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STEP Maths I,II,III 1987 Solutions

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  1. Offline

    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 ?
  2. Offline

    (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
  5. File Type: pdf 1987PAPER1.6pdf.pdf (28.5 KB, 310 views)
  6. Offline

    (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
  7. Offline

    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
  8. File Type: pdf 1987PAPER1.9.pdf (32.2 KB, 276 views)
  9. File Type: pdf 1987PAPER1.10.pdf (40.6 KB, 218 views)
  10. File Type: pdf 1987PAPER1.11.pdf (41.1 KB, 216 views)
  11. File Type: pdf 1987PAPER1.12.pdf (26.2 KB, 224 views)
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    STEP I numbers 13-16
    Again if anyone could check them I would be most grateful.
    Attached Files
  13. File Type: pdf 1987PAPER1.13.pdf (43.1 KB, 216 views)
  14. File Type: pdf 1987PAPER1.14.pdf (25.2 KB, 141 views)
  15. File Type: pdf 1987PAPER1.15.pdf (45.6 KB, 209 views)
  16. File Type: pdf 1987PAPER1.16.pdf (35.5 KB, 190 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?
  19. Offline

    The links for the papers don't work.
  20. Offline

    Still not working...
  21. Offline

    (Original post by DaveSimpson)
    Still not working...
    That site no longer "works", it hasn't for several months. Shame, it was really good.
  22. Offline


    That should work.
  23. Offline

    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
  25. File Type: pdf 1987PAPER fma.2.pdf (35.1 KB, 405 views)
  26. File Type: pdf 1987PAPER fma.3.pdf (28.9 KB, 325 views)
  27. File Type: pdf 1987PAPER fma.4.pdf (30.7 KB, 248 views)
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    198987 STEP Fma numbers 8,9 and 10
    Attached Files
  29. File Type: pdf 1987PAPER fma.8.pdf (43.7 KB, 211 views)
  30. File Type: pdf 1987PAPER fma.9.pdf (21.9 KB, 144 views)
  31. File Type: pdf 1987PAPER fma.10.pdf (35.5 KB, 158 views)
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    1987 STEP Fma numbers 12 -16
    Attached Files
  33. File Type: pdf 1987PAPER fma.12.pdf (51.6 KB, 164 views)
  34. File Type: pdf 1987PAPER fma.13.pdf (29.0 KB, 137 views)
  35. File Type: pdf 1987PAPER fma.14.pdf (41.1 KB, 152 views)
  36. File Type: pdf 1987PAPER fma.15.pdf (22.9 KB, 117 views)
  37. File Type: pdf 1987PAPER fma.16.pdf (15.4 KB, 163 views)
  38. Offline

    Those are the answers not the questions.
  39. Offline

    (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 .
  40. Offline

    Yeah but what's the point of answers if no one has the questions?


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