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HSC 2015 MX2 Marathon (archive) (1 Viewer)

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Sy123

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Re: HSC 2015 4U Marathon

Just an exercise in mathematical/logical argument and proof:









 

glittergal96

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Re: HSC 2015 4U Marathon

Just an exercise in mathematical/logical argument and proof:









What exactly are you allowed to assume about addition/subtraction in i) and ii)? On which sets is it defined and what does it map pairs of elements from this set into? Just saying that the sum of two positive integers is a positive integer doesn't seem to be enough. (Given that we don't have a rigorous definition of the integers or +,- here.)

iii) and iv) are easy enough because we can define rational addition/subtraction/multiplication in terms of the corresponding operations on the integers, but since the definition of integers and the elementary operations on them is really fundamental (close to the axiomatic base), I think you need to be clearer about what we are allowed to assume.
 

glittergal96

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Re: HSC 2015 4U Marathon

i) A straightforward induction. The key step:


ii) Directly replace each in the recurrence relation with and multiply out all negative powers of t.

iii) Lets introduce some additional assumptions so this thing converges. If , then i) shows that for every non-negative integer n. If as well, then the sum is absolutely convergent by comparison with the series We proceed under this assumption as it is certainly sufficient for iv). (There IS value in studying these things as an algebraic object even if they don't converge though.)



Solving for and using completes the proof.
 

glittergal96

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Re: HSC 2015 4U Marathon

Oh I forgot iv).

Well just sub in

to the identity proven in iii).

You can also find a surdic expression for x if you are that way inclined, by using double angle formulae for example.
 

Chlee1998

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Re: HSC 2015 4U Marathon

Oh I forgot iv).

Well just sub in

to the identity proven in iii).

You can also find a surdic expression for x if you are that way inclined, by using double angle formulae for example.
can you solve the questions in the current advanced thread?
 

braintic

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Re: HSC 2015 4U Marathon

For anyone who wants to picture the solid:

phpbyzuV5.jpg

(Not equilateral, but same idea)

Many people can't visualise the 'spine'. Others can't see that the curvature happens in only one dimension, like a cylinder.
 
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leehuan

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Re: HSC 2015 4U Marathon

Easy mechanics question from an HSC paper before you guys start confusing me again.





 
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