Does Hunter's Mark let you know if a target is alive or dead?

Here's the situation: a Ranger at my table had Hunter's Mark on a target.  Another character pushed the target into a hole in the floor.  As the DM, I knew the target died when it hit the bottom, but the PCs couldn't see what happened.

However...wouldn't the Ranger "know" if they could transfer the Hunter's Mark as a bonus action?  And therefore they would know whether or not the fall killed the target (or, technically, if the target dropped to zero hit points).  Right?

I mean, if you don't have some firm narrative value from the ambiguity, I'd say just tell them. Plus, even if the Ranger doesn't magically know immediately after the fall, by RAW they can now retarget Hunter's Mark, so they can get confirmation with a Bonus Action.

Given that hunter's mark includes a special ability to track the target, it seems unlikely that merely falling through a floor would be enough to confuse it.

I'd say the Ranger knows it can take a Bonus Action to move the Hunter’s Mark to a new creature it can see within range, which occur when the target drops to 0 Hit Points.

Bonus Action:  You can take a Bonus Action only when a special ability, a spell, or another feature of the game states that you can do something as a Bonus Action. You otherwise don’t have a Bonus Action to take.

If you don't want to outright say, you could always say that "you sense a not-unfamiliar shift in the nature of the Hunter's Mark you placed on your target," and then see if the player would ask a follow-up question, such as, "Do I know what this shift means?" You could then confirm "It is very much like the shift you get when the target of your mark drops to 0 HP. The mark is no longer bound to that target and can be moved as a bonus action."

The reason i wouldn't say as Ranger you know your quarry is alive or dead is because not all creatures die when reduced to 0 Hit Points, one can be Unconscious and alive while making Death Saving Throws for example.

But you'd know you can now move the Hunter’s Mark to a new creature.