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Author Topic: Mathematics Extension 2 Challenge Marathon  (Read 31637 times)  Share 

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RuiAce

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Mathematics Extension 2 Challenge Marathon
« on: February 28, 2016, 07:30:50 pm »
+4
Here, I will occasionally post challenge questions for the true maths brains to attempt.

Questions are not intended to reflect the scope of the difficulty in actual HSC questions, however may be completed using only knowledge taught in the course. Spoilers are intended to reveal what topics to draw knowledge from when a genuine, unaided attempt has been unsuccessful.

I invite everyone to also post their own questions at their own discretion, and for anyone who has completed 4U maths or equivalent to also answer. I invite collaboration, as from time to time, some questions may be, one will argue, ridiculous.



Spoiler
Required knowledge:
a) HSC 4U Complex Numbers
b) Preliminary 2U Tangent to a Curve and Derivative of a Function, HSC 4U Complex Numbers
c) Preliminary 2U Basic Arithmetic and Algebra, HSC 4U Complex Numbers, HSC 2U Trigonometric Functions

Edit: 3 years later I found a typo with part a). It has been fixed.
« Last Edit: January 12, 2019, 08:13:35 am by RuiAce »

RuiAce

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #1 on: March 07, 2016, 10:30:51 am »
+2
A devilishly scarring integral


Ali_Abbas

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #2 on: August 20, 2016, 10:17:26 pm »
0
For the integral of the square-root of tanx I got:

1/2*sin^-1[tan(x/2)] + C, C constant.

Is that correct ?

RuiAce

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #3 on: August 20, 2016, 10:25:45 pm »
+1
For the integral of the square-root of tanx I got:

1/2*sin^-1[tan(x/2)] + C, C constant.

Is that correct ?
Looks off.

What was your method?

birdwing341

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #4 on: August 21, 2016, 11:20:21 am »
+1
I got something stupefyingly difficult. This method does yield an answer, but I'm not sure if it is the most effective.

Sorry for unclear image

RuiAce

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Mathematics Extension 2 Challenge Marathon
« Reply #5 on: August 21, 2016, 11:31:08 am »
+1
I got something stupefyingly difficult. This method does yield an answer, but I'm not sure if it is the most effective.

Sorry for unclear image
Method's right. Don't remember the final answer off the top of my head but it takes a similar form.

The integral is famous. There's plenty of ways to do it and can be found everywhere on the Internet
« Last Edit: August 21, 2016, 11:34:59 am by RuiAce »

jamonwindeyer

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #6 on: August 21, 2016, 02:35:18 pm »
+2
For the integral of the square-root of tanx I got:

1/2*sin^-1[tan(x/2)] + C, C constant.

Is that correct ?

Welcome to the forums Ali! Happy to have you around ;D let me know if you need help finding anything ;D

Ali_Abbas

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #7 on: August 21, 2016, 03:29:35 pm »
+1
Welcome to the forums Ali! Happy to have you around ;D let me know if you need help finding anything ;D

Thanks for the welcome message Jamon :) If I require any assistance navigating my way around here I'll let you know.
Cheers

Ali_Abbas

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #8 on: August 22, 2016, 11:55:14 am »
0
Here's a question for people to try:

Suppose we have the hyperbola y = 1/x defined over the positive real numbers (first quadrant). Let P(p, 1/p) and Q(q, 1/q) be two arbitrarily fixed points along the curve, with p < q. Define M as the midpoint of the chord PQ. The line segment OM intersects the hyperbola at R(r, 1/r), where O is the origin (0,0).

Without expressing the coordinates of R in terms of p and q, i.e. without deriving the equation of the line OM, prove that the tangent to the hyperbola at R is parallel to the chord PQ.

Mod edit: Altered the language to make it a bit "easier" to comprehend with respect to the HSC 4U course
« Last Edit: August 23, 2016, 08:20:12 am by RuiAce »

RuiAce

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #9 on: August 23, 2016, 08:52:47 am »
+1
Gonna leave a diagram for anyone who attempts it.

Are you looking for a fully geometric proof? Because when I first glance at it, trying to prove pq=r2 is the easiest way to go about it but it feels pointless if the equation of OM is denied.

Ali_Abbas

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #10 on: August 23, 2016, 09:41:27 am »
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Gonna leave a diagram for anyone who attempts it.
(Image removed from quote.)
Are you looking for a fully geometric proof? Because when I first glance at it, trying to prove pq=r2 is the easiest way to go about it but it feels pointless if the equation of OM is denied.

Indeed, proving r^2 = pq (of which there are four ways to do so) is the most immediate way to achieve the desired result. However, as you have alluded to, I am in fact looking for a full geometric proof (not strictly Euclidean) just as a way to make it slightly more challenging.

Paradoxica

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #11 on: October 05, 2016, 06:32:06 pm »
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Consider an alphabet with n different available letters.

Let P(k) be the number of ways you can use exactly k different letters in an n letter word.

i) Explain why




ii) Show that



Let the Score of a word, X, be defined as 1/(1+ρ(X)), where ρ(X) is the number of letters that were not used by the word X.

iii) Show that the sum of all the Scores, S, over all possible n letter words, is given by:




iv) Hence, evaluate S in closed form.
« Last Edit: October 05, 2016, 06:35:42 pm by Paradoxica »

Paradoxica

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #12 on: October 05, 2016, 06:44:33 pm »
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'

Hint: It's a logarithm

 ;D

RuiAce

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #13 on: October 05, 2016, 07:15:14 pm »
+1
'

Hint: It's a logarithm

 ;D

Spoiler






Moderator Edit: Added spoiler to solution
« Last Edit: October 05, 2016, 07:17:03 pm by jamonwindeyer »

jamonwindeyer

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #14 on: October 05, 2016, 07:17:39 pm »
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Jesus Rui that was quick, save some for the students ;)