With reference to the figure of a cube of edge $a$ and mass $m$, state which of the following options are correct ($O$ is the centre of the cube):
$(a)$ The moment of inertia of the cube about the $z$-axis is $I_z = I_x + I_y$
$(b)$ The moment of inertia of the cube about the $A$-axis is $I_A = I_z + \frac{ma^2}{2}$
$(c)$ The moment of inertia of the cube about the $B$-axis is $I_B = I_z + \frac{ma^2}{2}$
$(d)$ $I_x = I_z$

If $I$ is the moment of inertia of a thin circular disc about an axis passing through the tangent of the disc and in the plane of the disc,what is the moment of inertia of the same circular disc about an axis perpendicular to the plane and passing through its centre?

$A$ solid sphere of mass $5 \ kg$ and radius $10 \ cm$ is kept in contact with another solid sphere of mass $10 \ kg$ and radius $20 \ cm$. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is . . . . . . $kg \cdot m^{2}$.

$(a)$ Find the moment of inertia of a sphere about a tangent to the sphere,given the moment of inertia of the sphere about any of its diameters to be $2 M R^{2} / 5,$ where $M$ is the mass of the sphere and $R$ is the radius of the sphere.
$(b)$ Given the moment of inertia of a disc of mass $M$ and radius $R$ about any of its diameters to be $M R^{2} / 4,$ find its moment of inertia about an axis normal to the disc and passing through a point on its edge.

As shown in the figure,two thin coplanar circular discs $A$ and $B$ each of mass $M$ and radius $r$ are attached to form a rigid body. The moment of inertia of this system about an axis perpendicular to the plane of disc $B$ and passing through its centre is (in $Mr^2$)

The moment of inertia of a uniform semicircular wire of mass $m$ and radius $r$,about an axis passing through its centre of mass and perpendicular to its plane is ..........