HSC 2016 MX2 Marathon ADVANCED (archive) (2 Viewers)

Sy123 · Oct 24, 2015

Post questions within the scope of Mathematics Extension 2 that are in general Q16 and beyond, focusing on problem solving and neat results within the reach of elementary mathematics.

InteGrand · Oct 24, 2015

Re: HSC 2016 4U Marathon - Advanced Level

Sy123 said:
Post questions within the scope of Mathematics Extension 2 that are in general Q16 and beyond, focusing on problem solving and neat results within the reach of elementary mathematics.

I remember the 2014 HSC 4U Advanced marathon was allowed to continue for a couple of years. Why not the 2015 one (or why did the 2014 one have to stop)? (Though the 2015 one probably wasn't as good or vibrant as the 2014 one, so it wouldn't be as fun to keep it going I suppose.)

InteGrand · Oct 24, 2015

Re: HSC 2016 4U Marathon - Advanced Level

Since 2016'ers probably have just started complex numbers recently, relatively easy Q to start off:

$Let $z$ and $w$ be complex numbers. Show that $\left|z-w \right| \geqslant \Big{|}|z|-|w| \Big{|}$. What is the geometric interpretation?$

Sy123 · Oct 24, 2015

Re: HSC 2016 4U Marathon - Advanced Level

$\\ $Show that the only pairs of positive integers$ \ x , y \ $so that$ \\ x^y = y^x \ $and$ \ x \neq y \\ $are$ \ x = 2, y = 4 \ $and$ \ x = 4, y = 2$

Sy123 · Oct 24, 2015

Re: HSC 2016 4U Marathon - Advanced Level

InteGrand said:
I remember the 2014 HSC 4U Advanced marathon was allowed to continue for a couple of years. Why not the 2015 one (or why did the 2014 one have to stop)? (Though the 2015 one probably wasn't as good or vibrant as the 2014 one, so it wouldn't be as fun to keep it going I suppose.)

2014 (actually 2013) stopped because mods wanted it to (I wanted it to keep going)

InteGrand · Oct 24, 2015

Re: HSC 2016 4U Marathon - Advanced Level

Sy123 said:
$\\ $Show that the only pairs of integers$ \ x , y \ $so that$ \\ x^y = y^x \ $and$ \ x \neq y \\ $are$ \ x = 2, y = 4 \ $and$ \ x = 4, y = 2$

(They do need to be positive integers of course (for that to be true).)

InteGrand · Oct 24, 2015

Re: HSC 2016 4U Marathon - Advanced Level

Sy123 said:
2014 (actually 2013) stopped because mods wanted it to (I wanted it to keep going)

Haha yeah, it was from 2013. And started in calendar year 2012 I think.

Drsoccerball · Oct 24, 2015

Re: HSC 2016 4U Marathon - Advanced Level

Here's a crappy one I made

$i) $ Prove that the equation $ 16x^4 -16x^2 +1 $ has a root $ x=\frac{1}{2}\sqrt{2-\sqrt{3}} $ by letting $ x= \frac{1}{2}\sqrt{2-\sqrt{3}}$

$ii) $ Using De Moivre's theorem find the quartic equation with a root $ cos(\frac{5 \pi}{12})$

$iii) $ Explain the significance of the results found above. $$

Paradoxica · Oct 24, 2015

Re: HSC 2016 4U Marathon - Advanced Level

$$The hyperbola $xy = 1$ is defined on the interval $x: {(1, \infty)}.$
$This segment of the function is rotated about the x-axis. This forms Gabriel's Horn.$
$a) Evaluate the volume of the resulting solid.$
$b) Evaluate the surface area of the resulting solid.$
$c) Compare the results obtained from a) and b). Explain the significance of these results.$

InteGrand · Oct 24, 2015

Re: HSC 2016 4U Marathon - Advanced Level

$(The surface area formula is $SA=\int _a ^b 2\pi f(x) \sqrt{1+\left( f^\prime (x)\right)^2}\text{ d}x$, where $SA$ is the area of the surface of revolution obtained by rotating the graph of $y=f(x)$ about the $x$-axis between $x=a$ and $x=b$.)$

$(Btw the brackets for the interval should be curved brackets, i.e. $(0,\infty)$).$

$(Also, the Gabriel's Horn part is taken to be the rotation of that hyperbola in the interval $[1,\infty)$.)$

Paradoxica · Oct 25, 2015

Re: HSC 2016 4U Marathon - Advanced Level

InteGrand said:
$(The surface area formula is $SA=\int _a ^b 2\pi f(x) \sqrt{1+\left( f^\prime (x)\right)^2}\text{ d}x$, where $SA$ is the area of the surface of revolution obtained by rotating the graph of $y=f(x)$ about the $x$-axis between $x=a$ and $x=b$.)$

$(Btw the brackets for the interval should be curved brackets, i.e. $(0,\infty)$).$

$(Also, the Gabriel's Horn part is taken to be the rotation of that hyperbola in the interval $[1,\infty)$.)$

Ah yes, fixed that.

calamebe · Oct 25, 2015

Re: HSC 2016 4U Marathon - Advanced Level

InteGrand said:
Since 2016'ers probably have just started complex numbers recently, relatively easy Q to start off:

$Let $z$ and $w$ be complex numbers. Show that $\left|z-w \right| \geqslant \Big{|}|z|-|w| \Big{|}$. What is the geometric interpretation?$

Proof: http://m.imgur.com/QApReFi

I'm guessing the geometric interpretation is that if you subtract w from z, it will be 'longer' (larger modulus) or be equal to the length of subracting the modulus of w from the modulus from z. Though that was a complete guess.

Small error: http://m.imgur.com/glJoL07

DatAtarLyfe · Oct 25, 2015

Re: HSC 2016 4U Marathon - Advanced Level

Oh cool theres a 4u section now for us. Now i have something to do when i'm bored

Drsoccerball · Oct 25, 2015

Re: HSC 2016 4U Marathon - Advanced Level

I think I'll recruit some of the new year 12s in my school to this forum.

InteGrand · Oct 25, 2015

Re: HSC 2016 4U Marathon - Advanced Level

calamebe said:
Proof: http://m.imgur.com/QApReFi

I'm guessing the geometric interpretation is that if you subtract w from z, it will be 'longer' (larger modulus) or be equal to the length of subracting the modulus of w from the modulus from z. Though that was a complete guess.

Small error: http://m.imgur.com/glJoL07

$The geometric interpretation is that if we have two fixed circles centred at the origin, and we pick one point each from them (representing complex numbers), the distance between the two points (which is $|z-w|$) is at least the width of the annulus formed by the two concentric circles (which is $\Big{|} |z|-|w|\Big{|}$, i.e. the difference of the radii).$

dan964 · Oct 25, 2015

Re: HSC 2016 4U Marathon - Advanced Level

InteGrand said:
$The geometric interpretation is that if we have two fixed circles centred at the origin, and we pick one point each from them (representing complex numbers), the distance between the two points (which is $|z-w|$) is at least the width of the annulus formed by the two concentric circles (which is $\Big{|} |z|-|w|\Big{|}$, i.e. the difference of the radii).$

Or in other terms the minimum distance between one point Z and another X on the two concentric circles i.e. |z-w| is the distance between the concentric circles itself ||z|-|w||. Equality occurs when arg z = arg w (it is a pointless statement if either z or w is 0)

It can simply be geometrically proved by using the triangle inequalities of the points O (origin), Z and W
|w|+|z-w|>= |z|
and |z|+|z-w|>= |w|

Rearranging gives
|z-w|>= |z|-|w|
and |z-w| >= |w|-|z|
which can leads to the generalised result:
|z-w|>= ||z|-|w||

Drsoccerball · Oct 25, 2015

Re: HSC 2016 4U Marathon - Advanced Level

Drsoccerball said:
Here's a crappy one I made

$i) $ Prove that the equation $ 16x^4 -16x^2 +1 $ has a root $ x=\frac{1}{2}\sqrt{2-\sqrt{3}} $ by letting $ x= \frac{1}{2}\sqrt{2-\sqrt{3}}$

$ii) $ Using De Moivre's theorem find the quartic equation with a root $ cos(\frac{5 \pi}{12})$

$iii) $ Explain the significance of the results found above. $$

.

lita1000 · Oct 25, 2015

Re: HSC 2016 4U Marathon - Advanced Level

Drsoccerball said:
.

What do you mean by using Demoivres theorem? It's not exactly clear what you're asking for, unless you mean like use the demoivre's theorem to derive the double angle formula for cos, and use that to find the exact value of cos (5pi/12), which is the squared value of the expression in part 1

Zen2613 · Oct 25, 2015

Re: HSC 2016 4U Marathon - Advanced Level

lita1000 said:
What do you mean by using Demoivres theorem? It's not exactly clear what you're asking for, unless you mean like use the demoivre's theorem to derive the double angle formula for cos, and use that to find the exact value of cos (5pi/12), which is the squared value of the expression in part 1

I think he wants us to use demoivre to find an expression for cos(4theta) in terms of cos(theta) because that will be a quartic which will probably resemble his 16x^4 - 16x^2 -1 = 0 thingo.

Zen2613 · Oct 25, 2015

Re: HSC 2016 4U Marathon - Advanced Level

Drsoccerball said:
.

I like that one. Nice introduction to 4 unit problems. But since you made the problem, how did you actually find the roots for that quartic equation in x ? The 1/2sqrt(2-sqrt(3)) ones.

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