STEP Maths I,II,III 1987 Solutions
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Re: STEP Maths I,II,III 1987 SolutionsAs 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.(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 ?
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.Last edited by Square; 18-09-2007 at 21:59. -
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Re: STEP Maths I,II,III 1987 SolutionsBeen 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(Original post by Dystopia)
STEP II, Q5.
i)
![[y - f(y)]^{n} = y^{n}(1-\alpha)^{n}](http://www.thestudentroom.co.uk/latexrender/pictures/1a/1a56451c0ccb1064e7e35c146bc1fdb1.png "[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}](http://www.thestudentroom.co.uk/latexrender/pictures/be/beede8cfc25dd333eb40b53ea7b5d6f4.png "\Rightarrow \frac{\mathrm{d}^{n-1}}{\mathrm{d}y^{n-1}} [y-f(y)]^{n} = n! y (1-\alpha)^{n}")
This is an infinite geometric series with
. Since 
, it converges to 
.
As expected.
ii) Let
![[y - f(y)]^{n} = \frac{y^{3n}}{2^{2n}}](http://www.thestudentroom.co.uk/latexrender/pictures/44/440fac36837f808f62806f83c865e593.png "[y - f(y)]^{n} = \frac{y^{3n}}{2^{2n}}")
But
, so
Note that
when n=0.
iii) Let
![[y - f(y)]^{n} = e^{\lambda n y}](http://www.thestudentroom.co.uk/latexrender/pictures/c0/c0a8217a016ca51f8c139b766ab40add.png "[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}](http://www.thestudentroom.co.uk/latexrender/pictures/6c/6c8cb707e629d0bc9c7f8576a31f55ee.png "\frac{\mathrm{d}^{n-1}}{\mathrm{d}y^{n-1}} [y-f(y)]^{n} = (\lambda n)^{n-1} e^{\lambda n y}")
But y = 0, so
Edit: Fixed. Didn't realise it was actually posting last night...
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Re: STEP Maths I,II,III 1987 SolutionsHere 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 13Last edited by brianeverit; 18-06-2011 at 12:06. Reason: corrected errors -
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Re: STEP Maths I,II,III 1987 SolutionsThank you for pointing that out; is it correct now?(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
Are you also in Y13 btw? -
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Re: STEP Maths I,II,III 1987 SolutionsLink
That should work.
.
