$A$ spherical surface of radius of curvature $10\,cm$ separates two media $X$ and $Y$ of refractive indices $3/2$ and $4/3$ respectively. The centre of the spherical surface lies in the denser medium. An object is placed in medium $X$. For the image to be real,the object distance must be:

Light from a point source in air falls on a spherical glass surface ($n = 1.5$ and radius of curvature $R = 20 \ cm$). The distance of the light source from the glass surface is $100 \ cm$. Find the image distance. (in $cm$)

$A$ fish is at the centre of a spherical water-filled $(\mu = 4/3)$ fish bowl. $A$ child stands in air at a distance $2R$ ($R$ is the radius of curvature of the sphere) from the centre of the bowl. At what distance from the centre would the child's nose appear to the fish situated at the centre (in $R$)?

What is the focal length of this curved surface in $cm$?

$A$ laser beam falls on a crystal ball of radius $R$ as shown in the figure. What is its refractive index?

$A$ spherical glass is attached to a rigid wall as shown in the figure. An observer located at point $O$ is looking at a point $A$ on the wall. The refractive index of the glass is $1.5$ and that of air is $1.0$. The distances are $OA = 8 \text{ cm}$, $XA = 3 \text{ cm}$. If the radius of curvature of the spherical glass surface is $R = 5 \text{ cm}$, then the apparent distance of $A$ from the observer $O$ is (in $\text{ cm}$)