This is a free lunch if you look closely to the fact that

is constant. Actually, it implies that it is constant
throughout the motion. Another point is that this speed is attached with the change of rocket mass, which is rocket's exhaust. Combining this two facts together, the answer should be (E). The rocket's exhaust in a stationary frame is unlikely to be constant since the rocket is moving with acceleration - as far as we concerned.
The correct answer is (E). 55 of 100 people answer this question correctly.