PhD Project proposal with plagiarism
Without any indication of university, I have to assume that yes, they will keep records of all the things you do. Furthermore, you can assume that your next application will go through the same persons, as they are probably the only ones working on your particular type of application, so even if they don't have records, they will likely remember it.

Plagiarism is extremely reprehensible in the research/academic world, hence you should never ever do it if you want to be considered worth doing research. Here (France), students can be banned up to 5 years from taking any examinations for these kind of faults.

If this is only a problem of time, have you considered asking for an extension for the submission? If you weren't aware of what had to be done for the submission, you might be able to get away with it by proving you only got the request 2 days ago. Otherwise, they might deny it because you could have prepared it without knowing the answer.

It's august, so university will probably be on holidays now they probably won't mind the delay. Bottom word: I can not answer for your university, but doing wrong things will impact your career, so you'd better refrain from doing it.