Different departments have varying terminologies and requirements for their programs, so the term "qualifying exam" may refer to different things depending on where you are. In the UCR math department, the quals refer to our written qualifying examinations. Currently, UCR offers six year-long graduate math core sequences, each consisting of three separate courses, denoted by A, B, and C, and offered during the fall, winter, and spring quarters, respectively. For each core sequence, the department offers an annual written exam that tests your knowledge of everything you learned in that sequence. The exam typically comprises of three parts, one for each course, with the professors who taught each course writing their respective portion of the exam. To advance to candidacy and graduate, you must pass two different quals. Each exam allows a maximum of two attempts, and some advisors may require that you pass the qual corresponding to their research area before working with them. Keep this in mind when deciding which qual to take. Note that all of these numbers have changed in the past and may vary among different departments.
The quals are offered at the end of summer during week 0 of the fall quarter (this also varies among departments). Ideally, you would have taken the course in the previous academic year and used the summer to study for the exams. However, you do not have to take the class to take the qual. In fact, you can even attempt the exam during the summer before you start your first year and possibly get one, if not both, quals out of the way before you even start graduate school! As mentioned above, the exam is broken up into multiple parts, and you will have three hours to complete all of it. Generally, it's one section for each course in the sequence. However, the real analysis exam often has a fourth section that covers undergraduate material as well. Each section will have about four to five questions of which you will answer some subset of them. For example, when I took the algebra qual, there were five questions on each section, and I had to answer four from each. However, in my topology qual, I only had to choose two out of the three questions for each section, so this can change drastically depending on the subject as well as who taught the course. The exam is typically a major time crunch; however, this also varies greatly. In a weird way, you sort of need to go into the exam knowing how to solve the problems before you start. There's lots of speculation on how the cutoffs and curves are determined, but no one really knows. I've seen a very low pass rate for a qual one year, as well as a 100% pass rate in the same subject given a different year, so the details are always changing. It's best not to worry about this too much and just put in a lot of dedication and effort into studying and try your hardest during the exam. You can find previous qualifying exams on the department's website
The written qualifying exams are unlike any other exams I've ever taken. The more time you put into it, the better your chances are at passing. There tends to be a direct correlation between the amount of time one spends studying and their pass rate. What makes the quals different than other exams in this regard is that you can be very intelligent but fail without studying. On the other hand, if you feel weak in the subject but you dedicate a lot of time studying for the exam, you will almost certainly pass. Take these exams seriously! You should treat this summer of studying as a full-time job. It's best to start studying early because you will need multiple months to prepare for this exam, and it's not an exam that you can cram all of your studying into the last couple of days. Anyone who has taken a qual will tell you that you don't really learn the material when taking the class, but rather when studying for the corresponding qual, so you'll need extra time to become an expert in the subject. I recommend creating a study plan as early as possible. I've run a qual prep seminar for the algebra exam a couple of times in the past and created a general schedule that may help you with your study plan. This 7-week long summer seminar would start 8 weeks before the exams, and I would assume that everyone has already looked over their notes and felt comfortable with the basics. I would spend a few weeks getting comfortable with the material and the exercises, and then spend the last month or so doing a bunch of practice quals. Specifically, I would aim for 1 week for section A, 1 week for section B, 1 week for section C, and 2 weeks for mock quals. I would make sure to have an additional 2 weeks buffer at least for when we inevitably don't stick to that timeline.
Since the professors making the problems for your qualifying exams are the same ones that taught the course, I highly recommend spending most of your time looking over anything like worksheets or notes that they created throughout the academic year. All resources are good resources, but in my opinion, the best material to study are the exams the professor gave during that course, as they will probably be the closest thing to the exam. Send them an email asking for any study material and share it amongst your colleagues. If the professors have written any previous qualifying exams, make sure you know how to do those problems as well, as there's a chance you may see the same problems on your exam. The main thing is to just try and get a feel for what kind of questions each professor likes to ask. As much as any educator may hate to admit it, a decent portion of the studying process, in general, is trying to predict what is going to be on the exam. I would argue that this is even more important for the qualifying exams, mostly due to the time constraint. That being said, you have all summer to study, so try to be prepared for anything and everything the professor can ask you. Most importantly, collaborate! I can't emphasize this enough. As much as it may feel like a competition, it is not! Everyone can pass (I've seen this happen many times), and everyone can fail, so work together and bring each other up! Working together will help keep everyone accountable and just make studying more fun, or at the very least, more breathable. Make a group chat, Discord, or Slack; somewhere where everyone can just talk math. People like to say that teaching a subject is the best way to learn something, so challenge yourself to see if you can summarize the whole sequence to a colleague in under an hour. I did this with my roommates, and I found it to be incredibly helpful. Take lots of mock qualifying exams, whether it's previous qualifying exams (which can be found on the department's website) or make your own and provide them to your colleagues to try. Practice qualifying exams will give you a good feel for the time you are allotted, and in the process, it will become very clear what areas you need to work on. Do this more and more often as the exam date gets closer.
Once it's time to take the qualifying exams, one should be at the point where one honestly feels like any more studying won't help anymore. The qualifying exams were definitely the hardest exams I've ever taken, and I'm not saying this to discourage you. In fact, I mean to do just the opposite. If you study hard and put lots of time into it, you will get there! As a previous graduate student once said, "You'll never know the material as well as you do the day you take your qualifying exam." So, good luck, and feel free to reach out if you have any questions or concerns!