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Should mathematics PhD students work through all of the past exams, if, say, the department makes the last 20 years of exam questions available to students?

Or should students mainly focus on the most recent years' exams instead? - and that perhaps one cannot do all of the exams, because it is almost not humanly possible.

Just seeking general advice.

Thanks.

  • Go through them linearly backwards after you've studies all the material, and see how far you get with however much time you have left. – Michael Angelo Jan 2 '16 at 7:52
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    You should go through all of them - as a math professor, I would assume that the students have gone through all the released exams, since they are publicly available. You will likely find that many questions are repeated in spirit, and so you get faster at solving the problems. That is the point. But you should not only do the released exams, you should also study the material directly, and be ready for problems that differ from the published exams. – Oswald Veblen Jan 13 '16 at 13:22
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Going through problems on old exams will give you some idea of the kinds of questions that are used in these exams, but the committees that write these exams will change over time. This means that very old exams might not be representative of what you'll be given this year. Even if you focus on more recent exams, the committee might have changed and you could well get an exam that looks very different from last year's exam.

In any case, solving problems from previous exams should not be the only way in which you study. You should also take time to review the subject and consolidate your understanding of key concepts, definitions, formulas, and theorems.

In my experience of grading these exams, I've seen that students who have difficulty typically fail for one of the following reasons:

  1. Not knowing some required definition or theorem. If a question is of the form "Show that A has the Frobnitz property" and you don't know what the Frobnitz property is, you simply won't be able to answer that question. You can avoid this by having read broadly on the subject and by making sure that you have committed important definitions and theorems to memory.

  2. Not having good problem solving skills. Although doing old comprehensive exam problems will certainly help with this, doing problems from many sources is likely to be just as helpful. The key here is to focus on solving hard problems that may involve several non-obvious steps rather than the trivial exercises that fill many textbooks. Keep in mind that with a textbook you've been given a lot of context- if the problem appears in the chapter on contour integration than you can assume that the problem involves contour integration. This won't be true of the problems on your comprehensive exam.

  3. Poorly written solutions. Sloppy logic in the solution to a homework problem might get substantial partial credit in an undergraduate class. In a comprehensive exam this is much less likely to happen. You need to give precise and rigorous solutions that cover all aspects of the problem, and there is no room for arithmetic or algebra errors. The difficulty in preparing for this is that you need someone competent to critique your solutions. If you have a study partner, I would suggest that you pick some problems to solve separately and then grade each other's solutions critically. If you find that you're making seemingly minor mistakes but usually have the basic idea for a correct solution, this is an indication that you need to check your work more carefully.

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Getting all worked up trying to solve all problems will just use up a huge lot of time, much of which might be better used otherwise.

Get really familiar with the areas covered. Particularly check on newish results that might be covered.

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