To solve this problem, we need to consider the conservation laws applicable to a rotating solid sphere.$\\$Given:$\\$• A solid sphere rotating freely about its symmetry axis in free space.$\\$• The radius of the sphere is increased while keeping its mass constant.$\\$We need to determine which physical quantity remains constant.$\\$Let's analyze each option:$\\$1. Rotational kinetic energy:$\\$The rotational kinetic energy $(K)$ is given by $K = \frac{1}{2} I \omega^2, \\$where $I$ is the moment of inertia and $\omega$ is the angular velocity.$\\$As the radius increases, the moment of inertia $(I)$ changes, which affects $K. \\$Thus, rotational kinetic energy is not constant.$\\$2. Moment of inertia:$\\$The moment of inertia of a solid sphere about its symmetry axis is $I = \frac{2}{5} m r^2, \\$where $m$ is the mass and $r$ is the radius.$\\$Since the radius $r$ is increased, $I$ will change.$\\$Thus, the moment of inertia is not constant.$\\$3. Angular velocity:$\\$The angular velocity $(\omega)$ is related to the moment of inertia and angular momentum.$\\$As the moment of inertia changes, the angular velocity will also change to conserve angular momentum.$\\$Thus, angular velocity is not constant.$\\$4. Angular momentum:$\\$Angular momentum $(L)$ is given by $L = I \omega. \\$In the absence of external torques, angular momentum is conserved.$\\$Therefore, even if the radius changes, the angular momentum remains constant.$\\$Conclusion:$\\$The physical quantity that remains constant is the angular momentum.$\\$This corresponds to Option 4.