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# STEP Maths I, II, III 1995 Solutions Tweet

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1. Re: STEP Maths I, II, III 1995 Solutions
(Original post by insparato)
Writing out STEP III Question 2 now.

ii)

Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
\displaystyle I_n = \frac{2}{3}(n+\frac{1}{2})\int (a-x)(a-x)^{\frac{1}{2}x^{n-\frac{1}{2}}

Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
\displaystyle I_n = \frac{2}{3}(n+\frac{1}{2})\int a(a-x)^{\frac{1}{2}x^{n-\frac{1}{2}}-(a-x)^{\frac{1}{2}}x^{n+\frac{1}{2} }

I think this answer is not finished. Here we are asked to evaluate I_n, but I think not in terms of I_n-1 but in terms of a.

I use the reduction formula to expand I_n as

(a^n+2)(Pi)(2n+1)!!/(2n+4)!!

Also if I am allowed to use double factorial notation here?
2. Re: STEP Maths I, II, III 1995 Solutions
Last part of III/2

3. Re: STEP Maths I, II, III 1995 Solutions
(Original post by insparato)
Writing out STEP III Question 2 now.

ii)

Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
\displaystyle I_n = \frac{2}{3}(n+\frac{1}{2})\int (a-x)(a-x)^{\frac{1}{2}x^{n-\frac{1}{2}}

Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
\displaystyle I_n = \frac{2}{3}(n+\frac{1}{2})\int a(a-x)^{\frac{1}{2}x^{n-\frac{1}{2}}-(a-x)^{\frac{1}{2}}x^{n+\frac{1}{2} }

4. Re: STEP Maths I, II, III 1995 Solutions
(Original post by DFranklin)
Of course, there's nothing special about painting the first post red (as opposed to blue or white), so the number of ways of painting n+1 posts so no two posts are the same colour and the first and last posts are coloured the same is just 3 times what we've just calculated. But then by identifying the 1st and last posts (which are the same colour), we are left with the same situation as n posts in a circle. (e.g. if we consider n=13, then we can think of having posts 1,2,...,12 as the hours on a clock, and we consider post 13 to be the same as post 1).

So the required answer is just .

Edit: Mistake spotted by DVS.
Sorry for bringing this post back from five years ago, but I just did this question and decided that the answer to the last part is given by because the arrangement can be rotated n times. If you try with the case n=3, there are two ways of doing it (RBW or RWB in any order). However, having thought about it further, I'm going to assume the question implies that this does not need to be considered, and my assumption about dividing by n is wrong anyway due to other concerns of a similar nature that would need to be factored in. I wonder how many marks I'd lose for putting it over n...
Last edited by und; 02-10-2012 at 22:13.
5. Re: STEP Maths I, II, III 1995 Solutions
STEP 1 QUESTION 9 ALTERNATE SOLUTION
(up) y = tsin(theta)sqrt(2gh) - 1/2(gt^2)
(side) x = tcos(theta)sqrt(2gh) so x^2 = t^2cos^2(theta)2gh
using pythagorean identity, x^2 = 2ght^2 - (y + 1/2(gt^2))^2
putting y into x^2 give quadratic in t^2
i.e. (1/4)(g^2)t^4 + (yg -2gh)t^2 + (y^2 + x^2) = 0
for t = real then (yg -2gh)^2 > (g^2)(y^2 + x^2)
which, after expanding and simplifying, gives us x^2 < 4h(h-y)