Given that ${{z}_{1}}=-\frac{\mathrm{i}}{2}$ is a root if the equation $2{{z}^{3}}+\left( \mathrm{i}

Question

Given that ${{z}_{1}}=-\frac{\mathrm{i}}{2}$ is a root if the equation $2{{z}^{3}}+\left( \mathrm{i}-8 \right){{z}^{2}}+az+13\mathrm{i}=0$, find the complex number $a$ and solve the equation, giving your answer in cartesian form $x+\mathrm{i}y$. [4] Hence find in cartesian form the roots of the equation $2{{w}^{3}}+\left( 1+8\mathrm{i} \right){{w}^{2}}-aw-13=0$ [2]

Accepted Answer

$a=26-4i$; roots $2 s3i$, $-\frac12$; transformed roots $\pm3-2i$, $-\frac12$

2016 HCI P2 Q2 (a)

2016 HCI P2 Q2 (a)

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