The complex number $w$ is such that $w=a+\mathrm{i}b$, where $a$ and $b$ are non-z

Question

The complex number $w$ is such that $w=a+\mathrm{i}b$, where $a$ and $b$ are non-zero real numbers. The complex conjugate of $w$ is denoted by $w^*$. Given that $\frac{{{w}^{2}}}{w^*}$ is purely imaginary, find the possible values of $w$ in terms of $a$.[5] The complex number $z$ is such that $z^5=-32\mathrm i$. Find the modulus and argument of each of the possible values of $z$.[4] Two of these values are $z_1$ and $z_2$, where $\frac12\pi<\arg z_1<\pi$ and $-\pi<\arg z_2<-\frac12\pi$. Find the exact value of $\arg(z_1-z_2)$ in terms of $\pi$ and show that $|z_1-z_2|=4\sin(\frac15\pi)$.[4]

Accepted Answer

(a) $w=a\pm\frac{a}{\sqrt3}\mathrm i$ (b)(i) $|z|=2$, $\arg z=-9\pi/10,-\pi/2,-\pi/10,3\pi/10,7\pi/10$ (ii) $\arg(z_1-z_2)=2\pi/5$, $|z_1-z_2|=4\sin(\pi/5)$.

N2015 P1 Q9

N2015 P1 Q9

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