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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?
    Attached Files
  1. File Type: pdf 1987PAPER1.6pdf.pdf (28.5 KB, 208 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 Files
  2. File Type: pdf 1987PAPER1.9.pdf (32.2 KB, 198 views)
  3. File Type: pdf 1987PAPER1.10.pdf (40.6 KB, 164 views)
  4. File Type: pdf 1987PAPER1.11.pdf (41.1 KB, 144 views)
  5. File Type: pdf 1987PAPER1.12.pdf (26.2 KB, 161 views)
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    STEP I numbers 13-16
    Again if anyone could check them I would be most grateful.
    Attached Files
  6. File Type: pdf 1987PAPER1.13.pdf (43.1 KB, 147 views)
  7. File Type: pdf 1987PAPER1.14.pdf (25.2 KB, 78 views)
  8. File Type: pdf 1987PAPER1.15.pdf (45.6 KB, 147 views)
  9. File Type: pdf 1987PAPER1.16.pdf (35.5 KB, 122 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
    Attached Files
  10. File Type: pdf 1987PAPER fma.2.pdf (35.1 KB, 309 views)
  11. File Type: pdf 1987PAPER fma.3.pdf (28.9 KB, 242 views)
  12. File Type: pdf 1987PAPER fma.4.pdf (30.7 KB, 176 views)
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    198987 STEP Fma numbers 8,9 and 10
    Attached Files
  13. File Type: pdf 1987PAPER fma.8.pdf (43.7 KB, 147 views)
  14. File Type: pdf 1987PAPER fma.9.pdf (21.9 KB, 100 views)
  15. File Type: pdf 1987PAPER fma.10.pdf (35.5 KB, 99 views)
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    1987 STEP Fma numbers 12 -16
    Attached Files
  16. File Type: pdf 1987PAPER fma.12.pdf (51.6 KB, 98 views)
  17. File Type: pdf 1987PAPER fma.13.pdf (29.0 KB, 93 views)
  18. File Type: pdf 1987PAPER fma.14.pdf (41.1 KB, 99 views)
  19. File Type: pdf 1987PAPER fma.15.pdf (22.9 KB, 75 views)
  20. File Type: pdf 1987PAPER fma.16.pdf (15.4 KB, 113 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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