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Thread: HSC 2018 MX2 Marathon

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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by mrbunton View Post
    z= x+iy, w = u+iv; w = z -1/z. , find locus of w if |z|=2
    edit:from patel textbook
    https://i.imgur.com/DOI8Tanr.jpg

    Hopefully im right, correct me if wrong

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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by si2136 View Post
    https://i.imgur.com/DOI8Tanr.jpg

    Hopefully im right, correct me if wrong
    u didnt fully complete it; the hardest part is finding the relationship between v and w.
    Last edited by mrbunton; 4 Feb 2018 at 7:25 PM.

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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by mrbunton View Post
    u didnt fully complete it; the hardest part is finding the relationship between v and w.




    Squaring, we have









    Replacing variables,



    or

    Is that right?
    HSC 2018 - [English Adv.] • [Maths Ext. 1] • [Maths Ext. 2] • [Chemistry] • [Software Design and Development]

    1(3√3) t2 dt cos(3π/9) = log(3√e) | Integral t2 dt, From 1 to the cube root of 3. Times the cosine, of three pi over nine, Equals log of the cube root of e.

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    Re: HSC 2018 MX2 Marathon

    yup
    Last edited by mrbunton; 4 Feb 2018 at 9:50 PM.

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    Re: HSC 2018 MX2 Marathon

    polynomial/complex number question:
    cot(x)= y+1
    a,b are roots of z^2-2z+2=0
    prove that ( (y+a)^n - (y+b)^n ) / (a-b) = sin(nx) / sinx^n
    Last edited by mrbunton; 12 Feb 2018 at 4:56 PM. Reason: error

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    Re: HSC 2018 MX2 Marathon

    This is an interesting question I found in the SGS notes.

    If is a complex number, find the locus of if

    HSC 2018 - [English Adv.] • [Maths Ext. 1] • [Maths Ext. 2] • [Chemistry] • [Software Design and Development]

    1(3√3) t2 dt cos(3π/9) = log(3√e) | Integral t2 dt, From 1 to the cube root of 3. Times the cosine, of three pi over nine, Equals log of the cube root of e.

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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by fan96 View Post
    This is an interesting question I found in the SGS notes.

    If is a complex number, find the locus of if

    Polar bash (mod-arg for the high schoolers) makes this problem trivial...
    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.

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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by fan96 View Post
    This is an interesting question I found in the SGS notes.

    If is a complex number, find the locus of if

    =Re(x+iy -(x-iy)/x^2+y^2)=0
    x - x/(x^2+y^2) = 0
    times by x^2 + y^2 then divide by x;
    0= x^2+y^2 -1;
    1 = x^2 + y^2

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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by mrbunton View Post
    =Re(x+iy -(x-iy)/x^2+y^2)=0
    x - x/(x^2+y^2) = 0
    times by x^2 + y^2 then divide by x;
    0= x^2+y^2 -1;
    1 = x^2 + y^2
    is not the correct answer.

    (Be careful of any restrictions you place on the locus)
    Last edited by fan96; 10 Feb 2018 at 9:34 AM.
    HSC 2018 - [English Adv.] • [Maths Ext. 1] • [Maths Ext. 2] • [Chemistry] • [Software Design and Development]

    1(3√3) t2 dt cos(3π/9) = log(3√e) | Integral t2 dt, From 1 to the cube root of 3. Times the cosine, of three pi over nine, Equals log of the cube root of e.

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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by fan96 View Post
    This is an interesting question I found in the SGS notes.

    If is a complex number, find the locus of if

















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    Re: HSC 2018 MX2 Marathon



    HSC 2018

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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by altSwift View Post
















    As I said above, that's not the correct answer.

    The locus is not .

    There are certain values of (specifically ) that need to be considered.

    (By the way, is not on the unit circle so it's not necessary to explicitly exclude it if the locus was just the unit circle)

    I'll post the solution tomorrow if nobody gets the right answer.
    Last edited by fan96; 10 Feb 2018 at 5:43 PM.
    HSC 2018 - [English Adv.] • [Maths Ext. 1] • [Maths Ext. 2] • [Chemistry] • [Software Design and Development]

    1(3√3) t2 dt cos(3π/9) = log(3√e) | Integral t2 dt, From 1 to the cube root of 3. Times the cosine, of three pi over nine, Equals log of the cube root of e.

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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by fan96 View Post
    As I said above, that's not the correct answer.

    The locus is not .

    There are certain values of (specifically ) that need to be considered.

    (By the way, is not on the unit circle so it's not necessary to explicitly exclude it if the locus was just the unit circle)

    I'll post the solution tomorrow if nobody can figure it out.
    rip, I'll have another go tonight
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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by altSwift View Post


    An exercise in Patel (I believe 4H, Q8) has a similar, but harder question.



















    HSC 2018 - [English Adv.] • [Maths Ext. 1] • [Maths Ext. 2] • [Chemistry] • [Software Design and Development]

    1(3√3) t2 dt cos(3π/9) = log(3√e) | Integral t2 dt, From 1 to the cube root of 3. Times the cosine, of three pi over nine, Equals log of the cube root of e.

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    Re: HSC 2018 MX2 Marathon

    For anyone interested, the Patel question I was talking about is:

    HSC 2018 - [English Adv.] • [Maths Ext. 1] • [Maths Ext. 2] • [Chemistry] • [Software Design and Development]

    1(3√3) t2 dt cos(3π/9) = log(3√e) | Integral t2 dt, From 1 to the cube root of 3. Times the cosine, of three pi over nine, Equals log of the cube root of e.

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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by fan96 View Post
    An exercise in Patel (I believe 4H, Q8) has a similar, but harder question.



















    Nice! Here is my solution:





















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    Re: HSC 2018 MX2 Marathon

    It definitely seems to be something that most people forget. For this type of question it tends to not matter (in terms of marks), but being able to spot when a variable's domain has been restricted has saved me a few marks in exams.

    (This might help with the locus problem I posted before...)
    HSC 2018 - [English Adv.] • [Maths Ext. 1] • [Maths Ext. 2] • [Chemistry] • [Software Design and Development]

    1(3√3) t2 dt cos(3π/9) = log(3√e) | Integral t2 dt, From 1 to the cube root of 3. Times the cosine, of three pi over nine, Equals log of the cube root of e.

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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by fan96 View Post
    For anyone interested, the Patel question I was talking about is:





























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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by fan96 View Post
    It definitely seems to be something that most people forget. For this type of question it tends to not matter (in terms of marks), but being able to spot when a variable's domain has been restricted has saved me a few marks in exams.

    (This might help with the locus problem I posted before...)
    Ooh these hints
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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by fan96 View Post
    This is an interesting question I found in the SGS notes.

    If is a complex number, find the locus of if

    Spoiler (rollover to view):

    let z= rcosθ+isinθ
    z-1/z = (r-1/r)cosθ + i(r+1/r)sinθ
    Re(z-1/z) = (r-1/r)cosθ

    If LHS = 0, then (r-1/r) = 0 or cosθ = 0
    hence, r=1 or θ= ±(2n+1)π/2

    Hence the locus is x^2+y^2 = 1 and x = 0 (excluding (0,0))
    Last edited by jathu123; 10 Feb 2018 at 9:07 PM.
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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by jathu123 View Post
    Spoiler (rollover to view):

    let z= rcosθ+isinθ
    z-1/z = (r-1/r)cosθ + i(r+1/r)sinθ
    Re(z-1/z) = (r-1/r)cosθ

    If LHS = 0, then (r-1/r) = 0 or cosθ = 0
    hence, r=1 or θ= ±(2n+1)π/2

    Hence the locus is x^2+y^2 = 1 and x = 0 (excluding (0,0))
    ahahaha I forgot to consider x = 0 jesus
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    Re: HSC 2018 MX2 Marathon

    A classic one

    2017
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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by jathu123 View Post
    Spoiler (rollover to view):

    let z= rcosθ+isinθ
    z-1/z = (r-1/r)cosθ + i(r+1/r)sinθ
    Re(z-1/z) = (r-1/r)cosθ

    If LHS = 0, then (r-1/r) = 0 or cosθ = 0
    hence, r=1 or θ= ±(2n+1)π/2

    Hence the locus is x^2+y^2 = 1 and x = 0 (excluding (0,0))
    Yep.

    If you used , you ended up with the locus



    Which is the correct answer.

    It's tempting to divide both sides by , but if you performed the division you were implicitly assuming that , which lost solutions.

    Clearly if , any value of is a solution, except for zero since .

    That can also be shown by considering

    which is zero for all .

    To graph , you could consider two separate cases: , and .

    This is the solution from the SGS notes:

    altSwift likes this.
    HSC 2018 - [English Adv.] • [Maths Ext. 1] • [Maths Ext. 2] • [Chemistry] • [Software Design and Development]

    1(3√3) t2 dt cos(3π/9) = log(3√e) | Integral t2 dt, From 1 to the cube root of 3. Times the cosine, of three pi over nine, Equals log of the cube root of e.

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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by mrbunton View Post
    polynomial/complex number question:
    cot(x)= y+1
    a,b are roots of z^2+2z-1=0
    prove that ( (y+a)^n - (y+b)^n ) / (a-b) = sin(nx) / sinx^n
    I tried taking out a factor of



    from the top two brackets but I still cant get to the LHS... any hints?
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    Re: HSC 2018 MX2 Marathon

    Quote Originally Posted by altSwift View Post
    I tried taking out a factor of



    from the top two brackets but I still cant get to the LHS... any hints?
    glad someone noticed my question
    make sure to use all the given equations.
    eg: a and b represent their own complex numbers (on the lhs), try finding them;
    its easier approaching it from the lhs to the rhs;without trying to take out any common factors immediately
    Last edited by mrbunton; 12 Feb 2018 at 1:32 AM.

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