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    (Original post by tigerz)
    LOOL you joker! I start A2 in2 weeks >.< not planning on buying anything though :P
    pfft, I started A2 in February
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    (Original post by mynameisntbobk)
    I have no idea what I was doing ugh, at least I won't make the mistake again.. and yeah, I meant 2 weeks as well, forgot it was half term
    LOOL, its cool I don't even use my books etc, all my books have been in my locker or borrowed by friends, yeah it doesn't really make a difference as we're on study leave, ready for A2 maths?

    (Original post by DJMayes)
    I probably would (I like most! ) but I've been out all day so haven't seen it.
    haha Felix covered me he seems to be getting there before everyone

    (Original post by justinawe)
    pfft, I started A2 in February
    someones keen I think concentrating on AS was challenging enough for me! Pretty cool
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    (Original post by tigerz)
    LOOL, its cool I don't even use my books etc, all my books have been in my locker or borrowed by friends, yeah it doesn't really make a difference as we're on study leave, ready for A2 maths?
    haha wish I was like you, I'm forever visiting my locker because my teachers are. well they like making sure we have all our stuff for lessons.. started leaving my S1 textbook and chem folder at home towards the end of the courses though :sexface:

    and I'm as ready as I'll ever be, not sure what to think yet. what about you?
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    (Original post by DJMayes)
    There don't seem to be many questions being thrown about tonight. Here's one for you all:

    Spoiler:
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    Let  x be an integer which can be written as the sum of two square numbers. Prove that  2^n x can be written as the sum of two square numbers for all natural numbers n.

    Not sure I'm right, so I haven't drawn a conclusion yet!
    Spoiler:
    Show


    Basis Case:

    Allow f(n)=2^n x

    When n=1, f(1)=2^1 (x) =2x=(1^2+1^2)x -> so true for n=1

    Assumption step

    Assume true for n=k, such that

    f(k)=2^k x

    Inductive step

    If true for n=k, then true for n=k+1 such that

    f(k+1)=2^{k+1} x
    f(k+1)=2(2^k) x

    Since x is an integer that can be written as the sum of two square numbers, and 2(2^k) i.e. (1^2+1^2)((1^2+1^2)^k) is also an integer that can be written as a sum of two square numbers therefore true for n=k+1 ? I'm not sure on this part

    Or would it be even simpler since we've assumed true for n=k, so all you'd really need to do is express 2 in the form of (1^2+1^2) > ?
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    (Original post by mynameisntbobk)
    haha wish I was like you, I'm forever visiting my locker because my teachers are. well they like making sure we have all our stuff for lessons.. started leaving my S1 textbook and chem folder at home towards the end of the courses though :sexface:

    and I'm as ready as I'll ever be, not sure what to think yet. what about you?
    You know, every year I get a locker, one of my friends gets unlucky so either has no locker or a badly placed locker, so I end up sharing it haha! In the end I hardly use it (half my books just stay in there, no worries about losing them eh?)
    My c1 & c2 book have been in there since our courses finished as we didn't use 'em in class lols
    Not doing any prep, just gonna get into it, trying to attempt some of the questions on here should be 'brain training' anyways
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    (Original post by tigerz)
    someones keen I think concentrating on AS was challenging enough for me! Pretty cool
    I finish in January (international students who finish in Jan can have Jan exams), so I had to start some A2 early :dontknow:
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    (Original post by tigerz)
    LOOL, its cool I don't even use my books etc, all my books have been in my locker or borrowed by friends, yeah it doesn't really make a difference as we're on study leave, ready for A2 maths?



    haha Felix covered me he seems to be getting there before everyone



    someones keen I think concentrating on AS was challenging enough for me! Pretty cool
    Is the solution  \frac{\pi r}{2v} ?

    (Original post by Robbie242)
    Not sure I'm right, so I haven't drawn a conclusion yet!
    Spoiler:
    Show


    Basis Case:

    Allow f(n)=2^n x

    When n=1, f(1)=2^1 (x) =2x=(1^2+1^2)x -> so true for n=1

    Assumption step

    Assume true for n=k, such that

    f(k)=2^k x

    Inductive step

    If true for n=k, then true for n=k+1 such that

    f(k+1)=2^{k+1} x
    f(k+1)=2(2^k) x

    Since x is an integer that can be written as the sum of two square numbers, and 2(2^k) i.e. (1^2+1^2)((1^2+1^2)^k) is also an integer that can be written as a sum of two square numbers therefore true for n=k+1 ? I'm not sure on this part
    I don't think this is quite true - x is the sum of two square numbers, not necessarily square itself - 2x is then the sum of four square numbers.
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    (Original post by justinawe)
    I finish in January (international students who finish in Jan can have Jan exams), so I had to start some A2 early :dontknow:
    Ahh I see, yeah I guess so, i'm planning on doing A2 revision in the Autumn half term How were your exams? (I already know the maths results ) Its so cool knowing international people I'm intrigued by many things
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    (Original post by tigerz)
    You know, every year I get a locker, one of my friends gets unlucky so either has no locker or a badly placed locker, so I end up sharing it haha! In the end I hardly use it (half my books just stay in there, no worries about losing them eh?)
    My c1 & c2 book have been in there since our courses finished as we didn't use 'em in class lols
    Not doing any prep, just gonna get into it, trying to attempt some of the questions on here should be 'brain training' anyways
    hahaha I know those kinds of situations the unlucky friend :') very nice of you though
    ahh, we actually never used our c1 and 2 textbooks in class at all, so I just realised I never actually needed it :eek:
    do you mean questions on this thread or past paper questions? I think I may have to join you as well
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    (Original post by DJMayes)
    Is the solution  \frac{\pi r}{2v} ?



    I don't think this is quite true - x is the sum of two square numbers, not necessarily square itself - 2x is then the sum of four square numbers.
    Yup, but you can take it one step further and conclude that was quick jheeze, I still need to finish chem, i'll start your question shortly

    (Original post by mynameisntbobk)
    hahaha I know those kinds of situations the unlucky friend :') very nice of you though
    ahh, we actually never used our c1 and 2 textbooks in class at all, so I just realised I never actually needed it :eek:
    do you mean questions on this thread or past paper questions? I think I may have to join you as well
    The funny thing is it happened twice last year in my old school, and then in collegel I made a new 'unlucky friend!'
    Ahh nah we used to do exercises from it all the time, the questions on this thread! They make you think....a lot! haha
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    (Original post by DJMayes)
    Is the solution  \frac{\pi r}{2v} ?



    I don't think this is quite true - x is the sum of two square numbers, not necessarily square itself - 2x is then the sum of four square numbers.
    Spoiler:
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    So could I say let x=a^2+b^2 where a and b are square numbers?

    And then use this to derive a proof, was my n=1 working wrong as well?

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    (Original post by tigerz)
    Ahh I see, yeah I guess so, i'm planning on doing A2 revision in the Autumn half term How were your exams? (I already know the maths results ) Its so cool knowing international people I'm intrigued by many things
    I've only had 2 exams and a practical so far... 7 of my 8 exams are A2, and the only one AS I was just retaking for fun I pretty much finished my AS this January - I finished Maths and Econ AS and finished Physics AS aside from the practical. Only leaving FM AS, which I'll finish now as well as Maths A2

    So far they've gone alright though. I made one silly mistake in S2 though, I think my streak of no half-arsed bull**** UMS might come to an end! :eek:
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    (Original post by justinawe)
    I've only had 2 exams and a practical so far... 7 of my 8 exams are A2, and the only one AS I was just retaking for fun I pretty much finished my AS this January - I finished Maths and Econ AS and finished Physics AS aside from the practical. Only leaving FM AS, which I'll finish now as well as Maths A2

    So far they've gone alright though. I made one silly mistake in S2 though, I think my streak of no half-arsed bull**** UMS might come to an end! :eek:
    LOOL you crazy person! I had no idea you did Econ! :five: Your murdering these exams, they aint got nothing on you!
    Ahh thats good, hmm its cool you may get lucky! Your UMS scores will never be tainted by 1 half-arsed bull**** score, its fine, i'll think of an excuse
    Its good to have the maths together :')
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    (Original post by tigerz)
    Yup, but you can take it one step further and conclude that was quick jheeze, I still need to finish chem, i'll start your question shortly
    My solution (Pretty much calculus free!):

    Spoiler:
    Show


    Draw out the circle, and let the fox be a distance x from the centre. We need to resolve the motion of the fox into two components, parallel and perpendicular to a radius of the circle. We know that the rabbit is moving with angular speed w given by:

     v=rw \Rightarrow w = \frac{v}{r}

    The component of the motion of fox perpendicular to the radius of the circle must equal this angular speed:

     vsin\theta = xw = \frac{vx}{r}

    So  x = rsin\theta , or  sin\theta = \frac{x}{r}

    The rate of change of x is the component of velocity parallel to the radius:

     vcos\theta = \frac{dx}{dt} or  cos\theta = \dfrac{\frac{dx}{dt}}{v}

    Now, use the identity  cos^2\theta + sin^2 \theta = 1 To get the equation:

     \dfrac{x^2}{r^2}+\dfrac{\frac{dx  }{dt}^2}{v^2} = 1

    This is the equation of an ellipse which can be parametrised as:

     x = rsin kt ,  \frac{dx}{dt} = vcoskt (We know that x=0 initially, so sine must be chosen for x)

    Upon inspection this yields the equation  x = rsin(\frac{v}{r}t)

    Setting x = r and solving then gives  t = \frac{\pi r}{2v}

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    (Original post by Robbie242)
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    So could I say let x=a^2+b^2 where a and b are square numbers?

    And then use this to derive a proof, was my n=1 working wrong as well?

    You could. And no, I don't think your n=1 working is correct.
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    (Original post by DJMayes)
    You could. And no, I don't think your n=1 working is correct.
    Spoiler:
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    How could I prove n=1 to be correct then, I have 2(a^2+b^2) at the moment, if I can prove it for n=1 then I will know what needs to be done for n=k+1

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    (Original post by DJMayes)
    My solution (Pretty much calculus free!):

    Spoiler:
    Show


    Draw out the circle, and let the fox be a distance x from the centre. We need to resolve the motion of the fox into two components, parallel and perpendicular to a radius of the circle. We know that the rabbit is moving with angular speed w given by:

     v=rw \Rightarrow w = \frac{v}{r}

    The component of the motion of fox perpendicular to the radius of the circle must equal this angular speed:

     vsin\theta = xw = \frac{vx}{r}

    So  x = rsin\theta , or  sin\theta = \frac{x}{r}

    The rate of change of x is the component of velocity parallel to the radius:

     vcos\theta = \frac{dx}{dt} or  cos\theta = \dfrac{\frac{dx}{dt}}{v}

    Now, use the identity  cos^2\theta + sin^2 \theta = 1 To get the equation:

     \dfrac{x^2}{r^2}+\dfrac{\frac{dx  }{dt}^2}{v^2} = 1

    This is the equation of an ellipse which can be parametrised as:

     x = rsin kt ,  \frac{dx}{dt} = vcoskt (We know that x=0 initially, so sine must be chosen for x)

    Upon inspection this yields the equation  x = rsin(\frac{v}{r}t)

    Setting x = r and solving then gives  t = \frac{\pi r}{2v}

    Awesome, I saw 2 ways of solving it but this is entirely different, i'm gonna start your question now, but first I gotta get rid of these hiccups lol and if the fox has a factor of  \pi/2 \approx 1.5 times slower
    The fox has completed half a circle of half the radius when the rabbit has completed a quarter of the full circle!
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    (Original post by tigerz)
    The funny thing is it happened twice last year in my old school, and then in collegel I made a new 'unlucky friend!'
    Ahh nah we used to do exercises from it all the time, the questions on this thread! They make you think....a lot! haha
    haha, that's so bad if anything I'm someone else's unlucky friend, just know I've been through a lot of embarassment this year
    ah right, fair enough, we never they really do, I just wish I could actually understand some of it, and use latex
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    (Original post by mynameisntbobk)
    haha, that's so bad if anything I'm someone else's unlucky friend, just know I've been through a lot of embarassment this year
    ah right, fair enough, we never they really do, I just wish I could actually understand some of it, and use latex
    LOOL, I don't need 'em, makes my bag heavier ^_^ Meh I always act embarrassing, used to it now!
    I don't understand half of them, but there's to occasional solvable one (which takes me half a day!), Its not too difficult, I learn't how to use LaTeX like 10 days ago!
    just use this website after a while you remember the shortcuts http://www.codecogs.com/latex/eqneditor.php
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    (Original post by Robbie242)
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    How could I prove n=1 to be correct then, I have 2(a^2+b^2) at the moment, if I can prove it for n=1 then I will know what needs to be done for n=k+1

    I'm sorry, if I told you that (At least my way of doing it) it makes the entire question trivial. However, you might find that trying it out with some values of x may give you a feel for the question.

    (Also, don't spend too much time on it; it's not really appropriate to A Level (Although I can assure you you know all the knowledge required to do it) and I wouldn't want it to get in the way of any important revision etc.)
 
 
 
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