$A$ uniform square plate is shown in the figure. Four identical small squares are removed from its corners. If squares $1, 2,$ and $3$ are removed,where will the center of mass $(C.M.)$ be located?

$A$ spherical cavity of radius $r$ is carved out of a uniform solid sphere of radius $R$ as shown in the figure below. The distance of the centre of mass of the resulting body from that of the solid sphere is given by

$A$ smaller cube with side $b$ (depicted by dashed lines) is excised from a bigger uniform cube with side $a$ as shown below,such that both cubes have a common vertex $P$. Let $X = a/b$. If the centre of mass of the remaining solid is at the vertex $O$ of the smaller cube,then $X$ satisfies:

$A$ uniform square plate of side $a$ is placed in the $xy$-plane with its center at the origin. If a circular portion of diameter $a$ is removed from the plate as shown in the figure,what is the position of the center of mass of the remaining portion?

$A$ spherical hollow is made in a lead sphere of radius $R,$ such that its surface touches the outside surface of the lead sphere and passes through the centre. What is the shift in the centre of mass of the lead sphere due to this?

$A$ thin uniform circular disc has mass $9M$ and radius $R$. $A$ small circular part of radius $R/3$ is cut from the disc as shown in the figure. Calculate the moment of inertia of the remaining part about an axis passing through the center $O$ and perpendicular to the plane of the disc.