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# Mathematics examination paper from 1970 watch

1. Q4:
first bit
Write c = cos t, s = sin t, z = c+is.

Then

and so the locus is the line Re z = 1.

(a)
Let z = x+iy, then (x-1)^2+y^2+(x+1)^2+y^2 = 4. So 2x^2 +2 + 2y^2 = 4, so x^2+y^2 = 1.

(b)
Again let z = x+iy. Then . So

So

So

(an ellipse. Probably an arithmetic error somewhere, but life is too short...)

(c)
We must have tan(arg(z-1)) = tan(arg(z+1)) and so

Im(z-1)/Re(z-1) = Im(z+1)/Re(z+1). Writing z = x+iy we get

y/(x-1) = y/(x+1), and so y = 0. So z lies on the real line. For the arguments to be equal, it must lie on same side of -1, 1. So either z<-1 or z>1.
2. (Original post by DFranklin)
Q4:
first bit
Write c = cos t, s = sin t, z = c+is.

Then

and so the locus is the line Re z = 1.

(a)
Let z = x+iy, then (x-1)^2+y^2+(x+1)^2+y^2 = 4. So 2x^2 +2 + 2y^2 = 4, so x^2+y^2 = 1.

(b)
Again let z = x+iy. Then . So

So

So

(an ellipse. Probably an arithmetic error somewhere, but life is too short...)

(c)
We must have tan(arg(z-1)) = tan(arg(z+1)) and so

Im(z-1)/Re(z-1) = Im(z+1)/Re(z+1). Writing z = x+iy we get

y/(x-1) = y/(x+1), and so y = 0. So z lies on the real line. For the arguments to be equal, it must lie on same side of -1, 1. So either z<-1 or z>1.
For b, is it not sufficient to note that |z + i| + |z - i| = constant => distance from (0,-1) + distance from (0,1) is constant, which defines an ellipse with foci at those points?
3. Generally looks alright, i.e there is nothing which stands out as completely undoable, having said this i haven't done maths for a year and therefore probably would actually struggle to get a solution to the question, and i am comparing it to the advanced higher course so i don't know what current A level is actually like
4. (Original post by Mr M)
yes s level was used for cambridge (and oxford) entry
I was talking to my dad about this. He said that many schools, especially state sector, didn't enter candidates for S level papers - I suppose in the same way as many now don't enter for STEP or AEA papers.

At my dad's school entering S level papers were optional, there was no additional tuition for them, and they were taken by any students who wanted to stretch themselves. It wasn't exclusively about Oxbridge admission though I suppose many who entered for S levels applied for Oxbridge. S levels were around til 1990 apparently.
5. Q5
first bit

.

So assume (obviously true for n = 1).

By the calc above, and the result follows by induction. Don't understand what they mean by "find the sum" - isn't that what we've just done?

(a)
.

So the integral I = But for any integer N, , and so I = 0.

(b)

.

So we want to find (*)

Now consider where A, B are +ve integers.

Suppose first A=B. Then we have

Suppose instead . Then 2 sin Ax sin Bx = cos(A-B)x - cos(A+B) x. Since for any non-zero integer N we have we must have .

It follows that in (*) the terms where m = n each add to the answer, and the other terms are all zero. So the answer required is .
6. (Original post by Scipio90)
For b, is it not sufficient to note that |z + i| + |z - i| = constant => distance from (0,-1) + distance from (0,1) is constant, which defines an ellipse with foci at those points?
Yeah, except I think you'd be still be expected to get to an actual equation, or at least values for major/minor axes (since you're expected to sketch each locus). And I couldn't be bothered to look up the ellipse equations to do that from the "sum of distance from focii" starting point.
7. (Original post by jj193)
taylor/maclauren series aren't on A-Level...
Pretty sure I did them for A-Level. Not sure what module, but was not that long ago, 2005/6.

The paper does not seem that bad, I am just very much out of practice with maths.

I am a physicist, don't need to do that complicated stuff too often.
8. (Original post by Mr M)
I have the other six (count 'em) 1970 A Level papers too. This is the hardest one. Graham at Edexcel has a library of old papers.
Would be interesting to see the other ones - I think the S-level/AEA/etc. has actually seen a smaller drop off in difficulty than the bog standard A-level.

I took my A Levels in 1986 and this is more difficult than my papers. I think the time pressure is the problem. I can answer most of the questions but I couldn't knock out 8 good answers in 3 hours!
Yeah, but don't forget you probably only needed 6 questions for a distinction. 8 in 3 hours looks tough, but not impossible; main problem for current generation is relative lack of coord geometry, since the last 4 questions all lean that way.
9. (Original post by JoshJGordon)
Pretty sure I did them for A-Level. Not sure what module, but was not that long ago, 2005/6.

The paper does not seem that bad, I am just very much out of practice with maths.

I am a physicist, don't need to do that complicated stuff too often.
Did Taylor/Maclaurin expansions in FP3 last year ,not sure if it's been taken out of the spec since then.

Pretty enlightening post seeing these questions , I could probably manage most of them but only by the fact I've finished my first year reading Mathematical Physics (we do a lot of the same maths modules as the maths students).I'd have definitely needed hints to finish most of those questions if I'd done them a year ago.
10. It says "special" A-level, which equivalent to today's AEA. My dad did one in Chemistry.

EDIT: looks like someone else beat me to it
11. (Original post by red_roses)
It says "special" A-level, which equivalent to today's AEA. My dad did one in Chemistry.

EDIT: looks like someone else beat me to it
In fairness, the OP said it in the very first post but most people didn't bother reading it
12. (Original post by OrmondDrone)
In fairness, the OP said it in the very first post but most people didn't bother reading it
Ahh I read the third post thinking it was still the OP's about it saying how the A-level syllabus had obviously been dumbed down. I fail at speed reading
13. I have quite a few questions from Oxford + Cambridge scholarship papers dated at the very latest 1963. Only problem is that I have no idea how many of them are per paper or what the time limit is. Here's one for now that I have wrote down:

1. (Oxford.) By means of the substitutions , or otherwise, find:

I'll try and bring some more later today.
14. (Original post by DFranklin)
Would be interesting to see the other ones - I think the S-level/AEA/etc. has actually seen a smaller drop off in difficulty than the bog standard A-level.
I will put some more up later.
15. (Original post by -G-a-v-)
Looking at some of those questions makes me feel incredibly bad at maths that said, i've not actually got a pen and paper out and tried to do them ... yet. lol

Anyone got access to any typical 'normal' A-level papers from back then?
i have a set of 1970s physics papers that are also a massive shock compared to current A-level papers.... I mean, they are a lot, lot harder
16. (Original post by Mr Nonsense)
i have a set of 1970s physics papers that are also a massive shock compared to current A-level papers.... I mean, they are a lot, lot harder
A Level physics is simply embarrassing these days. The problem comes from not requiring A Level physicists to study any mathematics beyond GCSE. I understand and accept the reason (the low number of candidates is a concern) but it means no topic can be examined in any meaningful depth.
17. Here are pdfs of the 1970 Mathematics A Level Pure Maths papers:

June 1970 Paper 1.pdf

June 1970 Paper 2.pdf
18. And pdfs of the 1970 Applied Mathematics A Level papers:

June 1970 Paper 3.pdf

June 1970 Paper 4.pdf
19. And finally pdfs of the Further Mathematics A Level papers:

June 1970 Paper 5.pdf

June 1970 Paper 6.pdf
20. (Original post by Mr Nonsense)
i have a set of 1970s physics papers that are also a massive shock compared to current A-level papers.... I mean, they are a lot, lot harder
Is there any chance you could upload them (physics section) ? I'd love to give them a try.

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