$A$ plano-convex lens $(f = 20 \ cm)$ is silvered at the plane surface. Now,the new focal length $F$ will be........$cm$.

$A$ converging mirror is placed on the right-hand side of a converging lens as shown in the figure. The focal length of the lens and the mirror are $15 \ cm$ and $20 \ cm$ respectively. The separation between the lens and the mirror is $40 \ cm$ and their principal axes coincide. $A$ point source is placed on the principal axis at a distance $d$ to the left of the lens. If the final beam comes out parallel to the principal axis,then the value of $d$ is: (in $cm$)

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

An object is placed at a distance of $12 \,{cm}$ from a convex lens. $A$ convex mirror of focal length $15 \,{cm}$ is placed on the other side of the lens at $8 \,{cm}$ as shown in the figure. The image of the object coincides with the object. When the convex mirror is removed,a real and inverted image is formed at a position. The distance of this image from the object will be ..... $(cm)$.

An upright object is placed at a distance of $40\, cm$ in front of a convergent lens of focal length $20\, cm$. $A$ convergent mirror of focal length $10\, cm$ is placed at a distance of $60\, cm$ on the other side of the lens. The position and size of the final image will be

The distance between a convex lens and a plane mirror is $10 \, cm$. The parallel rays incident on the convex lens after reflection from the mirror form an image at the optical centre of the lens. The focal length of the lens will be.....$cm$.