1. Solve in the set $\mathbb { C }$ of complex numbers the equation (E) with unknown $z$ : $$z ^ { 2 } - 8 z + 64 = 0$$
The complex plane is equipped with a direct orthonormal reference frame $( \mathrm { O } ; \vec { u } , \vec { v } )$.
2. We consider the points $\mathrm { A } , \mathrm { B }$ and C with affixes respectively $a = 4 + 4 \mathrm { i } \sqrt { 3 }$, $b = 4 - 4 \mathrm { i } \sqrt { 3 }$ and $c = 8 \mathrm { i }$. a. Calculate the modulus and an argument of the number $a$. b. Give the exponential form of the numbers $a$ and $b$. c. Show that the points $\mathrm { A } , \mathrm { B }$ and C lie on the same circle with center O whose radius will be determined. d. Place the points $\mathrm { A } , \mathrm { B }$ and C in the reference frame ( $\mathrm { O } ; \vec { u } , \vec { v }$ ).
For the rest of the exercise, you may use the figure from question 2. d. completed as the questions progress.
3. We consider the points $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ and $\mathrm { C } ^ { \prime }$ with affixes respectively $a ^ { \prime } = a \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } } , b ^ { \prime } = b \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } }$ and $c ^ { \prime } = c \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } }$. a. Show that $b ^ { \prime } = 8$. b. Calculate the modulus and an argument of the number $a ^ { \prime }$.
For the rest we admit that $a ^ { \prime } = - 4 + 4 \mathrm { i } \sqrt { 3 }$ and $c ^ { \prime } = - 4 \sqrt { 3 } + 4 \mathrm { i }$.
4. We admit that if $M$ and $N$ are two points in the plane with affixes respectively $m$ and $n$ then the midpoint $I$ of the segment $[ M N ]$ has affix $\frac { m + n } { 2 }$ and the length $M N$ is equal to $| n - m |$. a. We denote $r , s$ and $t$ the affixes of the midpoints respectively $\mathrm { R } , \mathrm { S }$ and T of the segments $\left[ \mathrm { A } ^ { \prime } \mathrm { B } \right] , \left[ \mathrm { B } ^ { \prime } \mathrm { C } \right]$ and $\left[ \mathrm { C } ^ { \prime } \mathrm { A } \right]$. Calculate $r$ and $s$. We admit that $t = 2 - 2 \sqrt { 3 } + \mathrm { i } ( 2 + 2 \sqrt { 3 } )$. b. What conjecture can be made about the nature of triangle RST? Justify this result.

We want to model in the plane the shell of a nautilus using a broken line in the form of a spiral. We are interested in the area delimited by this line.
We equip the plane with a direct orthonormal coordinate system $(O; \vec{u}; \vec{v})$. Let $n$ be an integer greater than or equal to 2. For all integer $k$ ranging from 0 to $n$, we define the complex numbers $z_k = \left(1 + \dfrac{k}{n}\right) \mathrm{e}^{\mathrm{i}\frac{2k\pi}{n}}$ and we denote by $M_k$ the point with affix $z_k$. In this model, the perimeter of the nautilus is the broken line connecting all the points $M_k$ with $0 \leqslant k \leqslant n$.
Part A: Broken line formed from seven points
In this part, we assume that $n = 6$. Thus, for $0 \leqslant k \leqslant 6$, we have $z_k = \left(1 + \dfrac{k}{6}\right) \mathrm{e}^{\mathrm{i}\frac{2k\pi}{6}}$.

1. Determine the algebraic form of $z_1$.
2. Verify that $z_0$ and $z_6$ are integers that you will determine.
3. Calculate the length of the altitude from $M_1$ in the triangle $OM_0M_1$ then establish that the area of this triangle is equal to $\dfrac{7\sqrt{3}}{24}$.

Part B: Broken line formed from $n+1$ points
In this part, $n$ is an integer greater than or equal to 2.

1. For all integer $k$ such that $0 \leqslant k \leqslant n$, determine the length $OM_k$.
2. For $k$ an integer such that $0 \leqslant k \leqslant n-1$, determine a measure of the angles $(\vec{u}; \overrightarrow{OM_k})$ and $(\vec{u}; \overrightarrow{OM_{k+1}})$. Deduce a measure of the angle $(\overrightarrow{OM_k}; \overrightarrow{OM_{k+1}})$.
3. For $k$ an integer such that $0 \leqslant k \leqslant n-1$, calculate the area of the triangle $OM_kM_{k+1}$ as a function of $n$ and $k$.

For any integer $k$, let $\alpha _ { k } = \cos \left( \frac { k \pi } { 7 } \right) + i \sin \left( \frac { k \pi } { 7 } \right)$, where $i = \sqrt { - 1 }$. The value of the expression $\frac { \sum _ { k = 1 } ^ { 12 } \left| \alpha _ { k + 1 } - \alpha _ { k } \right| } { \sum _ { k = 1 } ^ { 3 } \left| \alpha _ { 4 k - 1 } - \alpha _ { 4 k - 2 } \right| }$ is

If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg\left(\frac{1+z}{1+\bar{z}}\right)$ can be equal to (given $z \neq -1$)
(1) $\theta$
(2) $\pi - \theta$
(3) $-\theta$
(4) $\frac{\pi}{2} - \theta$

The least positive integer $n$ for which $\left( \frac { 1 + i \sqrt { 3 } } { 1 - i \sqrt { 3 } } \right) ^ { n } = 1$ is
(1) 2
(2) 5
(3) 6
(4) 3

Let $z = \left( \frac { \sqrt { 3 } } { 2 } + \frac { i } { 2 } \right) ^ { 5 } + \left( \frac { \sqrt { 3 } } { 2 } - \frac { i } { 2 } \right) ^ { 5 }$. If $R ( z )$ and $I ( z )$ respectively denote the real and imaginary parts of $z$, then
(1) $I ( z ) = 0$
(2) $R ( z ) < 0$ and $I ( z ) > 0$
(3) $R ( z ) > 0$ and $I ( z ) > 0$
(4) $R ( z ) = - 3$

The value of $\left( \frac{1 + \sin\frac{2\pi}{9} + i\cos\frac{2\pi}{9}}{1 + \sin\frac{2\pi}{9} - i\cos\frac{2\pi}{9}} \right)^{3}$ is
(1) $\frac{1}{2}(1 - i\sqrt{3})$
(2) $\frac{1}{2}(\sqrt{3} - i)$
(3) $-\frac{1}{2}(\sqrt{3} - i)$
(4) $-\frac{1}{2}(1 - i\sqrt{3})$

The value of $\left(\frac{-1+i\sqrt{3}}{1-i}\right)^{30}$ is:
(1) $6^5$
(2) $2^{15}\mathrm{i}$
(3) $-2^{15}$
(4) $-2^{15}\mathrm{i}$

Let $p , \quad q \in \mathbb { R }$ and $( 1 - \sqrt { 3 } i ) ^ { 200 } = 2 ^ { 199 } ( p + i q ) , i = \sqrt { - 1 }$. Then, $p + q + q ^ { 2 }$ and $p - q + q ^ { 2 }$ are roots of the equation.
(1) $x ^ { 2 } + 4 x - 1 = 0$
(2) $x ^ { 2 } - 4 x + 1 = 0$
(3) $x ^ { 2 } + 4 x + 1 = 0$
(4) $x ^ { 2 } - 4 x - 1 = 0$

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 $w_3 w_4 = \mathbf{XY}$.

$$z = 1 + i\sqrt{3}$$
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) $4\left(\cos\frac{\pi}{3} - i\sin\frac{\pi}{3}\right)$

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) $3$

$$\frac { 1 } { z } = \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$$
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) $\frac { 1 + i } { 2 }$

$$\alpha = \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 }$$
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) $\cos \pi + \mathrm { i } \sin \pi$