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 $\mathbb{R}^{*}$.

Exercise 4 - Candidates who have NOT followed the specialization course

For each of the five following propositions, indicate whether it is true or false and justify the answer chosen. One point is awarded for each correct answer correctly justified. An unjustified answer is not taken into account. An absence of answer is not penalized.

Proposition 1:

In the complex plane equipped with an orthonormal coordinate system, the points A, B and C with affixes respectively $z _ { \mathrm { A } } = \sqrt { 2 } + 3 \mathrm { i } , z _ { \mathrm { B } } = 1 + \mathrm { i }$ and $z _ { \mathrm { C } } = - 4 \mathrm { i }$ are not collinear.

Proposition 2:

There does not exist a non-zero natural integer $n$ such that $[ \mathrm { i } ( 1 + \mathrm { i } ) ] ^ { 2n }$ is a strictly positive real number.

Proposition 3:

ABCDEFGH is a cube with side 1. The point L is such that $\overrightarrow { \mathrm { EL } } = \frac { 1 } { 3 } \overrightarrow { \mathrm { EF } }$. The section of the cube by the plane (BDL) is a triangle.

Proposition 4:

ABCDEFGH is a cube with side 1. The point L is such that $\overrightarrow { \mathrm { EL } } = \frac { 1 } { 3 } \overrightarrow { \mathrm { EF } }$. The triangle DBL is right-angled at B.

Proposition 5:

We consider the function $f$ defined on the interval [2;5] and whose variation table is given below:

$x$	2	3	4	5
\begin{tabular}{ c } Variations
$\operatorname { of } f$

& 3 & & & 2 & & & 0 & 1 \end{tabular}
The integral $\int _ { 2 } ^ { 5 } f ( x ) \mathrm { d } x$ is between 1,5 and 6.

The complex plane is equipped with a direct orthonormal coordinate system $( \mathrm { O } ; \vec { u } ; \vec { v } )$. In what follows, $z$ denotes a complex number.
For each of the statements below, indicate on your answer sheet whether it is true or false. Justify. Any answer without justification earns no points. Statement 1: The equation $z - \mathrm { i } = \mathrm { i } ( z + 1 )$ has solution $\sqrt { 2 } \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 4 } }$. Statement 2: For every real $x \in ] - \frac { \pi } { 2 } ; \frac { \pi } { 2 } \left[ \right.$, the complex number $1 + \mathrm { e } ^ { 2 \mathrm { i } x }$ has exponential form $2 \cos x \mathrm { e } ^ { - \mathrm { i } x }$.
Statement 3: A point M with affix $z$ such that $| z - \mathrm { i } | = | z + 1 |$ belongs to the line with equation $y = - x$.
Statement 4: The equation $z ^ { 5 } + z - \mathrm { i } + 1 = 0$ has a real solution.

Exercise 3 (4 points) -- Common to all candidates
For each of the four following statements, indicate whether it 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 penalised.
The complex plane is equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). We consider the complex number $c = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } }$ and the points S and T with affixes respectively $c ^ { 2 }$ and $\frac { 1 } { c }$.

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 $n$, $$| c | + \left| c ^ { 2 } \right| + \ldots + \left| c ^ { n } \right| = 1 - \left( \frac { 1 } { 2 } \right) ^ { n } .$$

The complex plane is equipped with a direct orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). In what follows, $z$ denotes a complex number.
For each of the statements below, indicate on your answer sheet whether it is true or false. Justify. Any unjustified answer receives no points.
Statement 1: The equation $z - \mathrm{i} = \mathrm{i}(z + 1)$ has solution $\sqrt{2}\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$.
Statement 2: For all real $x \in ]-\frac{\pi}{2}; \frac{\pi}{2}[$, the complex number $1 + \mathrm{e}^{2\mathrm{i}x}$ has exponential form $2\cos x\, \mathrm{e}^{-\mathrm{i}x}$.
Statement 3: A point M with affix $z$ such that $|z - \mathrm{i}| = |z + 1|$ belongs to the line with equation $y = -x$.
Statement 4: The equation $z^5 + z - \mathrm{i} + 1 = 0$ has a real solution.

Exercise 3

The five questions of this exercise are independent. For each of the following statements, indicate whether it is true or false and justify the answer chosen. An unjustified answer is not taken into account. An absence of an answer is not penalized.

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: $15\ln(2) \leqslant \ln(A) \leqslant 16\ln(2)$

Exercise 4 — Candidates who have not followed the specialization course

For each of the following statements, indicate whether it is true or false, by justifying the answer. One point is awarded for each correct answer that is properly justified. An unjustified answer is not taken into account. An absence of an answer is not penalized.

1. Let $\left( u _ { n } \right)$ be the sequence defined by $$u _ { 0 } = 4 \text { and for all natural integer } n , u _ { n + 1 } = - \frac { 2 } { 3 } u _ { n } + 1$$ and let $( \nu _ { n } )$ be the sequence defined by $$\text { for all natural integer } n , v _ { n } = u _ { n } - \frac { 2 } { 3 }$$ Statement 1: The sequence $\left( v _ { n } \right)$ is a geometric sequence.
2. Let $( w _ { n } )$ be the sequence defined by, for all non-zero natural integer $n$, $$w _ { n } = \frac { 3 + \cos ( n ) } { n ^ { 2 } } .$$ Statement 2: The sequence $\left( w _ { n } \right)$ converges to 0.
3. Consider the following algorithm: $$\begin{aligned} & U \leftarrow 5 \\ & N \leftarrow 0 \end{aligned}$$ While $U \leqslant 5000$ $$\begin{aligned} & U \leftarrow 3 \times U - 8 \\ & N \leftarrow N + 1 \end{aligned}$$ End While Statement 3: At the end of execution, the variable $U$ contains the value 5000.
4. We denote $\mathbb { C }$ the set of complex numbers. We consider the equation (E) with unknown $z$ in $\mathbb { C }$ $$( z - \mathrm { i } ) \left( z ^ { 2 } + z \sqrt { 3 } + 1 \right) = 0$$ Statement 4: All solutions of equation (E) have modulus 1.
5. We consider the complex numbers $z _ { n }$ defined by $$z _ { 0 } = 2 \text { and for all natural integer } n , z _ { n + 1 } = 2 \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 2 } } z _ { n } .$$ We equip the complex plane with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). For all natural integer $n$, we denote $M _ { n }$ the point with affixe $z _ { n }$. Statement 5: For all natural integer $n$, the point O is the midpoint of the segment $\left[ M _ { n } M _ { n + 2 } \right]$.

For any complex number $z$ define $P ( z ) =$ the cardinality of $\left\{ z ^ { k } \mid k \text{ is a positive integer} \right\}$, i.e., the number of distinct positive integer powers of $z$. It may be useful to remember that $\pi$ is an irrational number.
(a) For each positive integer $n$ there is a complex number $z$ such that $P ( z ) = n$.
(b) There is a unique complex number $z$ such that $P ( z ) = 3$.
(c) If $| z | \neq 1$, then $P ( z )$ is infinite.
(d) $P \left( e ^ { i } \right)$ is infinite.

In this question $z$ denotes a non-real complex number, i.e., a number of the form $a + ib$ (with $a, b$ real) whose imaginary part $b$ is nonzero. Let $f(z) = z^{222} + \frac{1}{z^{222}}$.
Statements
(33) If $|z| = 1$, then $f(z)$ must be real. (34) If $z + \frac{1}{z} = 1$, then $f(z) = 2$. (35) If $z + \frac{1}{z}$ is real, then $|f(z)| \leq 2$. (36) If $f(z)$ is a real number, then $f(z)$ must be positive.

Consider the following four propositions
$P _ { 1 }$: If complex number $z$ satisfies $\frac { 1 } { z } \in \mathbf { R }$, then $z \in \mathbf { R }$;
$p _ { 2 }$: If complex number $z$ satisfies $z ^ { 2 } \in \mathbf { R }$, then $z \in \mathbf { R }$;
$p _ { 3 }$: If complex numbers $z _ { 1 }$, $z _ { 2 }$ satisfy $z _ { 1 } z _ { 2 } \in \mathbf { R }$, then $z _ { 1 } = \overline { z _ { 2 } }$;
$p _ { 4 }$: If $z + \bar { z } \in \mathbf { R }$, then $z \in \mathbf { R }$.
The true propositions are
A. $p _ { 1 } , p _ { 3 }$
B. $p _ { 1 } , p _ { 4 }$
C. $p _ { 2 } , p _ { 3 }$
D. $p _ { 2 } , p _ { 4 }$

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

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.

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.

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

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

The quadratic equation $p(x) = 0$ with real coefficients has purely imaginary roots. Then the equation
$$p(p(x)) = 0$$
has
(A) only purely imaginary roots
(B) all real roots
(C) two real and two purely imaginary roots
(D) neither real nor purely imaginary roots

Let $s , t , r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + i y ( x , y \in \mathbb { R } , i = \sqrt { - 1 } )$ of the equation $s z + t \bar { z } + r = 0$, where $\bar { z } = x - i y$. Then, which of the following statement(s) is (are) TRUE?
(A) If $L$ has exactly one element, then $| s | \neq | t |$
(B) If $| s | = | t |$, then $L$ has infinitely many elements
(C) The number of elements in $L \cap \{ z : | z - 1 + i | = 5 \}$ is at most 2
(D) If $L$ has more than one element, then $L$ has infinitely many elements

Let $S = \{ a + b \sqrt { 2 } : a , b \in \mathbb { Z } \} , T _ { 1 } = \left\{ ( - 1 + \sqrt { 2 } ) ^ { n } : n \in \mathbb { N } \right\}$, and $T _ { 2 } = \left\{ ( 1 + \sqrt { 2 } ) ^ { n } : n \in \mathbb { N } \right\}$.
Then which of the following statements is (are) TRUE?
(A) $\mathbb { Z } \bigcup T _ { 1 } \bigcup T _ { 2 } \subset S$
(B) $T _ { 1 } \cap \left( 0 , \frac { 1 } { 2024 } \right) = \phi$, where $\phi$ denotes the empty set.
(C) $T _ { 2 } \cap ( 2024 , \infty ) \neq \phi$
(D) For any given $a , b \in \mathbb { Z } , \cos ( \pi ( a + b \sqrt { 2 } ) ) + i \sin ( \pi ( a + b \sqrt { 2 } ) ) \in \mathbb { Z }$ if and only if $b = 0$, where $i = \sqrt { - 1 }$.

Given that the inverse trigonometric function assumes principal values only. Let $x , y$ be any two real numbers in $[ - 1,1 ]$ such that $\cos ^ { - 1 } x - \sin ^ { - 1 } y = \alpha , \frac { - \pi } { 2 } \leq \alpha \leq \pi$. Then, the minimum value of $x ^ { 2 } + y ^ { 2 } + 2 x y \sin \alpha$ is
(1) 0
(2) - 1
(3) $\frac { 1 } { 2 }$
(4) $- \frac { 1 } { 2 }$

On the complex plane, let $\bar { z }$ denote the complex conjugate of complex number $z$, and $i = \sqrt { - 1 }$. Select the correct options.
(1) If $z = 2 i$, then $z ^ { 3 } = 4 \bar { i } \bar { z }$
(2) If non-zero complex number $\alpha$ satisfies $\alpha ^ { 3 } = 4 i \bar { \alpha }$, then $| \alpha | = 2$
(3) If non-zero complex number $\alpha$ satisfies $\alpha ^ { 3 } = 4 i \bar { \alpha }$ and let $\beta = i \alpha$, then $\beta ^ { 3 } = 4 i \bar { \beta }$
(4) Among all non-zero complex numbers $z$ satisfying $z ^ { 3 } = 4 i \bar { z }$, the minimum possible value of the principal argument is $\frac { \pi } { 6 }$
(5) There are exactly 3 distinct non-zero complex numbers $z$ satisfying $z ^ { 3 } = 4 i \bar { z }$

Let $f(x)$ be a cubic polynomial with real coefficients. It is known that $f(-2 - 3i) = 0$ (where $i = \sqrt{-1}$), and the remainder when $f(x)$ is divided by $x^{2} + x - 2$ is 18. Select the correct options.
(1) $f(2 + 3i) = 0$
(2) $f(-2) = 18$
(3) The coefficient of the cubic term of $f(x)$ is negative
(4) $f(x) = 0$ has exactly one positive real root
(5) The center of symmetry of the graph $y = f(x)$ is in the first quadrant

Let $z$ be a nonzero complex number, and let $\alpha = |z|$ and $\beta$ be the argument of $z$, where $0 \leq \beta < 2\pi$ (where $\pi$ is the circumference ratio). For any positive integer $n$, let the real numbers $x_{n}$ and $y_{n}$ be the real and imaginary parts of $z^{n}$, respectively. Select the correct options.
(1) If $\alpha = 1$ and $\beta = \frac{3\pi}{7}$, then $x_{10} = x_{3}$
(2) If $y_{3} = 0$, then $y_{6} = 0$
(3) If $x_{3} = 1$, then $x_{6} = 1$
(4) If the sequence $\langle y_{n} \rangle$ converges, then $\alpha \leq 1$
(5) If the sequence $\langle x_{n} \rangle$ converges, then the sequence $\langle y_{n} \rangle$ also converges

Let the complex number $z$ have a nonzero imaginary part and $|z| = 2$. It is known that on the complex plane, $1, z, z^{3}$ are collinear. Select the correct options.
(1) $z \cdot \bar{z} = 2$
(2) The imaginary part of $\frac{z^{3} - z}{z - 1}$ is 0
(3) The real part of $z$ is $-\frac{1}{2}$
(4) $z$ satisfies $z^{2} - z + 4 = 0$
(5) On the complex plane, $-2, z, z^{2}$ are collinear

For complex numbers $z = a + b i ( b \neq 0 )$ and $w = c + d i$, if the sum $\mathbf { Z } + \mathbf { W }$ and the product $\mathbf { Z } \cdot \mathbf { W }$ are both real numbers, then I. $z$ and $w$ are conjugates of each other. II. $\mathrm { z } - \mathrm { w }$ is real. III. $z ^ { 2 } + w ^ { 2 }$ is real. Which of the following statements are true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III