Looking at solutions, I think I dropped 1 mark overall, in Question 13 (a) (ii). I used the formula for half of a circle instead of a quarter circle =(
BTW I know what the last question is in the 4U exam. It's a really awesome result and it is just so unexpected!
jk, but it's probably going to be a neat result anyway, knowing Carrot
Re: HSC 2013 4U Marathon
It is a good method, but why do you say this as if it is alien to you? You can imagine that the proof for this would be very similar to the proof of the existence of complex conjugate pairs in real polynomials, since the complex conjugate is analogous to the irrational...
Re: HSC 2013 4U Marathon
\\ k(k+1)(k+2)...(k+n) = \frac{(k+n)!}{(k-1)!}, where k>1 and we want to prove that \frac{(k+n)!}{(k-1)!n!} is an integer. Consider \binom{k+n}{k-1}, which we know is an integer K. \binom{k+n}{k-1} = \frac{(k+n)!}{(k-1)!(n+1)!}=K so \frac{(k+n)!}{(k-1)!n!}=K(n+1)...
Re: HSC 2013 4U Marathon
Normally would have taken the usual z' \mapsto \frac{1+z}{1-z}$ approach to observe that z = \frac{z'-1}{z'+1}, and then substitute into P(z). We acquire a polynomial that is easy to prove has all complex roots but proving that they are purely imaginary is so long...