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    For 10, can anyone confirm the answer to the last bit is

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    y=x


    Done the first part of 8, but it's not pretty...
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    (Original post by DFranklin)
    Well, if you want something not done yet, there's a distinct lack of sols for Q7-Q10. Q9, Q10 don't look too bad, I confess I'm not really seeing where to start on Q8, and Q7(b) looks horrible. (In my defense, I find coord geometry is one of those things you need to be in practice for to do well).
    I think I just did 7 but I could be deluding myself as it only took 5 mins!

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    Coordinates of P and Q are respectively (x_p,y_p) and (x_q,y_q)

    Midpoints are sum of x and y coords divided by 2:

    2h=x_p+x_q

    2k=y_p+y_q

    Using equation of ellipse:

    \frac{x_p^2}{a^2}+\frac{y_p^2}{b  ^2}=1

    \frac{x_q^2}{a^2}+\frac{y_q^2}{b  ^2}=1

    Both equal to 1 so equate them and rearrange:

    b^2(x_p^2-x_q^2)=a^2(y_q^2-y_p^2)

    Difference of 2 squares and use first two results:

    hb^2(x_p-x_q)=ka^2(y_q-y_p)

    Equaton of straight line in form y-y1 = m(x-x1) gives:

    m = \frac{-hb^2}{ka^2}

    Use point (h,k) as it is on the line:

    y-k=\frac{hb^2}{ka^2}(h-x)

    Just sub point (a,b) into this equation:

    (a) ka^2(b-k)=hb^2(h-a)

    Use gradient -1/m and same midpoint then sub (a,b) into new equation:

    (b) ka^2(a-h)=hb^2(b-k)

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    Re Q10: Yeah, I get that too. Comes out surprisingly simply.

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    If you take the midpoint to be (X,Y), then the two points can be written as (X-t1, Y-t2) and (X+t1, Y+t2). Needing the chord parallel to (x+y) gives t2=-t1, so we can write the points as (X-t, Y+t) and (X+t, Y-t). Sub both into the (x+y)^3 = 9xy equation and it drops out very quickly.


    Re Q7: Sorry, I can't follow a thing you've done past about the 5th line. Do you really tell your students to give that little explanation?
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    wow that's a good paper, that was linear a level days right? one/two papers?
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    (Original post by DFranklin)
    Re Q7: Sorry, I can't follow a thing you've done. Do you really tell your students to give that little explanation?
    No I was being lazy and assumed you would easily fill in the gaps. I will add a bit more.
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    (Original post by Mr M)
    No I was being lazy and assumed you would easily fill in the gaps. I will add a bit more.
    Sorry - the gaps are staying resolutely open. Not having done this much in 20 years is pretty killer. (There's usually ways of doing the questions without getting bogged down in pages of algebra, but you need to approach them in the right way. At the moment, 'the right way' is eluding me).

    Did you see a better way of doing 5(b), by the way? The approach I took was a bit brute force (and unlikely to occur to someone who hadn't encountered Fourier Series, I would say).
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    (Original post by DFranklin)
    Sorry - the gaps are staying resolutely open. Not having done this much in 20 years is pretty killer. (There's usually ways of doing the questions without getting bogged down in pages of algebra, but you need to approach them in the right way. At the moment, 'the right way' is eluding me).

    Did you see a better way of doing 5(b), by the way? The approach I took was a bit brute force (and unlikely to occur to someone who hadn't encountered Fourier Series, I would say).
    I haven't tried it. I will look at your solution now and see if I can think of anything.
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    (Original post by Mr M)
    I haven't tried it. I will look at your solution now and see if I can think of anything.
    It may well be there's some integration by parts you can do - I didn't try too hard because my integration is 'teh suck' these days...

    Re your solution to Q7: definitely falling into the "seems to work but I can't quite believe it - you don't seem to have done very much!" category. Can't get my head round it, I'm afraid. I'm sure it's right though.
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    (Original post by Mr M)
    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.
    How much maths was in the old physics A-level? Any calculus or anything?
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    Yeah, physics A-level was basically memorisation of useless definitions and year-9 level algebraic manipulation.
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    (Original post by -G-a-v-)
    How much maths was in the old physics A-level? Any calculus or anything?
    Definitely had stuff involving basic calculus when I did the A-level in 1987. (So we could have vague 'field/potential' discussions, for example).
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    (Original post by -G-a-v-)
    How much maths was in the old physics A-level? Any calculus or anything?
    Definitely some calculus. Basically, our A-level physics (this years OCR anyway...) is equivilent to GCSE physics about 25~30 years ago (according to my physics teacher, and the GCSE physics book).
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    (Original post by DFranklin)
    Did you see a better way of doing 5(b), by the way? The approach I took was a bit brute force (and unlikely to occur to someone who hadn't encountered Fourier Series, I would say).
    Sorry DF - I have had a good look at 5(b) and can't think of anything better than your succinct method.

    I tried to follow your hint by considering integration by parts and somehow after a number of dead ends and several pages of A4 ended up finding a reduction formula for:

    \int^{2 \pi}_0 \cos^n xdx

    As the integral can be expressed as a sum of cos^n x terms, it does actually work but cranking through it was utterly stupid (particularly as the students wouldn't have had a calculator).

    I agree there probably is a clever (and short) method but finding it has defeated me.
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    Interesting thread. Regarding the above post, would using de moivre's theorem to express powers of cosine as cos(nx) help?
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    (Original post by tingtongdingdong)
    Interesting thread. Regarding the above post, would using de moivre's theorem to express powers of cosine as cos(nx) help?
    Not really - it's actually fairly easy to work out I_n = \int_0^{2\pi} \cos^n x (int by parts to get a relation between I_n and I_{n-2}; it's easier than using DeMoivre.
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    I've only done AS maths + FM properly but have quite a bit of knowledge of A2 Maths/FM calculus and complex numbers and find the questions I've know the methods look decent on the normal A-level paper but the ones on the "special paper" look very difficult.
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    Anyone who can do that paper is an absolute mathematical beast.
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    I think it doesnt look very hard compared to AEA. I cant do either, but then again I detest maths.
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    (Original post by AsakuraMinamiFan)
    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 x \pm x^{-1} = t, or otherwise, find:

    \displaystyle \int \frac{x^2 + 1}{(x^4+x^2+1)^2} \, \mathrm{d}x \qquad \int \frac{x^2 - 1}{(x^4+x^2+1)^2} \, \mathrm{d}x  \qquad \int \frac{1}{x^4+x^2+1} \, \mathrm{d}x.

    I'll try and bring some more later today. :p:
    I might try them out later, but my algebra skills might be too rusty due to not having done any maths in 2 months though.
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    (Original post by mf2004)
    Is there any chance you could upload them (physics section) ? I'd love to give them a try.
    I'll try and do it tomorrow, although I've been having a few problems with the scanner recently, but I'll do my best (I need to sort out the stupid machine anyway...).
    Any topics of particular interest?
 
 
 
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