2019 NYJC P1 Q9
(a)
The complex numbers $z$ and $w$ satisfy the simultaneous equations
$\left| z \right|-{{w}^{*}}=-3-\sqrt{2}\mathbf{i}$ and ${{w}^{*}}+w+5z=1+20\mathbf{i}$,
where ${{w}^{*}}$ is the complex conjugate of $w$. Find the value of $z$ and the corresponding value of $w$.
[4]
(b)
It is given that $8\mathbf{i}$ is a root of the equation $\mathbf{i}{{z}^{3}}+\left( 8-2\mathbf{i} \right){{z}^{2}}+az+40=0$ where $a$ is a complex number.
(i) Find $a$.
[2]
(ii) Hence, find the other roots of the equation, leaving your answer in the form $a+b\mathbf{i}$ where $a$ and $b$ are real constants.
[3]
(iii) Deduce the number of real roots in the equation, ${{z}^{3}}-\left( 8-2\mathbf{i} \right){{z}^{2}}+a\mathbf{i}z+40=0$ has.
[1]
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