Could you, please, tell me what you call it when someone becomes so nervous in a situation (on an exam, for example) that they become ’frozen’ and are therefore unable to be themselves and to show what they know and what they are capable of?

Thank you!
Not all of these are verbs but they might be some alternatives:

paralyzed with fear

a paralyzing fear of test-taking

test-taking anxiety

test day panic

test anxiety

someone who is an excellent student but a poor test-taker
Thank you very much, Aerie! I love your expressions, they sound so real-life! Emotion: happy

May I ask if ’’to shut down’’, albeit a more general term, can mean the same thing?

I suspect, though, that if it is so, then it’s rather colloquial, so could I use ’’to become paralyzed’’ instead if I needed a more sophisticated, but similarly general term?

Thanks again!
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Absolutely! I would say that's a very descriptive choice of words! My brain certainly 'shuts down' when faced with any sort of math test!

And I would say you can use 'paralyzed' as long as you qualify it by using 'with fear' so that you convey the emotion rather than a physical paralysis...but that's only my humble opinion!
You can also use "freeze," or, more informally, "freeze up."

"I thought I was prepared for the exam, but when I say the first question /I just froze (up)/my mind just froze (up)."
Emotion: star
Great one Khoff.
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Aerie and Khoff, Thank You Very Much! [Y] Emotion: smile