(Course 2) Q2 Let $\alpha , \beta$ and $\gamma$ be three complex numbers representing three different points A, B and C on a complex plane. Also, $\alpha , \beta$ and $\gamma$ satisfy
$$\begin{aligned} & ( \gamma - \alpha ) ^ { 2 } + ( \gamma - \alpha ) ( \beta - \alpha ) + ( \beta - \alpha ) ^ { 2 } = 0 \quad \cdots (1)\\ & | \beta - 2 \alpha + \gamma | = 3 \quad \cdots (2) \end{aligned}$$
We are to find the area of the triangle ABC.
Since from (1)
$$\frac { \gamma - \alpha } { \beta - \alpha } = \frac { - \mathbf { M } \pm \sqrt { \mathbf { N } } i } { \mathbf { O } } ,$$
we have
$$\left| \frac { \gamma - \alpha } { \beta - \alpha } \right| = \mathbf { P } , \quad \arg \frac { \gamma - \alpha } { \beta - \alpha } = \pm \frac { \mathbf { Q } } { \mathbf { R } } \pi ,$$
where $- \pi < \arg \frac { \gamma - \alpha } { \beta - \alpha } < \pi$. Also, since
$$\beta - 2 \alpha + \gamma = ( \beta - \alpha ) \cdot \frac { \mathbf { S } \pm \sqrt { \mathbf { T } } } { \mathbf { U } } ,$$
we have from (2) that
$$| \beta - \alpha | = \mathbf { V } .$$

Answer the following questions.
(1) When we express the complex number $8 + 8\sqrt{3}i$ in polar form, we have $$\mathbf{MN}\left(\cos\frac{\pi}{\mathbf{O}} + i\sin\frac{\pi}{\mathbf{P}}\right).$$
(2) Consider the complex numbers $z$ that satisfy $z^4 = 8 + 8\sqrt{3}i$ in the range $0 \leqq \arg z < 2\pi$.
We see that $|z| = \mathbf{Q}$. There are 4 such complex numbers $z$. When these are denoted by $z_1, z_2, z_3, z_4$ in the ascending order of their arguments, we have $$\arg\frac{z_1 z_2 z_3}{z_4} = \frac{\pi}{\mathbf{R}}.$$
(3) Consider the complex numbers $w$ that satisfy $w^8 - 16w^4 + 256 = 0$ in the range $0 \leqq \arg w < 2\pi$. There are 8 such complex numbers $w$. Let us denote them by $w_1, w_2, w_3, w_4, w_5, w_6, w_7, w_8$ in the ascending order of their arguments. Then four of these coincide with numbers $z_1, z_2, z_3, z_4$ in (2). That is, $$w_{\mathbf{S}} = z_1, \quad w_{\mathbf{T}} = z_2, \quad w_{\mathbf{U}} = z_3, \quad w_{\mathbf{V}} = z_4.$$ Also, we have that $w_1 w_8 = \mathbf{W}$ and $w_3 w_4 = \mathbf{XY}$.

1. Suppose a quadratic equation with integer coefficients $a x ^ { 2 } + b x + c = 0$ has one root equal to $4 + 3 i$. If the two roots of this equation and the origin are marked on the complex plane, the area of the triangle formed by these three points is
(1) 5
(2) 6
(3) 12
(4) 16
(5) 24

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

Let $z _ { 1 }$、$z _ { 2 }$、$z _ { 3 }$、$z _ { 4 }$ be four distinct complex numbers whose corresponding points on the complex plane can be connected in order to form a parallelogram. Which of the following options must be real numbers?
(1) $\left( z _ { 1 } - z _ { 3 } \right) \left( z _ { 2 } - z _ { 4 } \right)$
(2) $z _ { 1 } - z _ { 2 } + z _ { 3 } - z _ { 4 }$
(3) $z _ { 1 } + z _ { 2 } + z _ { 3 } + z _ { 4 }$
(4) $\frac { z _ { 1 } - z _ { 2 } } { z _ { 3 } - z _ { 4 } }$
(5) $\left( \frac { z _ { 2 } - z _ { 4 } } { z _ { 1 } - z _ { 3 } } \right) ^ { 2 }$

On the complex plane, let $\bar { z }$ denote the complex conjugate of complex number $z$, and $i = \sqrt { - 1 }$. Select the correct options.
(1) If $z = 2 i$, then $z ^ { 3 } = 4 \bar { i } \bar { z }$
(2) If non-zero complex number $\alpha$ satisfies $\alpha ^ { 3 } = 4 i \bar { \alpha }$, then $| \alpha | = 2$
(3) If non-zero complex number $\alpha$ satisfies $\alpha ^ { 3 } = 4 i \bar { \alpha }$ and let $\beta = i \alpha$, then $\beta ^ { 3 } = 4 i \bar { \beta }$
(4) Among all non-zero complex numbers $z$ satisfying $z ^ { 3 } = 4 i \bar { z }$, the minimum possible value of the principal argument is $\frac { \pi } { 6 }$
(5) There are exactly 3 distinct non-zero complex numbers $z$ satisfying $z ^ { 3 } = 4 i \bar { z }$

Let the complex number $z$ have a nonzero imaginary part and $|z| = 2$. It is known that on the complex plane, $1, z, z^{3}$ are collinear. Select the correct options.
(1) $z \cdot \bar{z} = 2$
(2) The imaginary part of $\frac{z^{3} - z}{z - 1}$ is 0
(3) The real part of $z$ is $-\frac{1}{2}$
(4) $z$ satisfies $z^{2} - z + 4 = 0$
(5) On the complex plane, $-2, z, z^{2}$ are collinear

Let $i = \sqrt { - 1 }$. Given that the complex number $\frac { 1 - 7 i } { - 1 + i } = a + b i$, where $a, b$ are real numbers. Then $a =$ (10–1)(10–2), $b =$ (10–3).

Let $z _ { n }$ and $w _ { n } ( n = 0,1,2 , \ldots )$ be complex numbers. Consider a bag that contains two red cards and one white card. First, take one card from the bag and return it to the bag. $z _ { k + 1 } ( k = 0,1,2 , \ldots )$ is generated in the following manner based on the color of the card taken.
$$z _ { k + 1 } = \begin{cases} i z _ { k } & \text { if a red card was taken, } \\ - i z _ { k } & \text { if a white card was taken. } \end{cases}$$
Next, take one card from the bag again and return it to the bag. $w _ { k + 1 }$ is generated in the following manner based on the color of the card taken.
$$w _ { k + 1 } = \begin{cases} - i w _ { k } & \text { if a red card was taken, } \\ i w _ { k } & \text { if a white card was taken. } \end{cases}$$
Here, each card is independently taken with equal probability. The initial state is $z _ { 0 } = 1$ and $w _ { 0 } = 1$. Thus, $z _ { n } , w _ { n }$ are the values after repeating the series of the above two operations $n$ times starting from the state of $z _ { 0 } = 1$ and $w _ { 0 } = 1$. Here, $i$ is the imaginary unit.
Answer the following questions.
(1) Show that $\operatorname { Re } \left( z _ { n } \right) = 0$ if $n$ is odd, and that $\operatorname { Im } \left( z _ { n } \right) = 0$ if $n$ is even. Here, $\operatorname { Re } ( z )$ and $\operatorname { Im } ( z )$ represent the real part and the imaginary part of $z$ respectively.
(2) Let $P _ { n }$ be the probability of $z _ { n } = 1$, and $Q _ { n }$ be the probability of $z _ { n } = i$. Find recurrence equations for $P _ { n }$ and $Q _ { n }$.
(3) Find the probabilities of $z _ { n } = 1 , z _ { n } = i , z _ { n } = - 1$, and $z _ { n } = - i$ respectively.
(4) Show that the expected value of $z _ { n }$ is $( i / 3 ) ^ { n }$.
(5) Find the probability of $z _ { n } = w _ { n }$.
(6) Find the expected value of $z _ { n } + w _ { n }$.
(7) Find the expected value of $z _ { n } w _ { n }$.

2
For complex numbers $a, b, c$, consider the polynomial $f(z) = az^2 + bz + c$. Let $i$ be the imaginary unit.

1. [(1)] Let $\alpha$, $\beta$, $\gamma$ be complex numbers. When $f(0) = \alpha$, $f(1) = \beta$, $f(i) = \gamma$ hold, express $a$, $b$, $c$ in terms of $\alpha$, $\beta$, $\gamma$ respectively.
   
2. [(2)] When $f(0)$, $f(1)$, $f(i)$ are all real numbers satisfying $1 \leq f(0),\, f(1),\, f(i) \leq 2$, illustrate the range of values that $f(2)$ can take on the complex plane.

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Let $f(x) = \dfrac{x}{x^2+3}$, and let $C$ be the graph of $y = f(x)$. Let $l: y = g(x)$ be the tangent line to $C$ at the point $\mathrm{A}(1,\ f(1))$.

1. [(1)] Show that there exists exactly one point on $C$ that is common to $C$ and $l$ and is different from $\mathrm{A}$, and find the $x$-coordinate of that point.
   
2. [(2)] Let $\alpha$ be the $x$-coordinate of the common point found in (1). Compute the definite integral $\displaystyle\int_{\alpha}^{1} \{f(x) - g(x)\}^2\, dx$.

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Let $\bar{z}$ denote the conjugate of $z$. For the complex number $z = 2 + i$, $$\frac{z}{\bar{z}-1}$$ Which of the following is this expression equal to?
A) $\frac{1}{2} + \frac{3}{2}i$
B) $\frac{2}{3} - \frac{3}{2}i$
C) $1 + 3i$
D) $2 - 3i$
E) $3 + i$

$$z = 1 + i\sqrt{3}$$
Which of the following is this complex number equal to?
A) $2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)$
B) $2\left(\cos\frac{\pi}{6} - i\sin\frac{\pi}{6}\right)$
C) $2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$
D) $4\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$
E) $4\left(\cos\frac{\pi}{3} - i\sin\frac{\pi}{3}\right)$

For complex numbers $z = a + b i ( b \neq 0 )$ and $w = c + d i$, if the sum $\mathbf { Z } + \mathbf { W }$ and the product $\mathbf { Z } \cdot \mathbf { W }$ are both real numbers, then I. $z$ and $w$ are conjugates of each other. II. $\mathrm { z } - \mathrm { w }$ is real. III. $z ^ { 2 } + w ^ { 2 }$ is real. Which of the following statements are true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

The function f on the set of complex numbers is
$$f ( z ) = \sum _ { k = 0 } ^ { 101 } z ^ { k }$$
is defined in this form. Accordingly, what is the value of f(i)?
A) $1 + i$
B) $1 - \mathrm { i }$
C) i
D) - i
E) 1

If $\bar { z }$ denotes the conjugate of $z$, what is the non-zero complex number $z$ that satisfies the equation $z ^ { 2 } = \bar { z }$ and whose argument is between $\frac { \pi } { 2 }$ and $\pi$?
A) $\frac { - 1 } { 2 } + ( \sqrt { 3 } ) \mathrm { i }$
B) $\frac { - 1 } { 2 } + \left( \frac { \sqrt { 3 } } { 2 } \right) \mathrm { i }$
C) $\frac { - \sqrt { 2 } } { 2 } + \left( \frac { 1 } { 2 } \right) \mathrm { i }$
D) $\frac { - \sqrt { 2 } } { 2 } + \left( \frac { \sqrt { 2 } } { 2 } \right) i$
E) $\frac { - \sqrt { 3 } } { 2 } + \left( \frac { 1 } { 2 } \right) \mathrm { i }$

On the set of complex numbers
$$f ( z ) = 1 - 2 z ^ { 6 }$$
a function is defined. For $z _ { 0 } = \cos \left( \frac { \pi } { 3 } \right) + i \sin \left( \frac { \pi } { 3 } \right)$, what is $f \left( z _ { 0 } \right)$?
A) $1 + i$
B) $2i$
C) $1 - i$
D) $-1$
E) $3$

$$( | z | + z ) \cdot ( | z | - \bar { z } ) = i$$
Which of the following is the imaginary part of the complex number z that satisfies the equation?
A) $\frac { 2 } { | z | }$
B) $\frac { 1 } { | z | }$
C) $\frac { - | z | } { 2 }$
D) $\frac { 1 } { 2 | z | }$
E) $- | z |$

$z$ is a complex number, $\operatorname { Im } ( z ) \neq 0$ and $z ^ { 3 } = - 1$. Given this,
$$( z - 1 ) ^ { 10 }$$
Which of the following is this expression equal to?
A) $z + 1$
B) $z - 1$
C) $z$
D) $- z$
E) $- z - 1$

$$\frac { | z | ^ { 2 } + z } { \bar { z } } = z + i$$
Which of the following is the set of complex numbers z that satisfy this equality? (R is the set of real numbers.)
A) $\{ a + a i \mid a \in R , a \neq 0 \}$
B) $\{ a - a i \mid a \in R , a \neq 0 \}$
C) $\{ a + 2 a i \mid a \in R , a \neq 0 \}$
D) $\{ a - 2 a i \mid a \in R , a \neq 0 \}$
E) $\{ 2 a - a i \mid a \in R , a \neq 0 \}$

$$\frac { 1 } { z } = \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$$
Which of the following is the complex number z that satisfies this equation?
A) $\sqrt { 2 } ( 1 + i )$
B) $\sqrt { 2 } ( 1 - \mathrm { i } )$
C) $\frac { \sqrt { 2 } } { 2 } ( 1 + i )$
D) $\frac { \sqrt { 2 } } { 2 } ( 1 - \mathrm { i } )$
E) $\frac { 1 + i } { 2 }$

The functions $\mathrm { f } ( \mathrm { x } ) = \mathrm { x } + \mathrm { xi }$ and $\mathrm { g } ( \mathrm { x } ) = 2 \mathrm { x } - \mathrm { xi }$ are defined from the set of real numbers to the set of complex numbers and satisfy
$$f ( a ) + g ( b ) = 4 + 2 i$$
Accordingly, what is the sum $\mathbf { a } + \mathbf { b }$?
A) $\frac { 7 } { 2 }$
B) $\frac { 9 } { 2 }$
C) $\frac { 10 } { 3 }$
D) $\frac { 13 } { 3 }$
E) $\frac { 15 } { 4 }$

Let $z$ be a complex number and
$$z \cdot | \operatorname { Re } ( z ) | = - 4 + 3 i$$
Accordingly, what is $| \mathbf { z } |$?
A) $\frac { 5 } { 2 }$
B) $\frac { 7 } { 2 }$
C) $\frac { 9 } { 2 }$
D) $\frac { 8 } { 3 }$
E) $\frac { 10 } { 3 }$

$$\alpha = \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 }$$
Given that, which of the following is $\alpha ^ { 23 }$ equal to?
A) $\cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 }$
B) $\cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 }$
C) $\cos \frac { 4 \pi } { 3 } + i \sin \frac { 4 \pi } { 3 }$
D) $\cos \frac { 5 \pi } { 3 } + i \sin \frac { 5 \pi } { 3 }$
E) $\cos \pi + \mathrm { i } \sin \pi$

In the set of complex numbers, the result of the operation
$$( 3 - i ) ( 2 - i ) ( 1 + i ) ( 2 + i ) ( 3 + i )$$
is $\mathbf { a } + \mathbf { b i }$. What is the sum $a + b$?
A) 80
B) 84
C) 90
D) 96
E) 100

In the complex plane, the vertices of the quadrilateral formed by the roots of the equation
$$z ^ { 4 } = 16$$
have what area in square units?
A) 8
B) 12
C) 16
D) $4 \sqrt { 3 }$
E) $6 \sqrt { 2 }$