# Thread: HSC 2017 MX2 Marathon (archive)

1. ## Re: HSC 2017 MX2 Marathon

Originally Posted by seanieg89
Suppose you have n points on a circle such that no three distinct chords coincide at any single point.

How many regions do the nC2 chords divide the interior of the circle into?

(You don't have to provide rigorous proof for this question if you can guess the answer correctly).
2m where m is the number of chords. (2*nC2??)

Sorry thats not right, hold on.

2. ## Re: HSC 2017 MX2 Marathon

Originally Posted by frog1944
A question posted in the 2016 MX2 marathon by Paradoxica that wasn't answered:
$\lim_{x \to 0} \frac{\sqrt{1- \cos x^2}}{1 - \cos x}$
$\lim _{x \to 0}\frac{\sqrt{1-\cos{x^2}}}{1-\cos{x}} \\\\ =\lim _{x \to 0}\frac{x\sin{x^2}}{\sin{x}\sqrt{1-\cos{x^2}}} \text{ (L'Hospital's rule) } \\ \\\\ =\lim _{x \to 0}\frac{\sin{x^2}}{\sqrt{1-\cos{x^2}}} \text{ as }\lim _{x \to 0}\frac{x}{\sin{x}}=1 \\ \\\\ =\lim _{x \to 0}\frac{\sin{x^2}\sqrt{1+\cos{x^2}}}{\sqrt{1-\cos^2{x^2}}} \\\\ =\lim _{x \to 0}\sqrt{1+\cos{x^2}} \\ =\sqrt{1+1}=\sqrt{2}$

3. ## Re: HSC 2017 MX2 Marathon

Part (b) is a little long and requires good understanding of the problem, so keep that in mind

$\\ Consider the unit circle \ x^2 + y^2 =1 \ and the parameterisation \ C_t\left(\frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2} \right) \ and denote the point \ O(-1,0)$

$\\ Define a special kind of operation on the points of the unit circle so that if \ P , Q \ are points on the unit circle, then \ P + Q = R \ where \ R \ is the point such that \ RO \ is parallel to \ PQ$

$\\ In the case where \ P = Q \ then define \ P + P = R \ where \ RO \ is parallel to the tangent to the circle at \ P$
---------------------------------
$\\ a) Verify the following:$

$\\ i) \ P + Q = Q + P$

$\\ ii) \ P + O = O + P = P$

$\\ iii) For each point \ P \ on the circle, there is a point \ Q \ so that \ P + Q = O$

$\\ iv) \ P + (Q + R) = (P + Q) + R$
---------------------------------
$\\ b) If we define \ n \cdot P = \underbrace{P + P + \dots + P}_{n \ \text{times}}, \ then, other than the points \ (1,0) \ and \ (-1,0) \ are there any points \ P \ so that \ n \cdot P = O \ for some \ n \ ?$
---------------------------------
$\\ c) (Extension) Can a similar operation be defined on the general conic? In particular, investigate this operation on the conic \ P: x^2 - 2y^2 = 1 \ , choosing an appropriate \ O \ , and show that if \ (a,b) \ and \ (c,d) \ are points on \ P, then \\ (a, b) + (c,d) = (ac +2bd, ad + bc) \\ and hence show that addition of points on the conic \ P \ behaves like multiplication of numbers of the form \ a + b \sqrt{2} \ where \ a,b \ are rational$

4. ## Re: HSC 2017 MX2 Marathon

Originally Posted by seanieg89
Suppose you have n points on a circle such that no three distinct chords coincide at any single point.

How many regions do the nC2 chords divide the interior of the circle into?

(You don't have to provide rigorous proof for this question if you can guess the answer correctly).
something along the lines of the sum of three binomial coefficients of even lower argument...

I'll leave the details to the interested reader...

5. ## Re: HSC 2017 MX2 Marathon

Just a reminder this thread is mainly for Q1-15 material, and should be accessible to all Ext 2 students, not just the really talented ones.

I have moved the Lagrange multiplier & IMO question to the Advanced Marathon thread here...

If you have questions for help not related to questions posted here, it is advisable to post a separate thread rather than posting here in the marathon thread thanks. Although feel free to post a question here of course...
I have moved some of those also out of this thread.

here is another question to attempt
abbotsleigh20034u_q7b.PNG

6. ## Re: HSC 2017 MX2 Marathon

Just did my half yearly today... 30% of my internal mark... Calculating that im getting in the 70%'s
Why am i still in this course

7. ## Re: HSC 2017 MX2 Marathon

current question:
abbotsleigh20034u_q7b.PNG

8. ## Re: HSC 2017 MX2 Marathon

Originally Posted by dan964
current question:
abbotsleigh20034u_q7b.PNG
dat quality

9. ## Re: HSC 2017 MX2 Marathon

$\noindent Let n be a positive integer and let \zeta_1, \zeta_2, and \zeta_3 be the roots of the equation z^3 + 1 = 0. Here z \in \mathbb{C}.$

$\noindent If a> 0 show that \frac{1}{3} \sum^3_{k = 1} \frac{1}{|\zeta_k - a|^2} = \frac{1 + a^2 + a^4}{(1 + a^3)^2}.$

10. ## Re: HSC 2017 MX2 Marathon

HSC 2016 Q 10. I spent 5 minutes this morning on this question and couldn't work it out.

$Suppose that x+\frac{1}{x}=-1$

$What is the value of x^{2016}+\frac{1}{x^{2016}} ?$

Any tips? Advice? Short cuts? I assume 5 minutes would be too long for a question like this .

11. ## Re: HSC 2017 MX2 Marathon

Originally Posted by davidgoes4wce
HSC 2016 Q 10. I spent 5 minutes this morning on this question and couldn't work it out.

$Suppose that x+\frac{1}{x}=-1$

$What is the value of x^{2016}+\frac{1}{x^{2016}} ?$

Any tips? Advice? Short cuts? I assume 5 minutes would be too long for a question like this .

12. ## Re: HSC 2017 MX2 Marathon

Originally Posted by davidgoes4wce
HSC 2016 Q 10. I spent 5 minutes this morning on this question and couldn't work it out.

$Suppose that x+\frac{1}{x}=-1$

$What is the value of x^{2016}+\frac{1}{x^{2016}} ?$

Any tips? Advice? Short cuts? I assume 5 minutes would be too long for a question like this .
$\noindent Note that x must be non-real, since x+\frac{1}{x} is always at least 2 in absolute value for real x. Try substituting x = \mathrm{cis}\left(\theta\right). (If you initially substitute the more general x = r\,\mathrm{cis}\left(\theta\right), where r = |x|, you'll be able to show r must be 1.) Also recall the result that if x = \mathrm{cis}\left( \theta\right), then x^{n} + \frac{1}{x^{n}} = 2 \cos n\theta.$

13. ## Re: HSC 2017 MX2 Marathon

Originally Posted by davidgoes4wce
HSC 2016 Q 10. I spent 5 minutes this morning on this question and couldn't work it out.

$Suppose that x+\frac{1}{x}=-1$

$What is the value of x^{2016}+\frac{1}{x^{2016}} ?$

Any tips? Advice? Short cuts? I assume 5 minutes would be too long for a question like this .
we know that x^2+x+1=0. but this means that x is a cube root of unity ie x^3=1
therefore x^2016=1 and the solution is 2

14. ## Re: HSC 2017 MX2 Marathon

$\noindent Let n be a positive integer and let \zeta_1, \zeta_2, and \zeta_3 be the roots of the equation z^3 + 1 = 0. Here z \in \mathbb{C}.$

$\noindent If a> 0 show that \frac{1}{3} \sum^3_{k = 1} \frac{1}{|\zeta_k - a|^2} = \frac{1 + a^2 + a^4}{(1 + a^3)^2}.$
Is this by any chance just you taking a special case of the AMM Problem 11947 of 2016?

15. ## Re: HSC 2017 MX2 Marathon

$\noindent Let n be a positive integer and let \zeta_1, \zeta_2, and \zeta_3 be the roots of the equation z^3 + 1 = 0. Here z \in \mathbb{C}.$

$\noindent If a> 0 show that \frac{1}{3} \sum^3_{k = 1} \frac{1}{|\zeta_k - a|^2} = \frac{1 + a^2 + a^4}{(1 + a^3)^2}.$
Is this by any chance just you taking a special case of the AMM Problem 11947 of 2016?
Why indeed it is! As asked, the question makes for a very nice complex number question (and a bit of algebra) which is more than doable at the MX2 level.

16. ## Re: HSC 2017 MX2 Marathon

Hey guys, I have a question that I have never encountered before in 4U, how would I go about solving this? Thanks!
CodeCogsEqn.gif

17. ## Re: HSC 2017 MX2 Marathon

Originally Posted by si2136
Hey guys, I have a question that I have never encountered before in 4U, how would I go about solving this? Thanks!
CodeCogsEqn.gif
$\noindent It's of the form x = A\sec \theta, y = B\tan \theta, which as we know is a parametric representation of the hyperbola \frac{x^{2}}{A^{2}} - \frac{y^{2}}{B^{2}} = 1.$

18. ## Re: HSC 2017 MX2 Marathon

Just a curious question, is there a a faster way to prove the Recursive Integration Formulae without going through the long process?

19. ## Re: HSC 2017 MX2 Marathon

Originally Posted by si2136
Just a curious question, is there a a faster way to prove the Recursive Integration Formulae without going through the long process?
What's the long process? Integration by parts?

If you mean integration by parts, then almost all questions involve IBPs, but some can be done by manipulating the integrand.

20. ## Re: HSC 2017 MX2 Marathon

Originally Posted by si2136
Just a curious question, is there a a faster way to prove the Recursive Integration Formulae without going through the long process?
When you are asked to prove a given reduction formula, the efficient way is usually by differentiation.

21. ## Re: HSC 2017 MX2 Marathon

Given:

x+y+z = 3a

xy + yz + zx = 3b

Where x,y,z are all real, and ab>0, find the greatest and least possible values of any of x,y,z

22. ## Re: HSC 2017 MX2 Marathon

$\noindent A pack of sweets contains 25 sweets, a combination of mini-cupcakes and mini-macarons. If there are 18 varieties of mini-cupcakes, and 10 varieties of mini-macarons, how many ways can you fill a pack? (The order does not matter, just how many of each sweet type you have in the pack.)$

23. ## Re: HSC 2017 MX2 Marathon

$\noindent Given S_n =\sum_{k=1}^{n}k^2. Prove by mathematical induction that: \\ \\ \qquad nS_{n}-\sum_{r=1}^{n-1}S_r= \sum_{r=1}^{n}r^3 \quad \forall n\in \mathbb{Z}: n>1$

24. ## Re: HSC 2017 MX2 Marathon

Hey Guys!!

I am new around here, and not the best at this 4unit maths subject. When doing questions, I am not able to see the distinct ideas, and i get stuck, not knowing how to approach, or what the best and most efficient approach is. Any tips and help will be most appreciated! <3

25. ## Re: HSC 2017 MX2 Marathon

Originally Posted by EngineeringHelp
Hey Guys!!

I am new around here, and not the best at this 4unit maths subject. When doing questions, I am not able to see the distinct ideas, and i get stuck, not knowing how to approach, or what the best and most efficient approach is. Any tips and help will be most appreciated! <3
It will come with practice. Maybe post an example question so we can guide you through the thought process!

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