HSC 2013 MX2 Marathon (archive) (3 Viewers)

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btx3

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Re: HSC 2013 4U Marathon

i came to this thread for d babes as per sy's sig
where them bad bitches at?
Sorry man, you were too late. The one available chick in this thread was unfortunately taken by realise just before you, i'll let you know when more babes get here
 

Immortalp00n

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Re: HSC 2013 4U Marathon

Sorry man, you were too late. The one available chick in this thread was unfortunately taken by realise just before you, i'll let you know when more babes get here
im disappointed in you lot.
but thanks for the offer <3
 
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Re: HSC 2013 4U Marathon

Sy and Realise.. I think the lack of participation comes partly from the range of questions as well as - ITS THE HOLIDAYS! You guys probably forgot that bit ;)
 

Sy123

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Re: HSC 2013 4U Marathon

I was playing around with these sorts of problems on the bus the other day, but I ended up confusing myself and didn't get very far.

Say you had a polynomial with roots and you want to find a polynomial with roots (Sy's question would be a special case of this where ).

You can find an equation with these roots by replacing every instance of in the polynomial with (this is an interesting property to investigate/prove). In some cases, that equation can be algebraically manipulated into a polynomial without creating any new roots. This is why the trick of inverting coefficients works for -- it's a shortcut to replacing all terms with and multiplying out the resulting fractions.

When isn't monotonic (meaning is not a function), it still works; I'm pretty sure you can choose a maximal interval on its domain over which it's monotonic (probably the wrong terminology) and use that for the inverse, because the only necessity is that . Also, I'm not convinced that all equations generated in this manner can be manipulated into polynomials without creating new roots (for example, I had some trouble doing it with Sy's question, but I could have just messed up the algebra). Perhaps someone else will have more luck investigating this?
It is relatively simple to prove the first result, and yes it is interesting.
I actually did try this method with my poly question but it didn't seem to output any encouraging results. And yes they probably are not able to be manipulated into polynomial form, but if they aren't in polynomial form - they should be infinitely differentiable right (not rigorous to assume this, but from what I have encountered thus far, they are infinitely differentiable) and hence we can somehow employ Newton's Method of Approximation an infinite number of times right? I have seen a question regarding this.

Proof of the first result:





Create a function in y, such that the roots are





Which has roots alpha_k for all integers k 0< k <= n

If f is not monotonic increasing/decreasing then we must define the inverse f-1 such in a domain(s) such that this domain stretches over all the roots.

I will explore this further later
 

seanieg89

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Re: HSC 2013 4U Marathon

It is relatively simple to prove the first result, and yes it is interesting.
I actually did try this method with my poly question but it didn't seem to output any encouraging results. And yes they probably are not able to be manipulated into polynomial form, but if they aren't in polynomial form - they should be infinitely differentiable right (not rigorous to assume this, but from what I have encountered thus far, they are infinitely differentiable) and hence we can somehow employ Newton's Method of Approximation an infinite number of times right? I have seen a question regarding this.

Proof of the first result:





Create a function in y, such that the roots are





Which has roots alpha_k for all integers k 0< k <= n

If f is not monotonic increasing/decreasing then we must define the inverse f-1 such in a domain(s) such that this domain stretches over all the roots.

I will explore this further later

This method does work for your question although it is not all that fast. I will post this solution and more about this method and it's limitations tomorrow, it was one of my favourite ways of doing such questions.
 

RivalryofTroll

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Re: HSC 2013 4U Marathon

Also, not really an actual question but I need opinions.

For each of the following topics, which textbook would be better? (I have only SK PATEL Excel MX2 and CAMBRIDGE 4U available)

Categories may include: questions, worked examples, etc.

- Complex Numbers
- Curve Sketching
- Conics
- Polynomials
 
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Re: HSC 2013 4U Marathon

Also, not really an actual question but I need opinions.

For each of the following topics, which textbook would be better? (I have only SK PATEL Excel MX2 and CAMBRIDGE 4U available)

Categories may include: questions, worked examples, etc.

- Complex Numbers
- Curve Sketching
- Conics
- Polynomials
Get your hands on a Terry Lee for curve sketching and conics...
 

GoldyOrNugget

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Re: HSC 2013 4U Marathon

Syyy i'm bored post another question.

P.S. my reasoning for the roots thing was the same as yours i think. if is our original polynomial, then we can create one with the new roots as since our only requirement is that . But if this is the case, then it is also the case that , which means an equivalent equation is , and this is exactly the same as our original polynomial except .
 

Sy123

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Re: HSC 2013 4U Marathon

Another one I made - its original:





Now since I can't draw properly - you must draw it instead, for purposes of this question:







(ignore the scaling on the animation)



Ok, notation is out of the way, now for the actual question:




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


















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

If you can't be bothered observing the diagram for part 2, the relationship is:

alpha=-3theta
 
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barbernator

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Re: HSC 2013 4U Marathon

Unsure how difficult/easy this question is.

3 parts:

a) there are 10 rooms in a row. 3 people names allen, bill and chris are placed each in a room by themself. What is the chance that chris is more rooms away from allen than bill.

b) there are n rooms in a row. 3 people names allen, bill and chris are placed each in a room by themself. What is the chance that chris is more rooms away from allen than bill.

c) there are n rooms in a row. m people (m < n) , called a,c,b,d..... What is the chance that the intervals a-b, a-c, a-d, a-e .... are all in increasing distances from each other.


I'll remove and post later if u want sy
 
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Sy123

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Re: HSC 2013 4U Marathon

Unsure how difficult/easy this question is.

3 parts:

a) there are 10 rooms in a row. 3 people names allen, bill and chris are placed each in a room by themself. What is the chance that chris is more rooms away from allen than bill.

b) there are n rooms in a row. 3 people names allen, bill and chris are placed each in a room by themself. What is the chance that chris is more rooms away from allen than bill.

c) there are n rooms in a row. m people (m < n) , called a,c,b,d..... What is the chance that the intervals a-b, a-c, a-d, a-e .... are all in increasing distances from each other.


I'll remove and post later if u want sy
no problem lol - if you can notice that the thread is dieing then go right ahead. But I think I'm gonna pass on this question
 
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RealiseNothing

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Re: HSC 2013 4U Marathon

Unsure how difficult/easy this question is.

3 parts:

a) there are 10 rooms in a row. 3 people names allen, bill and chris are placed each in a room by themself. What is the chance that chris is more rooms away from allen than bill.

b) there are n rooms in a row. 3 people names allen, bill and chris are placed each in a room by themself. What is the chance that chris is more rooms away from allen than bill.

c) there are n rooms in a row. m people (m < n) , called a,c,b,d..... What is the chance that the intervals a-b, a-c, a-d, a-e .... are all in increasing distances from each other.


I'll remove and post later if u want sy
For part (i) I got
 

RealiseNothing

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Re: HSC 2013 4U Marathon

Unsure how difficult/easy this question is.

3 parts:

a) there are 10 rooms in a row. 3 people names allen, bill and chris are placed each in a room by themself. What is the chance that chris is more rooms away from allen than bill.

b) there are n rooms in a row. 3 people names allen, bill and chris are placed each in a room by themself. What is the chance that chris is more rooms away from allen than bill.

c) there are n rooms in a row. m people (m < n) , called a,c,b,d..... What is the chance that the intervals a-b, a-c, a-d, a-e .... are all in increasing distances from each other.


I'll remove and post later if u want sy
If my part (i) is right then part (ii) is:

 

Sy123

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Re: HSC 2013 4U Marathon





 
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Carrotsticks

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Re: HSC 2013 4U Marathon

I finally found a way to find the taylor series of tan-1 x without the non-rigour:





There's a bit more to it than that =p

Firstly, we must have x lying on the unit circle as that is the maximal radius for which convergence occurs.

Secondly, you should probably add a part where they evaluate the integral of the n+1'th term, generally called the Error Term E(x), which you will then need to prove converges to 0 as n gets large.

Thirdly, it would be in your best interest to use say t as a dummy variable, and then integrate from 0 to x, where |x|<1, so then you can avoid issues with the constant term etc.
 
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