A few issues to mention:


1 c): You were asked for the tangent to the level curve, not the tangent plane to the surface z = f(x,y).


3 a): 
- You were asked for a _necessary_ and _sufficient_ condition. Address both necessity and sufficience, please.
- You were not asked to calculate any solution.

3 b): I have been a bit strict on this: For AB to match the identity, every entry must match. That gives 8 conditions (9, but one is trivially satisfied) to check for four unknowns. Four of the conditions will determine the unknowns, and the rest should just be verified.
_However_: the wording of the problem is so that you can argue that you are not asked to verify, just to give necessary conditions on r, s, t and u -- that is probably the interpretation I would have used if I were part of an exam committee. So if this was totally clear to you when you did the problems, just disregard my remarks. 


4 a): 
- I have to apologize for giving this without a hint. Partial fractions is something I didn't have time to cover in class, so if anything like this will be given at the exam, some hint will be given as well.
- "+ C"!

4 b): again, "+ C".

4 c): This can be done without solving. At (1,1), the differential equation will say (1+e) x' = 2. Then use x - 1 = a (t - 1), with a being 2/(1+e).