Just one sec...
Hey! Sign in to get help with your study questionsNew here? Join for free to post

STEP Maths I,II,III 1987 Solutions

Announcements Posted on
GCSE grade boundaries are here - GO> 23-08-2016
    Offline

    ReputationRep:
    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 ?
    Offline

    ReputationRep:
    (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.
    Offline

    ReputationRep:
    thanks for the help and thanks for the thread, a great help
    Offline

    ReputationRep:
    Here is my attempt at Paper 1 number 6. Does anyone agree with it?
    Attached Images
  1. File Type: pdf 1987PAPER1.6pdf.pdf (28.5 KB, 347 views)
    Offline

    ReputationRep:
    (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
    Offline

    ReputationRep:
    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, 312 views)
  3. File Type: pdf 1987PAPER1.10.pdf (40.6 KB, 243 views)
  4. File Type: pdf 1987PAPER1.11.pdf (41.1 KB, 266 views)
  5. File Type: pdf 1987PAPER1.12.pdf (26.2 KB, 282 views)
    Offline

    ReputationRep:
    STEP I numbers 13-16
    Again if anyone could check them I would be most grateful.
    Attached Images
  6. File Type: pdf 1987PAPER1.13.pdf (43.1 KB, 248 views)
  7. File Type: pdf 1987PAPER1.14.pdf (25.2 KB, 182 views)
  8. File Type: pdf 1987PAPER1.15.pdf (45.6 KB, 240 views)
  9. File Type: pdf 1987PAPER1.16.pdf (35.5 KB, 229 views)
    Offline

    ReputationRep:
    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.
    Offline

    ReputationRep:
    (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?
    Offline

    ReputationRep:
    The links for the papers don't work.
    Offline

    ReputationRep:
    Still not working...
    Offline

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

    ReputationRep:
    Link

    That should work.
    Offline

    ReputationRep:
    lol it's no good, just gives me Page cannot be displayed
    Offline

    ReputationRep:
    STYEP 1987 Fma
    Questions 2,3 and 4
    Attached Images
  10. File Type: pdf 1987PAPER fma.2.pdf (35.1 KB, 467 views)
  11. File Type: pdf 1987PAPER fma.3.pdf (28.9 KB, 375 views)
  12. File Type: pdf 1987PAPER fma.4.pdf (30.7 KB, 283 views)
    Offline

    ReputationRep:
    198987 STEP Fma numbers 8,9 and 10
    Attached Images
  13. File Type: pdf 1987PAPER fma.8.pdf (43.7 KB, 252 views)
  14. File Type: pdf 1987PAPER fma.9.pdf (21.9 KB, 178 views)
  15. File Type: pdf 1987PAPER fma.10.pdf (35.5 KB, 179 views)
    Offline

    ReputationRep:
    1987 STEP Fma numbers 12 -16
    Attached Images
  16. File Type: pdf 1987PAPER fma.12.pdf (51.6 KB, 214 views)
  17. File Type: pdf 1987PAPER fma.13.pdf (29.0 KB, 174 views)
  18. File Type: pdf 1987PAPER fma.14.pdf (41.1 KB, 174 views)
  19. File Type: pdf 1987PAPER fma.15.pdf (22.9 KB, 159 views)
  20. File Type: pdf 1987PAPER fma.16.pdf (15.4 KB, 219 views)
    Offline

    ReputationRep:
    Those are the answers not the questions.
    Offline

    ReputationRep:
    (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 .
    Offline

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

Reply

Submit reply

Register

Thanks for posting! You just need to create an account in order to submit the post
  1. this can't be left blank
    that username has been taken, please choose another Forgotten your password?
  2. this can't be left blank
    this email is already registered. Forgotten your password?
  3. this can't be left blank

    6 characters or longer with both numbers and letters is safer

  4. this can't be left empty
    your full birthday is required
  1. Oops, you need to agree to our Ts&Cs to register
  2. Slide to join now Processing…

Updated: February 8, 2015
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

Poll
How do you sleep?

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22

Registered Office: International House, Queens Road, Brighton, BN1 3XE

Quick reply
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.