HSC 2013 MX2 Marathon (archive) (3 Viewers)

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Sy123

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Re: HSC 2013 4U Marathon

still editing




Edit: sorry, got to study for English now, someone feel free to take over lol.
Very good, a different approach to what I had in mind for the second one however. The third one is straightforward when b=3a then there is a horizontal point of inflection at x=1.

Can someone else post a question now? (preferably Complex and Polynomials, (or also anything that can be done with 2U))
 

SpiralFlex

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Re: HSC 2013 4U Marathon

^2U questions should be placed in the 2U thread. I'll post a question in a few minutes.
 

Sy123

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Re: HSC 2013 4U Marathon

2U questions should be placed in the 2U thread. I'll post a question in a few minutes.
Well the really hard ones should probably not be. For example my triangular numbers question should be in this thread imo (or 3U one)
 

SpiralFlex

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Re: HSC 2013 4U Marathon

Would you like circle geometry problems?

4U circle isn't any real extra theorems, it's just harder problems to 3U.
 

Sy123

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Re: HSC 2013 4U Marathon

Would you like circle geometry problems?

4U circle isn't any real extra theorems, it's just harder problems to 3U.
Err yeah sure I guess, it isnt my strength but practise makes perfect.
 
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Re: HSC 2013 4U Marathon

Very good, a different approach to what I had in mind for the second one however. The third one is straightforward when b=3a then there is a horizontal point of inflection at x=1.

Can someone else post a question now? (preferably Complex and Polynomials, (or also anything that can be done with 2U))
urgh so tired of belonging :( How come though?

Anyway here's a difficult complex/circle geo question that keeps propping up everywhere lol

 

RealiseNothing

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Re: HSC 2013 4U Marathon

i) Show that

ii) Consider the quadratic where the co-efficients are all positive.

If the roots are both complex, show that

iii) If one of the roots is , deduce that:

 

RealiseNothing

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Re: HSC 2013 4U Marathon

Actually I just remembered an awesome question (aye aye spiral :p)

If are all distinct positive integers, find what they are so that:

 

SpiralFlex

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Re: HSC 2013 4U Marathon

Actually I just remembered an awesome question (aye aye spiral :p)

If are all distinct positive integers, find what they are so that:

M8 this question cost you a generation of babies + you know.
 

Sy123

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Re: HSC 2013 4U Marathon

urgh so tired of belonging :( How come though?

Anyway here's a difficult complex/circle geo question that keeps propping up everywhere lol

I just found a mistake in my question, when I did it I divided the an inequality by b, which is not allowed since that is assuming b>0 there is indeed two domains for b such that there are 3 real roots.

Will do the question now
 

bobmcbob365

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Re: HSC 2013 4U Marathon

Lol, is the answer a = 479, b=478; c=31, d=2; a=45, b = 42; c=161, d=158
 
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Sy123

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Re: HSC 2013 4U Marathon

i) Show that

ii) Consider the quadratic where the co-efficients are all positive.

If the roots are both complex, show that

iii) If one of the roots is , deduce that:

Assuming a, and b real:

Construct:











Are you sure the last result is correct?
 

cutemouse

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Re: HSC 2013 4U Marathon

There's another way to do this problem using mappings in complex analysis. It's WAY more elegant.

EDIT: Here is my sketch solution:

Let 'c' be the complex number representing centre of the circle. Then the equation of the circle is |z-c| = |c| (since the circle passes through O).

Define the mapping w=1/z

Then |1/w - c| = |c|

This leads to |w-1/c| = |w|

This represents a line (in the w plane that is the right bisector of the join from O to 1/c).

As z_1, z_2, z_3 lie on |z-c|=|c| therefore 1/z_1, 1/z_2 and 1/z_3 lie on |w-1/c| = |w| and therefore are collinear.
 
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