bac-s-maths

Papers (167)
2025
bac-spe-maths__amerique-nord_j1 4 bac-spe-maths__amerique-nord_j2 5 bac-spe-maths__amerique-sud_j1 4 bac-spe-maths__amerique-sud_j2 7 bac-spe-maths__asie-sept_j1 4 bac-spe-maths__asie_j1 4 bac-spe-maths__asie_j2 4 bac-spe-maths__caledonie_j1 4 bac-spe-maths__caledonie_j2 4 bac-spe-maths__centres-etrangers_j1 6 bac-spe-maths__centres-etrangers_j2 4 bac-spe-maths__metropole-sept_j1 4 bac-spe-maths__metropole-sept_j2 5 bac-spe-maths__metropole_j1 4 bac-spe-maths__metropole_j2 5 bac-spe-maths__polynesie-sept_j1 4 bac-spe-maths__polynesie_j1 4 bac-spe-maths__polynesie_j2 4
2024
bac-spe-maths__amerique-nord_j1 5 bac-spe-maths__amerique-nord_j2 4 bac-spe-maths__amerique-sud_j1 4 bac-spe-maths__amerique-sud_j2 4 bac-spe-maths__asie_j1 7 bac-spe-maths__asie_j2 4 bac-spe-maths__centres-etrangers_j1 5 bac-spe-maths__centres-etrangers_j2 4 bac-spe-maths__metropole-sept_j1 4 bac-spe-maths__metropole-sept_j2 4 bac-spe-maths__metropole_j1 4 bac-spe-maths__metropole_j2 4 bac-spe-maths__polynesie-sept 4 bac-spe-maths__polynesie_j1 4 bac-spe-maths__polynesie_j2 4 bac-spe-maths__suede 4
2023
bac-spe-maths__amerique-nord_j1 4 bac-spe-maths__amerique-nord_j2 5 bac-spe-maths__amerique-sud_j1 6 bac-spe-maths__amerique-sud_j2 4 bac-spe-maths__asie_j1 4 bac-spe-maths__asie_j2 4 bac-spe-maths__caledonie_j1 4 bac-spe-maths__caledonie_j2 4 bac-spe-maths__centres-etrangers_j1 9 bac-spe-maths__centres-etrangers_j2 8 bac-spe-maths__europe_j1 4 bac-spe-maths__europe_j2 4 bac-spe-maths__metropole-sept_j1 7 bac-spe-maths__metropole-sept_j2 4 bac-spe-maths__metropole_j1 8 bac-spe-maths__metropole_j2 4 bac-spe-maths__polynesie-sept 4 bac-spe-maths__polynesie_j1 4 bac-spe-maths__polynesie_j2 4 bac-spe-maths__reunion_j1 4 bac-spe-maths__reunion_j2 4
2022
bac-spe-maths__amerique-nord_j1 4 bac-spe-maths__amerique-nord_j2 4 bac-spe-maths__amerique-sud_j1 4 bac-spe-maths__amerique-sud_j2 4 bac-spe-maths__asie_j1 4 bac-spe-maths__asie_j2 4 bac-spe-maths__caledonie_j1 4 bac-spe-maths__caledonie_j2 4 bac-spe-maths__centres-etrangers_j1 4 bac-spe-maths__centres-etrangers_j2 4 bac-spe-maths__madagascar_j1 4 bac-spe-maths__madagascar_j2 4 bac-spe-maths__metropole-sept_j1 9 bac-spe-maths__metropole-sept_j2 4 bac-spe-maths__metropole_j1 4 bac-spe-maths__metropole_j2 4 bac-spe-maths__polynesie-sept 4 bac-spe-maths__polynesie_j1 4 bac-spe-maths__polynesie_j2 4
2021
bac-spe-maths__amerique-nord 5 bac-spe-maths__asie_j1 5 bac-spe-maths__asie_j2 5 bac-spe-maths__centres-etrangers_j1 9 bac-spe-maths__centres-etrangers_j2 7 bac-spe-maths__metropole-juin_j1 5 bac-spe-maths__metropole-juin_j2 5 bac-spe-maths__metropole-sept_j1 8 bac-spe-maths__metropole-sept_j2 5 bac-spe-maths__metropole_j1 5 bac-spe-maths__metropole_j2 5 bac-spe-maths__polynesie 5
2020
antilles-guyane 9 caledonie 5 metropole 9 polynesie 9
2019
amerique-nord 5 amerique-sud 6 antilles-guyane 5 asie 6 caledonie 3 centres-etrangers 6 integrale-annuelle 4 liban 9 metropole 5 metropole-sept 5 polynesie 5
2018
amerique-nord 5 amerique-sud 5 antilles-guyane 6 asie 4 caledonie 5 centres-etrangers 17 liban 6 metropole 3 metropole-sept 5 polynesie 7 pondichery 7
2017
amerique-nord 6 amerique-sud 5 antilles-guyane 6 asie 8 caledonie 6 centres-etrangers 8 liban 5 metropole 5 metropole-sept 4 polynesie 7
2016
amerique-nord 5 amerique-sud 6 antilles-guyane 10 asie 5 caledonie 6 centres-etrangers 8 liban 6 metropole 7 metropole-sept 4 polynesie 5 pondichery 6
2015
amerique-nord 4 amerique-sud 8 antilles-guyane 4 asie 7 caledonie 7 centres-etrangers 9 liban 5 metropole 7 metropole-sept 9 polynesie 6 pondichery 5
2014
amerique-nord 4 amerique-sud 7 antilles-guyane 5 asie 4 caledonie 7 centres-etrangers 7 liban 7 metropole 5 metropole-sept 5 polynesie 5 pondichery 4
2013
amerique-nord 5 amerique-sud 4 antilles-guyane 9 asie 5 caledonie 5 centres-etrangers 8 liban 4 metropole 5 metropole-sept 5 polynesie 4 pondichery 4
2007
integrale-annuelle2 6
2019 metropole

5 maths questions

Q1 Hyperbolic functions View
Exercise 1 -- Part A
We consider the function $f$ defined on the set $\mathbb{R}$ of real numbers by: $$f(x) = \frac{7}{2} - \frac{1}{2}\left(\mathrm{e}^{x} + \mathrm{e}^{-x}\right)$$
  1. a. Determine the limit of the function $f$ at $+\infty$. b. Show that the function $f$ is strictly decreasing on the interval $[0; +\infty[$. c. Show that the equation $f(x) = 0$ admits, on the interval $[0; +\infty[$, a unique solution, which we denote $\alpha$.
  2. By noting that, for all real $x$, $f(-x) = f(x)$, justify that the equation $f(x) = 0$ admits exactly two solutions in $\mathbb{R}$ and that they are opposite.

Part B
The plane is given an orthonormal coordinate system with unit 1 meter. The function $f$ and the real number $\alpha$ are defined in Part A. In the rest of the exercise, we model a greenhouse arch by the curve $\mathscr{C}$ of the function $f$ on the interval $[-\alpha; +\alpha]$.
  1. Calculate the height of an arch.
  2. a. In this question, we propose to calculate the exact value of the length of the curve $\mathscr{C}$ on the interval $[0; \alpha]$. It is admitted that this length is given, in meters, by the integral: $$I = \int_{0}^{\alpha} \sqrt{1 + \left(f'(x)\right)^{2}} \, dx$$ Show that, for all real $x$, we have: $1 + \left(f'(x)\right)^{2} = \frac{1}{4}\left(\mathrm{e}^{x} + \mathrm{e}^{-x}\right)^{2}$ b. Deduce the value of the integral $I$ as a function of $\alpha$. Justify that the length of an arch, in meters, is equal to: $\mathrm{e}^{\alpha} - \mathrm{e}^{-\alpha}$.

Part C
We wish to build a garden greenhouse in the shape of a tunnel. We fix four metal arches to the ground, whose shape is that described in the previous part, spaced 1.5 meters apart. On the south facade, we plan an opening modeled by the rectangle $ABCD$ with width 1 meter and length 2 meters.
  1. Show that the quantity of sheet necessary to cover the south and north facades is given, in $m^2$, by: $$\mathscr{A} = 4\int_{0}^{\alpha} f(x)\,dx - 2$$
  2. We take 1.92 as an approximate value of $\alpha$. Determine, to the nearest $\mathrm{m}^2$, the total area of plastic sheet necessary to build this greenhouse.
Q2 5 marks Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View
Exercise 2 (5 points)
An online platform offers two types of video games: a game of type $A$ and a game of type $B$.
Part A
The durations of games of type $A$ and type $B$, expressed in minutes, can be modeled respectively by two random variables $X_A$ and $X_B$. The random variable $X_A$ follows the uniform distribution on the interval $[9; 25]$. The random variable $X_B$ follows the normal distribution with mean $\mu$ and standard deviation 3.
  1. a. Calculate the average duration of a game of type $A$. b. Specify using the graph the average duration of a game of type $B$.
  2. We choose at random, with equal probability, a game type. What is the probability that the duration of a game is less than 20 minutes? Give the result rounded to the nearest hundredth.

Part B
It is admitted that, as soon as the player completes a game, the platform proposes a new game according to the following model:
  • if the player completes a game of type $A$, the platform proposes to play again a game of type $A$ with probability 0.8;
  • if the player completes a game of type $B$, the platform proposes to play again a game of type $B$ with probability 0.7.
For a natural number $n$ greater than or equal to 1, we denote $A_n$ and $B_n$ the events: $A_n$: ``the $n$-th game is a game of type $A$.'' $B_n$: ``the $n$-th game is a game of type $B$.'' For any natural number $n$ greater than or equal to 1, we denote $a_n$ the probability of event $A_n$.
  1. a. Copy and complete the probability tree. b. Show that for any natural number $n \geqslant 1$, we have: $a_{n+1} = 0.5\,a_n + 0.3$.

In the rest of the exercise, we denote $a$ the probability that the player plays game $A$ during his first game, where $a$ is a real number belonging to the interval $[0; 1]$. The sequence $(a_n)$ is therefore defined by: $a_1 = a$, and for any natural number $n \geqslant 1$, $a_{n+1} = 0.5\,a_n + 0.3$.
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  1. Study of a particular case. In this question, we assume that $a = 0.5$. a. Show by induction that for any natural number $n \geqslant 1$, we have: $0 \leqslant a_n \leqslant 0.6$. b. Show that the sequence $(a_n)$ is increasing. c. Show that the sequence $(a_n)$ is convergent and specify its limit.
  2. Study of the general case. In this question, the real number $a$ belongs to the interval $[0; 1]$. We consider the sequence $(u_n)$ defined for any natural number $n \geqslant 1$ by $u_n = a_n - 0.6$. a. Show that the sequence $(u_n)$ is a geometric sequence. b. Deduce that for any natural number $n \geqslant 1$, we have: $a_n = (a - 0.6) \times 0.5^{n-1} + 0.6$. c. Determine the limit of the sequence $(a_n)$. Does this limit depend on the value of $a$? d. The platform broadcasts an advertisement inserted at the beginning of games of type $A$ and another inserted at the beginning of games of type $B$. Which advertisement should be the most viewed by a player intensively playing video games?
Q3 Complex Numbers Arithmetic True/False or Property Verification Statements View
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: $15\ln(2) \leqslant \ln(A) \leqslant 16\ln(2)$
Q4A Number Theory Linear Diophantine Equations View
Exercise 4 (For candidates who have followed the specialty course)
We denote $\mathbb{Z}$ the set of integers. In this exercise, we study the set $S$ of matrices that can be written in the form $A = \left(\begin{array}{cc} a & b \\ c & d \end{array}\right)$, where $a, b, c$ and $d$ belong to the set $\mathbb{Z}$ and satisfy: $ad - bc = 1$. We denote $I$ the identity matrix $I = \left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)$.
Part A
  1. Verify that the matrix $A = \left(\begin{array}{rr} 6 & 5 \\ -5 & -4 \end{array}\right)$ belongs to the set $S$.
  2. Show that there exist exactly four matrices of the form $A = \left(\begin{array}{ll} a & 2 \\ 3 & d \end{array}\right)$ belonging to the set $S$; state them explicitly.
  3. a. Solve in $\mathbb{Z}$ the equation $(E): 5x - 2y = 1$. We may note that the pair $(1; 2)$ is a particular solution of this equation. b. Deduce that there exist infinitely many matrices of the form $A = \left(\begin{array}{cc} a & b \\ 2 & 5 \end{array}\right)$ that belong to the set $S$. Describe these matrices.

Part B
In this part, we denote $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ a matrix belonging to the set $S$. We recall that $a$, $b$, $c$ and $d$ are integers such that $ad - bc = 1$.
  1. Show that the integers $a$ and $b$ are coprime.
  2. Let $B$ be the matrix: $B = \left(\begin{array}{rr} d & -b \\ -c & a \end{array}\right)$ a. Calculate the product $AB$. It is admitted that $AB = BA$. b. Deduce that the matrix $A$ is invertible and give its inverse matrix $A^{-1}$. c. Show that $A^{-1}$ belongs to the set $S$.
  3. Let $x$ and $y$ be two integers. We denote $x'$ and $y'$ the integers such that $\binom{x'}{y'} = A\binom{x}{y}$. a. Show that $x = dx' - by'$. It is admitted that likewise $y = ay' - cx'$. b. We denote $D$ the GCD of $x$ and $y$ and we denote $D'$ the GCD of $x'$ and $y'$. Show that $D = D'$.
  4. We consider the sequences of natural numbers $(x_n)$ and $(y_n)$ defined by: $x_0 = 2019$, $y_0 = 673$ and for any natural number $n$: $$\left\{\begin{array}{l} x_{n+1} = 2x_n + 3y_n \\ y_{n+1} = x_n + 2y_n \end{array}\right.$$ Using the previous question, determine, for any natural number $n$, the GCD of the integers $x_n$ and $y_n$.
Q4B Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 4 (For candidates who have not followed the specialty course)
We consider a cube $ABCDEFGH$ with edge length 1. We denote $I$ the midpoint of segment $[EF]$, $J$ the midpoint of segment $[EH]$ and $K$ the point of segment $[AD]$ such that $\overrightarrow{AK} = \frac{1}{4}\overrightarrow{AD}$. We denote $\mathscr{P}$ the plane passing through $I$ and parallel to the plane $(FHK)$.
Part A
In this part, the constructions requested will be performed without justification on the figure given in the appendix.
  1. The plane $(FHK)$ intersects the line $(AE)$ at a point which we denote $M$. Construct the point $M$.
  2. Construct the cross-section of the cube by the plane $\mathscr{P}$.

Part B
In this part, we equip the space with the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE})$. We recall that $\mathscr{P}$ is the plane passing through $I$ and parallel to the plane $(FHK)$.
  1. a. Show that the vector $\vec{n}\left(\begin{array}{c} 4 \\ 4 \\ -3 \end{array}\right)$ is a normal vector to the plane $(FHK)$. b. Deduce that a Cartesian equation of the plane $(FHK)$ is: $4x + 4y - 3z - 1 = 0$. c. Determine an equation of the plane $\mathscr{P}$. d. Calculate the coordinates of the point $M'$, the point of intersection of the plane $\mathscr{P}$ and the line $(AE)$.
  2. We denote $\Delta$ the line passing through point $E$ and perpendicular to the plane $\mathscr{P}$. a. Determine a parametric representation of the line $\Delta$. b. Calculate the coordinates of point $L$, the intersection of line $\Delta$ and plane $(ABC)$. c. Draw the line $\Delta$ on the figure provided in the appendix. d. Are the lines $\Delta$ and $(BF)$ intersecting? What about the lines $\Delta$ and $(CG)$? Justify.