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It is believed that the Room of Requirement is unplottable, it's known that the Ministry doesn't trace Unforgivable Curses performed by adult wizards and that although the Ministry can detect the use of magic near an underage wizard, they don't know who performed it.
My question is, if a student casts an Unforgivable Curse in Hogwarts does the Ministry detect it? And what about if the student casts it in the Room of Requirement? Could the Ministry even detect its use?
The Ministry does not track the Unforgivable Curses. The Trace is put on every underage wizard, but, as is established in the books, the Trace does not alert them if a wizard or witch does magic in a place like Hogwarts. Either the Trace is temporarily lifted for Hogwarts, or alerts from Hogwarts are ignored because underage magic is being taught there. [insert Moody quote.]
"The Trace, the Trace!" said Mad-Eye impatiently. "The charm that detects magical activity around under-seventeens, the way the ministry finds out about underage magic! If you, or anyone around you, casts a spell to get you out of here, Thicknesse is going to know about it, and so will the Death Eaters.
Emphasis mine
The Trace does not distinguish between Unforgivable Curses and other magic. Neither is there anything else that detects Unforgivable Curses (Moody/Crouch performed them many times in book 4) For this reason alone, they would not know. Also, Harry tried to use the Cruciatus Curse in the Ministry itself in book 5.
What's more, if they somehow tracked an Unforgivable Curse to being done at Hogwarts, there is no reason to suspect any one student.
Your question concerning the Room of Requirement is the same as the one as Hogwarts; the Room is unplottable, yes, but all of Hogwarts is also unplottable.
If somehow they found a student who had used on of these curses at school, they would have no power over him/her, as Dumbledore insists.
The Ministry doesn’t seem to watch the Trace when the students are at Hogwarts, and they don’t seem to monitor for Unforgivable Curses. With those factors combined, it’s highly unlikely that the Ministry would know if someone in Hogwarts had cast an Unforgivable Curse.
The Ministry has no authority to punish students for whatever they do at Hogwarts (which would include the Room of Requirement). That falls to the staff of Hogwarts, and the Ministry would be unable to do anything about any crime they detected. If the students do something ‘wrong’ but the staff at Hogwarts don’t choose to punish them for it, the Ministry can’t actually do anything about it. Because of this, in addition to the large number of spells that would be cast each day at Hogwarts because the students are learning magic, they likely don’t monitor it.
“But, as the Ministry has no authority to punish Hogwarts students for misdemeanours at school, Harry’s behaviour there is not relevant to this hearing,’ said Dumbledore, as politely as ever, but now with a suggestion of coolness behind his words.”
- Harry Potter and the Order of the Phoenix, Chapter 8 (The Hearing)
The Ministry can only punish students for magic they do outside school - not inside it. Therefore, it’d be a waste of time to monitor students at Hogwarts for illegal magic. They’d have a lot of spells to sort through, most of which would be fine, and if they do find illegal they can’t even do anything.
“Snape had removed his coat now; his odd smock looked less peculiar in the half-light. ‘… and the Ministry can punish you if you do magic outside school, you get letters.”
- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)
Therefore, it seems very likely that they don’t monitor the Trace when students are at Hogwarts.
There are many cases where the Ministry is seen to be unable to detect Unforgivable Curses. Over the course of the books (not counting when the Dark Lord controlled the Ministry), they’re used several times (for example, at Hogwarts, in the graveyard where the Dark Lord cast all three on Harry, at the Ministry itself by Bellatrix and attempted by Harry), and the Ministry didn’t seem to know about any of those times that they’d been used. The Dark Lord is also able to Imperius Ministry officials without them being removed - if they were able to detect Imperio being cast, they’d know as soon as someone used it and could go see who it was and know that individual had been compromised (it doesn’t seem to be easily removable, the Imperiused person has to fight it).
