Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body ?

In all the four cases, as the mass density is uniform, centre of mass is located at their respective geometrical centres.

i) Sphere - Centre of Sphere

ii) Cylinder - Middle Point on axis of cylinder

iii) Ring - At centre of ring (Outside the ring)

iv) Cube - At point of intersection of diagnols
No, it is not necessary that the centre of mass of a body should lie on the body. For example, in case of a circular ring, centre of mass is at the centre of the ring, where there is no mass.