The radii of curvature for spherical surfaces of a bi-convex lens are $20 \mathrm{cm}$ and $30 \mathrm{cm}$. The refractive index of the material of the lens with respect to air is $1.5$. If the lens is dipped in a liquid of refractive index $1.65$ with respect to air, calculate its effective focal length and state the nature of the lens.

The two surfaces of a concave glass lens are of radii of curvature $10 \mathrm{cm}$ and $30 \mathrm{cm}$. Find its focal length when immersed in water. The refractive indices of glass and water relative to air are $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ and $\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ respectively.

An object is placed at $0.06 \mathrm{m}$ from a convex lens of focal length $0.10 \mathrm{m}$. Find the position of the image.

A convergent beam of light passes through a diverging lens of focal length $0.2 \mathrm{m}$ and comes to a focus on the axis $0.3 \mathrm{m}$ behind the lens. Where would the beam have been focussed in the absence of the lens?

A beam of light converges to a point $P$. A concave lens of focal length $16 \mathrm{cm}$ is placed in the path of the beam at $12 \mathrm{cm}$ from $P$. At what point does the beam now converge?

A double convex lens has $10 \mathrm{cm}$ and $15 \mathrm{cm}$ as its two radii of curvature. The image of an object, placed $30 \mathrm{cm}$ from the lens, is formed at $20 \mathrm{cm}$ from the lens on the other side. Find the focal length and the refractive index of the material of the lens. What will be the focal length of the lens, if it is immersed in water of refractive index $1.33$?

An illuminated object and a screen are $90 \mathrm{cm}$ apart. Find the nature and focal length of a lens required to produce a clear image on the screen, twice the size of the object.