Moment of inertia of a disc about its own axis is $I$. Its moment of inertia about a tangential axis in its plane is

$A$ thin rod has a length of $1 \ m$ and a mass of $0.6 \ kg$. The moment of inertia of the rod about an axis perpendicular to its length and passing through a point $20 \ cm$ from one end is ................... $kg \cdot m^2$ (the width of the rod is negligible).

Match the items in Column-$I$ with those in Column-$II$.

Column-$I$	Column-$II$
$(1)$ Moment of inertia of a solid sphere about its diameter	$(a)$ $\frac{2}{3}MR^2$
$(2)$ Moment of inertia of a solid sphere about a tangent	$(b)$ $\frac{2}{5}MR^2$
	$(c)$ $\frac{7}{5}MR^2$

$(a)$ 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,$\sum m_{i} r_{i}=0$)

$A$ thin wire of length $l$ and uniform linear mass density $\rho$ is bent into a circular loop with centre $O$ and radius $r$ as shown in the figure. The moment of inertia of the loop about the axis $XX'$ is

Three identical spheres,each of mass $M$ and radius $R$,are placed as shown in the figure. Consider an axis $XX'$ which passes through the diameter of the top sphere and is tangent to the two bottom spheres. The moment of inertia of the system consisting of these three spheres about the $XX'$ axis is: