For the given uniform square lamina $ABCD$,whose centre is $O$,which of the following relations is correct?

For which of the following objects can the moment of inertia $NOT$ be found using the perpendicular axis theorem?

Find the moment of inertia of a solid cylinder of mass $M$ and radius $R$ about a line parallel to the axis of the cylinder and on the surface of the cylinder.

$A$ circular plate of mass $M$ and radius $R$ has its density varying as $\rho(r) = \rho_0 r$,where $\rho_0$ is a constant and $r$ is the distance from its center. The moment of inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is $I = aMR^2$. The value of the coefficient $a$ is:

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 uniform circular disc about a diameter is $I$. What is its moment of inertia about an axis perpendicular to its plane and passing through a point on its rim (in $I$)?