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Thread: Complex numbers question

  1. #1
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    Complex numbers question

    Question:
    a)Given that the complex numbers $\displaystyle w_{1} $ and $\displaystyle w_{2} $ are the roots of the equation $\displaystyle z^2-5-12i=0 $, express $\displaystyle w_{1} $ and $\displaystyle w_{2} $ in the form of $\displaystyle a +ib $ where a and b are real

    b)i)Indicate the point sets in an argand diagram corresponding to the sets of the complex numbers
    $\displaystyle A= \left \{z:\left | z \right | =3, z \epsilon \mathbb{C} \right \}$
    $\displaystyle B=\left \{z:\left | z \right | =2, z \epsilon \mathbb{C} \right \} $

    ii)Shade the region corresponding to the values of z for which the inequalities
    $\displaystyle 2 <\left | z \right |<3 $ and $\displaystyle 30^{\circ}<arg(z)<60^{\circ} $

    are simultaneously satisfied.

    My attempt:
    a)
    $\displaystyle z^2-5-12i=0 $
    $\displaystyle z^2= 5+12i $
    ]$\displaystyle z= \pm (5+12i)^{(0.5)}$
    note that $\displaystyle (3+2i)^2 = 5+12i $
    thus $\displaystyle z = \pm (3+2i) $
    $\displaystyle w_{1}= 3+2i $
    $\displaystyle w_{2}= -(3+2i) $

    b)i)

    $\displaystyle A={z:\left | z \right | =3, z \epsilon \mathbb{C}} $
    $\displaystyle B={z:\left | z \right | =2, z \epsilon \mathbb{C}} $
    Complex numbers question-c1.jpg
    ii)$\displaystyle 2 <\left | z \right |<3 $ and $\displaystyle 30^{\circ}<arg(z)<60^{\circ} $

    Complex numbers question-c2.jpg
    I shaded the region I thought would be the right answer in purple

    The part I am finding difficult is part b. Please help
    Last edited by bigmansouf; Apr 2nd 2019 at 04:25 PM.
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  2. #2
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    Re: Complex numbers question

    I'm not sure where you are getting the green ray from in your diagram.

    The shaded area should be between the two circles and between 30 and 60 degrees. Why have you gut it off shorter at 33.7 degrees?

    Also the shaded area should be bounded by dotted lines to indicate non-equality (eg a full line indicates < or = while a dotted line indicates <)
    Last edited by Debsta; Apr 2nd 2019 at 09:07 PM.
    Thanks from topsquark and bigmansouf
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  3. #3
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    Re: Complex numbers question

    I've just realised where you might have got the 33.7 degrees from. Is that the argument of 3 + 2i from part (a)? Part (b) is a separate question from part (a). There is no link.
    Thanks from bigmansouf
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    Re: Complex numbers question

    Quote Originally Posted by Debsta View Post
    I've just realised where you might have got the 33.7 degrees from. Is that the argument of 3 + 2i from part (a)? Part (b) is a separate question from part (a). There is no link.
    thank you that was the reason i did that but thanks for correcting me
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