From a solid sphere of mass $$M$$ and radius $$R$$ a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its center and perpendicular to one of its face is:

Given in the figure are two blocks $$A$$ and $$B$$ of weight 20 N and 100 N, respectively. These are being pressed against a wall by a force $$F$$ as shown. If the coefficient of friction between the blocks is 0.1 and between block $$B$$ and the wall is 0.15, the frictional force applied by the wall on block $$B$$ is :

A solid body of constant heat capacity $$1$$ $$J/{}^ \circ C$$ is being heated by keeping it in contact with reservoirs in two ways:
$$(i)$$ Sequentially keeping in contact with $$2$$ reservoirs such that each reservoir
$$\,\,\,\,\,\,\,\,$$supplies same amount of heat.
$$(ii)$$ Sequentially keeping in contact with $$8$$ reservoirs such that each reservoir
$$\,\,\,\,\,\,\,\,\,\,$$supplies same amount of heat.
In both the cases body is brought from initial temperature $${100^ \circ }C$$ to final temperature $${200^ \circ }C$$. Entropy change of the body in the two cases respectively is :

Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion, the average time of collision between molecules increases as $${V^q},$$ where $$V$$ is the volume of the gas. The value of $$q$$ is: $$\left( {\gamma = {{{C_p}} \over {{C_v}}}} \right)$$