What are three reasons why hamlet's mother is so fearful of him during this scene?
(act iii scene iv)

-she fears he will tell the king if she reveals anything.

-she is not sure whether he is mad or sane.

-hamlet loves ophelia.

-hamlet kills polonius in her room.

-hamlet speaks to a ghost she cannot see.

3 reasons:

Let me review a couple of  things about this scene.

Hamlet has just confronted his mother. I have seen this scene played out where when he opens the locket, he finds a small painting of Claudius and Hamlet's father each occupying 1/2 of the locket. The person playing Hamlet ripped the locket from around Gertrude's neck and screamed at her what the locket showed, and threw it forcibly in her midsection. The person playing Gertrude was 17 years old at the time. Do you think she expressed fear when that happened? Without saying a word, her face contorted and her eyes when wide and they had rehearsed that scene a dozen times. Still she was terrified. What son would do that to his mother? They both did that perfectly. We who watched were just as terrified.

Hamlet kills Polonius who was hiding behind a curtain. If Hamlet was not sane, what would stop him from killing her? He certainly didn't have a reason not to after some of the things he's said to her.

In scene I of Act III, Hamlet confronts Ophelia. He has just contemplated suicide (The To Be Or Not to Be speech), and Ophelia just walks in afterwards. Gertrude is not present but she undoubted knew what how brutal he was to her. If you have notes to this play, or you can find it on Google, look up what direction he is giving her when he tells her to get to a nunnery. Such brutality is not easily endured by a fiancee or a mother. So knowing that Hamlet did love Ophelia would add the rage he had in having to break up with her. That should terrify Gertrude, although I don't think it came up directly in Act III Scene IV. That makes the lines between Gertrude and Hamlet when he describes her marriage bed that much more fearful. He really makes it sound unclean.

He does speak to the ghost who tries to set him on the right path again. Imagine someone you know well suddenly talking to a ghost, and you not able to hear the other end of the conversation. What would you think? Would you wonder if they were sane? Not only that, Ghosts had a different meaning in Shakespeare's time. Part of Hamlet's problem with the ghost is that he has no idea from where the ghost comes from. Can you imagine thinking the ghost came from H*ll and what that would mean?

I'm sorry I don't remember whether she fears Claudius will find out anything or not. It might be so.