Do not use a graphing calculator for this question. The complex number $a+\mathbf{i}$ is a root of

Question

Do not use a graphing calculator for this question. The complex number $a+\mathbf{i}$ is a root of the equation ${{z}^{2}}-2\sqrt{2}z+b=0$ where $a$ and $b$ are positive real constants. Find the exact values of $a$ and $b$.[3] It is given that $\mathrm{f}(z)={{z}^{6}}-3{{z}^{4}}+11{{z}^{2}}-9$. Show that $\mathrm{f}(z)=\mathrm{f}(-z)$.[1] It is given that $a+\mathbf{i}$ is a root of the equation $\mathrm{f}(z)=0$. Using the results found in parts (i) and (ii), show that $\mathrm{f}(z)$ can be factorised into $3$ quadratic factors with real coefficients, leaving your answer in exact form.[4]

Accepted Answer

(i) $a=\sqrt2$, $b=3$ (iii) $\mathrm{f}(z)=(z^2-2\sqrt2z+3)(z^2+2\sqrt2z+3)(z^2-1)$

2022 HCI P1 Q8

2022 HCI P1 Q8

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