Recently, I took the PhD qualifier exams for my department. In my dept. there are 3 exams, each covering a subfield (say subfield A, B, and C). First you are given an hour to work through several problems, and then you have another hour to present to several professors, who critique your answers and ask supplemental questions.
I was well prepared for all of my exams, and did excellently on exams B and C. Exam A did not go as well. During the oral part of the exam, I was asked a question using term Z'. I stated that I am unfamiliar with this term and asked for clarification on its meaning. I was given a vague and unhelpful explanation, along with some chastisement about how this term was "fundamental" knowledge. This wasted perhaps 5 minutes or so of the exam and make the remainder of the oral part fairly awkward. Though I believe I solved the problems correctly, I left with a bad feeling about the terminology issue.
After the exam, I looked term Z' up on Google. Seems it's a very uncommon term for Z. The textbook I was being tested on used Z, as did all 3 of the courses I've had on subfield A.
A few days later I got the results of the exams. I failed exam A, but will be allowed to retake it in the future.
For the sake of identifying what I did wrong, I spoke with one of the professors who administered my exam. I asked what I should focus on for the next time. He said "the fundamentals." I asked whether the terminology issue was the deciding factor and it apparently was not. Instead, he thinks I don't understand absolutely fundamental concept Y. Oddly, he also complained that I used a more general form of Y. (So, which is it? Do I understand Y really well or not at all?) The most I can make of this is that the terminology issue made him and the other committee member think I'm an idiot, and then they were much less fair to me from that point on.
He then asked me to solve a problem which he made up on the fly. I used one version of Y to solve this problem, and he complained that (though correct) how I did this demonstrates that I don't know what I'm doing. The annoying thing about this is that I'm a TA for undergrad class A right now and the way I solved this problem is exactly the way the book the dept. uses does. It seems that he'll complain regardless of how I do things.
I now believe this exam was not fair, and this professor likely will be on my exam committee next time. I doubt I'll be able to pass if he continues being so unfairly critical of me.
At this point, what are my options? I am considering appealing to the dept. head, but I don't want to make things worse than they are right now. I could make enemies in the dept. if I appeal, and I'm not sure if there are other options I should consider.