$A$ spherical vessel of radius $10 \, cm$ is filled with water of refractive index $\mu = 4/3$. $A$ fish is at a distance of $4 \, cm$ from the center as shown in the figure. If viewed from end $E$,at what distance will the fish appear to be (in $, cm$)? (Neglect the thickness of the vessel)

$A$ spherical surface separates two media of refractive indices $1$ and $1.5$ as shown in the figure. Find the distance of the image of an object $O$ from the spherical surface. ($C$ is the center of curvature of the spherical surface and $R$ is the radius of curvature).

$A$ narrow parallel beam of light is incident paraxially on a solid transparent sphere of radius $r$ kept in air. What should be the refractive index if the beam is to be focused at the farther surface of the sphere?

As shown in the figure,an air bubble is located at a distance of $3 \, cm$ from a spherical surface of a glass $(\mu = 1.5)$ with a diameter of $10 \, cm$. If the surface is concave,at what distance from the surface will the bubble appear? (in $cm$)

Consider a point object $O$ situated at a distance of $30 \ cm$ from the centre of a sphere of radius $2 \ cm$ and refractive index $\mu_2 = 1.5$ as shown in the figure. If the refractive index of the region surrounding this sphere is $\mu_1 = 1.4$, then the position of the image due to refraction by the sphere with respect to the centre is:

Locate the image formed by refraction in the situation shown in the figure.