Four solids are shown in cross-section. The sections have equal heights and equal maximum widths. They have the same mass. The one which has the largest rotational inertia about a perpendicular axis through the centre of mass is:

The linear mass density of a thin rod $AB$ of length $L$ varies from $A$ to $B$ as $\lambda(x) = \lambda_{0}(1 + \frac{x}{L})$,where $x$ is the distance from $A$. If $M$ is the mass of the rod,then its moment of inertia about an axis passing through $A$ and perpendicular to the rod is $......ML^{2}$.

Match List-$I$ with List-$II$:

List-$I$	List-$II$
$(A)$ Moment of inertia of solid sphere of radius $R$ about any tangent	$(I)$ $\frac{5}{3} MR^2$
$(B)$ Moment of inertia of hollow sphere of radius $R$ about any tangent	$(II)$ $\frac{7}{5} MR^2$
$(C)$ Moment of inertia of circular ring of radius $R$ about its diameter	$(III)$ $\frac{1}{4} MR^2$
$(D)$ Moment of inertia of circular disc of radius $R$ about any diameter	$(IV)$ $\frac{1}{2} MR^2$

Choose the correct answer from the options given below:

$A$ metallic solid sphere is rotating about its diameter as an axis of rotation. If the temperature is increased by $200^{\circ} C$, the percentage increase in its moment of inertia is (Coefficient of linear expansion of the metal $= 10^{-5} /^{\circ} C$) (in $\%$)

Moment of inertia of a solid sphere about its diameter is $I$. It is then casted into $27$ small spheres of same diameter. The moment of inertia of each new sphere is

Four hollow spheres,each with a mass of $1\, kg$ and a radius $R = 10\, cm$,are connected with massless rods to form a square with a side of length $L = 50\, cm$. In case-$1$,the masses rotate about an axis that bisects two sides of the square. In case-$2$,the masses rotate about an axis that passes through the diagonal of the square,as shown in the figure. Compute the ratio of the moments of inertia $I_1/I_2$ for the two cases.