When torque applied on a system is zero,which of the following will be constant?

$A$ uniform rod of length $8 a$ and mass $6 m$ lies on a smooth horizontal surface. Two point masses $m$ and $2 m$ moving in the same plane with speed $2 v$ and $v$ respectively strike the rod perpendicularly at distances $a$ and $2 a$ from the mid point of the rod in the opposite directions and stick to the rod. The angular velocity of the system immediately after the collision is:

$A$ circular disc is rotating with an angular velocity $\omega$. If a child sits on the disc,which of the following quantities will be conserved?

Suppose all the people in the world line up at the equator and all start running at speed $v_{rel}$ relative to the surface of the Earth along the equatorial circle. The initial angular velocity of the Earth is $\omega_0$. Let the moment of inertia of the Earth be $I_E$,the moment of inertia of all people be $I_P$,and the radius of the Earth be $R$. Which of the following statements is correct?

$A$ mouse of mass $m$ jumps on the outside edge of a rotating ceiling fan of moment of inertia $I$ and radius $R$. The fractional loss of angular velocity of the fan as a result is

$A$ disc of mass $M$ and radius $R$ is moving with an angular velocity $\omega$ as shown in the figure. Find the angular momentum of the disc about the reference point $O$.