Send him a message about it and ask for a free Terry Lee 4U textbook as a reward.Had a brief glance, I think his Q16 (a) (iii) is incorrect.
 < g(x) $ then it's not necessarily the case that $ \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = 0. $ For example $ x^2 > x^2 + 1 $ but $ \lim_{x \rightarrow \infty} \frac{x^2}{x^2 + 1} = 1)
The answer to the question is
Would be wary about writing this solution in an MX2 exam.The answer to the question is
Since there are infinite spaces in each column, you can treat each column as a number (say 1, 2, ... , q).
The question is equivalent to "if I have a bag with the numbers 1 up to q in it, and I pick out q numbers with replacement, what is the probability that I pick out each number exactly once?"
The problem with that logic is ALL the columns now have infinitely many squares, not just one. Yes there are an infinite amount of squares in one column that the black counter can be put in, but there is also a infinite amount of squares NOT in that column. The logic breaks down when you are dealing with the competing infinities (not really sure what else to call it).
I would have thought that was the approach they were looking for though. It might not be completely rigorous but it is the HSC and they aren't always frantic about those things.
I guess we will have to wait till the exam notes come out from BOS to know for sure, but usually the questions at this point in the paper are written by people who know well enough that the answers they are designed to elicit are actually quite rigorous.
Yep, this is good and valid.
Knowing the HSC, I think they just wanted people to answer the Q. by finding the limit of a ratio of same degree polynomials. That's probably why they had the part (ii) (and gave you the expression for it, so you could still do the next part if you didn't manage to get (ii)). Then they just wanted or expected you to find the limit of this expression.
Is it ok to simplify it further and say (q-1)! / q^(q-1) ?Yep, this is good and valid.
Yes, you won't lose marks for doing that.
Yeah, sure. There's no way they would mark that wrong lol.
Under exam conditions I initially thought I was wrong, so I tried simplifying it as much as possible and eventually just ended up with saying (q-1)! / q^(q-1) as my final result, and so seeing that most solutions just kept it as q! / q^q, I got worried that my 'simplified' form may not be accepted lol.