Exercise 3 -- Common to all candidates
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 your answer. An unjustified answer earns no points. In questions 1. and 2., the plane is referred to the direct orthonormal coordinate system $(O, \vec{u}, \vec{v})$. Consider the points A, B, C, D and E with complex numbers respectively: $$a = 2 + 2\mathrm{i}, \quad b = -\sqrt{3} + \mathrm{i}, \quad c = 1 + \mathrm{i}\sqrt{3}, \quad d = -1 + \frac{\sqrt{3}}{2}\mathrm{i} \quad \text{and} \quad e = -1 + (2 + \sqrt{3})\mathrm{i}.$$

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.

Exercise 4 — For candidates who have NOT followed the specialization course
The plane is referred to an orthonormal direct coordinate system $( \mathrm { O } , \vec { u } , \vec { v } )$.
Let $\mathbb { C }$ denote the set of complex numbers.
For each of the following propositions, state whether it is true or false by justifying the answer.
1. Proposition: For every natural number $n$ : $( 1 + \mathrm { i } ) ^ { 4 n } = ( - 4 ) ^ { n }$.
2. Let (E) be the equation $( z - 4 ) \left( z ^ { 2 } - 4 z + 8 \right) = 0$ where $z$ denotes a complex number.
Proposition: The points whose affixes are the solutions, in $\mathbb { C }$, of (E) are the vertices of a triangle with area 8.
3. Proposition: For every real number $\alpha , 1 + \mathrm { e } ^ { 2 i \alpha } = 2 \mathrm { e } ^ { \mathrm { i } \alpha } \cos ( \alpha )$.
4. Let A be the point with affix $z _ { \mathrm { A } } = \frac { 1 } { 2 } ( 1 + \mathrm { i } )$ and $M _ { n }$ the point with affix $\left( z _ { \mathrm { A } } \right) ^ { n }$ where $n$ denotes a natural number greater than or equal to 2.
Proposition: if $n - 1$ is divisible by 4, then the points O, A and $M _ { n }$ are collinear.
5. Let j be the complex number with modulus 1 and argument $\frac { 2 \pi } { 3 }$.
Proposition: $1 + \mathrm { j } + \mathrm { j } ^ { 2 } = 0$.

This exercise is a multiple choice questionnaire. For each question, three answers are proposed and only one of them is correct. The candidate will write on the answer sheet the number of the question followed by the chosen answer and will justify their choice. One point is awarded for each correct and properly justified answer. An unjustified answer will not be taken into account. No points are deducted in the absence of an answer or in case of an incorrect answer.
For questions 1 and 2, space is equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. The line $\mathscr{D}$ is defined by the parametric representation $\left\{\begin{array}{rl} x &= 5-2t \\ y &= 1+3t \\ z &= 4 \end{array},\, t \in \mathbb{R}\right.$.

1. We denote by $\mathscr{P}$ the plane with Cartesian equation $3x + 2y + z - 6 = 0$. a. The line $\mathscr{D}$ is perpendicular to the plane $\mathscr{P}$. b. The line $\mathscr{D}$ is parallel to the plane $\mathscr{P}$. c. The line $\mathscr{D}$ is contained in the plane $\mathscr{P}$.
2. We denote by $\mathscr{D}'$ the line that passes through point A with coordinates $(3;1;1)$ and has direction vector $\vec{u} = 2\vec{i} - \vec{j} + 2\vec{k}$. a. The lines $\mathscr{D}$ and $\mathscr{D}'$ are parallel. b. The lines $\mathscr{D}$ and $\mathscr{D}'$ are secant. c. The lines $\mathscr{D}$ and $\mathscr{D}'$ are not coplanar.

For questions 3 and 4, the plane is equipped with a direct orthonormal coordinate system with origin O.

1. Let $\mathscr{E}$ be the set of points $M$ with affix $z$ satisfying $|z + \mathrm{i}| = |z - \mathrm{i}|$. a. $\mathscr{E}$ is the $x$-axis. b. $\mathscr{E}$ is the $y$-axis. c. $\mathscr{E}$ is the circle with center O and radius 1.
2. We denote by B and C two points in the plane whose respective affixes $b$ and $c$ satisfy the equality $\dfrac{c}{b} = \sqrt{2}\,\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$. a. The triangle OBC is isosceles with apex O. b. The points O, B, C are collinear. c. The triangle OBC is isosceles and right-angled at B.

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 $\left\{ \begin{array} { l } x = 2 + t \\ y = - 1 + t \\ z = 1 + 3 t \end{array} , t \in \mathbf { R } \right.$.

This exercise is a multiple choice questionnaire. No justification is required. For each question, only one of the four propositions is correct. Each correct answer is worth 1 point. An incorrect answer or no answer does not deduct any points.

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 $\mathscr { P }$.

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 $( - \sqrt { 3 } + \mathrm { i } ) ^ { 8 }$.

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.

Specify whether each of the following statements is true or false by justifying your answer.

1. Let $m$ be a real number and let the equation $( E )$ : $2 z ^ { 2 } + ( m - 5 ) z + m = 0$. a. Statement 1 : ``For $m = 4$, the equation ( $E$ ) admits two real solutions.'' b. Statement 2 : ``There exists only one value of $m$ such that ( $E$ ) admits two complex solutions that are pure imaginary numbers.''
2. In the complex plane, we consider the set $S$ of points $M$ with affixe $z$ satisfying: $$| z - 6 | = | z + 5 i |$$ Statement 3 : ``The set $S$ is a circle.''
3. We equip space with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). We denote $d$ the line with parametric representation: $$d : \left\{ \begin{aligned} x & = - 1 + t \\ y & = 2 - t \quad t \in \mathbb { R } . \\ z & = 3 + t \end{aligned} \right.$$ We denote $d ^ { \prime }$ the line passing through the point $\mathrm { B } ( 4 ; 4 ; - 6 )$ and with direction vector $\vec { v } ( 5 ; 2 ; - 9 )$. Statement 4 : ``The lines $d$ and $d ^ { \prime }$ are coplanar.''
4. We consider the cube ABCDEFGH. Statement 5 : ``The vector $\overrightarrow { \mathrm { DE } }$ is a normal vector to the plane (ABG).''

You are asked to take three distinct points $1 , \omega _ { 1 }$ and $\omega _ { 2 }$ in the complex plane such that $\left| \omega _ { 1 } \right| = \left| \omega _ { 2 } \right| = 1$. Consider the triangle T formed by the complex numbers $1 , \omega _ { 1 }$ and $\omega _ { 2 }$.
Statements
(5) There is exactly one such triangle T that is equilateral. (6) There are exactly two such triangles $T$ that are right angled isosceles. (7) If $\omega _ { 1 } + \omega _ { 2 }$ is real, the triangle T must be isosceles. (8) For any nonzero complex number $z$, the numbers $z , z \omega _ { 1 }$ and $z \omega _ { 2 }$ form a triangle that is similar to the triangle T.

This question is about complex numbers.
Statements
(9) The complex number $\left( e ^ { 3 } \right) ^ { i }$ lies in the third quadrant. (10) If $\left| z _ { 1 } \right| - \left| z _ { 2 } \right| = \left| z _ { 1 } + z _ { 2 } \right|$ for some complex numbers $z _ { 1 }$ and $z _ { 2 }$, then $z _ { 2 }$ must be 0. (11) For distinct complex numbers $z _ { 1 }$ and $z _ { 2 }$, the equation $\left| \left( z - z _ { 1 } \right) ^ { 2 } \right| = \left| \left( z - z _ { 2 } \right) ^ { 2 } \right|$ has at most 4 solutions. (12) For each nonzero complex number $z$, there are more than 100 numbers $w$ such that $w ^ { 2023 } = z$.

For a non-zero complex number $z$, let $\arg ( z )$ denote the principal argument with $- \pi < \arg ( z ) \leq \pi$. Then, which of the following statement(s) is (are) FALSE?
(A) $\arg ( - 1 - i ) = \frac { \pi } { 4 }$, where $i = \sqrt { - 1 }$
(B) The function $f : \mathbb { R } \rightarrow ( - \pi , \pi ]$, defined by $f ( t ) = \arg ( - 1 + i t )$ for all $t \in \mathbb { R }$, is continuous at all points of $\mathbb { R }$, where $i = \sqrt { - 1 }$
(C) For any two non-zero complex numbers $z _ { 1 }$ and $z _ { 2 }$, $$\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right) - \arg \left( z _ { 1 } \right) + \arg \left( z _ { 2 } \right)$$ is an integer multiple of $2 \pi$
(D) For any three given distinct complex numbers $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$, the locus of the point $z$ satisfying the condition $$\arg \left( \frac { \left( z - z _ { 1 } \right) \left( z _ { 2 } - z _ { 3 } \right) } { \left( z - z _ { 3 } \right) \left( z _ { 2 } - z _ { 1 } \right) } \right) = \pi$$ lies on a straight line

Let $Z$ and $W$ be complex numbers such that $| Z | = | W |$, and $\arg Z$ denotes the principal argument of $Z$. Statement 1: If $\arg Z + \arg W = \pi$, then $Z = - \bar { W }$. Statement 2: $| Z | = | W |$, implies $\arg Z - \arg \bar { W } = \pi$.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
(4) Statement 1 is false, Statement 2 is true.

Let $z$ satisfy $| z | = 1$ and $z = 1 - \bar { z }$. Statement $1 : z$ is a real number. Statement 2 : Principal argument of z is $\frac { \pi } { 3 }$
(1) Statement 1 is true Statement 2 is true; Statement 2 is a correct explanation for Statement 1.
(2) Statement 1 is false; Statement 2 is true.
(3) Statement 1 is true, Statement 2 is false.
(4) Statement 1 is true; Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.