The moment of inertia of a disc about its axis is $I$. What will be its moment of inertia about a tangent in its plane?

The moment of inertia of a ring about an axis perpendicular to its plane and passing through its center is $4 \,kg \,m^2$. Its moment of inertia about the tangent in the plane is

$(a)$ Prove the theorem of perpendicular axes. (Hint: Square of the distance of a point $(x, y)$ in the $x-y$ plane from an axis through the origin and perpendicular to the plane is $x^{2}+y^{2}$)
$(b)$ Prove the theorem of parallel axes. (Hint: If the centre of mass of a system of $n$ particles is chosen to be the origin,$\sum m_{i} r_{i}=0$)

Three rings each of mass $M$ and radius $R$ are arranged as shown in the figure. The moment of inertia of the system about the axis $YY'$ is:

Identical rings are arranged in a hexagonal plane pattern such that each ring touches its neighbouring rings. Each ring has mass $M$ and radius $R$. The moment of inertia of the system of seven rings about an axis passing through the centre of the central ring and normal to the plane of all rings is: (in $MR^2$)

Two spheres each of mass $M$ and radius $\frac{R}{2}$ are connected at the ends of a massless rod of length $2R$. What will be the moment of inertia of the system about an axis passing through the centre of one of the spheres and perpendicular to the rod?