HSC 2013 MX2 Marathon (archive) (3 Viewers)

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RealiseNothing

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Re: HSC 2014 4U Marathon

By using a suitable transformation, solve:

 
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Sy123

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Re: HSC 2014 4U Marathon

By using a suitable transformation, solve:

Seeking to transform this equation into a reducible quadratic, we substitute,



Now, we find the value q such that the co-efficient of t^3 and t are zero, by equating co-efficients, we find that if we let q=8, the co-efficient of t^3 is zero, and by co-incidence so is t (since the question is constructed in this way).

Applying the substitution,



 

VBN2470

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Re: HSC 2014 4U Marathon

Prove the following statement (i) using mathematical induction (ii) without using mathematical induction for all positive integer values of
Statement.PNG
 
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braintic

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Re: HSC 2014 4U Marathon

Define a 'composition' of a positive integer n to be an ordered list of positive integers whose sum is n.
For example, the compositions of 3 are:
1+1+1
1+2
2+1
3

Let c(n) be the number of compositions of n.
So c(3) = 4 (from the above example)

(i) Show that c(n) = c(n-1) + c(n-2) + ... + c(2) + c(1) + 1

(ii) Hence prove by mathematical induction that c(n) = 2^(n-1)

(iii) Now find a slick way to justify why c(n) = 2^(n-1) without using part (i) or induction.
 
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RealiseNothing

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Re: HSC 2014 4U Marathon

Suppose that boys and girls are going to the movies such that .

If no girl wants to sit next to each other, find the amount of possible arrangements that they can sit in a row in the cinema.
 

RealiseNothing

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Re: HSC 2014 4U Marathon

Let the above situation be A, and now consider the same scenario but instead they all sit around a circular dinner table. Call this B.

Define to be the amount of arrangements of a situation .

Show that, despite the girls refusing to sit next to each other, the ratio is independent on the amount of girls that attend.
 

RealiseNothing

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Re: HSC 2014 4U Marathon

And yes, me and Sy did go to the USYD computers during our break and sat down and posted questions together. Don't judge.
 

VBN2470

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Re: HSC 2014 4U Marathon

Someone please answer the following question (stuck mainly on parts (b) - (d)):
Question.PNG
 

Carrotsticks

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Re: HSC 2014 4U Marathon

I'm guessing so we still have a hyperbola.

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Sy123

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Re: HSC 2014 4U Marathon

Why the restriction that b can't be zero? Doesn't it still work?
I stole this question from a math tutorial on usyd, it had that restriction. I'm guessing because if b=0 then the function is a straight line and the only case for its own inverse is the trivial y=x
 

seanieg89

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Re: HSC 2014 4U Marathon

I stole this question from a math tutorial on usyd, it had that restriction. I'm guessing because if b=0 then the function is a straight line and the only case for its own inverse is the trivial y=x
(And y = -x + k, k being anything at all).
 
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