50 years of hurt this summer 1966-2016

They don't look too scary.

That's a bit dramatic, it's only an S level paper for A level - so the equivalent of an AEA.

That is a step question from step I. It has come up twice!


Posted from TSR Mobile

This is not a STEP question
this is from 1966 which predates STEP

Yes I do know that, just pointing out it came up on step after. But if it came up in step it is a step question.


Posted from TSR Mobile

still no actual attempt on the red ones apart from atsruser's answer (still have not done it myself)

Yes, I have not had time to try them yet, I probably won't because I am ona break for abit. Maybe I can't do it Maybe I can, I won't be bothered if I can't.


Posted from TSR Mobile

Do you expect to get into Cambridge with this attitude?
What if the admission tutor is here (undercover) and reads your last post?

I don't expect to get in again. If these are horrible beast questions, why would be bothered if I can't do one??????



Posted from TSR Mobile

you don't expect to get in?

did you have a bad interview?

Kind of, it was harder then I thought and chose the wrong questions in the test so my fault. The other interview went much better so hopefully that saves me. But for someone on a gap year it might not be enough. But im hoping for the best, preparing for the worst. It might not hurt as much if I prepare myself for not getting in. I just don't want it to consume my mind for the next month. Sorry if I come across as rude, I am just stressed from yesterday due to my stupid mistakes.


Posted from TSR Mobile

However your reply is at best "silly" and at worst "utter nonsense"

Well you did not across as rude and I do not know what happened yesterday.
(Most of the time I show logged in but rarely I am behind the screen to follow what is going on)

That is fine, I will try the questions though. Especially the tan one, that seems very interesting.


Posted from TSR Mobile

In the summer of 1966 England won the World Cup.
A few days before the final the best mathematicians in the country had to face this test
Any thoughts particularly on the "red" questions?

They are interesting questions. My solution for Q2 (i) is as follows.

Observe that $\displaystyle \tanh^{-1}x^2 = \sum^{\infty}_{n=0} \dfrac{(x^2)^{2n+1}}{2n+1} = \sum^{\infty}_{n=1} \dfrac{(x^2)^{2n-1}}{2n-1}$.

Hence we have that

$\displaystyle 2 \sum^{\infty}_{n=1} \dfrac{1}{2n-1}\left(\dfrac{\sqrt{p}-\sqrt{q}}{\sqrt{p}+\sqrt{q}} \right)^{4n-2} = 2 \tanh^{-1} \left( \left(\dfrac{\sqrt{p}-\sqrt{q}}{\sqrt{p}+\sqrt{q}} \right)^2\right)$.

Recall that $\tanh^{-1} x = \dfrac{1}{2} \ln\dfrac{1+x}{1-x}$. Hence with some algebraic manipulation we arrive at

$\displaystyle 2 \sum^{\infty}_{n=1} \dfrac{1}{2n-1}\left(\dfrac{\sqrt{p}-\sqrt{q}}{\sqrt{p}+\sqrt{q}} \right)^{4n-2} = \ln \left( \dfrac{p+q}{2 \sqrt{pq}} \right) = \ln \left(\dfrac{p+q}{2}\right) - \left( \dfrac{\ln p + \ln q}{2} \right)$ as required.

that is absolutely ludicrous...
I mean what possessed you to start from artanhx²?
I tried 5 minutes ago but from the Arithmetic means and tried in vain expansions such ln(1+x)

you must be truly amazing to see this

At first I started the same as you, but then I couldn't see where the 2n-1 on the denominator came from. Then I thought of the A-Level functions to see where I could get such a series expansion and artanh(x) seemed to fit .

At first I started the same as you, but then I couldn't see where the 2n-1 on the denominator came from. Then I thought of the A-Level functions to see where I could get such a series expansion and artanh(x) seemed to fit .

It isn't actually all that hard, provided you know some stuff. In this case, if you know the series expansions for the trig, inverse trig, hyperbolic and inverse hyperbolic functions, then you simply recognize the series and the rest is straightforward algebraic manipulation.

But the other point to recognize is that the series is there simply to catch those who haven't memorized their series! In those days of yore, one simply learned these sorts of series expansion off by heart.