For a cube of side $a$ and mass $M$,the moment of inertia is minimum about which axis?

$A$ uniform solid cylinder with radius $R$ and length $L$ has moment of inertia $I_1$ about the axis of the cylinder. $A$ concentric solid cylinder of radius $R/2$ and length $L/2$ is carved out of the original cylinder. If $I_2$ is the moment of inertia of the carved out portion of the cylinder, then $I_1/I_2$ is (Both $I_1$ and $I_2$ are about the axis of the cylinder). (in $: 1$)

Three thin rods,each of length $L$ and mass $M$,are placed along the $x, y,$ and $z-$ axes such that one end of each rod is at the origin. The moment of inertia of this system about the $z-$ axis is:

Three point masses $m_1, m_2$,and $m_3$ are placed at the vertices of an equilateral triangle of side $a$. Find the moment of inertia of this system about the altitude passing through $m_1$.

We have two spheres,one of which is a hollow shell and the other solid. They have identical masses and moments of inertia about their respective diameters. The ratio of their radii is given by

If a spherical planet of mass $M$ and radius $R$ suddenly shrinks to half its size,and its mass reduces to half,what is the new moment of inertia of the planet about its diameter?