These are some of the basic questions. Practice more such problems from mcq books like marvel or USS to strengthen your grip over the topic and the subject.

We know that momentum must be conserved .

Thus the momentum of that part of the rod is

mu .

And Angular momentum = rP

= (L/2)mu

= ωI = ω(∫( 2M/L)r² dr + m(L/2)²)

= ω((2M/L)(L/2)³/3 + m(L/2)²)

Solve the equation between the first and last line , and you will have your answer .

Best wishes. Thank you.