HSC Tips - Harder 3U (1 Viewer)

[McLake]

OK, this is the last topic-specific thread. I will include one last thread that is genral Trial/HSC Tips for 3/4U Maths later ....

Tips for Harder 3U:
- Harder 3U means just that, harder 3U questions. It can be from ANY area of 3U, but I will cover the most popular areas here (circle geo, inequalities, induction, probability, rates)

- Circle Geometry
-- As you may already know, there is no new geometry to learn for 4U, just harder question based on what you already know. Most of these questions you either get, or you don't, there are very few tips I can give.
-- One main tip is if it is taking too many steps then you probably have got it wrong. Even 4U querstions shouldn't be too complex.

- Inequalities
-- For inequalties you must know the arthimatic and geometric means identies off by heart, and be good at manipulating them. Arnold & Arnold have some good questions in there book.

- Induction
-- 4U inductions introduces the idea of recursive induction, using a formula to prove an induction statement. These arn't that much harder than regular induction, remeber to try and use the known result as a guide of how to manipulate the LHS of the equation to get the RHS.

- Probability
-- I HATE probability at the best of times, but 4U probability makes me scream. I have no good tips, so feel free to post some of your own.

- Rates
-- Rates cropped up in last year's 4U HSC as a harder 3U question, and they can be very tricky. Again, there is nothing new to learn, it's just tricker questions. In the case of rates questions it is really a matter of practice makes perfect.

 

[Affinity]

Probability -
Get yourself clear about when to add number and when to multiply.

You can get probaility involving intervals, not just discrete variables.

Inequalities:

The cauchy-schwarz inequality is another useful one.
(SUM {a_ib_i } )² <= SUM {a_i²} *SUM {b_i²}

Circle geometry:
Go through arnold.

 

[turtle_2468]

Inequalities: usually I don't think you are allowed to assume cauchy-schwarz, but someone correct me if that's not the case! A lot of the time, the things look deceptively simple, but if you really get stuck then just cross multiply everything out and cancel. Remember that you can split expressions up into a few parts, and prove that they are all larger than zero (squares for instance) to prove that the whole thing is greater than zero...

Induction: It may sometimes be helpful to start from RHS to try to get to LHS.

Circle geometry: All about finding cyclic quads. You know the two diagrams about supplementary angles and angles on same segment? LEARN THEM. If you do, then you'll see that any angle in a cyclic quad is "related" to another angle in there. Which means, if you need to prove two angles equal for instance, you can start from the first one, and figure out something it's equal to, then chase the angles around...

Hmm. I'm not sure whether much of that made sense, ask me if it didn't!

 

[Affinity]

oh.. forgot .. yes you have to prove the cauchy schwarz.

what about AM-GM?

 

[turtle_2468]

And... umm... proving cauchy-schwarz is a Q8 in itself, right?

AM-GM I think you might ahve to prove as well. In a v complicated question, don't bother, but I think for most purposes, you have to prove it.

Method: Prove for n=2, 4, 8, 2^k by induction.
Then sub in the AM of n-1 terms to go down a term. More later...

 

[Affinity]

oh :S didn't think about that...

would it be less writing just to prove AM-GM to 'usual way'
than to do 2^k then back tracing?

 

[turtle_2468]

That _is_ the usual way of proving AM-GM as far as I know! I don't think there's a short-enough-to-write-down-in-less-than-a-few-pages
way of doing it by straight induction.

 

[heath]

"For inequalties you must know the arthimatic and geometric means identies off by heart"

so what are these again?

 

[McLake]

Originally posted by heath
"For inequalties you must know the arthimatic and geometric means identies off by heart"

so what are these again?

AM = (a+b)/2
GM = sqrt(a^2+b^2/4) (<- right?)

 

[ND]

Originally posted by McLake
AM = (a+b)/2
GM = sqrt(a^2+b^2/4) (<- right?)

In the case where n=2, it's (a+b)/2 >= sqrt(ab). For the general case it's:
(a_1+a_2+...+a_n)/n >= (a_1*a_2*...*a_n)^(1/n), where a_1, a_2,...,a_n are positive integers.

 

[heath]

the arithmatic one is easy, knew that since ... a long time ago,

but how do you work out the gp one?

Tr in a geometric series is ar^n-1 right?

 

[heath]

nd i don't understand this:

(a_1+a_2+...+a_n)/n >= (a_1*a_2*...*a_n)^(1/n), where a_1, a_2,...,a_n are positive integers.

what the hell does underscore mean?

 

[ND]

nd i don't understand this:

(a_1+a_2+...+a_n)/n >= (a_1*a_2*...*a_n)^(1/n), where a_1, a_2,...,a_n are positive integers.

what the hell does underscore mean?

Subscript. It's just a way of denoting how many integers there are. For example, (a+b+c+d)/4 >= (abcd)^(1/4) can be written as (a_1+a_2+a_3+a_4)/4 >= (a_1*a_2*a_3*a_4)^(1/4).

 

[MyLuv]

Can we assume this or we need to prove it ( 2No. is ok but for n No... )
Btw,can we use stuff that they dont teach us here (since I've learned some good stuff in my country )

 

[heath]

nd that's not really u is it?

 

[ND]

Originally posted by heath
nd that's not really u is it?

Do you mean my avatar? Nope, i'm a guy (and no it's not my girlfriend either ).

 

[pigs_can_fly]

looks like jordan...am i rite?

 

[enak]

is that the chick from the uk? she's not bad

 

[withoutaface]

There are too many types of probability questions to list here. so I will just take a few common ones which seem to get people stuck.

1. When doing probability always look for binomial probability, as this is often the simplest way of doing a question, and you would be surprised by how many people in my class don't recognise a binomial question when they see it.

2. For arrangements such as 5 letter words from 'CRICKET' take it one step at a time:

2 C's, 3 non C's
1 C, 4 non C's
5 non C's

and find the number of combinations for each of these.


3. If a question asks you for example to find the number of three digit codes with the digits in ascending order from the numbers of 1-9, and each number can only be chosen once, remember that for each selection of numbers, only one is in ascending order, hence the answer would be 9C3.

Hopefully that will be of assistance to someone, but remember that often if you cannot see a probability question immediately you may never see it, so don't waste time, move on and save yourself some time for other more doable questions.

 

[HappyFeet]

And... umm... proving cauchy-schwarz is a Q8 in itself, right?

I don't think so, proving Cauchy-Schwarz's Inequality is quite simple actually. You can either prove it using vectors or simple quadratics.

Proving AM-GM is also quite easy .