Difficulty of each topic?

Viscass · Mar 10, 2013

hey i was just wondering what topics are considered harder than the others in 4U? i know it will just be opinions as some people prefer different things, i just want a rough idea. i found graphs the easiest by far and then i would say conics is looking alright but we havent quite finished the chapter yet and looking ahead it seems pretty hectic.

also what it takes to do well in each of them?

HeroicPandas · Mar 10, 2013

Viscass said:
also what it takes to do well in each of them?

Self-discipline and HSC questions

jenslekman · Mar 10, 2013

definitely hsc questions - both easy and hard - to train 1. accuracy 2. speed 3. problem solving. anyways, graphs can get tricky but in my opinion its the easiest. conics can get messy but generally straight forward. integration is fairly straight forward. mechanics - many struggle with this but practise and you'll be fine. polynomials, i find this the hardest topic imo but many find this easy

volumes is pretty straightforward. and harder 3u is probably hardest actually - thats why you should have plenty and plenty of time remaining in an exam to knock down these ones - so thats why speed and accuracy is so important for the easier questions.

tattoo2200 · Mar 10, 2013

yeah generally complex numbers, graphs, and integration are the easiest in my opinon. Conics is average difficulty since sometimes you can go off in a tangent (pun intended) and get nowhere doing 2pages of work. Volumes is ok. I havent dont the others yet but i assume the rest are difficult. And i fucken hate polynomials

Dan895 · Mar 10, 2013

What part of polynomials do you guys find hard, I think most of its pretty alright especically since my teacher taught roots of unity as a part of complex numbers. Graphs sometimes can stuff me up but they're not that hard I guess.

Viscass · Mar 14, 2013

a question on polynomials, not sure if this is a common type of question or if its from any past paper, i recieved it from an ext 2 workshop
$$The only roots of a polynomial P(x) are $\alpha$, $\beta$ and $\gamma$ where $\alpha$ is r-fold, $\beta$ is s-fold and $\gamma$ is t-fold.$\\$Show that P'(x) has two roots not equal to $\alpha$, $\beta$ or $\gamma$ and that the sum of these two roots is $\\ \frac{(s+t)\alpha+(t + r)\beta+(r+s)\gamma}{r+s+t}$

HeroicPandas · Mar 14, 2013

Uhh...

Obviously... using ur multiplicity knowledge, P(x) = ...

$P(x) = (x-\alpha)^r (x-\beta)^s (x-\gamma)^t$

Using the superior product rule...

$P'(x) = (x-\alpha)^{r-1} (x - \beta)^{s-1} (x-\gamma)^{t-1}[r(x-\beta)(x-\gamma)+s(x- \alpha)(x-\gamma)+t(x-\alpha)(x-\beta)]$

Using the some rule........

$$ If A.B.C...Z = 0 \\ \therefore A, B, C, ... , Z = 0$

So... Roots of P'(x)

Roots = alpha, beta, gamma???

and

Using the rule above,

$[r(x-\beta)(x-\gamma)+s(x- \alpha)(x-\gamma)+t(x-\alpha)(x-\beta)] = 0$

$x^2 (r + s+ t) - x[r(\gamma + \beta) + s(\gamma + \alpha) + t(\alpha + \beta)] + r\beta \gamma + s\alpha \beta + t \alpha \beta = 0$ ....[1]

Let roots of this quadratic be: WATER and FIRE (which =/= alpha and beta quadratic with roots alpha and beta is P(x) = x^2 - (alpha + beta)x + (alpha . beta)<---- this is not equivalent to equation [1] so roots of equation [1] is not alpha and beta?)

$\sum \alpha: $ water + fire = $\frac{r(\gamma + \beta) + s(\gamma + \alpha) + t(\alpha + \beta)}{r + s+ t}$

Re-arrange numerator and you will get the answer...

I don't know how to show that P'(x) has 2 roots lol

Viscass · Mar 14, 2013

HeroicPandas said:
Uhh...

Obviously... using ur multiplicity knowledge, P(x) = ...

$P(x) = (x-\alpha)^r (x-\beta)^s (x-\gamma)^t$

Using the superior product rule...

$P'(x) = (x-\alpha)^{r-1} (x - \beta)^{s-1} (x-\gamma)^{t-1}[r(x-\beta)(x-\gamma)+s(x- \alpha)(x-\gamma)+t(x-\alpha)(x-\beta)]$

Using the some rule........

$$ If A.B.C...Z = 0 \\ \therefore A, B, C, ... , Z = 0$

So... Roots of P'(x)

Roots = alpha, beta, gamma???

and

Using the rule above,

$[r(x-\beta)(x-\gamma)+s(x- \alpha)(x-\gamma)+t(x-\alpha)(x-\beta)] = 0$

$x^2 (r + s+ t) - x[r(\gamma + \beta) + s(\gamma + \alpha) + t(\alpha + \beta)] + r\beta \gamma + s\alpha \beta + t \alpha \beta = 0$ ....[1]

Let roots of this quadratic be: WATER and FIRE (which =/= alpha and beta quadratic with roots alpha and beta is P(x) = x^2 - (alpha + beta)x + (alpha . beta)<---- this is not equivalent to equation [1] so roots of equation [1] is not alpha and beta?)

$\sum \alpha: $ water + fire = $\frac{r(\gamma + \beta) + s(\gamma + \alpha) + t(\alpha + \beta)}{r + s+ t}$

Re-arrange numerator and you will get the answer...

Nice!
would this be a question you would expect in the HSC or is this more of a 'test your understanding' type question?

Viscass · Mar 14, 2013

HeroicPandas said:
I don't know how to show that P'(x) has 2 roots lol

to show there are two roots (beside alpha beta and gama) all you need to do is state that there are 2 solutions to the quadratic [1] =/= alpha, beta or gamma therefore there are two roots to P'(x) =/= alpha, beta or gamma

HeroicPandas · Mar 15, 2013

Viscass said:
Nice!
would this be a question you would expect in the HSC or is this more of a 'test your understanding' type question?

I'm not sure lol

Viscass said:
to show there are two roots (beside alpha beta and gama) all you need to do is state that there are 2 solutions to the quadratic [1] =/= alpha, beta or gamma therefore there are two roots to P'(x) =/= alpha, beta or gamma

But look at the P'(x) equation, in front of it, sub in alpha, beta and gamma one by one and P'(x) = 0, therefore alpha, beta, gamma are roots (by the factor theorem) of P'(x). hence P'(x) will have FIVE roots

yeh?

Viscass · Mar 15, 2013

HeroicPandas said:
But look at the P'(x) equation, in front of it, sub in alpha, beta and gamma one by one and P'(x) = 0, therefore alpha, beta, gamma are roots (by the factor theorem) of P'(x). hence P'(x) will have FIVE roots

yeh?

the questions states that P'(x) has two roots not equal to alpha beta or gamma it doesnt say there are only 2 roots, just that
2 of them are not aplha beta or gamma.

anomalousdecay · Mar 16, 2013

tattoo2200 said:
Conics is average difficulty since sometimes you can go off in a tangent (pun intended) and get nowhere doing 2pages of work. And i fucken hate polynomials

We did Polynomials before conics and this made all the tangents and normals work easy in conics.
I actually like Polynomials because we did 75% of the chapter in year 11 extension 1. What textbook did you use for year 11 extension 1. We used Maths in Focus.

Viscass · Mar 18, 2013

anomalousdecay said:
We did Polynomials before conics and this made all the tangents and normals work easy in conics.
I actually like Polynomials because we did 75% of the chapter in year 11 extension 1. What textbook did you use for year 11 extension 1. We used Maths in Focus.

yeah maths in focus, im pretty sure its the standard that most schools use. was the polynomial work you did in year 11 ext 1? i found it kind of imtimidating at first but now its simple crap

anomalousdecay · Mar 19, 2013

Viscass said:
yeah maths in focus, im pretty sure its the standard that most schools use. was the polynomial work you did in year 11 ext 1? i found it kind of imtimidating at first but now its simple crap

Yes. I did every question out of joy and ended up getting 100% in Polynomials in every exam. I always screwed up permutations and combinations and trigonometry though. I am hoping harder extension 1 will help me with permutations and combinations. Complex numbers made double angle trig easy, but all that 3d trig was hard to understand.

Difficulty of each topic?

Viscass

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Viscass

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HeroicPandas

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HeroicPandas

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