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I will be starting graduate school next semester in mathematics and I have no idea how I am going to fare. My goal for this summer and I am slowly making progress is to go over topics in real analysis, topology and algebra. I am also doing an individual study in multivariable calculus which also covers analysis and linear algebra as well. My big question, which is very broad is, how does one prepare for the quals?

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See if your department keeps copies of past exams (most do). If they do, then working the problems on the old exams is by far the best way to study.

You'll begin to notice the types of problems typically asked (there are some classic ones in each area that you should definitely know how to do), the average difficulty of problems, and most importantly help identify areas you need to brush up on.

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