$A$ 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 $cm$.

$A$ planoconvex lens becomes an optical system of $28 \, cm$ focal length when its plane surface is silvered and illuminated from left to right as shown in Fig $-A$. If the same lens is instead silvered on the curved surface and illuminated from the other side as in Fig. $-B$, it acts like an optical system of focal length $10 \, cm$. The refractive index of the material of the lens is

$A$ convex lens of focal length $15\,cm$ is placed in front of a convex mirror. When an object is placed at $O$,the image coincides with it. The radius of curvature of the mirror is.....$cm$.

An object is placed in front of a thin convex lens of focal length $30\, cm$ and a plane mirror is placed $15\, cm$ behind the lens. If the final image of the object coincides with the object,the distance of the object from the lens is.....$cm$.

$A$ point object is placed at a distance of $20 \, cm$ from a thin plano-convex lens of focal length $15 \, cm$. If the plane surface is silvered,where will the image form?

$A$ 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 $...\,cm$.