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STEP maths I, II, III 1991 solutions

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DFranklin
I think it comes out if you look at the real part of (1+e2iθ)m(1+e^{2i\theta})^m.


This type of trigonometric series came up again in the 1990 III paper, I encountered this example before in an outdated OCR maths textbook, so maybe it used to be in the syllabus? You then use a little trick to extract the real part if you don't know already.
khaixiang
This type of trigonometric series came up again in the 1990 III paper, I encountered this example before in an outdated OCR maths textbook, so maybe it used to be in the syllabus? You then use a little trick to extract the real part if you don't know already.
Quickest way I see of getting the real part is to "assume the answer". i.e. show that 2cosθ(cosθ+isinθ)=(1+cos2θ+isin2θ)2\cos \theta(\cos \theta + i \sin \theta) = (1+\cos 2\theta + i \sin 2\theta). Which is very short, but a bit of a leap if you don't know it already.

Is there a way of finding this without too much work?
DFranklin
Quickest way I see of getting the real part is to "assume the answer". i.e. show that 2cosθ(cosθ+isinθ)=(1+cos2θ+isin2θ)2\cos \theta(\cos \theta + i \sin \theta) = (1+\cos 2\theta + i \sin 2\theta). Which is very short, but a bit of a leap if you don't know it already.

Is there a way of finding this without too much work?


I've learnt from the old STEP booklet that they usually do the following factorisation, it can be very useful at times like this.

eniθ1eiθ1=eniθ2(eniθ2eniθ2)eiθ2(eiθ2eiθ2)\\ \displaystyle \Re{\frac{e^{ni\theta}-1}{e^{i\theta}-1}}\\ =\Re{\frac{e^{\frac{ni\theta}{2}}(e^{\frac{ni \theta}{2}}-e^{\frac{-ni\theta}{2}})}{e^{\frac{i\theta}{2}}(e^{\frac{i \theta}{2}}-e^{\frac{-i\theta}{2}})}}
khaixiang
I've learnt from the old STEP booklet that they usually do the following factorisation, it can be very useful at times like this.

eniθ1eiθ1=eniθ2(eniθ2eniθ2)eiθ2(eiθ2eiθ2)\\ \displaystyle \Re{\frac{e^{ni\theta}-1}{e^{i\theta}-1}}\\ =\Re{\frac{e^{\frac{ni\theta}{2}}(e^{\frac{ni \theta}{2}}-e^{\frac{-ni\theta}{2}})}{e^{\frac{i\theta}{2}}(e^{\frac{i \theta}{2}}-e^{\frac{-i\theta}{2}})}}
Neat. I've been wondering: does it make sense to start a "useful tricks for STEP / AEA" thread? There are a few things like this that do seem to come up a lot.
Reply 104
Khai, I don't quite understand the significance of that factorisation :frown:
Care to explain a little? I just woke up.... xD


Oh, that complex expansion works.
And (1+cos2T+isin2T)=(2cos²T+2isinTcosT) which is a one line proof! Thanks again.

Franklin
I've been wondering: does it make sense to start a "useful tricks for STEP / AEA" thread? There are a few things like this that do seem to come up a lot.

I would so benefit from that. I was sort of hoping there would be one around, but...no such luck. :p:
I wonder how much one could really put in though, since STEP questions tend to be more or less unique. (Or do they?)
Rabite
I would so benefit from that. I was sort of hoping there would be one around, but...no such luck. :p:
I wonder how much one could really put in though, since STEP questions tend to be more or less unique. (Or do they?)
I'm not sure how many formula you can give "exactly" (although if I were doing STEP, I think I would memorise the solutions to sinθ+sin2θ...+sinnθ\sin \theta + \sin 2\theta... + \sin n\theta and cosθ+cos2θ+...+sinnθ\cos \theta + \cos 2\theta + ... + \sin n\theta). But I think you can show enough "nice tricks" that (hopefully) in the exam you can think "I've seen something a bit like this before".

(In some senses, it's not that different from the solutions threads, but I think there are a lot of questions that really are "one-off" ideas that won't be repeated and so the stuff you really ought to make note of can get missed).
Rabite
Khai, I don't quite understand the significance of that factorisation :frown:
Care to explain a little? I just woke up.... xD


Oh, that complex expansion works.
And (1+cos2T+isin2T)=(2cos²T+2isinTcosT) which is a one line proof! Thanks again.


For III/8 second part:
(1+e2iθ)m=(eiθ)m(eiθ+eiθ)m=(emiθ)(2mcosmθ))=2mcosmθcosmθ\\ \Re{(1+e^{2i\theta})^m}\\=\Re{(e^{i\theta})^{m}(e^{-i\theta}+e^{i\theta})^m}\\ =\Re{(e^{mi\theta})(2^{m}\cos^m{\theta}))}\\=2^{m}\cos^m{\theta}\cos{m\theta}

For the case of (eiθ+e2iθ++eniθ)\Re{(e^{i\theta}+e^{2i\theta}+\dots+e^{ni\theta})} you can do similiar factorisation, which simplifies the algebra considerably.

eniθ1eiθ1=eniθ2(eniθ2eniθ2)eiθ2(eiθ2eiθ2)\\ \displaystyle \Re{\frac{e^{ni\theta}-1}{e^{i\theta}-1}}\\ =\Re{\frac{e^{\frac{ni\theta}{2}}(e^{\frac{ni \theta}{2}}-e^{\frac{-ni\theta}{2}})}{e^{\frac{i\theta}{2}}(e^{\frac{i \theta}{2}}-e^{\frac{-i\theta}{2}})}}

This trick seems to be used quite often in the solutions provided by Siklos
Reply 107
Ah hah, thanks for that. *sigh* I feel like I should be getting these things by now. :/

As for that little trick, I'll be hoping that they ask us to sum cosrθ\cos{r\theta} now :wink:
DFranklin
Neat. I've been wondering: does it make sense to start a "useful tricks for STEP / AEA" thread? There are a few things like this that do seem to come up a lot.

I think it might, if you have any useful tricks. I've never seen many, but maybe that's just me...
Reply 109
Are there a set of solutions for the STEP II specimen paper available online?
http://math.*you-can-guess-what-goes-here*.com/Data/Advanced%20Mathematics/STEP/STEP%20II/STEP%20II%20Sample.pdf
Speleo
Are there a set of solutions for the STEP II specimen paper available online?
http://math.*you-can-guess-what-goes-here*.com/Data/Advanced%20Mathematics/STEP/STEP%20II/STEP%20II%20Sample.pdf


All the questions in this specimen paper are from real STEP II papers around 2000, 2001 and 2002. (or maybe you know this already?) So you can find the solutions in meikleriggs.
Reply 111
Really? I only recognise a couple of them, which I assumed were from the Siklos book. Thanks :smile:
Speleo
Really? I only recognise a couple of them, which I assumed were from the Siklos book. Thanks :smile:


LOL, after checking through, the specimen paper is STEP II 2002.
Reply 113
So it is :biggrin:
Must have only glanced at that paper before (would explain why I hadn't done them but recognised a couple...).

That paper is very, very easy btw (got enough marks for an S with one hour's work unless my marking is very far out :tongue:), but the grade boundaries are as usual, must be because it's the first year of the specification :s-smilie:
Reply 114
III/6

God, this one took ages to type out.
No one's even gonna read it, but hey...

Could someone else at least try the question to see what the real answer is?
Rabite
Could someone else at least try the question to see what the real answer is?
I've skimmed what you've done and I don't see a problem; the question seems a bit long and unpleasant, but in my experience every STEP question mentioning a hyperbolic function turns out to be a bit long and unpleasant! Any particular reason you think you didn't get the right answer?
Reply 116
I've just never done a question with transformations before, and for some reason, I have this belief instilled in me that if a solution takes a ridiculous amount of algebra and multiple abbreviations, I'm probably doing it the wrong way :p:
That's all though.
Rabite
I've just never done a question with transformations before, and for some reason, I have this belief instilled in me that if a solution takes a ridiculous amount of algebra and multiple abbreviations, I'm probably doing it the wrong way :p:
That's all though.
The thing is, I've never really had solutions other than ones I've done myself to compare to. So possibly there are some nice tricks I'm just not seeing. But in my experience, if a question involves conic sections, change of coordinates or hyperbolic functions, it will tend to be long and heavy on the algebra. As this Q had all 3 of those, I didn't think your answer was longer or more complicated than I'd expect.

In practical terms, this is the kind of question I'd look at, think "I don't see me getting that out in a hurry" and move on to something else. Of course, sometimes you have to pick the best question of a bad lot...
Reply 118
I wouldn't have attempted it in an exam...
Also, is it too soon to start a 1990 thread? Or is there one...? I ran a search, but nothing came up, so I'm gussing not.
Then again, there are a lot of questions missing here.

I have mechanics phobia, so I won't be doing many more questions in this thread anyway, hehe...
Reply 119
You are welcome to start a 1990 thread, at this stage we are doing the questions mostly for our own benefit, so it doesn't really matter if one or two are missing from each paper.

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