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Thread: HSC 2017 MX2 Marathon (archive)

  1. #101
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    Re: HSC 2017 MX2 Marathon

    Not sure if its me or the book error but I had a complex number question

    (6+i)(a+bi)=2

    My solution for



    My book has a different solution to that.
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  2. #102
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    Re: HSC 2017 MX2 Marathon

    Quote Originally Posted by davidgoes4wce View Post
    Not sure if its me or the book error but I had a complex number question

    (6+i)(a+bi)=2

    My solution for



    My book has a different solution to that.
    I also got the same answer as you.
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    Re: HSC 2017 MX2 Marathon

    I spent 5 minutes on this and couldn't solve it.





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    Re: HSC 2017 MX2 Marathon

    I think my book has done another mistake. (The book has made a few mistakes so I'm just checking with people on here)
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    Re: HSC 2017 MX2 Marathon

    i got an

    x+yi


    my x-component was :



    I haven't solved the y-component yet. Want to get 'x' right before I do the y.
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    Re: HSC 2017 MX2 Marathon

    The book's answer is right, you can check it by squaring it to get 5 + i.

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    Re: HSC 2017 MX2 Marathon

    Quote Originally Posted by davidgoes4wce View Post
    I spent 5 minutes on this and couldn't solve it.





    x^2 - y^2 + 2xyi = 5+i

    Comparing real and imaginary
    2xy = 1
    y = 1/2x

    x^2 - y^2 = 5
    x^2 - 1/4x^2 = 5
    4x^4 - 20x^2 - 1 = 0
    x^2 = (20 +- sqrt(20^2 - 4(4)(-1))/2(4)
    x^2 = (20+- sqrt(416))/8
    x^2 = (20 +- 4sqrt(26))/8
    x^2 = (5 +- sqrt(26))/2

    But x is real so take the positive case only
    x = +- sqrt[(5+sqrt26)/2]
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  8. #108
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    Re: HSC 2017 MX2 Marathon

    Quote Originally Posted by davidgoes4wce View Post
    I spent 5 minutes on this and couldn't solve it.





    http://i.imgur.com/xWfCJry.jpg
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    Re: HSC 2017 MX2 Marathon



    apologies if the question isn't clear (hence my two examples)
    Last edited by jathu123; 10 Sep 2017 at 7:41 PM. Reason: Fixed typo
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    Re: HSC 2017 MX2 Marathon

    Quote Originally Posted by jathu123 View Post


    apologies if the question isn't clear (hence my two examples)
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    Re: HSC 2017 MX2 Marathon

    Last edited by seanieg89; 15 Sep 2017 at 5:29 PM.
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    Re: HSC 2017 MX2 Marathon

    Is there a formula for finding the sum of all points (angles where it's spiky) on an n pointed star? For example, for a 5 pointed star, it is 180 degrees
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    Re: HSC 2017 MX2 Marathon

    Quote Originally Posted by si2136 View Post
    Is there a formula for finding the sum of all points (angles where it's spiky) on an n pointed star? For example, for a 5 pointed star, it is 180 degrees
    For higher n, there becomes multiple ways to construct the star, each with their own angle sum. ie. the points of the star may create an n-gon, or another star, as shown for n = 7, 11. Note in each case, the first star creates an n-gon by joining every second vertex, while the others create other stars.



    Last edited by 1729; 25 Sep 2017 at 6:24 PM.

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    Re: HSC 2017 MX2 Marathon

    Quote Originally Posted by seanieg89 View Post
    This may not be in the spirit of the question, but if you had something different in mind with a more geometric proof, I'd like to see it. Let be the centre of circle . Consider the kite , and apply the cosine rule to the side from both and . Letting, . We obtain that, . A similar expression is obtained for . From this we obtain,

    . As we have that is monotonic increasing, and so is maximised if and only if is maximised. Using the appropriate formulae we obtain,

    , through calculus or the AM-GM inequality we obtain a maxima when that is unique and so on. However this sends suggesting that .
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    This too shall pass Sy123's Avatar
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    Re: HSC 2017 MX2 Marathon







    Last edited by Sy123; 6 Oct 2017 at 5:08 PM.

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    Re: HSC 2017 MX2 Marathon

    In how many ways can seven identical cats be put into three identical pens so that all of the pens are occupied? You must state reasoning. (2 marks)

    Why can we not use stars and bars? I.e. 9C2
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  17. #117
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    Re: HSC 2017 MX2 Marathon

    Quote Originally Posted by pikachu975 View Post
    In how many ways can seven identical cats be put into three identical pens so that all of the pens are occupied? You must state reasoning. (2 marks)

    Why can we not use stars and bars? I.e. 9C2


    Here is a useful article for placing n identical objects into r identical bins: https://brilliant.org/wiki/identical...identical-bins .
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    Re: HSC 2017 MX2 Marathon

    Quote Originally Posted by Sy123 View Post
    This may not be in the spirit of the question, but if you had something different in mind with a more geometric proof, I'd like to see it. Let be the centre of circle . Consider the kite , and apply the cosine rule to the side from both and . Letting, . We obtain that, . A similar expression is obtained for . From this we obtain,

    . As we have that is monotonic increasing, and so is maximised if and only if is maximised. Using the appropriate formulae we obtain,

    , through calculus or the AM-GM inequality we obtain a maxima when that is unique and so on. However this sends suggesting that .
    This is what I had in mind.

    On the topic of a more geometric solution, I suspect there is a way to kill it quicker using nonsyllabus techniques (inversive geometry perhaps), but it is not immediately apparent to me, I might have a crack later this week.

  19. #119
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    Re: HSC 2017 MX2 Marathon

    Quote Originally Posted by seanieg89 View Post
    I think LaTeX broke
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  20. #120
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    Re: HSC 2017 MX2 Marathon (archive)

    2017 marathon now locked.

    New 2018 MX2 Marathon here:
    HSC 2018 MX2 Marathon
    2014 HSC
    2015 Gap Year
    2016-2018 UOW Bachelor of Maths/Maths Advanced.

    Links:
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    Maths Notes - Free Sample

    thsconline: https://thsconline.github.io/s/
    For 2017 papers, check the thread in the Mathematics forum.

    Dan will be back 14/10/17

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