Examiners’ reports are very amusing. Here’s a paragraph from the 2010 Part A Mathematics:

Presentation skills were very mixed. Some candidates wrote to the standard of professional mathematicians, and their scripts were a pleasure to read. Others wrote as if grammar and style were far too costly to be lavished on mere examiners. Many scripts were unreasonably illiterate. Misuse of the symbol ⇒ to mean ‘then’ rather than ‘implies that’ was rampant, leading in a number of cases to assertions that could (and perhaps should) have been marked wrong. Too often crucial quantifiers, especially ‘there exists’, were missing (that is, left to the reader to supply).

And from 2009:

It also gave candidates the opportunity to show that they didn’t know what “singular” or “nonsingular” meant. Unfortunately, far too many candidates assumed that if the multiplicity of a root of the characteristic polynomial is r then there are r linearly independent eigenvectors. But then, somewhat surprisingly, those candidates did not deduce in one line that every linear transformation over ℂ is diagonalisable! Of course, if this should have been the case, then the problem, which leads to a proof of the Cayley-Hamilton theorem, would have been trivial (as would the whole of algebra…), but this didn’t stop those candidates from getting into even deeper water. …

This question, unwittingly, gave many candidates the opportunity to show lack of basic understanding. Leaving aside those who define W^⊥ = {w ∈ W^⊥ : ……} or worse … [f]or the next part, the few who realised that some geometry was going on fared best …

One candidate bravely wrote “in the lectures we were told that [if W is infinite dimensional, then] W^⊥⊥ = W″ – unfortunately, this is not universally true, for example when W = V !

On a computer screen yesterday I read this from a physics past paper (paraphrased): “This equation wasn’t done well. Perhaps we should have provided the auxiliary equation or shown the answer to be worked to? I didn’t think it was very difficult.”

Edit 28/vi/2011: Another someone quoted at me: “it would be an understatement to say that this question was badly answered: indeed, it caused utter carnage.” The report goes on for a page about how this happened.