1. Let $z _ { 1 } = \sqrt { 6 } \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 4 } }$ and $z _ { 2 } = \sqrt { 2 } \mathrm { e } ^ { - \mathrm { i } \frac { \pi } { 3 } }$. The exponential form of $\mathrm { i } \frac { z _ { 1 } } { z _ { 2 } }$ is: a. $\sqrt { 3 } \mathrm { e } ^ { \mathrm { i } \frac { 19 \pi } { 12 } }$ b. $\sqrt { 12 } \mathrm { e } ^ { - \mathrm { i } \frac { \pi } { 12 } }$ c. $\sqrt { 3 } \mathrm { e } ^ { \mathrm { i } \frac { 7 \pi } { 12 } }$ d. $\sqrt { 3 } \mathrm { e } ^ { \mathrm { i } \frac { 13 \pi } { 12 } }$
2. The equation $- z = \bar { z }$, with unknown complex number $z$, admits: a. one solution b. two solutions c. infinitely many solutions whose image points in the complex plane are located on a line. d. infinitely many solutions whose image points in the complex plane are located on a circle.
3. In a coordinate system of space, consider the three points $A ( 1 ; 2 ; 3 ) , B ( - 1 ; 5 ; 4 )$ and $C ( - 1 ; 0 ; 4 )$. The line parallel to the line $( A B )$ passing through point $C$ has the parametric representation: a. $\left\{ \begin{array} { l } x = - 2 t - 1 \\ y = 3 t \\ z = t + 4 \end{array} , t \in \mathbb { R } \right.$ b. $\left\{ \begin{array} { l } x = - 1 \\ y = 7 t \\ z = 7 t + 4 \end{array} , t \in \mathbb { R } \right.$ c. $\left\{ \begin{array} { l } x = - 1 - 2 t \\ y = 5 + 3 t \\ z = 4 + t \end{array} , t \in \mathbb { R } \right.$ d. $\left\{ \begin{array} { l } x = 2 t \\ y = - 3 t \\ z = - t \end{array} , t \in \mathbb { R } \right.$
4. In an orthonormal coordinate system of space, consider the plane $\mathscr { P }$ passing through point $D ( - 1 ; 2 ; 3 )$ and with normal vector $\vec { n } ( 3 ; - 5 ; 1 )$, and the line $\Delta$ with parametric representation $\left\{ \begin{array} { l } x = t - 7 \\ y = t + 3 \\ z = 2 t + 5 \end{array} , t \in \mathbb { R } \right.$. a. The line $\Delta$ is perpendicular to the plane $\mathscr { P }$. b. The line $\Delta$ is parallel to the plane $\mathscr { P }$ and has no common point with the plane $\mathscr { P }$. c. The line $\Delta$ and the plane $\mathscr { P }$ are secant. d. The line $\Delta$ is contained in the plane $\mathscr { P }$.

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. Statement 1: the points A, B and C are collinear.
2. Statement 2: the points B, C and D belong to the same circle with center E.
3. In this question, space is equipped with a coordinate system $(O, \vec{\imath}, \vec{\jmath}, \vec{k})$. Consider the points $I(1; 0; 0)$, $J(0; 1; 0)$ and $K(0; 0; 1)$. Statement 3: the line $\mathscr{D}$ with parametric representation $\left\{\begin{aligned} x &= 2 - t \\ y &= 6 - 2t \\ z &= -2 + t \end{aligned}\right.$ where $t \in \mathbb{R}$, intersects the plane (IJK) at point $E\left(-\frac{1}{2}; 1; \frac{1}{2}\right)$.
4. In the cube ABCDEFGH, the point T is the midpoint of segment $[HF]$. Statement 4: the lines (AT) and (EC) are orthogonal.

1. a) Calculate $z_{1}, z_{2}$ and $z_{3}$. b) Plot the points $A_{1}$ and $A_{2}$ on the graph given in the appendix, to be returned with your answer sheet. c) Write the complex number $\frac{1 + \mathrm{i}}{2}$ in trigonometric form. d) Prove that the triangle $\mathrm{OA}_{0}A_{1}$ is isosceles right-angled at $A_{1}$.
2. Prove that the sequence $(r_{n})$ is geometric, with common ratio $\frac{\sqrt{2}}{2}$.

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 $\cos\left(\frac{\pi}{12}\right)$.

Exercise 4 — Candidates who have not chosen the specialty course

Part A: properties of the number j

1. a. Solve in the set $\mathbb{C}$ of complex numbers the equation $$z^{2} + z + 1 = 0$$ b. Verify that the complex number j is a solution of this equation.
2. Determine the modulus and an argument of the complex number j, then give its exponential form.
3. Prove the following equalities: a. $j^{3} = 1$; b. $\mathrm{j}^{2} = -1 - \mathrm{j}$.
4. Let $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ be the images respectively of the complex numbers $1, \mathrm{j}$ and $\mathrm{j}^{2}$ in the plane.
   What is the nature of triangle PQR? Justify the answer.

Part B

1. Using question A-3.b., prove the equality: $a - c = \mathrm{j}(c - b)$.
2. Deduce that $\mathrm{AC} = \mathrm{BC}$.
3. Prove the equality: $a - b = \mathrm{j}^{2}(b - c)$.
4. Deduce that triangle ABC is equilateral.

Exercise 2

1. a. In this question, we assume that $z_{0} = 2$. Determine the numbers $z_{1}, z_{2}, z_{3}, z_{4}, z_{5}$ and $z_{6}$ b. In this question, we assume that $z_{0} = \mathrm{i}$. Determine the algebraic form of the complex numbers $z_{1}, z_{2}, z_{3}, z_{4}, z_{5}$ and $z_{6}$. c. In this question we return to the general case where $z_{0}$ is a given complex number. What can we conjecture about the values taken by $z_{3n}$ according to the values of the natural integer $n$? Prove this conjecture.
2. Determine $z_{2016}$ in the case where $z_{0} = 1 + \mathrm{i}$.
3. Are there values of $z_{0}$ such that $z_{0} = z_{1}$? What can we say about the sequence $(z_{n})$ in this case?

• 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.

Exercise 4 - Candidates who have NOT followed the specialization course

Proposition 1:

Proposition 2:

Proposition 3:

Proposition 4:

Proposition 5:

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.

Exercise 2 (4 points)

1. Solve in the set $\mathbb{C}$ of complex numbers the equation: $$-z^2 + 2z - 2 = 0$$ Deduce the affixes of the points whose image is the point with affixe 2.
   
2. Let $M$ be a point with affixe $z$ and $M'$ its image with affixe $z'$.
   We denote $N$ the point with affixe $z_N = z^2$. Show that $M$ is the midpoint of segment $[NM']$.
   
3. In this question, we assume that the point $M$ with affixe $z$ belongs to the circle $\mathscr{C}$ with center O and radius 1. We denote $\theta$ an argument of $z$. a. Determine the modulus of each of the complex numbers $z$ and $z_N$, as well as an argument of $z_N$ as a function of $\theta$. b. On the figure given in the appendix on page 7, a point $M$ on the circle $\mathscr{C}$ has been represented. Construct on this figure the points $N$ and $M'$ using a ruler and compass (leave the construction lines visible). c. Let A be the point with affixe 1. What is the nature of triangle $AMM'$?

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.

Exercise 4

Common to all candidates

1. Statement 1: The number $c$ can be written as $c = \frac { 1 } { 4 } ( 1 - \mathrm { i } \sqrt { 3 } )$.
2. Statement 2: For all natural integer $n$, $c ^ { 3 n }$ is a real number.
3. Statement 3: The points $\mathrm { O }$, $\mathrm { S }$ and T are collinear.
4. Statement 4: For all non-zero natural integer $n$, $$| c | + \left| c ^ { 2 } \right| + \ldots + \left| c ^ { n } \right| = 1 - \left( \frac { 1 } { 2 } \right) ^ { n } .$$

Exercise 3

1. In the set $\mathbb{C}$ of complex numbers, we consider the equation $(E): z^2 - 2\sqrt{3}\,z + 4 = 0$. We denote $A$ and $B$ the points of the plane whose affixes are the solutions of $(E)$. We denote O the point with affix 0. Statement 1: The triangle $OAB$ is equilateral.
   
2. We denote $u$ the complex number: $u = \sqrt{3} + \mathrm{i}$ and we denote $\bar{u}$ its conjugate. Statement 2: $u^{2019} + \bar{u}^{2019} = 2^{2019}$
   
3. Let $n$ be a non-zero natural number. We consider the function $f_n$ defined on the interval $[0; +\infty[$ by: $$f_n(x) = x\,\mathrm{e}^{-nx+1}$$ Statement 3: For any natural number $n \geqslant 1$, the function $f_n$ admits a maximum.
   
4. We denote $\mathscr{C}$ the representative curve of the function $f$ defined on $\mathbb{R}$ by: $f(x) = \cos(x)\,\mathrm{e}^{-x}$. Statement 4: The curve $\mathscr{C}$ admits an asymptote at $+\infty$.
   
5. Let $A$ be a strictly positive real number. We consider the algorithm: $$\begin{array}{|l} I \leftarrow 0 \\ \text{While } 2^I \leqslant A \\ \quad I \leftarrow I + 1 \\ \text{End While} \end{array}$$ We assume that the variable $I$ contains the value 15 at the end of execution of this algorithm. Statement 5: $15\ln(2) \leqslant \ln(A) \leqslant 16\ln(2)$