If a gas is filled in a stationary sphere,its molecules are in random motion due to thermal energy. Will the center of mass of these molecules be in motion?

$A$ man of mass $M$ stands at one end of a plank of length $\ell$,which lies at rest on a frictionless surface. The man walks to the other end of the plank. If the mass of the plank is $3M$,what is the distance moved by the man relative to the ground?

$A$ system consists of two particles of masses $m_1$ and $m_2$. If the particle of mass $m_1$ is moved towards the centre of mass through a distance $d$,then the distance the second particle should be moved,so as to keep the centre of mass at the same position is

$A$ body of mass $m_1 = 4 \text{ kg}$ moves at $5 \hat{i} \text{ m/s}$ and another body of mass $m_2 = 2 \text{ kg}$ moves at $10 \hat{i} \text{ m/s}$. The kinetic energy of the centre of mass is:

$A$ body $A$ of mass $M$ falls vertically downwards under gravity. It breaks into two parts: a body $B$ of mass $1/3 M$ and a body $C$ of mass $2/3 M$. The center of mass of bodies $B$ and $C$ taken together shifts from the direction of $A$ by:

$A$ system consists of two particles of masses $m_1$ and $m_2$ separated by a distance $r$. If the particle of mass $m_1$ is moved towards the centre of mass by a distance $d$,then the particle of mass $m_2$ must be moved by a distance $d'$ so that the centre of mass remains at the same position. The value of $d'$ is: