HSC 2014 MX2 Marathon ADVANCED (archive) (1 Viewer)

Trebla · Mar 19, 2014

Post any questions within the scope and level of Mathematics Extension 2 mainly targeting Q16 difficulty in the HSC.

Any questions beyond the scope of the HSC syllabus should be posted in the Extracurricular Topics forum:
http://community.boredofstudies.org/...icular-topics/

Once a question is posted, it needs to be answered before the next question is raised.

I encourage all current students in particular to participate in this marathon.

Original Context:
Guys I hate to do this but I'm gonna have to stamp my authority here because this is going on for far too long.

Given that I have given numerous warnings about deviating from HSC relevant material and to involve other users, we are still seeing excessively difficult or university level questions. Hence, I have decided to do the following:
- This thread will be renamed 'HSC 2014 4U marathon - Advanced level' and the questions here would be more advanced but should still be within the scope of the HSC syllabus (otherwise it defeats the title). If you wish to ask any questions which use university level concepts then create a thread in the 'Extracurricular topics' section (do not post them here)
- A new thread named 'HSC 2014 4U marathon' will be made which serves the purpose of involving other users who may not be so advanced in their mathematical knowledge or abilities. The questions asked in this thread should be more accessible to other users

integral95 · Mar 19, 2014

Re: HSC 2014 4U Marathon

Sy123 said:
$\\ $Define the notation$ \ f : \mathbb{R} \rightarrow \mathbb{R} \ $to mean that the function$ \ f \ $has inputs in the real numbers and outputs in the real numbers$$

$Note$ \ \mathbb{C} = \ $complex numbers$

$\\ $Also define a function$ \ f \ $ to have property$ \ S \ $if the entire output domain of the function can be arrived at through some input$ \\ \\ $i.e.$ \ f: \mathbb{R} \rightarrow \mathbb{R} \ , \ \ f(x) = x^3 \ \ $has the property$ \ S$

$\\ $Which of these functions have the property$ \ S$

$\\ $i)$ \ f: \mathbb{R} \rightarrow \mathbb{R} \ , \ \ f(x) = x^2 \\ \\ $ii)$ \ f: \mathbb{C} \rightarrow \mathbb{C} \ , \ f(z) = z^2 \\ \\ $iii)$ \ f: \mathbb{C} \rightarrow \mathbb{C} \ , \ \ f(z) = z+ \bar{z} \\ \\ $iv)$ \ f: \mathbb{C} \rightarrow \mathbb{R} \ , \ \ f(z) = z + \bar{z}$

i) No since the output only consist of positive numbers, which is only part of the output domain of the function
ii) I think yes haha since you can get a complex number after squaring a complex number
iii) No since you'll only result in real values as the output $f(z) = z+ \bar{z} = 2x [tex] where x is real iv) Yes ( same reason as iii) Lol i don't think my explanations are clear, could you tell me a better way to explain these?$

Sy123 · Mar 19, 2014

Re: HSC 2014 4U Marathon

Trebla said:
Guys I hate to do this but I'm gonna have to stamp my authority here because this is going on for far too long.

Given that I have given numerous warnings about deviating from HSC relevant material and to involve other users, we are still seeing excessively difficult or university level questions. Hence, I have decided to do the following:
- This thread will be renamed 'HSC 2014 4U marathon - Advanced level' and the questions here would be more advanced but should still be within the scope of the HSC syllabus (otherwise it defeats the title). If you wish to ask any questions which use university level concepts then create a thread in the 'Extracurricular topics' section (do not post them here)
- A new thread named 'HSC 2014 4U marathon' will be made which serves the purpose of involving other users who may not be so advanced in their mathematical knowledge or abilities. The questions asked in this thread should be more accessible to other users

Perhaps the domain co-domain question was a little too far, but I'm pretty sure every other question was a pretty decent level

Trebla · Mar 19, 2014

Re: HSC 2014 4U Marathon

Sy123 said:
Perhaps the domain co-domain question was a little too far, but I'm pretty sure every other question was a pretty decent level

Most questions were fine (just a bit beyond HSC difficulty), but there were a few examples which involved university level concepts. Going forward please only have questions within the scope of the HSC syllabus in this thread. The main issue was the difficulty of the questions which was intimidating for many users and preventing them from participating, which is what the second thread is for as a compromise.

seanieg89 · Mar 19, 2014

Re: HSC 2014 4U Marathon

Trebla said:
Most questions were fine (just a bit beyond HSC difficulty), but there were a few examples which involved university level concepts. Going forward please only have questions within the scope of the HSC syllabus in this thread. The main issue was the difficulty of the questions which was intimidating for many users and preventing them from participating, which is what the second thread is for as a compromise.

I think I have stuck fairly rigidly to these guidelines (difficulty aside).

The biggest reason why these questions are intimidating is that a lot of them are things that you would normally be guided through in baby-steps. Very few of these questions require any knowledge/techniques from outside syllabus as far as I can see, it is just that the way the syllabus techniques need to be strung together is more demanding than is the case in the actual HSC. (But very few even approach olympiad difficulty.)

In any case, I think that questions of this sort that are demanding in the creative sense rather than the technical sense are of immense benefit to students trying to improve their understanding of the syllabus theory. Far more so than routine integration drills. Of course, this is not to everyone's taste so it is probably for the best that the old marathon thread is split into two.

tl;dr I think that the majority of the problems in this thread as it stands now are more appropriate for this new "advanced marathon" thread vs extra-curricular.

Carrotsticks · Mar 20, 2014

Re: HSC 2014 4U Marathon - Advanced Level

I've looked at all the previously posted questions and of all the 'advanced' questions, few were actually 'extracurricular' as all the steps were within MX2.

But I strongly agree with the splitting of the thread. Other students should feel welcome to participate.

Sy123 · Mar 22, 2014

Re: HSC 2014 4U Marathon - Advanced Level

$\\ $In 3D space consider the points$ \ \ A(a,0,0) \ , \ B(0,b,0) \ , \ C(0,0,c) \ , $and$ \ O(0,0,0) \\ \\ $A tetrahedron is formed with these 4 points$ \\\\ $Let$ \ d \ $be the distance from$ \ O \ $to the triangle$ \ ABC \\ \\ $By considering volumes show that$ \\ \\ \frac{1}{d^2} = \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$

RealiseNothing · Mar 22, 2014

Re: HSC 2014 4U Marathon - Advanced Level

Is $d$ the perpendicular distance from O to the plane of the triangle ABC?

Sy123 · Mar 22, 2014

Re: HSC 2014 4U Marathon - Advanced Level

RealiseNothing said:
Is $d$ the perpendicular distance from O to the plane of the triangle ABC?

Yep

seanieg89 · Mar 22, 2014

Re: HSC 2014 4U Marathon - Advanced Level

In an MX2 level proof I would prove the cross-product formula for a triangles area (using perp distance formula etc), but here is the idea:

$V=\frac{abc}{6}=\frac{d|\triangle ABC|}{3}\\ \\ \Rightarrow a^2b^2c^2 = d^2|(B-A)\times (C-A)|=d^2(a^2b^2+a^2c^2+b^2c^2).\\ \\ $Rearrange to finish.$$

Sy123 · Mar 22, 2014

Re: HSC 2014 4U Marathon - Advanced Level

seanieg89 said:
In an MX2 level proof I would prove the cross-product formula for a triangles area (using perp distance formula etc), but here is the idea:

$V=\frac{abc}{6}=\frac{d|\triangle ABC|}{3}\\ \\ \Rightarrow a^2b^2c^2 = d^2|(B-A)\times (C-A)|=d^2(a^2b^2+a^2c^2+b^2c^2).\\ \\ $Rearrange to finish.$$

Is cross-product formula Heron's formula?

If not the alternative would be Heron's formula which is quite easy to prove

seanieg89 · Mar 22, 2014

Re: HSC 2014 4U Marathon - Advanced Level

Sy123 said:
Is cross-product formula Heron's formula?

If not the alternative would be Heron's formula which is quite easy to prove

No, it's http://en.wikipedia.org/wiki/Cross_product. It takes about as long to prove that it gives you area as it takes to prove Heron's.

Sy123 · Mar 22, 2014

Re: HSC 2014 4U Marathon - Advanced Level

seanieg89 said:
No, it's http://en.wikipedia.org/wiki/Cross_product. It takes about as long to prove that it gives you area as it takes to prove Heron's.

Ah alright

Here is a cool one:

$I_n = \int_0^1 x^n(1-x)^n \ dx$

$\\ $Find$ \ I_n \ $for integer$ \ n \ $hence evaluate$ \ \left(\frac{1}{2}\right)! \ $by extending the definition of the factorial$$

RealiseNothing · Mar 23, 2014

Re: HSC 2014 4U Marathon - Advanced Level

Sy123 said:
Ah alright

Here is a cool one:

$I_n = \int_0^1 x^n(1-x)^n \ dx$

$\\ $Find$ \ I_n \ $for integer$ \ n \ $hence evaluate$ \ \left(\frac{1}{2}\right)! \ $by extending the definition of the factorial$$

Notice the symmetry about $x=\frac{1}{2}$ as well as the "difference of two squares" nature of the integral. So use the substitution:

$x=\frac{1}{2}(1-sin(\theta))$

$I_n = \frac{1}{2^{2n+1}} \int_{-\frac{1}{2}}^{\frac{1}{2}} cos^{2n+1}(\theta) d\theta$

$I_n = \frac{1}{2^{2n}} \int_{0}^{\frac{1}{2}} cos^{2n+1}(\theta) d\theta$

We get the last line since $cos^{11}(\theta)$ is an even function. Don't think this goes anywhere though. My instinct is saying to play with the symmetry about $x=\frac{1}{2}$ more to get a really nice way of solving the integral.

Sy123 · Mar 23, 2014

Re: HSC 2014 4U Marathon - Advanced Level

RealiseNothing said:
Notice the symmetry about $x=\frac{1}{2}$ as well as the "difference of two squares" nature of the integral. So use the substitution:

$x=\frac{1}{2}(1-sin(\theta))$

$I_n = \frac{1}{2^{2n+1}} \int_{-\frac{1}{2}}^{\frac{1}{2}} cos^{2n+1}(\theta) d\theta$

$I_n = \frac{1}{2^{2n}} \int_{0}^{\frac{1}{2}} cos^{2n+1}(\theta) d\theta$

We get the last line since $cos^{11}(\theta)$ is an even function. Don't think this goes anywhere though. My instinct is saying to play with the symmetry about $x=\frac{1}{2}$ more to get a really nice way of solving the integral.

I think the limits of your integral are wrong, I didn't use this method however

RealiseNothing · Mar 23, 2014

Re: HSC 2014 4U Marathon - Advanced Level

Sy123 said:
I think the limits of your integral are wrong, I didn't use this method however

Yep I see what went wrong with the limits lol. Tried doing 2 substitutions at once - didn't end well did it.

seanieg89 · Mar 23, 2014

Re: HSC 2014 4U Marathon - Advanced Level

$For $m,n\geq 1$, let:\\ \\ $B(m,n)=\int_0^1 x^m(1-x)^n \, dx=\int_0^1 \frac{d}{dx}\left(\frac{x^{m+1}}{m+1}\right)(1-x)^n\, dx\\ \\ = \frac{n}{m+1}\int_0^1 x^{m+1}(1-x)^{n-1}\, dx = \frac{m}{n+1}B(m+1,n-1).\\ \\ $So, for positive integers m,n: $\\ \\ B(m,n)=\frac{m!n!}{(m+n)!}I(m+n,0)=\frac{m!n!}{(m+n+1)!}.\\ \\ $So using this expression formally to chuck in $m=n=1/2$, we get: $\\ \\ (1/2)!^2=2I(1/2,1/2)=\pi/4\\ \\ \Rightarrow \left(\frac{1}{2}\right)!=\frac{\sqrt{\pi}}{2}\\ \\ $ by a routine integration.$$

Carrotsticks · Mar 23, 2014

Re: HSC 2014 4U Marathon - Advanced Level

If you said that m,n are positive integers, how did you substitute in m=n=1/2?

seanieg89 · Mar 23, 2014

Re: HSC 2014 4U Marathon - Advanced Level

Carrotsticks said:
If you said that m,n are positive integers, how did you substitute in m=n=1/2?

We are just constructing a plausible definition for something that has been previously undefined (the factorials of half-integers), it is not a matter of using rigorous substitution to deduce the value of some previously defined and unknown quantity. (Hence the qualifier "formally".)

I can only imagine this is what Sy was asking for, if we are not going via the Gamma function (the Gamma function approach highlights the desirable properties of the "factorial extension" more clearly imo).

Carrotsticks · Mar 23, 2014

Re: HSC 2014 4U Marathon - Advanced Level

seanieg89 said:
We are just constructing a plausible definition for something that has been previously undefined (the factorials of half-integers), it is not a matter of using rigorous substitution to deduce the value of some previously defined and unknown quantity. (Hence the qualifier "formally".)

I can only imagine this is what Sy was asking for, if we are not going via the Gamma function (the Gamma function approach highlights the desirable properties of the "factorial extension" more clearly imo).

Yep, I figured that Sy was going for this approach of reduction formula --> factorials --> substitute in value.

Now that we've 'guessed' what the value of (1/2) factorial is, I think it'd be good to have a follow up question actually proving that (1/2)! = root(pi)/2 using the Gamma function. Might be a bit difficult (or even impossible) within MX2 constraints though since the reduction formulae used for the integral x^{n-1}*e^{-x} from 0 to 1 is not considered to be continuous.

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