STEP II 2010 Discussion Thread

Here's my take on the paper though. On the whole (compared to STEP I mainly), I thought it had a lot of long and tedious sequences of algebra (where I've probably made stupid errors), but compared to other STEP II, the insight required for questions wasn't particularly difficult.

Question 1 I got a solution to, but the centre of my circle had a horrible a+b(root(c)) form, so I'm not convinced by my solution. 12-20 marks .
Question 2 I almost completed, got the quartic, took out the factor they gave us, but couldn't solve the cubic. Pretty sure I factorised wrong now. I reckon 16/17ish.
I had a full solution to Question 4, which is surely one of the best integration questions ever. When you realise you have to expand the sine in the denominator and write cosx=sin(90-x) is brilliant. Then the substitution for the last one is even better. Quite long, but really fun.
I almost solved question 8, but had 2 negative signs when I should have had positive signs. For the sum, I used their results, factorised out the first bit and used the formula for sum of a geometric progresstion. 16/20?
Also had the smallest fragments in a mad rush at the end for 3 and 7, maybe 2-5 marks from them.

Hopefully enough for a 2, what do you guys think?

Oh I could cry.

LOVED Q2, LOVED Q3, LOVED Q7, LOVED Q8.
COULDN'T EVEN DO THEM.

I really wanted to start Q1, but I didn't know what osculating meant. D:
I assumed it wasn't a vibrating circle so I moved on.

Q2, amazing. Got to useless Eustace, and tried finding a reduction formulae for Ssin^nxdx, then I remembered that II doesn't require further maths after I saw I was getting nowhere. Well, I got somewhere, but I couldn't find sinx from cosx. So embarrassing.
Now that I've seen Unbounded' solution, I'm ******* kicking myself.

I got a quartic for the last part, and tried c = 1, -1, 2, -2, 4, -4, 8, -8... None worked. Turns out I had a slightly wrong quadratic. I simply wrote "well, clearly I can't solve this, but a = cos^-1(c).
Can't believe I didn't realise one of the factors was (c + 1/6) too.

Q4, omg, luv luv luv. It was amazing. I spotted the substitution and was like, , but then I couldn't get the log integral in terms of f(x) and f(a-x).
I don't know what I was doing. I was rushing I think. I was doing 1 - 2 + x, 1 - x - 1... ugh. Again, looking at Unbounded's solution... Ugh.
The trig one was fine.
OMG at part ii! I used u = 1/x, but then I was so focussed on getting rid of the x in the denominator and trring to use the method at the top, I didn't think of adding 2 of them together. Well I did, but at first glance it seemed that it wouldn't do any good (I had written my integral slightly differently somehow). Oh well!

Q7... Looked so easy. It was NOT. D:
I don't think I even showed that the curve only crossed the x axis once. I found the stationary points, and y < 0 for one of them. I think I showed that the other one gave y < 0 too... something about 1 > 2root2?

Anyway, found the quadratic in u^3 or p or whatever the **** it was, eventually. Then I got NOWHERE. Did a small part of the last part but gave up.

Start Q8. Looked lovely. Lovely graph. Found the points of contact. Tried to integrate between xn and xn+1, completely failed. Gave up.

Q12 looked lovely too but I saw it asked for median values, and I don't know how to do them so I didn't even try. Didn't fancy 'thinking' for Q13 either!

Tried the projectiles question. Quickly gave up. I can't do mechanics unless it's one that leads you through it.

Can't remember if I attempted a fifth or sixth question... I can't remember the questions.

I was serious scarce on time for this one. Nearly two hours had gone past and I was flicking between 2 4 and 7. Rushed 8.
I ignored geometry. I probably would have tried the Fibonacci question if I'd had time. Oh well!

Looks like 2,3 or 3,3 for me!

You didn't need to know what osculating means for Q1 (they gave a, slightly disjointed, explanation of it)

If they did, I didn't understand it.

I'm usually good with circles too.

If they did, I didn't understand it.

I'm usually good with circles too.

I have the paper so the exact wording was:
"1. Let P be a given point on a given curve C. The osculating circle to C at P is defined to be the circle that satisfies the following 2 conditions at P: it touches C; and the rate of change of it's gradient and is equal to the rate of change of the gradient at C.
Find the centre and radius of the osculating circle to the curve $y=1-x+tanx$ at point on the curve with x-coordinate $\dfrac{\pi}{4}$
So let the circle be $g(x)=(x+a)^2 + (y+b)^2=r^2$. All you need to do is put $\dfrac{d^2y}{dx^2}g(x=\dfrac{\pi}{4})=\dfrac{d^2y}{dx^2}(1-x+tanx)$ and solve the circle with $y=1-x+tanx$ to ensure that there was only the one solution at $x=\dfrac{\pi}{4}$.
I had a go at the question but got an incredibly horrendous $\dfrac{d^2y}{dx^2}$ for the circle so gave up, heh.

Yeh Q4 was very easy for me.

IF I KNEW HOW TO INTEGRATE 1/x PROPERLY

how many marks do you think I would lose for that?

Yeah my $\dfrac{d^2y}{dx^2}$ was pretty hideous too, but I thought I've got this far so I might as well finish it.
The key with this questions is you get three simultaneous equations (from y, dy/dx and the second derivative), to satisfy 3 variables, (the x co-ordinate of the circle centre, the y co-ordinate and the radius).

Am I right in assuming that the dy/dx supposed to be the same for both at P?

The question says it touches C => tangency => equal dy/dx.

So yes you were right.

Yeah I now see that rather than differentiating my dy/dx, I should have differentiated the previous step (ty Unbounded) as this gives a much much nicer product rule.

Yeah sort of trick that would have been helpful. Guess that's he going to get an S and I'm going to be scraping around a 2

If you got to the stage where you had the integral of 1/x, I would say you've done all the hard bit, so I would guess 1/2 dropped. I hate making mistakes like that.
Out of interest what did you integrate 1/x to?

Yep! I'll hopefully scrap a 3 (I was doing it for fun though hehe)