Practice section 5 of 9

The complex numbers $z$ and $w$ are given by $z=\sqrt3+i$ and $w=-1+i\sqrt3$.

Sketch an Argand diagram, with origin $O$, showing the points $Z$, $W$ and $P$ representing $z$, $w$ and $z+w$ respectively. Show that $OWPZ$ is a square.

The appendix works with $z=1+i$ and, where two numbers are needed, $z_1=1+i$ and $z_2=\sqrt3+i$. Follow its nine examples: (a) $zz^*$; (b) $\lvert z^*\rvert$; (c) $\arg z^*$; (d) $\lvert z_1z_2\rvert$; (e) $\lvert z^4\rvert$; (f) $\lvert z_1/z_2\rvert$; (g) $\arg(z_1z_2)$; (h) $\arg(z^4)$; (i) $\arg(z_1/z_2)$.

A graphic calculator is not to be used in answering this question.

Solve the equation $z^2=3-4i$, giving your answers in the form $a+ib$. Hence, by expressing $\frac13i-\frac14$ in the form $k(3-4i)$, where $k$ is a constant, find the roots of the equation

$w^2=\frac13i-\frac14$.

The complex number $w$ is such that $kw^2+kww^*+iw-iw^*-1=0$, where $w^*$ is the complex conjugate of $w$ and $k$ is a real and non-zero constant.

(i) For $w=a+bi$ where $a$ and $b$ are real numbers, obtain an expression for $b$ in terms of $a$ and $k$. Explain why $w$ is either purely real or purely imaginary.

(ii) Using your result in part (i), or otherwise, find the real roots of the equation $2w^2+2ww^*+iw-iw^*-1=0$.

Solve the equation $zz^*-2z+2z^*=5-4i$.

Solve the following simultaneous equations:

(a) $4iz-3w=1+5i$ and $2z+(1+i)w=2+6i$,

(b) $iz+2w=1$ and $4z+(3-i)w^*=-6$,

giving $z$ and $w$ in the form $a+bi$ where $a$ and $b$ are real.