The moment of inertia of a square loop made of four uniform solid cylinders, each having radius $R$ and length $L$ $(R < L)$, about an axis passing through the midpoints of opposite sides, is (Take the mass of the entire loop as $M$):

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 $I_B = I_z + \frac{ma^2}{2}$
$(d)$ $I_x = I_z$

The moment of inertia of a solid cylinder of mass $M$, length $L = 2R$ and radius $R$ about an axis passing through the centre of mass and perpendicular to the axis of the cylinder is $I_1$, and about an axis passing through one end of the cylinder and perpendicular to the axis of the cylinder is $I_2$. Then:

The moment of inertia of a rod of mass $M$ and length $L$ about an axis passing through a point midway between the center and the end is:

The moment of inertia of a uniform hollow hemisphere about the given axes $I_A$ and $I_B$ is:

Will the theorem of perpendicular axis be applicable to a solid sphere?