A point-object is situated at the centre of a solid glass sphere of radius $6 \mathrm{cm}$ and refractive index $1.5$. The distance of its virtual image from the surface of the sphere is:

A spherical surface of radius of curvature $R$ separates air (refractive index $1.0$) from glass (refractive index $1.5$). The centre of curvature is in the glass. A point-object P placed in air is found to have a real image $Q$ in the glass. The line $PQ$ cuts the surface at a point $O$ and $PO=OQ$. The distance $PO$ is equal to:

A thin convex lens made from crown glass $(n=\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)$ has focal Iength $f$. When it is measured in two different liquids having refractive indices $\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ and $\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$, it has the focal lengths $f_1$ and $f_2$ respectively. The correct relation between the focal lengths is:

A bi-convex lens of focal length $15 \mathrm{cm}$ is in front of a plane mirror. The distance between the lens and the mirror is $10 \mathrm{cm}$. A small object is kept at a distance of $30 \mathrm{cm}$ from the lens. The final image is:

A lens is formed by pressing mutually the plane faces of two identical plano-convex lenses, each of focal length $40 \mathrm{cm}$. It is used to obtain a real, inverted image of the same size as the object. The object is placed from the lens at a distance of:

Two thin lenses of focal lengths $f_1$ and $f_2$ are in contact and coaxial. The power of the combination is:

The image of an object, formed by a plano-convex lens at a distance of $8 \mathrm{m}$ behind the lens, is real and is one-third the size of the object. The wavelength of light inside the lens is $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ times the wavelength in free space. The radius of the curved surface of the lens is: