I like Serena said:
Exams are often carefully set up in different difficulty levels.
Questions verifying you know the definitions.
A couple of checks if you can reproduce an example.
If you can do both, you may already have a sufficient grade.
Then more questions for which you may need to know a proof.
And more questions, requiring deeper understanding of the matter.
You can only get a perfect grade if you can do all.
Interesting, you should give that piece of advice to my teachers. I haven't seen such a set up for my exam in a LONG while. Right now, most of my examinations require fierce memorization of lists of exercises set forth, and those who fail to memorize all the details score poorly. In fact, that happened in cases where we had the teacher choose a set of problems.
When I took multivariable analysis the set was "all exercises from 'Calculus on Manifolds' by Spivak"; when I took real analysis (measure theory) the set was "all problems from 'Elements of Real Analysis' by Bartle". It also happened that both teachers did not know how to solve said exercises. The first constantly used the solutions manual found in the internet, the second didn't care. We didn't know topology when we took measure theory, and the teacher made constant use of it. At some point he realized people weren't exactly following his 'explanations' (read: copying the book on the board). He promptly asked: "Everyone knows point-set topology, right?" The answer was an unanimous "no." He replied with "Well, Bartle does," followed by a shrug and continuing to copy the book on board.
I am taking Metric Spaces Topology this semester. My lecturer has decided that this course is useless and instead chose to use Munkres' General Topology as the textbook. Frequently he does not know what he is doing in front of the class and claims absurdities until somebody points out the huge flaws in them (he once said that projections weren't continuous in the product topology). He confessed that while he chose some problems of the book he didn't know how to do most of them.
Yet, can you guess what happened on the test? That's right: rote memorization of certain problems he had chosen on the lists. And not quite so easy ones: one of the questions reminded you about the uniform topology on $\mathbb{R}^{\omega}$ (the cartesian product of the real line with itself "real" times), and defined the metric as $$d(x,y) = \left( \sum_{i=1}^{\infty} (x_i - y_i)^2 \right)^{\frac{1}{2}},$$ assuming that $\sum_{i=1}^{\infty} x^2_i \leq \infty$. The problem: show that this defines a metric on $\mathbb{R}^{\omega}$ and that it produces a topology finer than the uniform topology but coarser than the box topology.
There were 4 other questions to be done within a period of 2 hours. Can you imagine proving this in necessary detail and still having feasible time to answer the other four? I could tell you them, but I believe this already gives quite an idea of what is common around here.
tl;dr All this ranting is to say one thing: if the exam isn't "carefully set up in different difficulty levels" but rather "sloppily set up with arbitrary criteria", how does one get by? I have a hard time not getting pissed off about all of this. It goes against my nature to just see things hitting the fan and going unpunished. (Swearing) Later, the same incompetent professor will be the one to evaluate your skills, and when he sees the numbers without the accompanying background he'll be likely to say "well, not so skilled, are we?" (Fubar) (Angry)
