STEP Maths I,II,III 1987 Solutions

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    Just started looking at some of these and not really up to standard. Can anyone just explain on STEP I question 2 the first part, why the angle is pi/n ?
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    (Original post by thestudent)
    Just started looking at some of these and not really up to standard. Can anyone just explain on STEP I question 2 the first part, why the angle is pi/n ?
    As DFranklin said, you take a right angled triangle, hypotenuse R, small side r consider the angle of this triangle theta such that sin(theta)=r/R.

    Now if you think of it like this, there are 2 'thetas' for every smaller circle and there are n smaller circles, so n=2pi/2theta so theta=pi/n.

    It's kinda hard to explain without a diagram.
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    thanks for the help and thanks for the thread, a great help
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    Here is my attempt at Paper 1 number 6. Does anyone agree with it?
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  1. File Type: pdf 1987PAPER1.6pdf.pdf (28.5 KB, 379 views)
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    (Original post by Dystopia)
    STEP II, Q5.

    i) f(x) = \alpha x

    [y - f(y)]^{n} = y^{n}(1-\alpha)^{n}

    \Rightarrow \frac{\mathrm{d}^{n-1}}{\mathrm{d}y^{n-1}} [y-f(y)]^{n} = n! y (1-\alpha)^{n}

    \displaystyle \sum_{n=0}^{\infty}\frac{1}{n!}n  !y(1 - \alpha)^{n}

    This is an infinite geometric series with a=y(1-\alpha), \; r=(1-\alpha). Since 0 < \alpha < 2, it converges to \frac{y}{\alpha} - y.

    \Rightarrow f^{-1}(y) = y + \frac{y}{\alpha} - y = \frac{y}{\alpha}

    As expected.

    ii) Let y = f(x) = x - \frac{x^{3}}{4}

    [y - f(y)]^{n} = \frac{y^{3n}}{2^{2n}}

    \frac{\mathrm{d}^{n-1}}{\mathrm{d}y^{n-1}} y^{3n} = \frac{(3n)!}{(2n+1)!}y^{2n+1}

    \displaystyle f^{-1}(y) = y + \sum_{n=1}^{\infty} \frac{(3n)!y^{2n+1}}{n!(2n+1)!2^  {2n}}

    But y=\frac{1}{2}, so

    \displaystyle x = \frac{1}{2} + \sum_{n=1}^{\infty}\frac{(3n)!}{  n!(2n+1)!2^{4n+1}}

    Note that \frac{(3n)!}{n!(2n+1)!2^{4n+1}}=  \frac{1}{2} when n=0.

    \displaystyle x = \sum_{n=0}^{\infty}\frac{(3n)!}{  n!(2n+1)!2^{4n+1}}

    iii) Let y = f(x) = x - e^{\lambda x}

    [y - f(y)]^{n} = e^{\lambda n y}

    \frac{\mathrm{d}^{n-1}}{\mathrm{d}y^{n-1}} [y-f(y)]^{n} = (\lambda n)^{n-1} e^{\lambda n y}

    \displaystyle f^{-1}(y) = y + \sum_{n=1}^{\infty} \frac{(\lambda n)^{n-1}}{n!} e^{\lambda n y}

    But y = 0, so

    \displaystyle x = \sum_{n=1}^{\infty} \frac{(\lambda n)^{n-1}}{n!}

    Edit: Fixed. Didn't realise it was actually posting last night...
    Been wondering what was wrong with this. Finally realised that in first part you have lower limit of sum as n=0 when it should be n=1
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    Here are my attempts at numbers 9-12 on 1987 STEP Paper I
    I would be grateful for any confirmation or reports of errors.

    Some errors have been discopvered. For revised solutions
    see post no.237 for question 10
    Post 238 for question 11 and post 239 for question 13
    Attached Images
  2. File Type: pdf 1987PAPER1.9.pdf (32.2 KB, 350 views)
  3. File Type: pdf 1987PAPER1.10.pdf (40.6 KB, 271 views)
  4. File Type: pdf 1987PAPER1.11.pdf (41.1 KB, 294 views)
  5. File Type: pdf 1987PAPER1.12.pdf (26.2 KB, 313 views)
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    STEP I numbers 13-16
    Again if anyone could check them I would be most grateful.
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  6. File Type: pdf 1987PAPER1.13.pdf (43.1 KB, 280 views)
  7. File Type: pdf 1987PAPER1.14.pdf (25.2 KB, 200 views)
  8. File Type: pdf 1987PAPER1.15.pdf (45.6 KB, 267 views)
  9. File Type: pdf 1987PAPER1.16.pdf (35.5 KB, 258 views)
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    I think you need to update the links to the papers on the front page, as when I click on them I get sent to some random advertising webiste and am bombarded by pop-ups for 5 minutes.
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    (Original post by brianeverit)
    Been wondering what was wrong with this. Finally realised that in first part you have lower limit of sum as n=0 when it should be n=1
    Thank you for pointing that out; is it correct now?

    Are you also in Y13 btw?
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    The links for the papers don't work.
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    Still not working...
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    (Original post by DaveSimpson)
    Still not working...
    That site no longer "works", it hasn't for several months. Shame, it was really good.
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    Link

    That should work.
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    lol it's no good, just gives me Page cannot be displayed
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    STYEP 1987 Fma
    Questions 2,3 and 4
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  10. File Type: pdf 1987PAPER fma.2.pdf (35.1 KB, 501 views)
  11. File Type: pdf 1987PAPER fma.3.pdf (28.9 KB, 403 views)
  12. File Type: pdf 1987PAPER fma.4.pdf (30.7 KB, 309 views)
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    198987 STEP Fma numbers 8,9 and 10
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  13. File Type: pdf 1987PAPER fma.8.pdf (43.7 KB, 281 views)
  14. File Type: pdf 1987PAPER fma.9.pdf (21.9 KB, 203 views)
  15. File Type: pdf 1987PAPER fma.10.pdf (35.5 KB, 200 views)
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    1987 STEP Fma numbers 12 -16
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  16. File Type: pdf 1987PAPER fma.12.pdf (51.6 KB, 239 views)
  17. File Type: pdf 1987PAPER fma.13.pdf (29.0 KB, 199 views)
  18. File Type: pdf 1987PAPER fma.14.pdf (41.1 KB, 197 views)
  19. File Type: pdf 1987PAPER fma.15.pdf (22.9 KB, 191 views)
  20. File Type: pdf 1987PAPER fma.16.pdf (15.4 KB, 245 views)
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    Those are the answers not the questions.
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    (Original post by DaveSimpson)
    Those are the answers not the questions.
    The purpose of this thead is to provide answers to the 1987 paper questions .
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    Yeah but what's the point of answers if no one has the questions?
 
 
 
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