Two thin similar convex lenses are joined together front to front with the rear portion silvered such that a sharp image of an object at infinity is formed $20 \, cm$ from the mirror. When the air between the lenses is replaced by water $(\mu_w = 4/3)$,then the image formed by the mirror is at a distance of ........ $cm$.

Two identical thin plano-convex glass lenses (refractive index $1.5$) each having a radius of curvature of $20\, cm$ are placed with their convex surfaces in contact at the centre. The intervening space is filled with oil of refractive index $1.7$. The focal length of the combination is.......$cm$.

$A$ convex lens of focal length $20\,cm$ is placed in front of a convex mirror with their principal axes coinciding. The distance between the lens and the mirror is $10\,cm$. $A$ point object is placed on the principal axis at a distance of $60\,cm$ from the convex lens. The image formed by the combination coincides with the object itself. The focal length of the convex mirror is $...\,cm$.

When the plane surface of a plano-convex lens is silvered,it behaves as a concave mirror of focal length $60 \ cm$. However,when the convex surface is silvered,it behaves as a concave mirror of focal length $20 \ cm$. What is the refractive index of the lens?

Given is a thin convex lens of glass (refractive index $\mu$) and each side having radius of curvature $R$. One side is polished for complete reflection. At what distance from the lens,an object be placed on the optic axis so that the image gets formed on the object itself?

$A$ pin is placed at a distance of $10 \, cm$ in front of a convex lens of focal length $20 \, cm$. The lens is made of a material with a refractive index of $1.5$. The surface of the lens farther from the pin is silvered,and its radius of curvature is $22 \, cm$. Find the position of the final image in $cm$.