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 mirror is placed on a horizontal table with its axis pointing vertically upwards. Let $O$ be the pole of the mirror and $C$ be its center of curvature. $A$ point object is placed at $C$. It forms a real image at $C$. If the mirror is now filled with water,the image will be formed . . . . . .

An equiconvex glass lens $(a)$ has a focal length $f$ and power $P$. It is cut into two symmetrical halves $(b)$ by a plane containing the principal axis. The two pieces are recombined as shown in figure $(c)$. The power of the new combination is

The figure shows an equiconvex lens of focal length $f$. If the lens is cut along $AB$,the focal length of each half will be

$A$ thin biconvex lens is prepared from glass $(mu = 1.5)$ where both curved surfaces have equal radii of $20 \text{ cm}$ each. The left side surface of the lens is silvered from the outside to make it reflecting. To have the position of the image and the object at the same place,the object should be placed from the lens at a distance of . . . . . . $\text{cm}$.

$A$ plano-convex lens $(\mu = 1.5)$ has a radius of curvature of $10 \; cm$. It is silvered on its plane surface. Find the focal length after silvering. (in $; cm$)