Given that imaginary unit $\mathbf{i}$ is one of the roots of the equation \({{z}^{3}}-2{{z}^{2}}+

Question

Given that imaginary unit $\mathbf{i}$ is one of the roots of the equation ${{z}^{3}}-2{{z}^{2}}+az+1+3\mathbf{i}=0$, find the complex number ${a}$ and the other roots. The root that has the largest modulus in (a) is denoted as ${w}$ and represented by the point ${A}$. The points ${B, C}$ and ${D}$ represent the complex numbers ${{w}^{*}}$, $\mathbf{i}w$ and ${{\mathbf{i}}^{2}}w$ respectively. On the same diagram, plot the points ${A, B, C}$ and ${D}$ and state each transformation that maps the point ${A}$ onto the point ${B, C}$ and ${D}$ respectively and state the angle ${OAC}$.

Accepted Answer

(a) ${a = -2+3\mathbf{i}\,,\; z = -1 	ext{ or } 3-\mathrm{i}}$

2014 HCI P1 Q3 [Modified]

2014 HCI P1 Q3 [Modified]

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