The complex number $z$ is given by $z=r{{\mathrm{e}}^{\mathrm{i} heta }}$ where $r0$ and $-\frac{2

Question

The complex number $z$ is given by $z=r{{\mathrm{e}}^{\mathrm{i}	heta }}$ where $r>0$ and $-\frac{2\pi }{3}\le 	heta \le \frac{2\pi }{3}$. Given that $w=\left( \frac{\sqrt{3}}{4}-\frac{1}{4}\mathrm{i} ight)z$, find the modulus and argument of $w$, leaving your answers in terms of $r$ and $	heta $. [3] Find the possible values of $	heta $ if $w$ is purely imaginary. [3] Two of the roots of the equation ${{x}^{4}}+{{x}^{3}}+a{{x}^{2}}+bx+c=0$ are $2$ and $\mathrm{i}\sqrt{5}$, where $a$, $b$ and $c\in \mathbb{R}$. The sum of all the roots is $-1$. Find the other roots of the equation and the values of $a$, $b$ and $c$. [5]

Accepted Answer

(a) (i) $\frac{r}{2}{{\mathrm{e}}^{\left( 	heta -\frac{\pi }{6} ight)\mathrm{i}}}$ (a)(ii) $-\frac{\pi }{3}$ 	ext{ or } $\frac{2\pi }{3}$ (b) $a=-1$; $b=5$; $c=-30$

2022 CJC J2 MYE P1 Q7

2022 CJC J2 MYE P1 Q7

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