$A$ pin is placed at a distance of $10 \, cm$ in front of a convex lens of focal length $20 \, cm$. The lens is made of a material with a refractive index of $1.5$. The surface of the lens farther from the pin is silvered,and its radius of curvature is $22 \, cm$. Find the position of the final image in $cm$.

$A$ thin convex lens of focal length $f$ is placed on a plane mirror as shown in the figure. When an object is kept at a distance $a$ from the lens-mirror combination,its image is formed at a distance $a/3$ in front of the combination. The value of $a$ is

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 $^{\circ}$)

$A$ biconvex lens of focal length $15 \,cm$ is in front of a plane mirror. The distance between the lens and the mirror is $10 \,cm$. $A$ small object is kept at a distance of $30 \,cm$ from the lens. The final image is

The two surfaces of a biconvex lens have the same radii of curvature. This lens is made of glass of refractive index $1.5$ and has a focal length $10 \ cm$ in air. The lens is cut into two equal halves along a plane perpendicular to its principal axis to yield two plano-convex lenses. The two pieces are glued such that the convex surfaces touch each other. If this combination lens is immersed in water (refractive index $= 4/3$), its focal length (in $cm$) is:

$A$ convex lens has a focal length $f$. It is cut into two parts along the dotted line as shown in the figure. The focal length of each part will be