STEP Prep Thread 2016 (Mark. II)

No, of course not, why would they?

For step ii 2015 question 5 i've just proved by induction S(n)=arctan(n/(n+1)) for the first bit but then i realised that I should have proved tanS(n)=n/(n+1) for the first bit and S(n)=arctan(n/(n+1)) for the second bit, are they not the same thing? surely if one is true then the other is true? also how would you prove anS(n)=n/(n+1) without using S(n)=arcgan(n/(n+1))? sorry the mark scheme was a bit cryptic so i just asked on here

Well, no - tan S_n = a and S_n = arctan a are two different things, remember that tangent is a pi-periodic function, so how do you know that it's know S_n = arctan a + pi, for example? You need to prove that S_n is bounded between 0 and pi for you to assert that S_n = arctan a.

Well, no - tan S_n = a and S_n = arctan a are two different things, remember that tangent is a pi-periodic function, so how do you know that it's know S_n = arctan a + pi, for example? You need to prove that S_n is bounded between 0 and pi for you to assert that S_n = arctan a.

so do i just have to state that 0=<arctanx=<pi/2 ?
also how do i even prove tanS(n)=n/(n+1) without using the tan formula to show that S(k)= arctann/(n+1) + arctan1/2k^2 ?

No, of course not. You have to prove it.

Well, you do it by induction.

sorry so do i start by showing tan(arctan.5 + arctan.125...+arctan1/2k^2) =n/(n+1) then do the k+1 induction
i realise i must sound like a moron but i'm just really pants at induction proof

sorry so do i start by showing tan(arctan.5 + arctan.125...+arctan1/2k^2) =n/(n+1) then do the k+1 induction
i realise i must sound like a moron but i'm just really pants at induction proof

the question tells you that arctan is in interval (0, pi/2)

No, you assume that tan (arctan 0.5 + ...) = n/n+1 and then use that assumption to prove that the k+1 case is also true.



Exactly, hence why you need to prove that S_n is bounded above by pi/2 in order to apply the arctan function.

get it now thx for your help

So here is how I did:

Started off with question 2 (because calculus is usually my best and it looked easy) and I completed it.

Then I took up question 4 (more calculus ) did the first bit; used the first result in the second bit to give another differential equation and after squaring tried to find f(x) cant remember what I got but it was something weird not sure whether its right. Didnt give a geometric interpretation (or whatever was asked)

Question 1 and 3 came next - 1 was good i proved all the results then got factors in (ii) which I think are correct. As for question three the graphs were fine i saw sin3<sin2 which was good and i showed open and coloured circles where needed and I hope I got it completely.

Then I went for Q10 cuz it looked good but I couldnt complete it - i got the first result and then attempted (ii) after a lot of horrible algebra i came up with a weird cubic for lambda which i couldnt solve.

The final question was the real mess - I had roughly 20-25 mins left and I just went for 11 but i couldnt get the first result - i did resolve and write down equations and stuff but the answer didnt come. I did write differentiate then equalise to zero to get maximum. I also did Q9 bc 11 wasnt going far but i think i made an error with reaction but i wrote down as many resolving equations and moments as was needed - that reaction error didnt give me the result i think.

So how much do u guys think I would get? not really feeling very confident after messing up the last Q - looking back i think i shoulda done 8 instead of scratching at bits of mechanics questions at the end...

20, 16, 20, 15, 4, 8 = 86

I did 7 Qs btw (1,2,3,4,9,10,11) so wouldnt 1,2,3,4,10 and one of 9 or 11 be my best six? so wouldnt it be more like Q1=20, Q2=20, Q3=20? Q4=16, Q10=15, Q11=4? Q10=8? giving like more than 90?

yeah, you put two questions in the same paragraph so I double counted.

yeah sry about that - and no worries. btw in my last post i meant Q10=15, Q11=4 and Q9(not 10) = 8 haha

so yeah depends on where the S boundary would go - any guesses?

100+ i'd say

thought so too - so yeah i guess ive gotten a good 1 (if all went as expected and i didnt make some ridiculously large error and didnt notice it )

Well, good luck in STEP 2

Yeah, reckon I got the same. Incredibly annoyed about that.

Thanks, you too(?).