$A$ plano-convex lens,when silvered on the plane side,behaves like a concave mirror of focal length $30 \ cm$. However,when silvered on the convex side,it behaves like a concave mirror of focal length $10 \ cm$. The refractive index of its material is:

$A$ point source of light is placed at $2f$ from a converging lens of focal length $f$. $A$ flat mirror is placed on the other side of the lens at a distance $d$ such that rays reflected from the mirror are parallel after passing through the lens again. If $f=30 \, cm$,then $d$ is equal to ............. $cm$.

Consider a concave mirror and a convex lens (refractive index $=1.5$) of focal length $10 \ cm$ each,separated by a distance of $50 \ cm$ in air (refractive index $=1$) as shown in the figure. An object is placed at a distance of $15 \ cm$ from the mirror. Its erect image formed by this combination has magnification $M_1$. When the setup is kept in a medium of refractive index $7/6$,the magnification becomes $M_2$. The magnitude $\left|\frac{M_2}{M_1}\right|$ is

$A$ planoconvex lens,when silvered at its plane surface,is equivalent to a concave mirror of focal length $28 \ cm$. When its curved surface is silvered and the plane surface is not silvered,it is equivalent to a concave mirror of focal length $10 \ cm$. The refractive index of the material of the lens is:

$A$ convex lens forms a real image of a point object at a distance of $50 \, cm$ from the convex lens. $A$ concave lens is placed $10 \, cm$ behind the convex lens on the image side. On placing a plane mirror on the image side and facing the concave lens,it is observed that the final image now coincides with the object itself. The focal length of the concave lens is .............. $cm$.

One face of a rectangular glass plate $6 \, cm$ thick is silvered. An object held $8 \, cm$ in front of the first face forms an image $12 \, cm$ behind the silvered face. The refractive index of the glass is