If two objects have the same momentum, but different mass, their kinetic energies cannot be equal, unless they are both at rest. This is seen mathematically by setting m1v1 = m2v2, where m2 = nm1, where n ≠ 1. Then m1v1 = nm1v2, which leads to v2 = v1/n. But now if you try to equate kinetic energies, you can't, because m2v2^2/2 = nm1(v1/n)^2/2 = nm1v1^2/2n^2 = m1v1^2/2n. In fact if n > 1, then object 2, which is heavier (m2 = nm1) has the same momentum as object 1, but less kinetic energy. Conversely, if you had two objects with the same kinetic energy, then the heavier object will have more momentum.
I will use (1) to denote the heavier mass and (2) to denote the other. In B, all of the initial kinetic energy is due to (1). After collision, the momentum of (2) has to equal the momentum of (1) before collision. But as shown before, the kinetic energy of the lighter object has to be greater, so the collision is not elastic (in fact, this is physically impossible). D is easily ruled out because this is completely inelastic collision. For A, remember that momentum is a vector. If the final momentum of the system is equal to the initial momentum of the system, then the p(2)f + p(1)f = p(1)i. If you consider right to be positive, then p(1) < 0. This means that p(2)f > p(1)i. Again, this means that the kinetic energy of (2) after collision is greater than that of (1) before collision, which is not possible. Note that unlike momentum, kinetic energy has to be positive. (1) can have negative momentum, but it cannot have negative kinetic energy. Does that help?
