We have a triangle ABC on the complex plane whose vertices are the three points $\mathrm { A } ( \alpha )$, $\mathrm { B } ( \beta )$ and $\mathrm { C } ( \gamma )$ that satisfy
$$\frac { \gamma - \alpha } { \beta - \alpha } = 1 - i$$
(In the following, the range of an argument $\theta$ is $0 \leqq \theta < 2 \pi$.)
(1) When we express the complex number $\frac { \gamma - \alpha } { \beta - \alpha }$ in polar form, we have
$$\frac { \gamma - \alpha } { \beta - \alpha } = \sqrt { \mathbf { N } } \left( \cos \frac { \mathbf { O } } { \mathbf { P } } \pi + i \sin \frac { \mathbf { O } } { \mathbf { P } } \pi \right) .$$
Hence we see that point C is the point resulting from rotating point B by $\frac { \square \mathbf { Q } } { \mathbf{R} } \pi$ around point A and then changing its distance from point A to its distance multiplied by $\sqrt { \mathbf { S } }$. From this we also see that the absolute value and the argument of the complex number $w = \frac { \gamma - \beta } { \alpha - \beta }$ are
$$| w | = \mathbf { T } \quad \text { and } \quad \arg w = \frac { \mathbf { U } } { \mathbf { 4 } } \pi .$$
(2) If $\alpha + \beta + \gamma = 0$, then we have that
$$| \alpha | : | \beta | : | \gamma | = \sqrt { \mathbf { W } } : \sqrt { \mathbf { X } } : \sqrt { \mathbf { Y } } .$$

In the complex plane, let $O$ be the origin, and let $A$ and $B$ represent points with coordinates corresponding to complex numbers $z$ and $z + 1$ respectively. Given that both points $A$ and $B$ lie on the unit circle centered at $O$, select the correct options.
(1) Line $AB$ is parallel to the real axis
(2) $\triangle OAB$ is a right triangle
(3) Point $A$ is in the second quadrant
(4) $z^{3} = 1$
(5) The point with coordinate $1 + \frac{1}{z}$ also lies on the same unit circle

In the complex plane, a complex number $z$ is in the first quadrant and satisfies $|z| = 1$ and $\left|\frac{-3+4i}{5} - z^3\right| = \left|\frac{-3+4i}{5} - z\right|$, where $i = \sqrt{-1}$. If the real part of $z$ is $a$ and the imaginary part is $b$, then $a = \dfrac{\sqrt{\phantom{0}}}{\sqrt{\phantom{0}}}$ and $b = \dfrac{\sqrt{\phantom{0}}}{\sqrt{\phantom{0}}}$. (Express in simplest radical form)

Consider the complex function $M(z) = \frac{mz}{mz - z + 1}$, where $m$ is a complex number such that $|m| = 1$ and $m \neq 1$.

1. Find all fixed points of $M(z)$ which satisfy $M(z) = z$.
2. Express the derivative of $M(z)$ at $z = 0$ by using $m$.
3. Find $m$ for which the circle $\left| z - \frac{1-i}{2} \right| = \frac{1}{\sqrt{2}}$ on the complex $z$ plane is mapped onto the real axis through $M(z)$.

Deduce the conditions for $z$ and, on the complex $z$ plane, draw the area of $z$ in which the imaginary part of the complex function $J(z) = e^{-i\alpha} z + e^{i\alpha} z^{-1}$ is positive. Here, $\alpha$ is a real number and $0 < \alpha < \pi/2$.

In the complex number plane $$|z-1| = |z+2|$$ Which of the following does this equation represent?
A) The line $x = 1$
B) The line $x = \frac{-1}{2}$
C) The line $x = 2$
D) The circle $(x-1)^{2} + y^{2} = 1$
E) The circle $x^{2} + (y+2)^{2} = 1$

For the complex number $z = a + ib$ whose distance to the number 1 is 2 units and whose distance to the number i is 3 units, what is the difference $a - b$?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 7 } { 2 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 7 } { 3 }$

Let z be a complex number such that
$$\begin{aligned} & | z - 1 | = | z - 2 | \\ & | z | = \sqrt { 3 } \end{aligned}$$
What is the value of $| z - 3 |$?
A) 2
B) $\sqrt { 2 }$
C) $\sqrt { 3 }$
D) $1 + \sqrt { 2 }$
E) $\sqrt { 3 } - 1$

Below, line segments $[ A B ]$ and $[ C D ]$ are given in the complex number plane.
For each complex number z taken on these line segments, the number $\mathrm { w } = \mathrm { z } \cdot \overline { \mathrm { z } }$ is defined.
Accordingly, in which of the following are the minimum and maximum values that w can take given respectively?
A) 5 and 20
B) 5 and 25
C) 5 and 30
D) 10 and 20
E) 10 and 25