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HSC 2018-2019 MX2 Marathon (1 Viewer)

fan96

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Re: HSC 2018 MX2 Marathon

i)


i.e. . But,



Therefore the locus of is the line , which is linear and passes through .

The exception is when , where the locus of is all complex numbers . Going off the question, I will assume is nonzero.

ii)


Using a substitution obtainable from the locus from i), , we have



Simplify and we get the quadratic equation



Let the roots be . Then, using the product of roots formula,



Which I will call equation . Now,



Using equation ,



Therefore,

, for .

___________________________________________________

Considering the case where :



So, either of or or both are zero.

If and it can be proved in a similar manner that .

If , which I addressed in part i).
 
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altSwift

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Re: HSC 2018 MX2 Marathon

I can't load any latex in this thread anymore, I've tried on 3 computers, is anyone else getting this issue or just me?
 

fan96

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Re: HSC 2018 MX2 Marathon

Yep, it's broken for me as well.
 

Paradoxica

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Re: HSC 2018 MX2 Marathon

Find separate expressions for the real and imaginary parts of

Hence determine as a sum of independent real and imaginary parts
 

fan96

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Re: HSC 2018 MX2 Marathon

where did you get that part from? nice solution btw


and satisfies the locus in i), so



which gives us



therefore

 
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fan96

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Re: HSC 2018 MX2 Marathon

Find separate expressions for the real and imaginary parts of

Hence determine as a sum of independent real and imaginary parts
Let .

(Note that if then , and we have . Otherwise,)

Expanding, we get .

Equating real and imaginary parts gives us



Squaring,



Adding and using the identity with , we get



Since and are both positive, we can take the positive square roots of both sides to get





and since ,



So cleaning up, we have



and we can get rid of that second sign if desired:



Where is the sign function.
 
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Paradoxica

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Re: HSC 2018 MX2 Marathon

Let .

(Note that if then , and we have . Otherwise,)

Expanding, we get .

Equating real and imaginary parts gives us



Squaring,



Adding and using the identity with , we get



Since and are both positive, we can take the positive square roots of both sides to get





and since ,



So cleaning up, we have



and we can get rid of that second sign if desired:



Where is the sign function.
 

altSwift

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Re: HSC 2018 MX2 Marathon

No idea how to do ii):

capture.PNG

I know this isn't the best place to put this but i didn't want to dedicate a thread to a 1 mark question that's apparently simple
 
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CapitalSwine

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Re: HSC 2018 MX2 Marathon

I dont really understand why you have to find the stationary points (says so in the answers), would appreciate an expanation on how to solve this problem.
 

pikachu975

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Re: HSC 2018 MX2 Marathon

I dont really understand why you have to find the stationary points (says so in the answers), would appreciate an expanation on how to solve this problem.
If you know the stationary points you can get a rough sketch of the graph and hence find the k values (which shift the graph up and down) which satisfy the equation having one real root
 

fan96

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Re: HSC 2018 MX2 Marathon

I dont really understand why you have to find the stationary points (says so in the answers), would appreciate an expanation on how to solve this problem.
For simplicity, let .

First, sketch the graph of (the quickest way to do this is to find the turning points of the graph). The graph of can be obtained by shifting the graph of by units up or down.

Finally, a polynomial has as many real roots as its graph has -intercepts. So a polynomial has exactly one real root if and only if it crosses the -axis exactly once.
 

Paradoxica

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Re: HSC 2018 MX2 Marathon

For which a ∈ R is the sum of the squares of the zeroes of x for
x² − (a − 2)x − a − 1 minimal
P(√x)=0 has roots α², β²

Simplify the equation to form a standard quadratic equation.
x - (a+1) = (a-2)√x

x² - 2(a+1)x + (a+1)² = (a-2)²x

x² - 2(a+1)x + (a+1)² = (a²-4a+4)x

x² - (a²-4a+4+2a+2)x + (a+1)² = 0

x² - (a²-2a+6)x + (a+1)² = 0

α²+β² = a²-2a+6 = (a-1)²+5

clearly the minimum is 5, for a=1
 
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