For the adjoining diagram,the incorrect relation between $I_{1}, I_{2}$ and $I_{3}$ is ($I$ = moment of inertia).

$A$ solid sphere of radius $10 \ cm$ is rotating about an axis which is at a distance $15 \ cm$ from its centre. The radius of gyration about this axis is $\sqrt{n} \ cm$. The value of $n$ is

Two circular loops $A$ and $B$ of radii $R$ and $NR$ respectively are made from a uniform wire. The moment of inertia of $B$ about its axis is $3$ times that of $A$ about its axis. The value of $N$ is:

$A$ non-uniform rod $OM$ (of length $l$) is kept along the $x$-axis and is rotating about an axis $AB$,which is perpendicular to the rod as shown in the figure. The rod has a linear mass density that varies with the distance $r$ from the left end $O$ of the rod according to $\lambda = \lambda_0 \left( \frac{r^3}{l^3} \right)$,where $\lambda_0$ is a constant. If the axis $AB$ is at a distance $x$ from the end $O$,what is the value of $x$ such that the moment of inertia of the rod about the axis $AB$ $(I_{AB})$ is minimum?

$A$ massless equilateral triangle $EFG$ of side $'a'$ (as shown in the figure) has three particles of mass $m$ situated at its vertices. The moment of inertia of the system about the line $EX$ perpendicular to $EG$ in the plane of $EFG$ is $\frac{N}{20} ma^{2}$,where $N$ is an integer. The value of $N$ is

The figure shows a thin metallic triangular sheet $ABC.$ The mass of the sheet is $M.$ The moment of inertia of the sheet about side $AC$ is