HSC 2013 MX2 Marathon (archive) (1 Viewer)

Sy123 · Jan 8, 2013

Re: HSC 2013 4U Marathon

asianese said:
$\text{i)} A=\int_1^n \frac{1}{x} dx\\ = [\ln x]^n_1 \\ = \ln(n) u^2\\ \text{ii)} V= \lim_{\delta x \to 0}\sum_{x=1}^n \pi [f(x)]^2 \delta x\\ =\pi \lim_{\delta x \to 0} \sum_{x=1}^n \frac{1}{x^2} \delta x\\ = \pi \int_1^n x^{-2} dx \\ = -\pi \left[ \frac{1}{x} \right]^n_1\\=\pi \left( 1-\frac{1}{n} \right) u^3\\\\ \text{iii)} \text{As n approaches infinity, the area underneath the curve increases indefinitely, whilst the volume generated by a full revolution of this area converges to } \pi u^3.$

very simple question but explores something quite cool, how we can have infinite volume but the area of the cross-section is infinite.
Similar to:

http://en.wikipedia.org/wiki/Gabriel's_Horn

HeroicPandas · Jan 8, 2013

Re: HSC 2013 4U Marathon

$$ Let w = $cos\frac{2\pi}{9} +isin\frac{2\pi}{9}$ \\ \\$ $ (i) Show that w^k$ $ is a solution of z^9 - 1 = 0,$ $ where k is an integer \\ \\ $ $ (ii) Prove that w + w^2 +...+w^8 = -1 \\ \\$ $ (iii) Hence show that $cos\frac{\pi}{9}cos\frac{2\pi}{9}cos\frac{4\pi}{9} = \frac{1}{8}$

without equating coefficients

GoldyOrNugget · Jan 9, 2013

Re: HSC 2013 4U Marathon

Sy your summation of products of triangular numbers & sines question is raping me. Hint plz?

so far I've been working with Re( (z+z^2+...+z^n) + 2(z^2 + ... + z^n) + ... + n(z^n) ), then simplified that with geometric series, but it's not getting anywhere.

Sy123 · Jan 9, 2013

Re: HSC 2013 4U Marathon

GoldyOrNugget said:
Sy your summation of products of triangular numbers & sines question is raping me. Hint plz? so far I've been working with Re( (z+z^2+...+z^n) + 2(z^2 + ... + z^n) + ... + n(z^n) ), then simplified that with geometric series, but it's not getting anywhere.

You got this right?

$\sum_{k=1}^{n} \cos k \theta = \frac{\cos \theta -1 + \cos n\theta - \cos (n+1)\theta}{2-2\cos \theta}$

$\sum_{k=1}^{n} \sin k \theta = \frac{\sin \theta + \sin n\theta - \sin(n+1)\theta}{2-2\cos \theta}$

Then differentiation

Magical Kebab · Jan 9, 2013

Re: HSC 2013 4U Marathon

A bag contains 12 red marbles and 12 yellow marbles. Six marbles are selected
at random without replacement.
(i) Calculate the probability that exactly three of the selected marbles 1 are red. Give your answer correct to two decimal places.
(ii) Hence, or otherwise, calculate the probability that more than three 2 of the selected marbles are red. Give your answer correct to two
decimal places.

RivalryofTroll · Jan 9, 2013

Re: HSC 2013 4U Marathon

If the polynomial:
x^3 + 3px^2 + 3qx + r = 0 has a double root. Show that the double roots must be (pq-r)/(2q-p^2).

Omed62 · Jan 10, 2013

Re: HSC 2013 4U Marathon

HEY GUYS!

PERFECT EXAM QUESTIONS!! Right here.

Guys see the below solutions, feel free to correct it, with full worked out solutions.

Practice Question: PROBABILITY:

I will start...

Q1. In how many ways can three groups of three numerals be chosen without replacement from the nine numerals 1,2,3,4,5,6,7,8,9 ?

9!/3!x3!x3!=?

Q2. In how many ways can a group of three letters be chosen without replacement from the nine letters of the world AUSTRALIA?

9C3= 84

PRACTICE QUESTIONS: COMPLEX Numbers:

Q1. In an Argand diagram the vectors OP and OQ represent the complex numbers z and w repectively such that Triangle OPQ is equilateral.
What is an expression for z^2 +w^2.

I planned to keep on posting questions to help other students for their upcoming half yearly exams, so please make sure you post your answers.

Immortalp00n · Jan 10, 2013

Re: HSC 2013 4U Marathon

awesome initiative there fagmed62, keep us posted with the questions !!~~~@@@!!!

twinklegal19 · Jan 10, 2013

Re: HSC 2013 4U Marathon

Omed62 said:
HEY GUYS!

PERFECT EXAM QUESTIONS!! Right here.

Guys see the below solutions, feel free to correct it, with full worked out solutions.

Practice Question: PROBABILITY:

I will start...

Q1. In how many ways can three groups of three numerals be chosen without replacement from the nine numerals 1,2,3,4,5,6,7,8,9 ?

9!/3!x3!x3!=?

Q2. In how many ways can a group of three letters be chosen without replacement from the nine letters of the world AUSTRALIA?

9C3= 84

PRACTICE QUESTIONS: COMPLEX Numbers:

Q1. In an Argand diagram the vectors OP and OQ represent the complex numbers z and w repectively such that Triangle OPQ is equilateral.
What is an expression for z^2 +w^2.

I planned to keep on posting questions to help other students for their upcoming half yearly exams, so please make sure you post your answers.

http://community.boredofstudies.org/showthread.php?t=298062

nice word for word copied questions...

Immortalp00n · Jan 10, 2013

Re: HSC 2013 4U Marathon

twinklegal19 said:
http://community.boredofstudies.org/showthread.php?t=298062

nice word for word copied questions...

it's to help other students for their upcoming half yearly exams bro!

HeroicPandas · Jan 10, 2013

Re: HSC 2013 4U Marathon

Magical Kebab said:
A bag contains 12 red marbles and 12 yellow marbles. Six marbles are selected
at random without replacement.
(i) Calculate the probability that exactly three of the selected marbles 1 are red. Give your answer correct to two decimal places.
(ii) Hence, or otherwise, calculate the probability that more than three 2 of the selected marbles are red. Give your answer correct to two
decimal places.

do u do 4U?

RivalryofTroll said:
If the polynomial:
x^3 + 3px^2 + 3qx + r = 0 has a double root. Show that the double roots must be (pq-r)/(2q-p^2).

I got $\frac{pq-r}{2q-2p^2}$

HeroicPandas · Jan 10, 2013

Re: HSC 2013 4U Marathon

Omed62 said:
Q1. In an Argand diagram the vectors OP and OQ represent the complex numbers z and w repectively such that Triangle OPQ is equilateral.
What is an expression for z^2 +w^2.

z^2 + w^2 = 0 using rotation

GoldyOrNugget · Jan 10, 2013

Re: HSC 2013 4U Marathon

Omed62 said:
HEY GUYS!

PERFECT EXAM QUESTIONS!! Right here.

Guys see the below solutions, feel free to correct it, with full worked out solutions.

Practice Question: PROBABILITY:

I will start...

Q1. In how many ways can three groups of three numerals be chosen without replacement from the nine numerals 1,2,3,4,5,6,7,8,9 ?

9!/3!x3!x3!=?

Q2. In how many ways can a group of three letters be chosen without replacement from the nine letters of the world AUSTRALIA?

9C3= 84

PRACTICE QUESTIONS: COMPLEX Numbers:

Q1. In an Argand diagram the vectors OP and OQ represent the complex numbers z and w repectively such that Triangle OPQ is equilateral.
What is an expression for z^2 +w^2.

I planned to keep on posting questions to help other students for their upcoming half yearly exams, so please make sure you post your answers.

this guy...

GoldyOrNugget · Jan 10, 2013

Re: HSC 2013 4U Marathon

Omed62 said:
HEY GUYS!

PERFECT EXAM QUESTIONS!! Right here.

Guys see the below solutions, feel free to correct it, with full worked out solutions.

Practice Question: PROBABILITY:

I will start...

Q1. In how many ways can three groups of three numerals be chosen without replacement from the nine numerals 1,2,3,4,5,6,7,8,9 ?

9!/3!x3!x3!=?

Q2. In how many ways can a group of three letters be chosen without replacement from the nine letters of the world AUSTRALIA?

9C3= 84

PRACTICE QUESTIONS: COMPLEX Numbers:

Q1. In an Argand diagram the vectors OP and OQ represent the complex numbers z and w repectively such that Triangle OPQ is equilateral.
What is an expression for z^2 +w^2.

I planned to keep on posting questions to help other students for their upcoming half yearly exams, so please make sure you post your answers.

this guy...

SpiralFlex · Jan 10, 2013

Re: HSC 2013 4U Marathon

Has anyone done integration yet? Don't wanna post anything you guys haven't done. Actually this is quite appropriate, I think it's around Q14 level

$$Suppose$\; 0 \leq a < b.\; $Let$\;f(x)=\frac{1}{1+x^2},\;x \in [a,b]$

$$a) \;Prove that$\; \frac{1}{1+b^2}<\frac{\tan^{-1}b-\tan^{-1}a}{b-a}<\frac{1}{1+a^2}$

$$b) \;i) Show that$\; \tan \frac{3\pi}{8}=1+\sqrt{2}$

$$\;ii) \;Hence evaluate$\;\tan \frac{\pi}{24}$

$$c)\;Deduce$\;3(\sqrt{6}-\sqrt{2})<\pi< \24(\sqrt{6}+\sqrt{2}-\sqrt{3}-2)$

Note: You are NOT allowed to use the mean value theorem for part a) only HSC methods. However a hint would be to use integration.

RealiseNothing · Jan 10, 2013

Re: HSC 2013 4U Marathon

Omed62 said:
PRACTICE QUESTIONS: COMPLEX Numbers:

Q1. In an Argand diagram the vectors OP and OQ represent the complex numbers z and w repectively such that Triangle OPQ is equilateral.
What is an expression for z^2 +w^2.

Ok first we want to know what this looks like, it is an argand diagram with complex numbers $z$ and $w$ .

Now since it is an equilateral triangle, $z=w$ due to all sides being the same length. Hence $z^2+w^2=z^2+z^2=2z^2$

Now let $z=x+iy$ , then by substitution we get:

$2z^2 = 2(x+iy)^2 = 2(x^2-y^2+2xyi)$

So we end up with:

$z^2 + w^2 = 2z^2 = 2(x^2-y^2+2xyi)$

A really big thanks to your contributions for helping me and other students for our upcoming half yearlies!

I'm really looking forward to future questions that you have to offer!

cutemouse · Jan 10, 2013

Re: HSC 2013 4U Marathon

RealiseNothing said:
Now since it is an equilateral triangle, $z=w$

With respect, that is wrong... |z|=|w| is, however, correct.

bleakarcher · Jan 10, 2013

Re: HSC 2013 4U Marathon

cutemouse said:
With respect, that is wrong... |z|=|w| is, however, correct.

lol, he was trolling.

cutemouse · Jan 10, 2013

Re: HSC 2013 4U Marathon

Actually, my bad. It's actually correct. Stop trolling bleakarcher

Sorry, it's been a while since I've done complex numbers.

Omed62 · Jan 10, 2013

Re: HSC 2013 4U Marathon

RealiseNothing said:
Ok first we want to know what this looks like, it is an argand diagram with complex numbers $z$ and $w$ .

Now since it is an equilateral triangle, $z=w$ due to all sides being the same length. Hence $z^2+w^2=z^2+z^2=2z^2$

Now let $z=x+iy$ , then by substitution we get:

$2z^2 = 2(x+iy)^2 = 2(x^2-y^2+2xyi)$

So we end up with:

$z^2 + w^2 = 2z^2 = 2(x^2-y^2+2xyi)$

A really big thanks to your contributions for helping me and other students for our upcoming half yearlies!

I'm really looking forward to future questions that you have to offer!

Thanks RealiseNothing, your one of the best guys out their, so nice and keen on helping students.

I hope god be with you always... lol could you please answer the probability questions... please

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