Higher Maths, Paper One

subway_ie

I've kind of neglected my maths revision for the past week or so, concentrating on english the whole time, so now I'm just starting to get back into revising for it. So, I was thinking we could follow Discharger Snake's lead with the applied maths papers and make out a list of the hardest questions asked since 1994, or just the more unusual ones and any tips in general along with an explanation of why you think they're difficult. Should make an interesting paper to work on.

In 2003, the factor theorem was asked, so it probably won't come up again. But there was no differentiation by first principles, so I'd say that's a given - hopefully CosX or SinX - they could ask the sum or the quotient rule, which wouldn't be very nice. Also, they didn't give an area question in Q8, the first time since 1999, so that'll probably come up too. I've got a feeling they'll ask for the proof of De Moivre's theorem, since the omega stuff came up in Q3 (c) in 2003.

Hardest Questions:
Q.1 - 1994 - Question 1 generally isn't hard, but those part (c) type are easy to go wrong in.
Q.2 - 1995 - Again, generally an easy question - part (c) was long and difficult.
Q.3 - 1999 - One of those long, drawn out ones with a De Moivre's theorem application.
Q.5 - 2003 - not so much hard, as long and tedius. Proof by induction AND binomial expansion.
Q.6 - 2000 - Again, long and drawn out, a first principles proof and an "interesting" (c) part (iii).
Q.7 - 1996 - Nothing new or unusual, but (c) part (ii) is a bit intimidating at first.
Q.8 - 1995 - The only difficult part is the volume of the cone, which shouldn't really be difficult at all.

DS

Q.2 - 1995 - Again, generally an easy question - part (c) was long and difficult.

I'm very interested in how you approached that part c. I used differentiation to find that f(x) is at a minimum at x=k/root(3), and is decreasing from x=0 to that point, therefore between x=0 and x=k/root(3), as x increases, f(x) decreases i.e. if p < q, f(p) > f(q). I used the same logic to do the second bit, and it seemed like a nice solution to me but it's obviously not the one in the marking scheme (which I'd love to see).

When they say "divide f(q) - f(p) by q - p" they're obviously telling you to write down an equation for the slope of the tangent to the curve f(x) at any point, q being a tiny bit ahead of p, say, ala first principles. But I wasn't too sure how to proceed from there once I had simplified it, so I went off differentiating. Is it the same sort of principle if you do it algebraically? Or how did you end up doing it?

subway_ie

Well, since it's question 2, you definetly shouldn't be using calculus - I've seen the marking scheme, and there was no mention of differentiation at all. Although, it didn't mention my method either:
First part, just standard division, with "q^2 + pq + p^2 - k^2"
Next part, I proved that K is positive using the info given, then manipulated it a bit to show that "3q^2 - k^2" is less than or equal to 0. Then using the fact that "q > p", I showed that f(q) < f(p).
The last part, I proved that 3p^2 - k^2 >= 0 and did some (a lot) more manipulation to finally get f(q) > f(p).

Definetly not a very nice Q2. Especially since you're supposed to spend about 23 minutes doing the whole question - usually Q2 will take about 12-15 mins, for this one I think you'd need the full 23 minutes (maybe more?). Not very fair at all.

DS

First part, just standard division, with "q^2 + pq + p^2 - k^2"

But you didn't use that to prove the next part no? I'm sure the official solution does, because it's obviously a hint. I know I shouldn't have used calculus, but surely if it's mathematically correct they'd have to give me the marks right? Anyway, I'm sure I won't have the same dilemma on Thursday Did you make any reference to k/root(3) in your proof?

subway_ie

Yeah I used "q^2 + pq + p^2 - k^2" to prove the next part - I basically used the previous answer for each proof, the very last section needed nearly all of the previous ones.
If your solution was mathematicaly correct, then they might give you the marks - BUT, if it only works when you make assumptions that you don't have any actual basis for (eg. that it's asking you to write down an equation for the slope of the tangent to the curve f(x) at any point, q being a tiny bit ahead of p, say, ala first principles) then they might be reluctant - especially if it's an examiner who sticks rigidly to the marking schemes (which most do - much, much faster). You'd probably get it on an appeal though if they didn't give it to you first time round.

Mutant_Fruit

2000 - Q.7. (C) ii

That in my opinion was one of the nastiest parts ever. I just could not get it, no matter what i tried, in the end i had to look up the solution online. I'd say only 5 marks were going for it though, so its not too bad...

DS

(eg. that it's asking you to write down an equation for the slope of the tangent to the curve f(x) at any point, q being a tiny bit ahead of p, say, ala first principles)

Well that wasn't part of my proof, I didn't refer to the first part of the question at all. It'd be insteresting to see how they'd mark it though.

2000 - Q.7. (C) ii

Yeah it threw me at first, but the key to those "Hence," questions is that you know you have to use the result you got in the previous part, and if you just concentrate on that you've a much better chance of cracking it.

Mutant_Fruit

i knew that hence meant i was supposed to use what i already had, but for the life of me i couldn't figure it. Even when i was reading the solution i thought to myself "What were they smoking". I never would have gotten it.

Ah well, its the only thing i couldn't do in the whole 2000 paper... so thats a good thing. And i know how to do it in future.

subway_ie

2000 was actually supposed to be the easiest paper on the new course - all the papers since then have supposed to follow the same model of letting nearly everybody get Cs and having a failue rate of about 9%. They've also started giving more marks for the part a's and b's, and the early bits of part c's. That means you can still get your A1 without getting every (or any) question out fully.

PrecariousNuts

I proved 2000 part c by induction, how were you supposed to do it via the hence part?

DS

You've shown that f(x) is a max at f(x)=1/e
And you know f(x) = lnx/x
So, simple as you like:
lnx/x <= 1/e <- that's the hard bit, now it's just simplifying
elnx <= x
ln(x^e) <= x
e^(ln(x^e)) <= e^x
x^e <= e^x

Induction would have been great if they had said "Hence, or otherwise", but when they don't say otherwise they penalise you if you don't use the previous result. No idea how many marks you'd lose though. Could only be 3.

Funky

If a really really really nasty part to a part C is on the paper don't waste your time trying to get it forever move on , cause if its really bad chances are it;ll only be worth 5 marks in the end...

subway_ie

Originally posted by Funky
If a really really really nasty part to a part C is on the paper don't waste your time trying to get it forever move on , cause if its really bad chances are it;ll only be worth 5 marks in the end...

Only do that if you've made an attempt at it at least - NEVER skip over a question without writing something, no matter how stupid/fundamental/obvious you think it is. At least then you'll probably get the attempt marks (usually 3/10).

PrecariousNuts

I've wondered, where do the other 15 marks go then? Isn't it a bit harsh on those that manage to get it out?

Mutant_Fruit

If you get it out, you get the satisfaction of getting it out, and the remaining few marks.

Always remember, the last parts in a partc question are worth very few marks, you can still get an A and not do half of em!

I worked it out before, and you can get an A1 by getting everything right, but leaving out 3 part C's and one or two part B's completely. So there is some leeway there! Don't worry if you cant get something, just drop it immediately and continue on, then come back at the end.

I did that for my mock, and it was the ets decision i ever made.

DS

More numbers for people gunning for an A1: you can afford 20 blunders.

In regards to that 2000 7c, it was 20 marks, 10 for each part, and according to my solutions book thing they were very strict on marking part ii i.e. induction might have only been worth an attempt. Induction only proves it for natural numbers anyway right? Not "for all x > 0"