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

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 $z_{\mathrm{A}'}$.

• 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 $S = \sigma \circ h$.

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.

Exercise 3 -- Candidates who have NOT followed the specialization course

1. Solve the equation $(E)$ in the set $\mathbb{C}$ of complex numbers.
2. We consider the sequence $(M_n)$ of points with affixes $z_n = 2^n \mathrm{e}^{\mathrm{i}(-1)^n \frac{\pi}{6}}$, defined for $n \geqslant 1$. a. Verify that $z_1$ is a solution of $(E)$. b. Write $z_2$ and $z_3$ in algebraic form. c. Plot the points $M_1, M_2, M_3$ and $M_4$ on the figure provided in the appendix and draw, on the figure provided in the appendix, the segments $[M_1, M_2]$, $[M_2, M_3]$ and $[M_3, M_4]$.
3. Show that, for every integer $n \geqslant 1$, $z_n = 2^n\left(\frac{\sqrt{3}}{2} + \frac{(-1)^n \mathrm{i}}{2}\right)$.
4. Calculate the lengths $M_1M_2$ and $M_2M_3$.

1. Solve in $\mathbb { C }$ the equation $Z ^ { 2 } + 4 Z + 16 = 0$. Write the solutions of this equation in exponential form.
2. We denote by $a$ the complex number whose modulus is equal to 2 and one of whose arguments is equal to $\frac { \pi } { 3 }$. Calculate $a ^ { 2 }$ in algebraic form. Deduce the solutions in $\mathbb { C }$ of the equation $z ^ { 2 } = - 2 + 2 \mathrm { i } \sqrt { 3 }$. Write the solutions in algebraic form.
3. Organized presentation of knowledge We assume it is known that for every complex number $z = x + \mathrm { i } y$ where $x \in \mathbb { R }$ and $y \in \mathbb { R }$, the conjugate of $z$ is the complex number $\bar{z}$ defined by $\bar{z} = x - \mathrm { i } y$. Prove that:
   • For all complex numbers $z _ { 1 }$ and $z _ { 2 } , \overline { z _ { 1 } z _ { 2 } } = \overline { z _ { 1 } } \cdot \overline { z _ { 2 } }$.
   • For every complex number $z$ and every non-zero natural integer $n , \overline { z ^ { n } } = ( \bar { z } ) ^ { n }$.
4. Prove that if $z$ is a solution of equation (E) then its conjugate $\bar { z }$ is also a solution of (E). Deduce the solutions in $\mathbb { C }$ of equation (E). We will assume that (E) has at most four solutions.

1. Give the exponential form of the complex number $\frac { 3 } { 4 } + \frac { \sqrt { 3 } } { 4 }$ i.
2. a. Show that the sequence ( $r _ { n }$ ) is geometric with common ratio $\frac { \sqrt { 3 } } { 2 }$. b. Deduce the expression of $r _ { n }$ as a function of $n$. c. What can be said about the length $\mathrm { O } A _ { n }$ as $n$ tends to $+ \infty$ ?
3. Consider the following algorithm:

Part A

1. Calculate $u_0$.
2. Prove that $(u_n)$ is a geometric sequence with common ratio $\sqrt{2}$ and first term 2.
3. For every natural number $n$, express $u_n$ as a function of $n$.
4. Determine the limit of the sequence $(u_n)$.
5. Given a positive real number $p$, we wish to determine, using an algorithm, the smallest value of the natural number $n$ such that $u_n > p$. Copy the algorithm below and complete it with the processing and output instructions, so as to display the sought value of the integer $n$. \begin{verbatim} Variables : u is a real number p is a real number n is an integer Initialization : Assign to n the value 0 Assign to u the value 2 Input : Request the value of p Processing : Output : \end{verbatim}

Part B

1. Determine the algebraic form of $z_1$.
2. Determine the exponential form of $z_0$ and of $1 + \mathrm{i}$. Deduce the exponential form of $z_1$.
3. Deduce from the previous questions the exact value of $\cos\left(\frac{\pi}{12}\right)$.

1. a. Determine the coordinates of the points $A _ { 0 } , A _ { 1 }$ and $A _ { 2 }$. b. Copy and complete the following algorithm so that it constructs the points $A _ { 0 }$ to $A _ { 20 }$: \begin{verbatim} Variables : i,x,y,t: real numbers Initialization : x takes the value -3 y takes the value 4 Processing : For i ranging from 0 to 20 Construct the point with coordinates (x;y) t takes the value x x takes the value .... y takes the value .... End For \end{verbatim} c. Identify the points $A _ { 0 } , A _ { 1 }$ and $A _ { 2 }$ on the point cloud figure. What appears to be the set to which the points $A _ { n }$ belong for every natural integer $n$?
2. The purpose of this question is to construct geometrically the points $A _ { n }$ for every natural integer $n$. In the complex plane, we denote, for every natural integer $n$, $z _ { n } = x _ { n } + \mathrm { i } y _ { n }$ the affix of the point $A _ { n }$. a. Let $u _ { n } = \left| z _ { n } \right|$. Show that, for every natural integer $n , u _ { n } = 5$. What geometric interpretation can be made of this result? b. We admit that there exists a real number $\theta$ such that $\cos ( \theta ) = 0,8$ and $\sin ( \theta ) = 0,6$. Show that, for every natural integer $n , \mathrm { e } ^ { \mathrm { i } \theta } z _ { n } = z _ { n + 1 }$. c. Prove that, for every natural integer $n , z _ { n } = \mathrm { e } ^ { \mathrm { i } n \theta } z _ { 0 }$. d. Show that $\theta + \frac { \pi } { 2 }$ is an argument of the complex number $z _ { 0 }$. e. For every natural integer $n$, determine, as a function of $n$ and $\theta$, an argument of the complex number $z _ { n }$. Explain, for every natural integer $n$, how to construct the point $A _ { n + 1 }$ from the point $A _ { n }$.

1. A point $M$ is called invariant when it coincides with the associated point $M ^ { \prime }$.
   Prove that there exist two invariant points. Give the affixe of each of these points in algebraic form, then in exponential form.
2. Let A be the point with affixe $\frac { - 3 - \mathrm { i } \sqrt { 3 } } { 2 }$ and B the point with affixe $\frac { - 3 + \mathrm { i } \sqrt { 3 } } { 2 }$.
   Show that OAB is an equilateral triangle.
3. Determine the set $\mathcal { E }$ of points $M$ with affixe $z = x + \mathrm { i } y$ where $x$ and $y$ are real, such that the associated point $M ^ { \prime }$ lies on the real axis.
4. In the complex plane, represent the points A and B as well as the set $\mathcal { E }$.

• the five sides have the same length;
• the points $A _ { 0 } , A _ { 1 } , A _ { 2 } , A _ { 3 }$ and $A _ { 4 }$ belong to the unit circle;
• for any integer $k$ belonging to $\{ 0 ; 1 ; 2 ; 3 \}$ we have $\left( \overrightarrow { O A _ { k } } ; \overrightarrow { O A _ { k + 1 } } \right) = \frac { 2 \pi } { 5 }$.

1. We consider the points $B$ with affix $-1$ and $J$ with affix $\frac { \mathrm { i } } { 2 }$.
   The circle $\mathscr { C }$ with center $J$ and radius $\frac { 1 } { 2 }$ intersects the segment $[ B J ]$ at a point $K$. Calculate $B J$, then deduce $B K$.
2. a. Give in exponential form the affix of point $A _ { 2 }$. Justify briefly. b. Prove that $B A _ { 2 } { } ^ { 2 } = 2 + 2 \cos \left( \frac { 4 \pi } { 5 } \right)$. c. A computer algebra system displays the results below, which may be used without justification:
   

   \multicolumn{2}{|l|}{Formal calculation}
   1	\begin{tabular}{ l } $\cos \left( 4 ^ { * } \mathrm { pi } / 5 \right)$
   $\rightarrow \frac { 1 } { 4 } ( - \sqrt { 5 } - 1 )$

   \hline 2 & $\operatorname { sqrt } ( ( 3 - \operatorname { sqrt } ( 5 ) ) / 2 )$ \hline & $\rightarrow \frac { 1 } { 2 } ( \sqrt { 5 } - 1 )$ \hline \end{tabular}
   ``sqrt'' means ``square root'' Deduce, using these results, that $B A _ { 2 } = B K$.
3. In the coordinate system ( $\mathrm { O} , \vec { u } , \vec { v }$ ) provided in the appendix, construct a regular pentagon with straightedge and compass. Do not use a protractor or the ruler's graduations and leave the construction lines visible.

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.

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.

1. Let ( $u _ { n }$ ) be the sequence defined for all natural number $n$ by $u _ { n } = z _ { n } - z _ { \mathrm { A } }$. a) Show that, for all natural number $n , u _ { n + 1 } = \frac { 1 } { 2 } \mathrm { i } \times u _ { n }$. b) Prove that, for all natural number $n$:
   $$u _ { n } = \left( \frac { 1 } { 2 } \mathrm { i } \right) ^ { n } ( - 4 - 2 \mathrm { i } )$$
   
2. Prove that, for all natural number $n$, the points $\mathrm { A } , M _ { n }$ and $M _ { n + 4 }$ are collinear.

1. Give an integer solution of ( $E$ ).
2. Prove that, for every complex number $z$, $$z ^ { 4 } + 2 z ^ { 3 } - z - 2 = \left( z ^ { 2 } + z - 2 \right) \left( z ^ { 2 } + z + 1 \right) .$$
3. Solve equation ( $E$ ) in the set of complex numbers.
4. The solutions of equation ( $E$ ) are the affixes of four points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ in the complex plane such that ABCD is a non-crossed quadrilateral. Is quadrilateral ABCD a rhombus? Justify.

1. For all natural integer $n$, we denote $A _ { n }$ the point with affix $z _ { n }$. a. Prove that, for all natural integer $n , \frac { z _ { n + 4 } } { z _ { n } }$ is real. b. Prove then that, for all natural integer $n$, the points O , $A _ { n }$ and $A _ { n + 4 }$ are collinear.
2. For which values of $n$ is the number $z _ { n }$ real?

1. Give the exponential and trigonometric forms of the complex numbers $1 + \mathrm{i}$ and $1 - \mathrm{i}$.
2. For every natural number $n$, we define $$S_{n} = (1 + \mathrm{i})^{n} + (1 - \mathrm{i})^{n}.$$ a. Determine the trigonometric form of $S_{n}$. b. For each of the two following statements, say whether it is true or false by justifying your answer. An unjustified answer will not be taken into account and the absence of an answer is not penalised.
   Statement A: For every natural number $n$, the complex number $S_{n}$ is a real number. Statement B: There exist infinitely many natural numbers $n$ such that $S_{n} = 0$.

1. Solve in $\mathbb{C}$ the equation $\left(z^{2} - 2z + 4\right)\left(z^{2} + 4\right) = 0$.
2. We consider the points A and B with complex numbers $z_{\mathrm{A}} = 1 + \mathrm{i}\sqrt{3}$ and $z_{\mathrm{B}} = 2\mathrm{i}$ respectively. a. Write $z_{\mathrm{A}}$ and $z_{\mathrm{B}}$ in exponential form and justify that the points A and B lie on a circle with centre O, whose radius you will specify. b. Draw a figure and place the points A and B. c. Determine a measure of the angle $(\overrightarrow{\mathrm{OA}}, \overrightarrow{\mathrm{OB}})$.
3. We denote by F the point with complex number $z_{\mathrm{F}} = z_{\mathrm{A}} + z_{\mathrm{B}}$. a. Place the point F on the previous figure. Show that OAFB is a rhombus. b. Deduce a measure of the angle $(\overrightarrow{\mathrm{OA}}, \overrightarrow{\mathrm{OF}})$ then of the angle $(\vec{u}, \overrightarrow{\mathrm{OF}})$. c. Calculate the modulus of $z_{\mathrm{F}}$ and deduce the expression of $z_{\mathrm{F}}$ in trigonometric form. d. Deduce the exact value of: $$\cos\left(\frac{5\pi}{12}\right)$$
4. Two calculator models from different manufacturers give for one: $$\cos\left(\frac{5\pi}{12}\right) = \frac{\sqrt{2 - \sqrt{3}}}{2}$$ and for the other: $$\cos\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6} - \sqrt{2}}{4}$$ Are these results contradictory? Justify your answer.

1. We consider the three points $\mathrm{A}^{\prime}$, $\mathrm{B}^{\prime}$ and $\mathrm{C}^{\prime}$ with affixes respectively $a^{\prime} = \mathrm{j}a$, $b^{\prime} = \mathrm{j}b$ and $c^{\prime} = \mathrm{j}c$ where j is the complex number $-\frac{1}{2} + \mathrm{i}\frac{\sqrt{3}}{2}$. a. Give the trigonometric form and the exponential form of j. Deduce the algebraic and exponential forms of $a^{\prime}$, $b^{\prime}$ and $c^{\prime}$. b. The points $\mathrm{A}$, $\mathrm{B}$ and C as well as the circles with center O and radii 2, 3 and 4 are represented on the graph provided in the Appendix. Place the points $\mathrm{A}^{\prime}$, $\mathrm{B}^{\prime}$ and $\mathrm{C}^{\prime}$ on this graph.
2. Show that the points $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ are collinear.
3. We denote M the midpoint of segment $[\mathrm{A}^{\prime}\mathrm{C}]$, N the midpoint of segment $[\mathrm{C}^{\prime}\mathrm{C}]$ and P the midpoint of segment $[\mathrm{C}^{\prime}\mathrm{A}]$. Prove that triangle MNP is isosceles.

1. a. Show that $z _ { 1 } = - \mathrm { i }$ and that $z _ { 2 } = 1 - 2 \mathrm { i }$. b. Calculate $z _ { 3 }$. c. On your answer sheet, plot the points $\mathrm { B } , A _ { 1 } , A _ { 2 }$ and $A _ { 3 }$ in the direct orthonormal coordinate system $( \mathrm { O } ; \vec { u } , \vec { v } )$. d. Prove that the triangle $\mathrm { B } A _ { 1 } A _ { 2 }$ is isosceles right-angled.
2. For every natural number $n$, set $u _ { n } = \left| z _ { n } - 1 \right|$. a. Prove that for every natural number $n$, we have $u _ { n + 1 } = \sqrt { 2 } u _ { n }$. b. Determine from which natural number $n$ the distance $\mathrm { B } A _ { n }$ is strictly greater than 1000. Detail the approach chosen.
3. a. Determine the exponential form of the complex number $1 + \mathrm { i }$. b. Prove by induction that for every natural number $n$, $z _ { n } = 1 - ( \sqrt { 2 } ) ^ { n } \mathrm { e } ^ { \mathrm { i } \frac { n \pi } { 4 } }$. c. Does the point $A _ { 2020 }$ belong to the x-axis? Justify.