If in a plano-convex lens,the radius of curvature of the convex surface is $10 \,cm$ and the focal length of the lens is $30 \,cm$,the refractive index of the material of the lens will be ..........

An object is placed at a distance of $f/2$ from a convex lens. The image will be

$A$ convex lens has its radii of curvature equal. The focal length of the lens is $f$. If it is divided vertically into two identical plano-convex lenses by cutting it,then the focal length of the plano-convex lens is ($\mu =$ the refractive index of the material of the lens).

Explain how the image is formed by a thin convex lens and derive the lens maker's formula: $\frac{1}{f} = (n_{21} - 1) \left[ \frac{1}{R_1} - \frac{1}{R_2} \right]$.

Double convex lenses are to be manufactured from a glass of refractive index $1.55$ with both faces of the same radius of curvature. What is the radius of curvature required if the focal length is to be $20 \ cm$ (in $cm$)?

The dotted part of the lens is cut and kept on the $x$-axis as shown in the diagram. If parallel paraxial rays are falling on this system,then the coordinate of the image formed after refraction from both the lenses is $(30, -1)$. If $x = 2.5 \, cm$,then $y = .......... \, cm$. (Assume the lens has no spherical aberration)