Question

A smooth uniform rod of length l and mass m has two identical beads

Answer 1

Negligible size, each of mass m, which can slide freely along the rod.initially the two beads are at the centre of the rod and angular vel. of the system is omega about axis per. to plane and passing through the center . When the beads reach the end of the rods,ang. vel. is?no ext. force.  

the question is why do we apply conservation of angular momentum .

The answer to your question is there is no such external force  so there is a conservation of angular  momentum .It is basically very easy to find the inertia of the  rod  which is equals to l that is l * omega =((2*ml^2)/4+l)w

and hence the answer is W.