The ratio of the moments of inertia of two uniform rings of radii $R$ and $nR$ about an axis passing through their centers and perpendicular to their planes is $1 : 8$. What is the value of $n$?

Two discs have the same thickness. Their radii are $R_1$ and $R_2$ and their densities are $d_1$ and $d_2$. The moment of inertia of the second disc is greater than the first if:

Three balls of masses $2 \,kg$,$4 \,kg$,and $6 \,kg$ are placed at the midpoints of the sides of an equilateral triangle of side length $2 \,m$. The moment of inertia of the system about an axis passing through the centroid and perpendicular to the plane of the triangle is . . . . . .$kg \,m^2$.

$Assertion$ : Radius of gyration of a body is a constant quantity.
$Reason$ : The radius of gyration of a body about an axis of rotation may be defined as the root mean square distance of the particles from the axis of rotation.

$A$ circular disc has radius $R_1$ and thickness $T_1$. Another circular disc made of the same material has radius $R_2$ and thickness $T_2$. If the moment of inertia of both discs are same and $\frac{R_1}{R_2}=2$, then $\frac{T_1}{T_2}=\frac{1}{\alpha}$. The value of $\alpha$ is . . . . . . .

In a rectangle $ABCD$ where $BC = 2AB$,the moment of inertia will be minimum about which axis?