HSC 2013 MX2 Marathon (archive) (2 Viewers)

Omed62 · Jan 12, 2013

Re: HSC 2013 4U Marathon

Immortalp00n said:
thanks heaps bro
do u have any practise questions for 4unit? my half yearlies are soon man
god bless u so much my friend

God bless you my friends,

Thanks for your replay... I will start to organise some questions just for you and all other students sa well to help you guys...

Thanks for being so kind my friend... sure I have questions ... I am more than happy to post them up for you...

but could you please answer these questions becuse I had some trouble with them...

As you guys no that probability is one of the hardest topics that some students struggle, so I have posted some probability questions...

Eg. 4: A group of 12 people are to be divided into discussion groups. In how many ways
can the discussion groups be formed if there are 3 groups containing 4 people each?
Your working:

Solution:

1. Select four people from the twelve to go into the rst group. There are 12C4 ways of doing this.
2. Select four people from the remaining eight to go into the second group. There are 8C4 ways of doing this.
3. Select four people from the remaining four to make the last group. There are 4C4 ways of doing this.
4. These groups are not dierent from each other, so we must divide by 3! to avoid counting some ways more than once.

Finally ans. ?

Eg. 5: (MX1 2000 (6)

A standard pack of 52 cards consists of 13 cards of each of the four suits: spades, hearts, clubs and diamonds.

(a) In how many ways can six cards be selected without replacement so that
exactly two are spades and four are clubs?
(b) In how many ways can six cards be selected without replacement if at least
ve cards must be of the same suit?

1. ( MX1 2001 (2c) part ii)
How many arrangements of the letters of the word ALGEBRAIC are possible if
the vowels must occupy the 2nd, 3rd, 5th and 8th positions?

2. ( MX2 2002 (4c) part ii)
From a pack of nine cards numbered 1, 2, 3, : : : , 9, three cards are drawn at
random and placed from left to right. What is the probability that the digits are
drawn in descending order? [Hint: for example, 9, 5 and 1 in that order]

3. ( MX2 2003 (4c) part i)
A hall has n doors. Suppose that n people choose any door at random to enter the hall. In how many ways can this be done?

Eg. 8: (MX2 2004 (5b) part ii)
In how many ways can ve students be placed in three distinct rooms so that no
room is empty?

Enjoy guys

Immortalp00n · Jan 12, 2013

Re: HSC 2013 4U Marathon

Omed62 said:
God bless you my friends,

Thanks for your replay... I will start to organise some questions just for you and all other students sa well to help you guys...

Thanks for being so kind my friend... sure I have questions ... I am more than happy to post them up for you...

but could you please answer these questions becuse I had some trouble with them...

thanks so much bro
ur a good guy
yeah ill try my best answering them

RealiseNothing · Jan 12, 2013

Re: HSC 2013 4U Marathon

twinklegal19 said:
That theorem still requires FOG to be collinear and since F and G lie on the intervals AO and OP by construction then you're saying that AOP are collinear

oh wai-

You're loving this aren't you.

Omed62 · Jan 12, 2013

Re: HSC 2013 4U Marathon

who ever is reading my post... the question I gave up some of my own time... just to organise the questions for you guys to help you guys succeed because I really understand How stressful HSC can be sometimes...

So from now own... I will start to post questions for all the topics for the upcoming exams...

just to let you guys no that I am also a HSC student this year 2013... guys I just posted some question that I needed help with below about complex number and probability. So if you have time could you please answer them please... I would be happy..

Examine · Jan 12, 2013

Re: HSC 2013 4U Marathon

lol jesus

RealiseNothing · Jan 12, 2013

Re: HSC 2013 4U Marathon

Immortalp00n said:
thank u so much my friend, i really appreciate it.
u are one of the nicest people i have seen on the whole wide forum

This is so nice, it's like we are one big family.

cutemouse · Jan 12, 2013

Re: HSC 2013 4U Marathon

Omed62 said:
Hey bro...

Yes I have more questions

But can you please post the answers these questions up becuase I am need help with them... also try to explain each of them...

Practice Question: PROBABILITY:

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 ?

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

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.

To be fair I feel compelled to say that they're not exactly easy questions. I think I would've had considerable difficulty with those questions when I was first beginning with Extension 2. I would be probably correct in saying that most students just beginning with Extension 2 would percieve those questions to be considerably difficult as well. On the contrary they're not impossibly difficult questions either. That is, they probably wouldn't be Q16 difficulty of the HSC exam (or Q8 for the predecessor exams).

I do understand that you are finding those questions difficult. But I don't think you helped your cause by posting your entire (assessable) assignment for other members to provide full solutions to, which they declined, for moral and ethical reasons, to do so. It seems you dug a bigger hole for yourself by then attempting to decieve others into providing full solutions to selected assignment questions in this thread...

In any case, I'll try and provide a starting point for your questions. For the probability questions, they're the slighty harder standard types. You should be able to learn how to do such problems from any reputable textbook (like Coroneos). For the complex numbers question, draw a diagram. Then you'll see that the angle between the vectors OP and OQ would be 60deg. Then w=z*(cos +/- 60 deg + i sin +/- 60 deg). The +/- is because OP could be above OQ, or the other way around. The rest should follow easily (using DeMovire's theorem etc). If you don't understand this then you'd probably be best revising rotations of points/vectors about the origin in the Complex plane.

Hope that helps.

Sy123 · Jan 12, 2013

Re: HSC 2013 4U Marathon

I've had enough of this, this is a marathon thread, I don't want to see your assignment questions again, they are your assignment questions, not ours. I also suggest that if you are going to cheat like this, don't put your real name in your username, makes you very easy to catch.

==========================

$Define the function$

$\Gamma (n) = \lim_{t \to \infty} \int_{0}^{t} x^{n-1}e^{-x} \ dx$

$$i) Show that$ \ \ \ \Gamma (1) = 1 \ \ \ \ \ \ \fbox{1}$

$ii) Using integration by parts, show that$

$\Gamma (n+1) = n\Gamma (n) \ \ \ \ \fbox{2}$

$iii) Hence deduce that$

$\Gamma (n+1) = n! \ \ \ \ \fbox{1}$

n does not have to be a natural number (; this way we can define non-integral factorials, or even negative factorials if we want, but that is out of the syllabus.

Trebla · Jan 12, 2013

Re: HSC 2013 4U Marathon

Sy123 said:
I've had enough of this, this is a marathon thread, I don't want to see your assignment questions again, they are your assignment questions, not ours. I also suggest that if you are going to cheat like this, don't put your real name in your username, makes you very easy to catch.

==========================

$Define the function$

$\Gamma (n) = \lim_{t \to \infty} \int_{0}^{t} x^{n-1}e^{-x} \ dx$

$$i) Show that$ \ \ \ \Gamma (1) = 1 \ \ \ \ \ \ \fbox{1}$

$ii) Using integration by parts, show that$

$\Gamma (n+1) = n\Gamma (n) \ \ \ \ \fbox{2}$

$iii) Hence deduce that$

$\Gamma (n+1) = n! \ \ \ \ \fbox{1}$

n does not have to be a natural number (; this way we can define non-integral factorials, or even negative factorials if we want, but that is out of the syllabus.

lol this is almost identical to the question I pitched for Carrot's BOS Trial. You will need to assume without proof that the exponential dominates the power function as the t gets large though

Kurosaki · Jan 12, 2013

Re: HSC 2013 4U Marathon

Sy123 said:
I've had enough of this, this is a marathon thread, I don't want to see your assignment questions again, they are your assignment questions, not ours. I also suggest that if you are going to cheat like this, don't put your real name in your username, makes you very easy to catch.

==========================

$Define the function$

$\Gamma (n) = \lim_{t \to \infty} \int_{0}^{t} x^{n-1}e^{-x} \ dx$

$$i) Show that$ \ \ \ \Gamma (1) = 1 \ \ \ \ \ \ \fbox{1}$

$ii) Using integration by parts, show that$

$\Gamma (n+1) = n\Gamma (n) \ \ \ \ \fbox{2}$

$iii) Hence deduce that$

$\Gamma (n+1) = n! \ \ \ \ \fbox{1}$

n does not have to be a natural number (; this way we can define non-integral factorials, or even negative factorials if we want, but that is out of the syllabus.

Sorry for stupid question, what does that shifty symbol in front of (n+1) mean?

GoldyOrNugget · Jan 12, 2013

Re: HSC 2013 4U Marathon

It's the name of the function he defined.

Trebla · Jan 12, 2013

Re: HSC 2013 4U Marathon

Kurosaki said:
Sorry for stupid question, what does that shifty symbol in front of (n+1) mean?

It is the capital gamma. Think of it like f(x) where f is replace by capital gamma and x is replaced by n + 1.

GoldyOrNugget · Jan 12, 2013

Re: HSC 2013 4U Marathon

Sy123 said:
I've had enough of this, this is a marathon thread, I don't want to see your assignment questions again, they are your assignment questions, not ours. I also suggest that if you are going to cheat like this, don't put your real name in your username, makes you very easy to catch.

==========================

$Define the function$

$\Gamma (n) = \lim_{t \to \infty} \int_{0}^{t} x^{n-1}e^{-x} \ dx$

$$i) Show that$ \ \ \ \Gamma (1) = 1 \ \ \ \ \ \ \fbox{1}$

$ii) Using integration by parts, show that$

$\Gamma (n+1) = n\Gamma (n) \ \ \ \ \fbox{2}$

$iii) Hence deduce that$

$\Gamma (n+1) = n! \ \ \ \ \fbox{1}$
[/COLOR]

$$i) $\Gamma(1) = \lim_{t \to \infty} \int_{0}^{t}e^{-x} \ dx = \lim_{t \to \infty} \left[ -e^{-x} \right]_{0}^{t} = -e^{-\infty} - (-e^{0}) = 0 - (-1) = 1.$

$$ii) $ \Gamma(n+1) = \lim_{t \to \infty} \int_{0}^{t} x^{n}e^{-x} \ dx = \lim_{t \to \infty} \left( \left[ -x^ne^{-x} \right]_{0}^{t} - (-n)\int_{0}^{t} e^{-x}x^{n-1} \ dx \right) = \lim_{t \to \infty} n\int_{0}^{t} x^{n-1}e^{-x} \ dx = n\Gamma(n).$

$$iii) $\Gamma(n+1) = n\Gamma(n) = n(n-1)\Gamma(n-1) = n(n-1)(n-2)...\Gamma(1) = n(n-1)(n-2)...1 = n!.$

SpiralFlex · Jan 12, 2013

Re: HSC 2013 4U Marathon

$1.\int \frac{1}{1+x^4}\;$d$x$

$2. \int e^{3x}\sqrt{1+e^{2x}}\;$d$x$

$3. \int \frac{1}{(3x+7)\sqrt{x+2}}\;$d$x$

$4. \int \cot (x-a)\cot (x-b)\;$d$x$

These are quite challenging from easier to harder. But they are great practice.

twinklegal19 · Jan 12, 2013

Sy123 said:
Yeah I made the wrong assumption that angle TPC = angle TPE
My method is invalid

Don't give up lol, just prove those two angles are equal by using a few congruent triangles proofs

RealiseNothing said:
You're loving this aren't you.

Yep

Sy123 · Jan 12, 2013

Re: HSC 2013 4U Marathon

SpiralFlex said:
$1.\int \frac{1}{1+x^4}\;$d$x$

$2. \int e^{3x}\sqrt{1+e^{2x}}\;$d$x$

$3. \int \frac{1}{(3x+7)\sqrt{x+2}}\;$d$x$

$4. \int \cot (x-a)\cot (x-b)\;$d$x$

These are quite challenging from easier to harder. But they are great practice.

Ok, I attempted the first one with my limited integration experience/knowledge and I was able to get it in the end, I probably took the long way but:

1. part i
1. part ii
1. part iii

I will attempt the others ones now, then go back to the circle geo one :L

Kurosaki · Jan 12, 2013

Re: HSC 2013 4U Marathon

Sy123 said:
Ok, I attempted the first one with my limited integration experience/knowledge and I was able to get it in the end, I probably took the long way but:

1. part i
1. part ii
1. part iii

I will attempt the others ones now, then go back to the circle geo one :L

Are you still stuck on part A?

Sy123 · Jan 12, 2013

Re: HSC 2013 4U Marathon

Kurosaki said:
Are you still stuck on part A?

I did part B, but part A I going to attempt later (yes it is the one I am stuck on)

though for Spiral's integrals, here is

Q2:

$J=\int e^{3x} \sqrt{1+e^{2x}} \ dx$

Make the substitution:

$e^x = \tan u$

$e^x \ dx = \sec^2 u \ du$

$J= \int \sec^3 u \tan^2 u \ du$

$J= \int \sec^3 u \ du + \int \sec^5 u \du$

So all we need to do is evaluate these integrals, which I will now attempt to do as whole by finding a recurrence formula via by parts

$I_{2n+1} = \int \sec^{2n+1} u \ du$

$I_{2n+1} = \tan u \sec^{2n-1}u - (2n-1) \int \sec^{2n+1} u - \sec^{2n-1} u \du$

$2I_3= \tan u \sec u + I_1 \\ \\ 4I_5=\tan u \sec^{3} u + 3I_3$

$J=I_3+I_5 = \frac{7}{8} \tan u \sec u + \frac{1}{4} \tan u \sec^3 u + \frac{5}{4} I_1$

$I_1 = \ln |\sec u + \tan u|$

$J= \frac{7}{8} \tan (\tan^{-1}e^x) \sec (\tan^{-1}e^x)+ \frac{1}{4} \tan{\tan^{-1}e^x} \sec^3 (\tan^{-1}e^x) + \frac{5}{4} \ln |\sec{\tan^{-1}x} + \tan{\tan^{-1}x}|$

$J=\frac{7}{8} e^x \sqrt{1+e^{2x}} + \frac{1}{4} e^x (\sqrt{(1+e^{2x})^3}) + \frac{5}{4} \ln |e^x+ \sqrt{1+e^{2x}}| `+ C$

lol not sure if I made a mistake somewhere.

==================

EDIT: Q3 is more simple:

$I=\int \frac{1}{(3x+7)\sqrt{x+2}} \ dx$

$u^2=x+2 \ \ \ \ 2u \ du= dx$

$\int \frac{2u \du}{(3u^2+1)u}$

$=\frac{2}{\sqrt{3}} \int \frac{\sqrt{3}}{1+3u^2} \ du$

$I= \frac{2}{\sqrt{3}}\tan^{-1} (\sqrt{3}u) + C$

$I = \frac{2}{\sqrt{3}} \tan^{-1} (\sqrt{3(x+2)}) + C$

Streetboy · Jan 12, 2013

Re: HSC 2013 4U Marathon

Sy123 said:
Here, post questions that you have made/found for Extension 2 Maths, so we can all benefit from new experience to different questions

Guys, post questions that you have made/found for Extension 2 Maths, so we can all benefit from new experience to different questions.

Streetboy · Jan 12, 2013

Re: HSC 2013 4U Marathon

Here I will post some questions;

(a)
(i) In how many ways can m identical yellow discs and n identical black discs be arranged in a row?
(ii) In how many ways can 10 identical coins be allocated to 4 different boxes?

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