# HSC 2013 MX2 Marathon (archive) (2 Viewers)

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#### Sy123

##### This too shall pass
Re: HSC 2014 4U Marathon

Yep well done

$\bg_white \\ Let \ P_n(x) \ be a sequence of polynomials for non-negative integers \ n \ and real \ x \\ \\ Such that \ \ \ P_n(\cos \theta) = \cos(n \theta) \\ \\ Show that \\ \\ P_{n+1}(x) = 2xP_{n}(x) - P_{n-1}(x)$

(I like the new latex font!)

#### anomalousdecay

##### Premium Member
Re: HSC 2014 4U Marathon

We needed a new LaTex renderer because of the problems we experienced before.

So inevitably the font changed!

#### Sy123

##### This too shall pass
Re: HSC 2014 4U Marathon

$\bg_white \\ Find \ b \ if the equation \ \ \ x^4 + ax^3 + bx^2 + cx + d = 0 \ \ \ has rational coefficients and 4 complex roots, two with sum 3 + 4i and the other two with product 13 + i.$

#### Sy123

##### This too shall pass
Re: HSC 2014 4U Marathon

$\bg_white \\ If \ \alpha , \beta \ are roots of the equation \ x^2 + px + 1=0 \ \\ \\ and \ \gamma , \delta \ are roots of the equation \ x^2+qx+1=0 \\ \\ Prove that \\ \\ (\alpha - \gamma)(\beta - \gamma)(\alpha + \delta)(\beta + \delta) = q^2-p^2$

#### study1234

##### Member
Re: HSC 2014 4U Marathon

#### theprofitable95

##### New Member
Re: HSC 2014 4U Marathon

Polys/Complex. Difficulty 3/5.

$\bg_white Find all real polynomials p(x) such that:\\ \\ p(x+1)p(x-1)=p(x^2+1)\\ \\ for all real x.$
These type of questions aren't even in the four unit syllabus are they? I mean I have no idea how to even to begin attempting this problem.

#### seanieg89

##### Well-Known Member
Re: HSC 2014 4U Marathon

These type of questions aren't even in the four unit syllabus are they? I mean I have no idea how to even to begin attempting this problem.
Well, they only aren't in the syllabus in the sense that some of them are harder than any question the bos would set in an actual HSC paper.

They don't need any theory from outside the mx2 course to do.

#### Sy123

##### This too shall pass
Re: HSC 2014 4U Marathon

Let f be a real-valued function defined on R. For which values of the real constant p does the inequality below imply that f is twice differentiable?

|f(x)-f(y)| =< C|x-y|^p for all x,y. C a constant.

Justify your answer with proof.
Just checking, but is the answer p=2?

#### Sy123

##### This too shall pass
Re: HSC 2014 4U Marathon

$\bg_white Prove for positive integers \ n \\ \\ \sqrt[n]{n!} \leq \sqrt[n+1]{(n+1)!}$

In case latex is down it is asking to prove:

(n!)^(1/n) <= ((n+1)!)^(1/(n+1))

for positive integral n

#### Sy123

##### This too shall pass
Re: HSC 2014 4U Marathon

A (2 dimensional) ball with radius 'r' has a fixed red marker on it. It is rolled along the flat ground and the red marker leaves a trail in the air.
If the ball was initially above the origin with the marker being at the origin. Find the Cartesian equation describing the path of the marker.

#### seanieg89

##### Well-Known Member
Re: HSC 2014 4U Marathon

Just checking, but is the answer p=2?
p=2 is not quite good enough to get second differentiability (consider f(x)=x^3/|x|, f(0)=0. this func is not twice diffble at 0, yet obeys the p=2 bound), (also, there is more than value of p that works).

#### jyu

##### Member
Re: HSC 2014 4U Marathon

$\bg_white Prove for positive integers \ n \\ \\ \sqrt[n]{n!} \leq \sqrt[n+1]{(n+1)!}$

In case latex is down it is asking to prove:

(n!)^(1/n) <= ((n+1)!)^(1/(n+1))

for positive integral n

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#### vafa

##### Member
Re: HSC 2014 4U Marathon

These type of questions aren't even in the four unit syllabus are they? I mean I have no idea how to even to begin attempting this problem.
Hint: Use the fundamental theorem of calculus: a polynomial of degree n, has n roots. Let set A be all the roots and let set B be A-1. From that you can see that B^2 is always a root. If B is a complex number and its modulus is bigger than one, you get an infinite number of solutions; this can not happen because you only should have n roots. If the modulus of that complex number is between 0 and 1 same thing happens therefore that the modulus has got to be 0 or 1 and continue this logic ...

#### braintic

##### Well-Known Member
Re: HSC 2014 4U Marathon

A (2 dimensional) ball with radius 'r' has a fixed red marker on it. It is rolled along the flat ground and the red marker leaves a trail in the air.
If the ball was initially above the origin with the marker being at the origin. Find the Cartesian equation describing the path of the marker.
The red marker has me stumped. But I think I can do it if it is green.

#### Sy123

##### This too shall pass
Re: HSC 2014 4U Marathon

$\bg_white \\ Using complex numbers, prove that if on a quadrilateral, two sides are opposite, equal and parallel, then the other two sides are parallel and equal.$

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#### seanieg89

##### Well-Known Member
Re: HSC 2014 4U Marathon

$\bg_white \\ Using complex numbers, prove that if on a quadrilateral, two sides are opposite and equal, then the other two sides are parallel and equal.$
Perhaps the assumption should instead be:

"Two opposite sides are equal and parallel."

#### seanieg89

##### Well-Known Member
Re: HSC 2014 4U Marathon

$\bg_white n lines are drawn in a plane, cutting it into regions. Prove that each of these regions can be coloured white or black such that no two adjacent regions have the same colour.$

Hint: Induction is one way to do this.

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#### Sy123

##### This too shall pass
Re: HSC 2014 4U Marathon

Perhaps the assumption should instead be:

"Two opposite sides are equal and parallel."
Yep thanks for the correction.

#### jyu

##### Member
Re: HSC 2014 4U Marathon

Polys/Complex. Difficulty 3/5.

$\bg_white Find all real polynomials p(x) such that:\\ \\ p(x+1)p(x-1)=p(x^2+1)\\ \\ for all real x.$
p(x)=1

#### dragerx

##### New Member
Re: HSC 2014 4U Marathon

Graph y= x

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