A train engine sounds a whistle at frequency . The frequency heard...

The passenger and the train engine are moving with the same velocity, so there is no relative motion between the source of sound (engine) and the observer (passenger).

According to the Doppler effect, the apparent frequency is given by f' = \nu \left(\dfrac{v - v_o}{v - v_s} \right).

Since the velocity of the observer $v_o$ is equal to the velocity of the source $v_s$, we get $f' = \nu$.

Thus, the frequency heard by the passenger is equal to $\nu$.