$A$ 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.....$cm$.

$A$ luminous object is placed at a distance of $30 \,cm$ from a convex lens of focal length $20 \,cm$. On the other side of the lens,at what distance from the lens should a convex mirror of radius of curvature $10 \,cm$ be placed in order to have an upright image of the object coincident with it (in $,cm$)?

$A$ convex lens has power $P$. It is cut into two halves along its principal axis. Further,one piece (out of two halves) is cut into two halves perpendicular to the principal axis as shown in the figure. Choose the incorrect option for the reported lens pieces.

An equiconvex lens of glass of focal length $0.1 \text{ m}$ is cut along a plane perpendicular to the principal axis into two equal parts. The ratio of the focal lengths of the new lenses formed is

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.....$cm$.

When a thin convex lens is placed in contact with a concave mirror of the same focal length,the resultant focal length of the system is ......