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STEP Prep Thread 2018

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Original post by Bangz
STEP
This question refers to the 1994 STEP1 paper question 8. I have done the first two parts but I dont know how they relate to the last part.

so far I have tried manipulating (1+sinx)/(1+cosx) to get 1 + tan(x). I tried splitting the integral, using the property of logarithms, and I have gotten nowhere.

what am I not seeing?

p.s. please can you nudge and not give the whole question away.

thanks


I just told you on another thread that you need to use a similar substitution...

Further hint if necessary:

Spoiler

(edited 6 years ago)
Original post by Bangz
STEP
This question refers to the 1994 STEP1 paper question 8. I have done the first two parts but I dont know how they relate to the last part.

To get the idea of what's going on, sketch the graphs of 1+sin(x) and 1+cos(x) and think about how the graphs of ln(1+cos(x)) and ln(1+sin(x)) will be related. You should be able to see a) useful symmetries and b) the value of the integral.

Then try to apply a similar trick to the first part, but based on the symmetries of sin and cos rather than tan, to evaluate the integral more rigorously.
Could someone please help with the very last part of this question? I got as far as showing that angle ECF is 30 degrees, and I was just stuck there for a long time, but according to Siklos' answer, this completes the proof! I don't understand how showing that angle FCB = 120 - 2a would show that angle ACB has been trisected, since showing this doesn't show that angles ECB and FCE are equal necessarily?
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Original post by I hate maths
Could someone please help with the very last part of this question? I got as far as showing that angle ECF is 30 degrees, and I was just stuck there for a long time, but according to Siklos' answer, this completes the proof! I don't understand how showing that angle FCB = 120 - 2a would show that angle ACB has been trisected, since showing this doesn't show that angles ECB and FCE are equal necessarily?


To trisect an angle a at a point P means to construct an angle a/3 at P.

We must have here that

ACB=180−3α=3(60−α)⇒2ACB/3=120−2α=FCB,ACB/3=60−α=FCAACB = 180-3\alpha=3(60-\alpha) \\ \Rightarrow 2ACB/3=120-2\alpha = FCB, ACB/3=60-\alpha=FCA

So FCA is the trisection of ACB
Original post by atsruser
To trisect an angle a at a point P means to construct an angle a/3 at P.

We must have here that

ACB=180−3α=3(60−α)⇒2ACB/3=120−2α=FCB,ACB/3=60−α=FCAACB = 180-3\alpha=3(60-\alpha) \\ \Rightarrow 2ACB/3=120-2\alpha = FCB, ACB/3=60-\alpha=FCA

So FCA is the trisection of ACB


Just that first line did it for me - for some reason I thought the lines FC and CE had to split the angle into three equal parts. But that's not what trisecting is - it's just making splitting the angle so you get one new angle that's a third of the original. Thanks so much!
Original post by I hate maths
Just that first line did it for me - for some reason I thought the lines FC and CE had to split the angle into three equal parts. But that's not what trisecting is - it's just making splitting the angle so you get one new angle that's a third of the original. Thanks so much!


Right. I only put the rest of the working there as I wasn't entirely sure what you didn't understand, and was wondering if you were misreading the angles or something.
Hey guys sorry If I am being redundant, but when will the 2017 papers be released? Also does anyone know any other examinations (possibly from outside the UK), like the MAT or AEA? I am considering moving unto AIME papers and the BMO, but its easy to see that the lack of calculus (integration is a very weak point of mine) being a major disadvantage.
(edited 6 years ago)
Original post by Electric-man7
Hey guys sorry If I am being redundant, but when will the 2017 papers be released? Also does anyone know any other examinations (possibly from outside the UK), like the MAT or AEA? I am considering moving unto AIME papers and the BMO, but its easy to see that the lack of calculus (integration is a very weak point of mine) being a major disadvantage.


Have you done all the STEP from 87?
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Original post by Electric-man7
Hey guys sorry If I am being redundant, but when will the 2017 papers be released? Also does anyone know any other examinations (possibly from outside the UK), like the MAT or AEA? I am considering moving unto AIME papers and the BMO, but its easy to see that the lack of calculus (integration is a very weak point of mine) being a major disadvantage.


Sometime later this month tends to be the norm.
I have completed STEP II ans STEP I and working on the remaining STEP III papers.
Original post by Electric-man7
I have completed STEP II ans STEP I and working on the remaining STEP III papers.


how's that a good thing? you dont have anymore practice material for step and the exam is in 9 months...
Original post by Student1256
how's that a good thing? you dont have anymore practice material for step and the exam is in 9 months...

I never claimed it was a good thing, but I have been preparing for a long time now (atleast 11 months) so its easy to see how I ran out of practice material. Its partially my schools fault for forcing me to take all 4 A-levels a year early.
Original post by Electric-man7
I never claimed it was a good thing, but I have been preparing for a long time now (atleast 11 months) so its easy to see how I ran out of practice material. Its partially my schools fault for forcing me to take all 4 A-levels a year early.


What subjects did you do and what did you get ?????
Also......what are you doing this year then?

P.S........start panic when seeing someone has already done all the papers.....!
I am doing enrichment mathematics and participating in informatics contests.
I took Further Maths, Maths, Physics and Biology. With an A*A*A*B, will my B in biology affect my chances?
(edited 6 years ago)
I would not really panic though, you're still incredibly early. In fact my problem is possibly much worse than yours.
Original post by Electric-man7
I would not really panic though, you're still incredibly early. In fact my problem is possibly much worse than yours.


I planned to start it in Nov (before interview).....now still doing things that interest me (Fourier series...complex analysis) lol.... but somehow I still found Step questions not straightforward. How do you find them after going through all the papers????
Some of them become more straightforward than others, geometry and number theory become so much easier if you solve enough olympiad questions ,differentiation gets easier after a lot of practice, however my dream of being good at integration and statistics is still far, but I am starting to see an improvement in stats after studying extra modules (I took virtually no statistics in Maths or further maths).
Original post by Electric-man7
Some of them become more straightforward than others, geometry and number theory become so much easier if you solve enough olympiad questions ,differentiation gets easier after a lot of practice, however my dream of being good at integration and statistics is still far, but I am starting to see an improvement in stats after studying extra modules (I took virtually no statistics in Maths or further maths).


Which college are you applying to? Trinity...?
I highy doubt fourier series will help with STEP though, STEP questions arenot really susceptible to higher maths.

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