Consider the following two statements:
$1.$ Linear momentum of a system of particles is zero.
$2.$ Kinetic energy of a system of particles is zero.
Then:

Given below are two statements:
Statement $I$: For a mechanical system of many particles,the total kinetic energy is the sum of the kinetic energies of all the particles.
Statement $II$: The total kinetic energy can be expressed as the sum of the kinetic energy of the center of mass with respect to the origin and the kinetic energy of all the particles with respect to the center of mass as the reference frame.
In the light of the above statements,choose the correct answer from the options given below:

Separation of motion of a system of particles into motion of the centre of mass and motion about the centre of mass:
$(a)$ Show $p = \sum p_{i}^{\prime} + M V$,where $p$ is the total momentum of the system,$p_{i}^{\prime} = m_{i} v_{i}^{\prime}$,and $v_{i}^{\prime}$ is the velocity of the $i^{th}$ particle relative to the centre of mass. Also,prove using the definition of the centre of mass that $\sum p_{i}^{\prime} = 0$.
$(b)$ Show $K = K^{\prime} + \frac{1}{2} M V^{2}$,where $K$ is the total kinetic energy of the system,$K^{\prime}$ is the total kinetic energy of the system relative to the centre of mass,and $\frac{1}{2} M V^{2}$ is the kinetic energy of the translation of the system as a whole.
$(c)$ Show $L = L^{\prime} + R \times M V$,where $L^{\prime} = \sum r_{i}^{\prime} \times p_{i}^{\prime}$ is the angular momentum of the system about the centre of mass. Note $r_{i}^{\prime} = r_{i} - R$.
$(d)$ Show $\frac{d L^{\prime}}{d t} = \sum r_{i}^{\prime} \times \frac{d p_{i}^{\prime}}{d t}$. Further,show that $\frac{d L^{\prime}}{d t} = \tau_{ext}^{\prime}$,where $\tau_{ext}^{\prime}$ is the sum of all external torques acting on the system about the centre of mass.

$A$ uniform rod of length $L$ and mass $M$ is placed on a smooth horizontal surface. $A$ particle of mass $m$ moving with velocity $v$ strikes the rod at one end perpendicularly and comes to rest. The velocity of the centre of mass of the rod after the collision is:

$A$ system of $N$ particles is free from any external forces. Which of the following must be true for the sum of the magnitudes of the momenta of the individual particles in the system?

$A$ uniform rod of mass $M$ and length $L$ is placed on a smooth horizontal surface. $A$ particle of mass $m$ moving with velocity $v$ strikes the rod at one end perpendicularly and comes to rest. The velocity of the centre of mass of the rod after the collision is: