Two bodies of mass $1 \mathrm{kg}$ and $3 \mathrm{kg}$ have position vectors $\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $-3\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\hat{\mathrm{k}}$ respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector :

Match List I with List II.

Choose the correct answer from the options given below:

Solid sphere $A$ is rotating about an axis $PQ$. If the radius of the sphere is $5 \mathrm{cm}$, then its radius of gyration about $PQ$ will be $\sqrt{x} \mathrm{cm}$. The value of $x$ is _____.

A uniform solid cylinder with radius $R$ and length $L$ has moment of inertia $I_1$, about the axis of cylinder. A concentric solid cylinder of radius $R^{\text{'}}=\frac{R}{2}$  and length $L^{\text{'}}=\frac{L}{2}$ is carved out of the original cylinder. If $I_2$ is the moment of inertia of the carved out portion of the cylinder then $\frac{I_1}{I_2}=$  __________.

(Both $I_1$ and $I_2$ are about the axis of the cylinder)   

An object of mass $8 \mathrm{kg}$ is hanging from one end of a uniform rod $CD$ of mass $2 \mathrm{kg}$ and length $1 \mathrm{m}$ pivoted at its end $C$ on a vertical wall as shown in figure. It is supported by a cable $AB$ such that the system is in equilibrium. The tension in the cable is:

(Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

${\mathrm{I}}_{\mathrm{CM}}$ is moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc. ${\mathrm{I}}_{\mathrm{AB}}$ is its moment of inertia about an axis AB perpendicular to plane and parallel to axis CM at a distance $\frac{2}{3}\mathrm{R}$ from center, where $\mathrm{R}$ is the radius of the disc. The ratio of ${\mathrm{I}}_{\mathrm{AB}}$ and ${\mathrm{I}}_{\mathrm{CM}}$ is $x:9$. The value of $x$ is ______.