Let $\alpha$ and $\beta$ be the two solutions of the quadratic equation $x ^ { 2 } + \sqrt { 3 } x + 1 = 0$, where $0 < \arg \alpha < \arg \beta < 2 \pi$. Consider the complex numbers $z$ satisfying the following three conditions:
$$\begin{cases} \arg \dfrac { \alpha - z } { \beta - z } = \dfrac { \pi } { 2 } & \ldots\ldots\ldots (1)\\ ( 1 + i ) z + ( 1 - i ) \bar { z } + k = 0 & \ldots\ldots\ldots (2)\\ \dfrac { \pi } { 2 } < \arg z < \pi , & \ldots\ldots\ldots (3) \end{cases}$$
where $k$ is a real number.
Let us denote the points on the complex number plane which express $\alpha$, $\beta$ and $z$ by $\mathrm{ A }$, $\mathrm{ B }$ and P.
(1) The arguments of $\alpha$ and $\beta$ are
$$\arg \alpha = \frac { \mathbf { A } } { \mathbf { B } } \pi \quad \text{ and } \quad \arg \beta = \frac { \mathbf { C } } { \mathbf { D } } \pi .$$
(2) For each of $\mathbf { E }$ $\sim$ $\mathbf { Q }$ in the following sentences, choose the correct answer from among (0) $\sim$ (9) below.
Since $\mathbf { E } = \dfrac { \pi } { 2 }$ from (1), the point P is located on the circumference of the circle with the center $-\dfrac{ \sqrt{\mathbf{F}} }{ \mathbf{G} }$ and the radius $\dfrac { \mathbf { H } } { \mathbf { I } }$.
On the other hand, from (2), the point P is on the straight line which has the slope $\mathbf{J}$ and passes through a certain point.
From these, we see that when $n$ is the number of complex numbers $z$ which simultaneously satisfy (1), (2) and (3), the maximum value of $n$ is $\mathbf { M }$, and in this case the range of values of $k$ is
$$\mathbf { N } + \sqrt { \mathbf { O } } < k < \sqrt { \mathbf { P } } + \sqrt { \mathbf { Q } }$$
where $\mathbf { P } < \mathbf { Q }$.
(0) 0
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5 (6) 6 (7) $\angle \mathrm{ PAB }$ (8) $\angle \mathrm{ PBA }$ (9) $\angle \mathrm{ APB }$

Let $z$ be a complex number satisfying $| z | = 2$. In the complex number plane with the origin O, let A and B be the points representing $1 + z$ and $1 - \frac { 1 } { 2 } z$, respectively.
First of all, we can express the complex number $z$ as
$$z = \mathbf { M } ( \cos \theta + i \sin \theta ) \quad ( - \pi \leqq \theta < \pi ) .$$
(1) If $z$ is not a real number, then the area $S$ of the triangle OAB is $S = \mathbf { N }$. For $\mathbf{N}$, choose the correct answer from among (0) $\sim$ (8) below.
Hence, when $\theta = \pm \frac { \mathbf { O } } { \mathbf { P } } \pi$, $S$ is maximized.
(0) $\frac { 1 } { 2 } \left| \sin \left( \theta + \frac { 1 } { 3 } \pi \right) \right|$ (1) $\frac { 1 } { 2 } | \sin \theta |$ (2) $\frac { 1 } { 2 } \left| \sin \left( \theta - \frac { 1 } { 3 } \pi \right) \right|$ (3) $\left| \sin \left( \theta + \frac { 1 } { 3 } \pi \right) \right|$ (4) $| \sin \theta |$ (5) $\left| \sin \left( \theta - \frac { 1 } { 3 } \pi \right) \right|$ (6) $\frac { 3 } { 2 } \left| \sin \left( \theta + \frac { 1 } { 3 } \pi \right) \right|$ (7) $\frac { 3 } { 2 } | \sin \theta |$ (8) $\frac { 3 } { 2 } \left| \sin \left( \theta - \frac { 1 } { 3 } \pi \right) \right|$
(2) When the triangle OAB is an isosceles triangle where $\mathrm { OA } = \mathrm { OB }$, we see that
$$| 1 + z | = \left| 1 - \frac { 1 } { 2 } z \right| = \sqrt { \mathbf { Q } }$$
and
$$\arg ( 1 + z ) = \pm \frac { \mathbf { R } } { \mathbf { S } } \pi , \quad \arg \left( 1 - \frac { 1 } { 2 } z \right) = \mp \frac { \mathbf { T } } { \mathbf{U} } \pi ,$$
where the right-hand sides of the equations are of opposite signs, and where $- \pi \leqq \arg ( 1 + z ) < \pi$ and $- \pi \leqq \arg \left( 1 - \frac { 1 } { 2 } z \right) < \pi$.

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
$$| \alpha | : | \beta | : | \gamma | = \sqrt { \mathbf { W } } : \sqrt { \mathbf { X } } : \sqrt { \mathbf { Y } } .$$

6. If $\Gamma = \{z \mid z \text{ is a complex number and } |z - 1| = 1\}$, which of the following points lie on the graph $\Omega = \{w \mid w = iz, z \in \Gamma\}$?
(1) $2i$
(2) $-2i$
(3) $1 + i$
(4) $1 - i$
(5) $-1 + i$

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)

Problem 3

Answer the following questions. Here, for any complex value $z , \bar { z }$ is the complex conjugate of $z$, arg $z$ is the argument of $z , | z |$ is the absolute value of $z$, and $i$ is the imaginary unit.
I. Sketch the region of $z$ on the complex plane that satisfies the following:
$$z \bar { z } + \sqrt { 2 } ( z + \bar { z } ) + 3 i ( z - \bar { z } ) + 2 \leq 0$$
II. Answer the following questions on the complex valued function $f ( z )$ below.
$$f ( z ) = \frac { z ^ { 2 } - 2 } { \left( z ^ { 2 } + 2 i \right) z ^ { 2 } }$$

1. Find all the poles of $f ( z )$ as well as the orders and residues at the poles.
2. By applying the residue theorem, find the value of the following integral $I _ { 1 }$. Here, the integration path $C$ is the circle on the complex plane in the counterclockwise direction which satisfies $| z + 1 | = 2$. $$I _ { 1 } = \oint _ { C } f ( z ) \mathrm { d } z$$

III. Answer the following questions.

1. Let $g ( z )$ be a complex valued function, which satisfies $$\lim _ { | z | \rightarrow \infty } g ( z ) = 0$$ for $0 \leq \arg z \leq \pi$. Let $C _ { R }$ be the semicircle, with radius $R$, in the upper half of the complex plane with the center at the origin. Show $$\lim _ { R \rightarrow \infty } \int _ { C _ { R } } e ^ { i a z } g ( z ) \mathrm { d } z = 0$$ where $a$ is a positive real number.
2. Find the value of the following integral, $I _ { 2 }$ : $$I _ { 2 } = \int _ { 0 } ^ { \infty } \frac { \sin x } { x } \mathrm {~d} x$$

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