M208 and M337 Exams Debriefing

Well what a day I started with M337 I have to say it was a fair exam even though I probably wont get higher than grade 3.
Part 1 has 8 questions the topics were

1) Allegedly simple questions on algebraic manipulation of complex numbers. However the last part was a fiendish raising a number to a complex power. In general for a complex number z
$$z^{\alpha} = exp(\alpha Log z) $$
where Log z = log|z| + iArg z

So it should have been simple to work out but I got bogged down

2) Sketching some sets and their difference and deciding whether or not the sets are regions, compact etc
Straightforward but I froze and for the life of me couldn't after about 2 attempts sketch the difference between the two sets.
It took me about 1/2 an hour to do these questions which should have been a simple warm up was not feeling very happy.

Then the core 3 questions on Complex Integration my favourite topic think I managed to do justice to all three parts and finished these in the next half hour so calmed down and felt a bit better,

Question 6 involved Rouche's theorem which is a method of finding zero's of a complex function within a given interval managed to get the first part out which is quite straightforward but unable to answer the last part as you had to work out how many of the zero's lay in the upper half and I couldn't see how to do it.

Question 7 was a standard one on fluid flows and their complex potentials and sketching the stream lines
I got the complex potential but couldn't sketch the stream line as it involved a complicated circle and I couldn't remember where the centre of a circle of the form

    $$x^2 + y^2 + y = c$$ was

Question 8 was a question on Complex Iteration and determination of whether or not a point lay in the Mandelbrot set got about 3/4 of this question out

So reckon probably about 3/4 of part 1 answered correctly

Part II there were 4 questions
9 Part 1 was on the Cauchy Riemann conditions which I think I managed to answer more or less correctly
  Part II was on curves and the effects of a transformation. Normally the curves are quite straightforward but this time they weren't and so only managed to answer about 1/2 of the second part

Then my last question 10 should have been a straightforward derivation of a Laurent Series for the function

$$\frac{z}{1-cosz} $$

however I couldn't get it out all the usual tricks didn't work and I couldn't see how to proceed

I abandoned this and was able to answer a question on the Complex Integration of the series as they gave the answer. The second part was about the singularities of the function but by this time my brain was frazzled so I gave up.

So about 1/2 of part 2 answered. So not a satisfactory paper. I've done enough to pass probably grade 3
but certainly no more.

M208 This went much better
Part 1 had twelve questions
1 Sketch of a graph.
2 Exercise on proof involving the converse of a statement
3 Question on whether or not two sets were groups
4 Another question on groups and their cosets
5 A question on row reduction
6 A question on basis for a linear transformation I did not answer this as I hadn't revised this so left it
7 A question on the solution to an inequality
8 A question on whether or not two sequences were convergent or not
9 A question on the symmetry groups of a hexagon
10 A question on homomorphisms
11 A question on L'Hopital's Rule
12 A question on showing that an Integral was less than a certain value
    Think I got most of this correct 
So apart from question 6 managed to give reasonably full answers to most of the questions but may have lost one or two marks here and there

Part II had 5 questions of which I answered 2
13 A question on the diagonalisation of a 3x3 matrix. Seemed to come out so confident that I got most of the marks for this question
17 A question on the Taylor polynomial of a function which I got most of

     A question on the epsilon delta definition of continuity. By this time it was the last 15 minutes and I was feeling quite tired. Also as I did not write out a sample problem of this type in my handbook unlike some people I didn't phrase the answer properly so will probably only get a few marks for this question.

Still having answered most of part 1 and 3/4 of part 2 I think I've done enough to get grade 2 and if the examiners are feeling generous might get distinction.

Some people might think I'm being perverse in refusing to annotate the handbook. My argument is that the exam is a test of how much you know without any crutches. Not an ability to spot model answers to a question and then simply copying them out. There was much heated debate on this in the forum. In my opinion annotation should not be allowed however no doubt other people will think differently.

So a bit disappointed by my performance on M337 but reasonably satisfied with my performance on M208
Will try and review both courses in the next week or two and also how I plan to use down time productively.