$th the angle of prism. The angle of deviation is

The radii of curvature of both the surfaces of a convex lens of focal length $$f$$ and power $$P$$ are equal. One of the surfaces is made by plane grinding. The new focal length and focal power of the lens is

A spherical surface of radius of curvature '$$R$$' separates air from glass of refractive index 1.5. The centre of curvature is in the glass. A point object $$\mathrm{P}$$ placed in air forms a real image $$\mathrm{Q}$$ in the glass. The line $$P Q$$ cuts the surface at point '$$O$$' and $$\mathrm{PO}=\mathrm{OQ}=\mathrm{x}$$. Hence the distance '$$\mathrm{x}$$' is equal to