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

$A$ convex lens of focal length $30 \ cm$ is placed in contact with a concave mirror of focal length $20 \ cm$. What is the focal length of the system in $cm$?

$A$ double convex lens of focal length $F$ is cut into two equal parts along the vertical axis. The focal length of each part will be

$A$ thin convex lens $L$ (refractive index $= 1.5$) is placed on a plane mirror $M$. When a pin is placed at $A$,such that $OA = 18\, cm$,its real inverted image is formed at $A$ itself,as shown in the figure. When a liquid of refractive index $\mu_l$ is put between the lens and the mirror,the pin has to be moved to $A'$,such that $OA' = 27\, cm$,to get its inverted real image at $A'$ itself. The value of $\mu_l$ will be

Consider a glass slab which is silvered at one side and the other side is transparent. The refractive index of the glass slab is $1.5$. If a ray of light is incident at an angle of $45^{\circ}$ on the transparent side,then the deviation of the ray of light from its initial path,when it comes out of the slab,is (in $^{\circ}$)

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