STEP Maths I, II, III 1993 Solutions

Reply 80

2

What I done was said $T_n(r) = (1+r^2+r^4 + .. + r^{6n}) + (1+r^3 + r^6 + ... +r^{6n}) - (1+r^6 + r^{12} + ... + r^{6n})$
$= \frac{1-r^{2(3n+1)}}{1-r^2} + \frac{1-r^{3(2n+1)}}{1-r^3} - \frac{1-r^{6(n+1)}}{1-r^6}$
(probably counted the number of terms wrong, I can't be arsed working it out again), but just as I went looking for a nice way to put arrows in in latex, I clicked refresh and lost it all :rolleyes:

And for r=-1, ignore the first term and thereafter for every four terms we have 3 positive and one negative, so every four terms is a +2, so it sums to (2n+1)...I think.

Reply 81

Rabite

8

I think I got Q5 for Paper A...I need to draw a diagram though. :frown:

I'll post it later. Unless someone already has?

Reply 82

generalebriety

16

STEP III Q1 done. :smile:

$\displaystyle[br]P: \; x = \sin\theta,\; y=\cos 2\theta = 1 - 2\sin^2\theta = 1-2x^2.\\[br]-\pi /2 < \theta < \pi /2 \Rightarrow -1 < x < 1. \\$
(See diagram.)

$\displaystyle[br]\bigg( \frac{\text{d}s}{\text{d}x} \bigg)^2 = 1 + \bigg( \frac{\text{d}y}{\text{d}x} \bigg)^2 = 1 + (-4x)^2 = 1 + 16x^2.\\[br]s = 2\int_0^1 (1+16x^2)^{1/2} \text{d}x.\\[br]\text{Put } 4x = \sinh u \Rightarrow \text{d}x = \frac{1}{4}\cosh u \text{d}u.\\[br]s = 2\int_0^{\sinh^{-1}4} \frac{1}{4} \cosh^2 u\text{d}u = \frac{1}{4} \int_0^{\sinh^{-1}4} (\cosh 2u + 1)\text{d}u\\[br]= \frac{1}{4} \bigg[\frac{1}{2} \sinh 2u + u\bigg]_0^{\sinh^{-1}4}.\\[br]\sinh u = 4 \Rightarrow \cosh u = \sqrt{17},\; \sinh 2u = 8\sqrt{17}.\\[br]\therefore s = = \frac{1}{4}(4\sqrt{17} + \sinh^{-1}4) = \sqrt{17} + \frac{1}{4} \sinh^{-1}4.\\ \\[br][br]V = \int \pi y^2 \text{d}x = \pi \int_{-1}^1 (2 - 2x^2)^2\text{d}x = 4\pi \int_{-1}^1 (1-2x^2+x^4) \text{d}x\\[br]= 8\pi \bigg[x - \frac{2}{3}x^3 + \frac{1}{5}x^5\bigg]_0^1 = \frac{64}{15}\pi.$

q1.gif3.8KB

Reply 83

generalebriety

16

By the way, you all might have missed my post before - are these going on the wiki? And if not, why not?

Reply 84

nota bene

15

generalebriety

By the way, you all might have missed my post before - are these going on the wiki? And if not, why not?

I think they are being planned on being Latex-ed into pdf's so they are all readable and neat. However that will take some time I guess (I might have some time during summer to do a few years, but I'm nowhere near Aaron's Latex skills...)

edit: So to answer your question; I don't know if they're going onto the wiki, I guess they easily could once they are latexed although I don't see the need for it if we have them as pdf's.

And Speleo, no the bus question is not finished, part c is not ready I'm trying to look at it now. I'll let you know when I finish it (if I do!)

Reply 85

generalebriety

16

Ah. I suspected something might be done with them, which is why I've LaTeX-ed most of my solutions. :smile:

Alright, thanks.

Reply 86

insparato

15

That'll save time for me :p:

. Ill start doing these after June 29th. See how it goes then.

Reply 87

Speleo

7

Which STEPs are you taking insparato?

Reply 88

insparato

15

None, not bright enough. I am however taking the AEA in maths just for the fun of it.

Reply 89

generalebriety

16

insparato

None, not bright enough. I am however taking the AEA in maths just for the fun of it.

Not bright enough? **** off, you've done two STEP questions here in the past two days, that's more than I could do when I started. :p:

Reply 90

Speleo

7

Links to all markschemes 1994-2006 added to original post.

You've done loads of questions insparato, you should go enter yourself now if you still can :tongue:

Reply 91

generalebriety

16

Speleo

Links to all markschemes 1994-2006 added to original post.

You've done loads of questions insparato, you should go enter yourself now if you still can :tongue:

Want me to start a 1992 thread? I might as well share the load (updating your first post must take quite a while each day :wink:

).

Or should I wait a while?

Reply 92

Speleo

7

Yeah two running simultaneously would be good, make it clear in the OP that this one still isn't finished.

Here's the code for the past threads with the URL= replaced by UR=

TSR threads:
[UR="http://www.thestudentroom.co.uk/showthread.php?t=367072"]1994[/UR] [UR="http://www.thestudentroom.co.uk/showthread.php?t=359567"]1995[/UR] [UR="http://www.thestudentroom.co.uk/showthread.php?t=358891"]1996[/UR] [UR="http://www.thestudentroom.co.uk/showthread.php?t=358386"]1997[/UR] [UR="http://www.thestudentroom.co.uk/showthread.php?t=357961"]1998[/UR]
External sites:
[UR="http://www.priorexams.com/"]1999-2003[/UR]
[UR="http://www.meikleriggs.org.uk/"]2001-2006[/UR] Requires sign-up.

Reply 93

generalebriety

16

Thanks.

Reply 94

Kolya

17

I'm sure Speleo will polish off the 1993 STEP III Q5-11 and Q13-16 for us, by 6am on Thursday morning. :smile:

Reply 95

Speleo

7

Lusus_Naturae

I'm sure Speleo will polish off STEP III Q5-11 and Q13-16 by 6am on Thursday morning for us.

I don't really like the old papers but I can't refuse a challenge! Might have to give the stats ones a miss though if it's ok with you :tongue:

Also, repeat question alert! 1992 STEP II has the 'diamond/derivative' question that cropped up on STEP III a year or two back IIRC. There was also a very similar one in the thread that I did with INT xf(sinx) dx that was repeated very recently but with slightly different later parts.

Btw, the 1992 STEPs are mislabelled in the exact same fashion as these.

Reply 96

generalebriety

16

Speleo

I don't really like the old papers but I can't refuse a challenge! Might have to give the stats ones a miss though if it's ok with you :tongue:

Also, repeat question alert! 1992 STEP II has the 'diamond/derivative' question that cropped up on STEP III a year or two back IIRC. There was also a very similar one in the thread that I did with INT xf(sinx) dx that was repeated very recently but with slightly different later parts.

Btw, the 1992 STEPs are mislabelled in the exact same fashion as these.