HSC 2013 MX2 Marathon (archive) (1 Viewer)

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Re: HSC 2013 4U Marathon

Conics is quite bland I find... The proof of certain properties are memory based (some only, don't jump on me). Also there aren't that many applications nowadays and it isn't explored in depth in university so it has a lot less motivation than perhaps complex numbers or polynomials (and calculus :p)
 

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Re: HSC 2013 4U Marathon

Conics is quite bland I find... The proof of certain properties are memory based (some only, don't jump on me). Also there aren't that many applications nowadays and it isn't explored in depth in university so it has a lot less motivation than perhaps complex numbers or polynomials (and calculus :p)
I wish they put Real Analysis as a topic and remove conics :D
 

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Re: HSC 2013 4U Marathon

wtf guys, conics is awesome! Probably one of the easiest topics in MX2 imo, and a lot of fun! :D
I find polynomials pree boring, complex numbers was very fun, and will be doing integration then curve sketching soon.
 

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Re: HSC 2013 4U Marathon

wtf guys, conics is awesome! Probably one of the easiest topics in MX2 imo, and a lot of fun! :D
I find polynomials pree boring, complex numbers was very fun, and will be doing integration then curve sketching soon.
Conics is a cool topic but in my opinion it is the most boring topic compared to the rest of MX2 topics.
Polynomials is considered to be one of the simplest topics by most school teachers but the questions can get very difficult and interesting. Most school teachers go over relation between roots and roots of multiplicity which are very simple and leave out the hard end...
 
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Sy123

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Re: HSC 2013 4U Marathon







Now notice that the function:
So its an even function, hence the roots are opposites of each other.

So I will only take one of the above cases, namely:









is hence a solution

(pm there for the extra solution)





Hence those are the 6 solutions. Note how I skipped some cases for the pm and stuff only because they will just give the same answers that I have before.
There is probably a much easier way to do this question, and that I did it the long way, but oh well.
 
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Re: HSC 2013 4U Marathon

Good observation on the cos3x expansion Sy!

I would definitely like to see it 'solved in 1-2 lines'...but the 'long way' would be expanding it all and factorising l0l
 

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Re: HSC 2013 4U Marathon

i) 0, 1, 1, 2, 3, 5, 8

ii) and iii)









iv) To find alpha and beta just solve the quadratic, to get A and B, just do common denominator and simultaneously solve for A and B. It is a long process, the end yields something hidden within my notes and I can't be bothered doing it again:

v)

Note how:



To find F_k we must equate the co-efficient of z^k on both sides



Now, just sub in alpha, beta, A and B, rationalise denominators and some arithmetic, it should yield the final result.

======================================================

As for conics, I find it a little dry considering there are no advanced conics and the questions are relatively simple. i.e. no rotated ellipses hyperbola etc.
But its a pretty good topic in my opinion (and in spirit I made a question for it)













 
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Sy123

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Re: HSC 2013 4U Marathon

This can also be proved using mappings in complex numbers/analysis.
This is true (as had been demonstrated previous pages back) but I'm pretty sure the parametric equations of a hyperbola are out of syllabus (hyperbolic sine and cosine).

EDIT: I did it and it is still applicable to use complex numbers by simply using x+iy (disregard the above part)
But the question still stands lol
 
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Sy123

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Re: HSC 2013 4U Marathon

let be a root of z^n-1=0
Lets form a polynomial with roots, 1-1, 1-z_1, 1-z_2, ....



Therefore the polynomial with roots:



is





Now we will eliminate one solution, namely 0, or 1-1



The above equation has roots (1-z_1), (1-z_2), .... , (1-z_(n-1))

The constant term is -n ??
 
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