Exercise 2 (Candidates who have not followed the specialization course)
The complex plane is referred to the direct orthonormal frame $(\mathrm{O}, \vec{u}, \vec{v})$. Let $R$ be the rotation of the plane with centre $\Omega$, with affix $\omega$ and angle of measure $\theta$. The image by $R$ of a point in the plane is therefore defined as follows:

• $R(\Omega) = \Omega$
• for any point $M$ in the plane, distinct from $\Omega$, the image $M'$ of $M$ is defined by $$\Omega M' = \Omega M \text{ and } (\overrightarrow{\Omega M}, \overrightarrow{\Omega M'}) = \theta \quad [2\pi].$$

We recall that, for points $A$ and $B$ with affixes $a$ and $b$ respectively, $$AB = |b - a| \text{ and } (\vec{u}, \overrightarrow{AB}) = \arg(b - a) \quad [2\pi]$$

1. Show that the affixes $z$ and $z'$ of any point $M$ in the plane and its image $M'$ by the rotation $R$ are related by the relation $$z' - \omega = \mathrm{e}^{\mathrm{i}\theta}(z - \omega).$$
2. We consider the points I and B with affixes $z_{\mathrm{I}} = 1 + \mathrm{i}$ and $z_{\mathrm{B}} = 2 + 2\mathrm{i}$ respectively. Let $R$ be the rotation with centre B and angle of measure $\frac{\pi}{3}$. a. Give the complex form of $R$. b. Let A be the image of I by $R$. Calculate the affix $z_{\mathrm{A}}$ of A. c. Show that O, A and B lie on the same circle with centre I. Deduce that OAB is a right-angled triangle at A. Give a measure of the angle $(\overrightarrow{\mathrm{OA}}, \overrightarrow{\mathrm{OB}})$. d. Deduce a measure of the angle $(\vec{u}, \overrightarrow{\mathrm{OA}})$.
3. Let $T$ be the translation of vector $\overrightarrow{\mathrm{IO}}$. We set $\mathrm{A}' = T(\mathrm{A})$. a. Calculate the affix $z_{\mathrm{A}'}$ of $\mathrm{A}'$. b. What is the nature of the quadrilateral OIAA'? c. Show that $-\frac{\pi}{12}$ is an argument of $z_{\mathrm{A}'}$.

Exercise 2 (Candidates who have followed the specialization course)
We assume the following results are known:

• the composition of two plane similarities is a plane similarity;
• the inverse transformation of a plane similarity is a plane similarity;
• a plane similarity that leaves three non-collinear points of the plane invariant is the identity of the plane.

1. Let A, B and C be three non-collinear points in the plane and $s$ and $s'$ be two similarities of the plane such that $s(\mathrm{A}) = s'(\mathrm{A})$, $s(\mathrm{B}) = s'(\mathrm{B})$ and $s(\mathrm{C}) = s'(\mathrm{C})$. Show that $s = s'$.
2. The complex plane is referred to the orthonormal frame $(\mathrm{O}, \vec{u}, \vec{v})$. We are given the points A with affix $2$, E with affix $1 + \mathrm{i}$, F with affix $2 + \mathrm{i}$ and G with affix $3 + \mathrm{i}$. a. Calculate the lengths of the sides of the triangles OAG and OEF. Deduce that these triangles are similar. b. Show that OEF is the image of OAG by an indirect similarity $S$, by determining the complex form of $S$. c. Let $h$ be the homothety with centre O and ratio $\frac{1}{\sqrt{2}}$. We set $\mathrm{A}' = h(\mathrm{A})$ and $\mathrm{G}' = h(\mathrm{G})$, and we call I the midpoint of $[\mathrm{EA}']$. We denote by $\sigma$ the orthogonal symmetry with axis (OI). Show that $S = \sigma \circ h$.

Exercise 2 (For candidates who did not choose the mathematics speciality)
The complex plane is equipped with a direct orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$ (graphical unit: 4 cm). Let A be the point with affixe $z_{\mathrm{A}} = \mathrm{i}$ and B the point with affixe $z_{\mathrm{B}} = \mathrm{e}^{-\mathrm{i}\frac{5\pi}{6}}$.

1. Let $r$ be the rotation with centre O and angle $\frac{2\pi}{3}$. Let C denote the image of B by $r$. a. Determine a complex expression for $r$. b. Show that the affixe of C is $z_{\mathrm{C}} = \mathrm{e}^{-\mathrm{i}\frac{\pi}{6}}$. c. Write $z_{\mathrm{B}}$ and $z_{\mathrm{C}}$ in algebraic form. d. Plot the points A, B and C.
2. Let D be the centroid of points A, B and C with respective coefficients $2, -1$ and $2$. a. Show that the affixe of D is $z_{\mathrm{D}} = \frac{\sqrt{3}}{2} + \frac{1}{2}\mathrm{i}$. Plot point D. b. Show that A, B, C and D lie on the same circle.
3. Let $h$ be the homothety with centre A and ratio 2. Let E denote the image of D by $h$. a. Determine a complex expression for $h$. b. Show that the affixe of E is $z_{\mathrm{E}} = \sqrt{3}$. Plot point E.
4. a. Calculate the ratio $\frac{z_{\mathrm{D}} - z_{\mathrm{C}}}{z_{\mathrm{E}} - z_{\mathrm{C}}}$. Write the result in exponential form. b. Deduce the nature of triangle CDE.

For each of the four propositions below, indicate whether it is true or false and justify your chosen answer. One point is awarded for each correct answer with proper justification. An answer without justification is not taken into account. No answer is not penalized.

1. Proposition 1: In the plane with an orthonormal coordinate system, the set of points $M$ whose affix $z$ satisfies the equality $| z - \mathrm { i } | = | z + 1 |$ is a line.
2. Proposition 2: The complex number $( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 4 }$ is a real number.
3. Let ABCDEFGH be a cube. Proposition 3: The lines (EC) and (BG) are orthogonal.
4. Space is equipped with an orthonormal coordinate system ($\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$). Let the plane $\mathscr { P }$ with Cartesian equation $x + y + 3z + 4 = 0$. We denote S the point with coordinates $( 1 , - 2 , - 2 )$. Proposition 4: The line passing through S and perpendicular to the plane $\mathscr { P }$ has parametric representation $\left\{ \begin{array} { l } x = 2 + t \\ y = - 1 + t \\ z = 1 + 3 t \end{array} , t \in \mathbf { R } \right.$.

The complex plane is equipped with a direct orthonormal coordinate system $( \mathrm { O } , \vec { u } , \vec { v } )$. We denote by i the complex number such that $\mathrm { i } ^ { 2 } = - 1$. We consider the point A with affixe $z _ { \mathrm { A } } = 1$ and the point B with affixe $z _ { \mathrm { B } } = \mathrm { i }$. To any point $M$ with affixe $z _ { M } = x + \mathrm { i } y$, with $x$ and $y$ two real numbers such that $y \neq 0$, we associate the point $M ^ { \prime }$ with affixe $z _ { M ^ { \prime } } = - \mathrm { i } z _ { M }$. We denote by $I$ the midpoint of the segment [AM]. The purpose of the exercise is to show that for any point $M$ not belonging to (OA), the median $( \mathrm { O } I )$ of the triangle $\mathrm { OA} M$ is also a height of the triangle $\mathrm { OB } M ^ { \prime }$ (property 1) and that $\mathrm { B } M ^ { \prime } = 2 \mathrm { O } I$ (property 2).

1. In this question and only in this question, we take $z _ { M } = 2 \mathrm { e } ^ { - \mathrm { i } \frac { \pi } { 3 } }$. a. Determine the algebraic form of $z _ { M }$. b. Show that $z _ { M ^ { \prime } } = - \sqrt { 3 } - \mathrm { i }$. Determine the modulus and an argument of $z _ { M ^ { \prime } }$. c. Place the points $\mathrm { A } , \mathrm { B } , M , M ^ { \prime }$ and $I$ in the coordinate system $( \mathrm { O } , \vec { u } , \vec { v } )$ using 2 cm as the graphical unit. Draw the line $( \mathrm { O } I )$ and quickly verify properties 1 and 2 using the graph.
2. We return to the general case by taking $z _ { M } = x + \mathrm { i } y$ with $y \neq 0$. a. Determine the affixe of point $I$ as a function of $x$ and $y$. b. Determine the affixe of point $M ^ { \prime }$ as a function of $x$ and $y$. c. Write the coordinates of points $I$, B and $M ^ { \prime }$. d. Show that the line $( \mathrm { O } I )$ is a height of the triangle $\mathrm { OB } M ^ { \prime }$. e. Show that $\mathrm { B } M ^ { \prime } = 2 \mathrm { O } I$.

The four questions in this exercise are independent. For each question, a statement is proposed. Indicate whether each of them is true or false, by justifying the answer.
One point is awarded for each correct answer that is properly justified. An unjustified answer earns no points. An absence of an answer is not penalized.
In questions 1. and 2., the plane is referred to the direct orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$. We denote by $\mathbb{R}$ the set of real numbers.
Statement 1: The point with affix $(-1+i)^{10}$ is located on the imaginary axis.
Statement 2: In the set of complex numbers, the equation $$z - \bar{z} + 2 - 4\mathrm{i} = 0$$ admits a unique solution.
Statement 3: $$\ln\left(\sqrt{\mathrm{e}^{7}}\right) + \frac{\ln\left(\mathrm{e}^{9}\right)}{\ln\left(\mathrm{e}^{2}\right)} = \frac{\mathrm{e}^{\ln 2 + \ln 3}}{\mathrm{e}^{\ln 3 - \ln 4}}$$
Statement 4: $$\int_{0}^{\ln 3} \frac{\mathrm{e}^{x}}{\mathrm{e}^{x}+2}\,\mathrm{d}x = -\ln\left(\frac{3}{5}\right)$$
Statement 5: The equation $\ln(x-1) - \ln(x+2) = \ln 4$ admits a unique solution in $\mathbb{R}^{*}$.

For each of the four following statements, indicate whether it is true or false by justifying your answer.
One point is awarded for each correct answer with proper justification. An answer without justification is not taken into account. An absence of answer is not penalized.

1. In the plane with an orthonormal coordinate system, let $S$ denote the set of points $M$ whose affix $z$ satisfies the two conditions: $$|z - 1| = |z - \mathrm{i}| \quad \text{and} \quad |z - 3 - 2\mathrm{i}| \leqslant 2.$$ In the figure below, we have represented the circle with center at the point with coordinates $(3;2)$ and radius 2, and the line with equation $y = x$. This line intersects the circle at two points A and B.
   Statement 1: the set $S$ is the segment $[AB]$.
   
2. Statement 2: the complex number $(\sqrt{3} + \mathrm{i})^{1515}$ is a real number.
   
3. For questions 3 and 4, consider the points $\mathrm{E}(2; 1; -3)$, $\mathrm{F}(1; -1; 2)$ and $\mathrm{G}(-1; 3; 1)$ whose coordinates are defined in an orthonormal coordinate system of space.
   Statement 3: a parametric representation of the line $(EF)$ is given by: $$\left\{\begin{array}{rlr} x & = & 2t \\ y & = & -3 + 4t, \quad t \in \mathbb{R} \\ z & = 7 - 10t \end{array}\right.$$
   
4. Statement 4: a measure in degrees of the geometric angle $\widehat{\mathrm{FEG}}$, rounded to the nearest degree, is $50°$.

We denote by $\mathbb{C}$ the set of complex numbers. In the complex plane equipped with an orthonormal coordinate system $(O; \vec{u}, \vec{v})$ we have placed a point $M$ with affixe $z$ belonging to $\mathbb{C}$, then the point $R$ intersection of the circle with center $O$ passing through $M$ and the half-axis $[O; \vec{u})$.
Part A

1. Express the affixe of point $R$ as a function of $z$.
2. Let the point $M'$ with affixe $z'$ defined by $$z' = \frac{1}{2}\left(\frac{z + |z|}{2}\right).$$ Reproduce the figure on the answer sheet and construct the point $M'$.

Part B
We define the sequence of complex numbers $(z_n)$ by a first term $z_0$ belonging to $\mathbb{C}$ and, for every natural integer $n$, by the recurrence relation: $$z_{n+1} = \frac{z_n + |z_n|}{4}$$ The purpose of this part is to study whether the behavior at infinity of the sequence $(|z_n|)$ depends on the choice of $z_0$.

1. What can we say about the behavior at infinity of the sequence $(|z_n|)$ when $z_0$ is a negative real number?
2. What can we say about the behavior at infinity of the sequence $(|z_n|)$ when $z_0$ is a positive real number?
3. We now assume that $z_0$ is not a real number. a. What conjecture can we make about the behavior at infinity of the sequence $(|z_n|)$? b. Prove this conjecture, then conclude.

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 $k$.

We equip the complex plane with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ).
For each of the three following propositions, indicate whether it is true or false and justify the chosen answer. One point is awarded for each correct answer properly justified. An unjustified answer is not taken into account.
Proposition 1 The set of points in the plane with affixe $z$ such that $| z - 4 | = | z + 2 \mathrm { i } |$ is a line that passes through the point A with affixe 3i.
Proposition 2 Let ( $E$ ) be the equation $( z - 1 ) \left( z ^ { 2 } - 8 z + 25 \right) = 0$ where $z$ belongs to the set $\mathbb { C }$ of complex numbers. The points in the plane whose affixes are the solutions in $\mathbb { C }$ of the equation ( $E$ ) are the vertices of a right triangle.
Proposition 3 $\frac { \pi } { 3 }$ is an argument of the complex number $( - \sqrt { 3 } + \mathrm { i } ) ^ { 8 }$.

The complex plane is given an orthonormal direct coordinate system $(\mathrm{O}; \vec{u}, \vec{v})$. We consider the point A with affixe 4, the point B with affixe $4\mathrm{i}$ and the points C and D such that ABCD is a square with centre O. For any non-zero natural number $n$, we call $M_n$ the point with affixe $z_n = (1 + \mathrm{i})^n$.

1. Write the number $1 + \mathrm{i}$ in exponential form.
2. Show that there exists a natural number $n_0$, which we will determine, such that, for any integer $n \geqslant n_0$, the point $M_n$ is outside the square ABCD.

We place ourselves in the complex plane with coordinate system $(O ; \vec { u } , \vec { v })$. Let $f$ be the transformation that associates to any non-zero complex number $z$ the complex number $f ( z )$ defined by:
$$f ( z ) = z + \frac { 1 } { z }$$
We denote by $M$ the point with affixe $z$ and $M ^ { \prime }$ the point with affixe $f ( z )$.

1. We call A the point with affixe $a = - \frac { \sqrt { 2 } } { 2 } + \mathrm { i } \frac { \sqrt { 2 } } { 2 }$. a. Determine the exponential form of $a$. b. Determine the algebraic form of $f ( a )$.
2. Solve, in the set of complex numbers, the equation $f ( z ) = 1$.
3. Let $M$ be a point with affixe $z$ on the circle $\mathscr { C }$ with center O and radius 1. a. Justify that the affixe $z$ can be written in the form $z = \mathrm { e } ^ { \mathrm { i } \theta }$ with $\theta$ a real number. b. Show that $f ( z )$ is a real number.
4. Describe and represent the set of points $M$ with affixe $z$ such that $f ( z )$ is a real number.

Exercise 4 (3 points)
In the complex plane equipped with a direct orthonormal coordinate system $(\mathrm{O}; \vec{u}, \vec{v})$, we consider the points A and B with complex numbers respectively $z_{\mathrm{A}} = 2\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$ and $z_{\mathrm{B}} = 2\mathrm{e}^{\mathrm{i}\frac{3\pi}{4}}$.

1. Show that OAB is a right isosceles triangle.
2. We consider the equation $$(E): z^2 - \sqrt{6}\, z + 2 = 0$$ Show that one of the solutions of $(E)$ is the complex number of a point located on the circumscribed circle of triangle OAB.

Consider in $\mathbb{C}$ the equation: $$\left(4z^2 - 20z + 37\right)(2z - 7 + 2i) = 0$$ Statement 4: the solutions of the equation are the affixes of points belonging to the same circle with centre the point P with affix 2. Indicate whether Statement 4 is true or false, justifying your answer.

In this exercise, $x$ and $y$ are real numbers greater than 1. In the complex plane equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ), we consider the points $\mathrm { A} , \mathrm { B }$ and C with affixes respectively $z_{\mathrm{A}} = 1 + \mathrm{i}$, $z_{\mathrm{B}} = x + \mathrm{i}$, $z_{\mathrm{C}} = y + \mathrm{i}$.
Problem: we seek the possible values of real numbers $x$ and $y$, greater than 1, for which :
$$\mathrm { OC } = \mathrm { OA } \times \mathrm { OB } \quad \text { and } ( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$$

1. Prove that if $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$, then $y ^ { 2 } = 2 x ^ { 2 } + 1$.
2. Reproduce on your answer sheet and complete the following algorithm so that it displays all couples $( x , y )$ such that : \begin{verbatim} For x going from 1 to ... do For... If... Display x and y End If End For End For \end{verbatim} When this algorithm is executed, it displays the value 2 for variable $x$ and the value 3 for variable $y$.
3. Study of a particular case: in this question only, we take $x = 2$ and $y = 3$. a. Give the modulus and an argument of $z _ { \mathrm { A } }$. b. Show that $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$. c. Show that $z _ { \mathrm { B } } z _ { \mathrm { C } } = 5 z _ { \mathrm { A } }$ and deduce that $( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$.
4. We return to the general case, and we seek whether there exist other values of real numbers $x$ and $y$ such that points $\mathrm { A } , \mathrm { B }$ and C satisfy the two conditions : $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$ and $( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$. Recall that if $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$, then $y ^ { 2 } = 2 x ^ { 2 } + 1$ (question 1.). a. Prove that if $( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$, then $\arg \left[ \frac { ( x + \mathrm { i } ) ( y + \mathrm { i } ) } { 1 + \mathrm { i } } \right] = 0 \bmod 2 \pi$.
   Deduce that under this condition : $x + y - x y + 1 = 0$. b. Prove that if the two conditions are satisfied and moreover $x \neq 1$, then :
   $$y = \sqrt { 2 x ^ { 2 } + 1 } \quad \text { and } y = \frac { x + 1 } { x - 1 }$$
   
5. We define the functions $f$ and $g$ on the interval $] 1 ; + \infty [$ by :
   $$f ( x ) = \sqrt { 2 x ^ { 2 } + 1 } \quad \text { and } g ( x ) = \frac { x + 1 } { x - 1 }$$
   Determine the number of solutions to the initial problem. We may use the function $h$ defined on the interval $] 1 ; + \infty [$ by $h ( x ) = f ( x ) - g ( x )$ and rely on the screenshot of a computer algebra software given below.

The plane is equipped with a direct orthonormal coordinate system $(\mathrm{O}; \vec{u}, \vec{v})$.
The purpose of this exercise is to determine the non-zero complex numbers $z$ such that the points with affixes $1$, $z^2$ and $\dfrac{1}{z}$ are collinear. On the graph provided in the appendix, point A has affix 1.
Part A: study of examples
1. A first example
In this question, we set $z = \mathrm{i}$. a. Give the algebraic form of the complex numbers $z^2$ and $\dfrac{1}{z}$. b. Plot the points $N_1$ with affix $z^2$, and $P_1$ with affix $\dfrac{1}{z}$ on the graph given in the appendix. We note that in this case the points $\mathrm{A}$, $N_1$ and $P_1$ are not collinear.
2. An equation
Solve in the set of complex numbers the equation with unknown $z$: $z^2 + z + 1 = 0$.
3. A second example
In this question, we set: $z = -\dfrac{1}{2} + \mathrm{i}\dfrac{\sqrt{3}}{2}$. a. Determine the exponential form of $z$, then those of the complex numbers $z^2$ and $\dfrac{1}{z}$. b. Plot the points $N_2$ with affix $z^2$ and $P_2$, with affix $\dfrac{1}{z}$ on the graph given in the appendix. We note that in this case the points $\mathrm{A}$, $N_2$ and $P_2$ are collinear.
Part B
Let $z$ be a non-zero complex number. We denote by $N$ the point with affix $z^2$ and $P$ the point with affix $\dfrac{1}{z}$.

1. Establish that, for every complex number different from 0, we have: $$z^2 - \frac{1}{z} = \left(z^2 + z + 1\right)\left(1 - \frac{1}{z}\right)$$
2. We recall that if $\vec{U}$ is a non-zero vector and $\vec{V}$ is a vector with affixes respectively $z_{\vec{U}}$ and $z_{\vec{V}}$, the vectors $\vec{U}$ and $\vec{V}$ are collinear if and only if there exists a real number $k$ such that $z_{\vec{V}} = k z_{\vec{U}}$. Deduce that, for $z \neq 0$, the points $\mathrm{A}$, $N$ and $P$ defined above are collinear if and only if $z^2 + z + 1$ is a real number.
3. We set $z = x + \mathrm{i}y$, where $x$ and $y$ denote real numbers. Justify that: $z^2 + z + 1 = x^2 - y^2 + x + 1 + \mathrm{i}(2xy + y)$.
4. a. Determine the set of points $M$ with affix $z \neq 0$ such that the points $\mathrm{A}$, $N$ and $P$ are collinear. b. Trace this set of points on the graph given in the appendix.

Let $z \in \mathbb{C}$. We denote $C_z$ (respectively $\Omega_z$) the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| = 1$ (respectively $|Z(Z-2z)| < 1$). In this question we assume that $z$ is a real number denoted $a$. We work in the orthonormal frame $\mathcal{R}'$ with center $O'$ with affixe $a$, obtained from $\mathcal{R}$ by translation. Show that an equation of the curve $C_a$ in ``polar coordinates $(\rho, \theta)$'' in the frame $\mathcal{R}'$ is $$\left(\rho^2 + a^2\right)^2 - 4a^2 \rho^2 \cos^2\theta = 1$$

Let $z \in \mathbb{C}$. We denote $\Omega_z$ the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| < 1$. Justify that $\Omega_z$ is a bounded subset of the plane. Is it open? closed? compact?

Let $z \in \mathbb{C}$. We denote $\Omega_z$ the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| < 1$. Justify that the origin $O$ is an interior point of $\Omega_z$.

Let $w \in \mathbb{C}$ be a number that is neither real nor purely imaginary.
(a) Show that the equation $$\left|\frac{z-1}{z+1}\right| = \left|\frac{w-1}{w+1}\right|$$ defines a circle in the complex plane, which passes through $w$. Verify that the interval $]-1,1[$ intersects this circle at a unique point; we denote this point by $y$. We will express $y$ in terms of the number $$\lambda = \left|\frac{w-1}{w+1}\right|.$$
(b) Show the inequality $$\left|\frac{1-w}{1-y}\right| > 1.$$
(c) Show that the equation $$\left|\frac{z-w}{z-y}\right| = \left|\frac{1-w}{1-y}\right|$$ defines a circle in the complex plane, which passes through $1$ and through $-1$.
Deduce that, for all $x \in [-1,1] \setminus \{y\}$, we have $$\left|\frac{w-x}{y-x}\right| \geqslant \left|\frac{w-1}{y-1}\right| = \left|\frac{w+1}{y+1}\right|$$

Recall that, for any non-zero complex number $w$ which does not lie on the negative real axis, $\arg(w)$ denotes the unique real number $\theta$ in $(-\pi, \pi)$ such that $w = |w|(\cos\theta + i\sin\theta)$. Let $z$ be any complex number such that its real and imaginary parts are both non-zero. Further, suppose that $z$ satisfies the relations $\arg(z) > \arg(z+1)$ and $\arg(z) > \arg(z+i)$. Then $\cos(\arg(z))$ can take
(a) Any value in the set $(-1/2, 0) \cup (0, 1/2)$ but none from outside
(b) Any value in the interval $(-1, 0)$ but none from outside
(c) Any value in the interval $(0, 1)$ but none from outside
(d) Any value in the set $(-1, 0) \cup (0, 1)$ but none from outside.

Suppose $z$ is a complex number with $| z | < 1$. Let $w = ( 1 + z ) / ( 1 - z )$. Which of the following is always true? [$\operatorname { Re } ( w )$ is the real part of $w$ and $\operatorname { Im } ( w )$ is the imaginary part of $w$]
(a) $\operatorname { Re } ( w ) > 0$
(b) $\operatorname { Im } ( w ) \geq 0$
(c) $| w | \leq 1$
(d) $| w | \geq 1$

Find the locus of $z$ satisfying $|z - ia| = \text{Im}(z) + 1$, where $a$ is a real constant.

Let $\mathbb { C }$ denote the set of all complex numbers. Define $$\begin{aligned} & A = \{ ( z ,w ) \mid z , w \in \mathbb { C } \text { and } | z | = | w | \} \\ & B = \left\{ ( z , w ) \mid z , w \in \mathbb { C } , \text { and } z ^ { 2 } = w ^ { 2 } \right\} \end{aligned}$$ Then,
(a) $A = B$
(b) $A \subset B$ and $A \neq B$
(c) $B \subset A$ and $B \neq A$
(d) none of the above.