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(i) Express the complex number z = 4i/(1-i√3)in Cartesian form.
(ii) Determine the modulus and argument of z, and hence express z in
polar form.
(b) Solve the equation z4 − z = 0, expressing your solutions in Cartesian form.

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  1. 4i/(1-√3 i)

    = 4i(1+√3 i)/(1+3) = -√3 + i
    Now, tanθ = -1/√3, so z = (2,-π/6)

    z^4-z = 0
    z(z^3-1) = 0
    z(z-1)(z^2+z+1) = 0
    and so on

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  2. b) form a) z = -√3 + i
    which is in quadrant II if graphed in the Argand plane
    so if tanθ = -1/√3 , θ = 5π/6
    modulus of z = √(3 + 1) = 2
    argument of z = 5π/5
    z = 2cos 5π/6 + 2sin 5π/6

    check: 2cos 5π/6 + 2sin 5π/6 i = -1.73205.. + (1)i or -√3 + i

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