Mathematics Extension 2

A shaded region on a complex plane is shown. Which relation best describes the region shaded on the complex plane?

A. $|z-i|>2|z-1|$
B. $|z-i|<2|z-1|$
C. $|z-1|>2|z-i|$
D. $|z-1|<2|z-i|$

Sketch the intersection of the following:

$z_1$ 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.

Find $z_3$ in terms of $z_1$ and $z_2$.

$z_1$ 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 $z_2$.

Which of the following is the locus of $z$ such that
$\arg(z+3i)-\arg(z-i)=\frac\pi2?$

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:

In the diagram, the vectors $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ represent the complex numbers $z_1$ and $z_2$ respectively. If
$\frac{z_2}{z_1}=\sqrt3\,i,$
what is the value of
$\left|\frac{z_1+z_2}{z_2-z_1}\right|?$

A. $1$
B. $\sqrt3$
C. $\frac1{\sqrt3}$ D.

In the Argand diagram below, $\overrightarrow{OA}=z_1$, $\overrightarrow{OB}=z_2$, $OA=OB$ and $\overrightarrow{AC}=\overrightarrow{CB}$. Express $\overrightarrow{OC}$ in terms of $z_1$ and $z_2$.

Given the complex number $z=e^{i\theta}$, show that
$w=\frac{\bar z^2-1}{\bar z^2+1}$
is purely imaginary.

On the Argand diagram, $z$ and $zw$ are represented by $A$ and $B$. Given $z=re^{i\theta}$ and $w=e^{i\pi/3}$, where $r>0$, explain why $OAB$ is an equilateral triangle.

The complex number $z$ satisfies
$\left|z-\frac4z\right|=2.$

Using the triangle inequality, or otherwise, show that
$|z|\le\sqrt5+1.$

Find all the complex numbers $z_1,z_2,z_3$ that satisfy the following three conditions simultaneously:
$|z_1|=|z_2|=|z_3|,$
$z_1+z_2+z_3=1,$
$z_1z_2z_3=1.$

A square in the Argand plane has vertices $5+5i$, $5-5i$, $-5-5i$ and $-5+5i$. The complex numbers $z_A=5+i$, $z_B$ and $z_C$ lie on the square and form the vertices of an equilateral triangle, as shown in the diagram.

Find the exact value of the complex number $z_B$.

Consider the two non-zero complex numbers $z$ and $w$ as vectors. Which expression is the projection of $z$ onto $w$?

A. $\frac{\operatorname{Re}(zw)}{|w|}w$
B. $\left|\frac zw\right|w$
C. $\operatorname{Re}\left(\frac zw\right)w$
D. $\frac{\operatorname{Re}(z)}{|w|}w$

The diagram shows the complex number $z$ on the Argand diagram.

Which of the following diagrams best shows the position of $\dfrac{z^2}{|z|}$?

What is the maximum value of $|e^{i\theta}-2|+|e^{i\theta}+2|$ for $0\leq\theta\leq2\pi$?

A. $\sqrt5$
B. $4$
C. $2\sqrt5$ D.

Let $z_1$ be a complex number and let
$z_2=e^{i\pi/3}z_1.$
The diagram shows points $A$ and $B$, which represent $z_1$ and $z_2$, respectively, in the Argand plane.

(i) Explain why triangle $OAB$ is an equilateral triangle.

Prove that $z_1^2+z_2^2=z_1z_2.$