Two thin similar convex lenses are joined together front to front with the rear portion silvered such that a sharp image of an object at infinity is formed $20 \, cm$ from the mirror. When the air between the lenses is replaced by water $(\mu_w = 4/3)$,then the image formed by the mirror is at a distance of ........ $cm$.

An object is placed in front of a symmetrical convex lens with refractive index $1.5$ and radius of curvature $40 \, cm$. The surface of the lens further away from the object is silvered. Under auto-collimation condition,the object distance is.......$cm$.

$A$ concave lens of focal length $20 \ cm$ placed in contact with a plane mirror acts as a

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

Two identical symmetric double convex lenses of focal length $f$ are cut into two equal parts $L_1, L_2$ by the $AB$ plane and $L_3, L_4$ by the $XY$ plane as shown in the figure respectively. The ratio of the focal lengths of lenses $L_1$ and $L_3$ is

$A$ concave mirror of focal length $f_{1}$ is placed at a distance $d$ from a convex lens of focal length $f_{2}$. $A$ parallel beam of light coming from infinity parallel to the principal axis falls on the convex lens and then after refraction falls on the concave mirror. If the light is to retrace its path,the distance $d$ should be: