$A$ uniform disc of radius $R$ lies in the $x-y$ plane with its centre at the origin. Its moment of inertia about the $z$-axis is equal to its moment of inertia about the line $y = x + c$. The value of $c$ is:

Moment of inertia of a body about two perpendicular axes $X$ and $Y$ in the plane of the lamina are $20 \text{ kg m}^2$ and $25 \text{ kg m}^2$ respectively. Its moment of inertia about an axis perpendicular to the plane of the lamina and passing through the point of intersection of $X$ and $Y$ axes is:

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 $M$):

From a disc of mass $M$ and radius $R$,a circular hole of diameter $R$ is cut whose rim passes through the centre. The moment of inertia of the remaining part of the disc about a perpendicular axis passing through the centre is

$A$ 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 . . . . . . $kg \cdot m^{2}$.

The moment of inertia of a uniform circular disc of radius $R$ and mass $M$ about an axis touching the disc at its diameter and normal to the disc is