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Mathematics examination paper from 1970

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What level is this question on anyway? It's the first integral question ive actually ever done so I don't know anything in terms of difficulty.

It's hard to say. With the guidance given, it's a normal application of integration by substitution and partial fractions with some standard integrals (atan and log). Of course, it's a bit disingenuous to say that it's straightforward because it only requires elementary methods. A good example of this is computing ∫0∞ln⁡xx2+a2 dx\displaystyle \int^{\infty}_{0} \frac{\ln x}{x^2 + a^2} \, \mathrm{d}x, which only requires elementary considerations but is rather difficult from scratch: you have to take the right route.

Without the guidance given (that is, the first two integrals and the substitution), I think the 3rd one would waste a lot of time by taking the "routine" method and therefore be a hard question.
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Without the guidance given (that is, the first two integrals and the substitution), I think the 3rd one would waste a lot of time by taking the "routine" method and therefore be a hard question.


So the third one is just the 1st one minus the 2nd one and then multiplied by 2...so you just stick the integrals in?
ShortRef
So the third one is just the 1st one minus the 2nd one and then multiplied by 2...so you just stick the integrals in?

Yes. I now have the book again and I double-checked, and there was indeed a square in all of the denominators. Perhaps it was a printing mistake?

As promised, attached is some more questions from Cambridge's scholarship level examinations dated at the very latest 1961. Again, I have no idea how many questions or how long is expected to do them. I've picked one from each chapter with no regard to difficulty or elegance.
bah.pdf57.5KB
The sin series one can be done by considering that =S and a similar series with cos = C, then do C+jS, you get a geometric sum after De Moivre. These are commonly put in modern FP2 papers as "C+jS".
Any idea on Q1???
Original post by Mr M
I thought some of you might enjoy having a go at the A Level Special Paper from 1970. This was the equivalent of the AEA and was designed for the top 15% of A Level candidates. You had to answer 8 questions out of 10 in 3 hours.



This is what A-level maths should be!

At present, A-level maths is far to simple and uninteresting, and we're falling behind the likes of China where students are tested very early on.
Original post by Indeterminate
This is what A-level maths should be!

At present, A-level maths is far to simple and uninteresting, and we're falling behind the likes of China where students are tested very early on.


A-Level Maths shouldn't be like this, otherwise it would: put a lot of people off taking the subject or convince people to drop the subject, and thus, they will only have an education of up to GCSE Maths for when they go to university. So the only people who will benefit are the elite mathematicians. Also note, the demand for A-Level Maths teachers will decline because less people are choosing to study it, hence less jobs for mathematics graduates.

However, for the elite, 'we' are already challenging ourselves with AEA/STEP/MAT and the paper provided by Mr M is no harder than AEA.

When 90% of an average year group can master STEP III, then it will be appropriate to increase the difficulty of the A-Level exams.
Original post by iMadRichard
The sin series one can be done by considering that =S and a similar series with cos = C, then do C+jS, you get a geometric sum after De Moivre. These are commonly put in modern FP2 papers as "C+jS".
Any idea on Q1???


The discriminant, d, of a cubic

f(x)=ax3+bx2+cx+df(x) = ax^3 + bx^2 + cx + d

is

d=18abcd−4b3d+b2c2−4ac3−27a2d2d=18abcd - 4b^3 d + b^2 c^2 - 4ac^3 - 27a^2 d^2.

If

d≥0 d \geq 0

then the cubic, f(x), has roots that are all real.

You can probably see that a lot of these terms vanish, so there you go :smile:
(edited 11 years ago)
This seems easier than STEP, and a bit harder than standard Pure Further Maths stuff, but not massively...

It does seem like a fairly standard extension thing, and not that different to the way we have it now imo
Original post by Manchester United
A-Level Maths shouldn't be like this, otherwise it would: put a lot of people off taking the subject or convince people to drop the subject, and thus, they will only have an education of up to GCSE Maths for when they go to university. So the only people who will benefit are the elite mathematicians. Also note, the demand for A-Level Maths teachers will decline because less people are choosing to study it, hence less jobs for mathematics graduates.

However, for the elite, 'we' are already challenging ourselves with AEA/STEP/MAT and the paper provided by Mr M is no harder than AEA.

When 90% of an average year group can master STEP III, then it will be appropriate to increase the difficulty of the A-Level exams.


There's a lot of truth in the following report

https://docs.google.com/viewer?a=v&q=cache:TGAz84uls4UJ:www.conservatives.com/news/news_stories/2011/08/~/media/files/downloadable%2520files/vorderman%2520maths%2520report.ashx+&hl=en&gl=uk&pid=bl&srcid=ADGEESgXNvqRQMB2c_2kyPLOBcDoZl5iI4WEnEmH0V7OPIUeCzkNhrtqTezGMs4PXEz2hlfYFQ-tun56GuA3j9_cXzWtb_Mhku7y9JgpMUk8i7v731dolQe2guaNtdz7RC5jsIrtZ_RW&sig=AHIEtbQK-Hhgb0TYAJiYij7cFufsVqtmzA
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Original post by Indeterminate
This is what A-level maths should be!

At present, A-level maths is far to simple and uninteresting, and we're falling behind the likes of China where students are tested very early on.


Then you'll like this, a present day A-Level Maths paper.

http://www.scribd.com/doc/52489790/al-pure-2011
Original post by Indeterminate
The discriminant, d, of a cubic

f(x)=ax3+bx2+cx+df(x) = ax^3 + bx^2 + cx + d

is

d=18abcd−4b3d+b2c2−4ac3−27a2d2d=18abcd - 4b^3 d + b^2 c^2 - 4ac^3 - 27a^2 d^2.

If

d≥0 d \geq 0

then the cubic, f(x), has roots that are all real.

You can probably see that a lot of these terms vanish, so there you go :smile:


Thanks a lot but were we supposed to know that???
For one I suggest people people differentiate and find the coordinates of the stationary points. Find and inequality for when they are above and below the axis and think about what this implies.

Original post by iMadRichard
The sin series one can be done by considering that =S and a similar series with cos = C, then do C+jS, you get a geometric sum after De Moivre. These are commonly put in modern FP2 papers as "C+jS".
Any idea on Q1???
Original post by iMadRichard
Thanks a lot but were we supposed to know that???


I'd imagine you would have in 1970.

At present, things don't go further than quadratics at A-level.
Original post by Indeterminate
I'd imagine you would have in 1970.

At present, things don't go further than quadratics at A-level.


For question 1, you don't need to know that.
Original post by bananarama2
For question 1, you don't need to know that.


It's the most obvious way for me, at least.

But yes, there are other ways to do it.
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Original post by Indeterminate
I'd imagine you would have in 1970.

At present, things don't go further than quadratics at A-level.


I imagine you wouldn't.

for a start, why WOULD you memorise the cubic determinant? so you can answer a bunch of questions on cubic equations?
Original post by Indeterminate
It's the most obvious way for me, at least.

But yes, there are other ways to do it.


What is the other way?
Original post by around
I imagine you wouldn't.

for a start, why WOULD you memorise the cubic determinant? so you can answer a bunch of questions on cubic equations?


I memorised the cubic discriminant in Y10, 4 years ago, when learning about quadratics and b2−4acb^2 - 4ac

I don't know if that's weird or not :frown:
Original post by iMadRichard
What is the other way?


Consider the stationary points, and think of how they relate to the points at which the polynomial is zero.

When I was preparing for STEP III last year, I encountered a very interesting question involving the same cubic on a past paper (I'm pretty sure it's on the 2011 paper).
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Original post by Indeterminate
I memorised the cubic discriminant in Y10, 4 years ago, when learning about quadratics and b2−4acb^2 - 4ac

I don't know if that's weird or not :frown:


it's mod odd, i'll definitely say that much at least

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