JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Rotational Mechanics

$A$ is the centre of the circular path. Taken about $A$, the position vector of the mass lies in the plane of the circle and the velocity is perpendicular to it, so

which is fixed in magnitude and points permanently along $+z$.
Taken about $B$, the position vector of the mass has a component along the axis as well as the radial component. The axial part contributes an angular momentum of magnitude $M\left(\ell \cos \alpha \right)v$ that lies in the horizontal plane and turns with the mass, while the radial part contributes the same $M{r}^2\omega$ along $+z$. The resultant ${\vec{L}}_B$ therefore keeps a fixed magnitude but sweeps round a cone about the axis.
Hence $L_A$ is constant both in magnitude and in direction, while $L_B$ is constant in magnitude only.