Mathematics Extension 2

A complex number $z$ lies on the unit circle in the complex plane, as shown in the diagram. Which of the following complex numbers is equal to $\bar z$?

A. $-z$
B. $z^2$
C. $-z^3$
D. $z^4$

Which of the following statements about complex numbers is true?

A. For all real numbers $x,y,\theta$ with $x\ne0$, $\tan\theta=\frac yx\Rightarrow x+iy=re^{i\theta}$ for some real number $r$.
B. For all non-zero complex numbers $z_1,z_2$, if $\operatorname{Arg}(z_1)=\theta_1$ and $\operatorname{Arg}(z_2)=\theta_2$, then $\operatorname{Arg}(z_1z_2)=\theta_1+\theta_2$.
C. For all real $r_1,r_2,\theta_1,\theta_2$ with $r_1,r_2>0$, $r_1e^{i\theta_1}=r_2e^{i\theta_2}\Rightarrow r_1=r_2$ and $\theta_1=\theta_2$.
D. For all real $x,y,r,\theta$ with $r>0$ and $x\ne0$, $x+iy=re^{i\theta}\Rightarrow\theta=\arctan(y/x)$.

Sketch the intersection of the following:

Use De Moivre’s Theorem to solve the equation

$z_1$ and $z_2$ are two complex numbers representing the two points $A$ and $B$ in the Argand diagram. $z_3$ is a complex number representing the point $C$ such that $|AB|:|AC|=1:4$. $z_4$ is a complex number representing the point $D$, such that $|OB|:|OD|=1:k$, for some constant $k$ and $O$ is the origin. The points $A$, $B$ and $C$ are collinear.

Given that $z_2-z_1$ and $z_4-z_3$ are perpendicular, prove that

where $\theta$ is the angle between $z_1$ and $z_2$.

Let $R$ be the region in the complex plane defined by $1<\operatorname{Re}(z)\le3$ and $\frac{\pi}{6}\le\operatorname{Arg}(z)<\frac{\pi}{3}$. Which diagram best represents $R$?

It is known that a particular complex number $z$ is NOT a real number. Which of the following could be true for this number $z$?

A. $\bar z=iz$

B. $\bar z=|z^2|$

C. $\operatorname{Re}(iz)=\operatorname{Im}(z)$

D. $\operatorname{Arg}(z^3)=\operatorname{Arg}(z)$

If the complex number $z$ has $|z|>1$ and $\frac\pi2<\operatorname{Arg}(z)<\pi$, and $\overrightarrow{OA}=z$, then if the two square roots of $z$ are $\overrightarrow{OP}$ and $\overrightarrow{OQ}$, $P$ and $Q$ could be at:

Express $-1+i\sqrt3$ and $1+i$ in modulus/argument form.

Hence find the exact value of $\tan\frac{5\pi}{12}$.

If $z=-\sqrt3+i$ and $w=1+i$, find $\operatorname{Arg}(zw)$ in terms of $\pi$.

Find the two square roots of $-i$, giving the answers in the form $x+iy$, where $x$ and $y$ are real numbers.

Consider the complex numbers $w=-1+4i$ and $z=2-i$. Evaluate $|w|$.