The complex numbers $z$ and $w$ satisfy the simultaneous equations \(z+{{w}^{*}}+5\mathbf{i}=10\

Question

The complex numbers $z$ and $w$ satisfy the simultaneous equations $z+{{w}^{*}}+5\mathbf{i}=10$ and ${{\left| w \right|}^{2}}=z+18+\mathbf{i}$. Find $z$ and $w$.[4] It is given that $2+\mathbf{i}$ is a root of the equation ${{z}^{2}}-5z+7+\mathbf{i}=0$. Find the second root of the equation in Cartesian form, showing your working clearly.[2] Hence find the roots of the equation $-\mathbf{i}{{w}^{2}}+5w+7\mathbf{i}-1=0$.[2] The complex number $z$ is given by $z=-a+a\,\mathbf{i}$, where $a$ is a positive real number. It is given that $w=-{{\frac{\sqrt{2}z}{{{z}^{4}}}}^{*}}$. Express $w$ in the form $r{{\mathrm{e}}^{i\theta }}$, in terms of $a$, where $r>0$ and $-\pi <\theta \le \pi $.[4] Find the two smallest positive whole number values of $n$ such that $\operatorname{Re}\,({{w}^{n}})=0$.[3]

Accepted Answer

(a) $z=14-i$ (b)(i) $k=3-i$ (b)(ii) $w=1-2i$ or $-1-3i$ (c)(i) $\frac{1}{2a^3} e^{\mathrm{i}\left(-\frac{3\pi}{4}\right)}$ (c)(ii) $n = -\frac{2}{3}(1 + 2k)$, where $k \in \mathbb{Z}$ Three smallest positive whole number values of $n$ are $2, 6$.

2017 CJC P2 Q4

2017 CJC P2 Q4

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