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
2023 bac-spe-maths__amerique-nord_j2

5 maths questions

Q1A Applied differentiation MCQ on derivative and graph interpretation View
The plane is equipped with an orthogonal coordinate system. We consider a function $f$ defined and differentiable on $\mathbb{R}$. We denote $f^{\prime}$ its derivative function. The representative curve of the derivative function $f^{\prime}$ is given.
In this part, results will be obtained by graphical reading of the representative curve of the derivative function $f^{\prime}$. No justification is required.
  1. Give the direction of variation of the function $f$ on $\mathbb{R}$. Use approximate values if necessary.
  2. Give the intervals on which the function $f$ appears to be convex.
Q1B Differentiating Transcendental Functions Full function study with transcendental functions View
We admit that the function $f$ from part $\mathbf{A}$ is defined on $\mathbb{R}$ by
$$f(x) = \left(x^{2} - 5x + 6\right)\mathrm{e}^{x}$$
We denote $\mathscr{C}$ the representative curve of the function $f$ in a coordinate system.
  1. a. Determine the limit of the function $f$ at $+\infty$. b. Determine the limit of the function $f$ at $-\infty$.
  2. Show that, for all real $x$, we have $f^{\prime}(x) = \left(x^{2} - 3x + 1\right)\mathrm{e}^{x}$.
  3. Deduce the direction of variation of the function $f$.
  4. Determine the reduced equation of the tangent line $(\mathscr{T})$ to the curve $\mathscr{C}$ at the point with abscissa 0.

We admit that the function $f$ is twice differentiable on $\mathbb{R}$. We denote $f^{\prime\prime}$ the second derivative function of $f$. We admit that, for all real $x$, we have $f^{\prime\prime}(x) = (x+1)(x-2)\mathrm{e}^{x}$.
5. a. Study the convexity of the function $f$ on $\mathbb{R}$. b. Show that, for all $x$ belonging to the interval $[-1; 2]$, we have $f(x) \leqslant x + 6$.
Q2 Geometric Sequences and Series Applied Geometric Model with Contextual Interpretation View
We study a group of 3000 athletes who practice either athletics in club A or basketball in club B. In 2023, club A has 1700 members and club B has 1300. We decide to model the number of members of club A and club B respectively by two sequences $(a_{n})$ and $(b_{n})$, where $n$ denotes the rank of the year starting from 2023. The year 2023 corresponds to rank 0. We then have $a_{0} = 1700$ and $b_{0} = 1300$. For our study, we make the following assumptions:
  • during the study, no athlete leaves the group;
  • each year, 15\% of the athletes in club A leave this club and join club B;
  • each year, 10\% of the athletes in club B leave this club and join club A.

  1. Calculate the number of members of each club in 2024.
  2. For all natural integer $n$, determine a relation linking $a_{n}$ and $b_{n}$.
  3. Show that the sequence $(a_{n})$ satisfies the following relation for all natural integer $n$: $$a_{n+1} = 0{,}75\, a_{n} + 300.$$
  4. a. Prove by induction that for all natural integer $n$, we have: $$1200 \leqslant a_{n+1} \leqslant a_{n} \leqslant 1700.$$ b. Deduce that the sequence $(a_{n})$ converges.
  5. Let $\left(v_{n}\right)$ be the sequence defined for all natural integer $n$ by $v_{n} = a_{n} - 1200$. a. Prove that the sequence $\left(v_{n}\right)$ is geometric. b. Express $v_{n}$ as a function of $n$. c. Deduce that for all natural integer $n$, $a_{n} = 500 \times 0{,}75^{n} + 1200$.
  6. a. Determine the limit of the sequence $(a_{n})$. b. Interpret the result of the previous question in the context of the exercise.
  7. a. Copy and complete the Python program below so that it returns the smallest value of $n$ from which the number of members of club A is strictly less than 1280. \begin{verbatim} def seuil() : n = 0 A = 1700 while... : n=n+1 A = ... return... \end{verbatim} b. Determine the value returned when the seuil function is called.
Q3 5 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
In space equipped with an orthonormal coordinate system with unit 1 cm, we consider the points
$$\mathrm{D}(3;1;5), \quad \mathrm{E}(3;-2;-1), \quad \mathrm{F}(-1;2;1), \quad \mathrm{G}(3;2;-3).$$
  1. a. Determine the coordinates of the vectors $\overrightarrow{\mathrm{EF}}$ and $\overrightarrow{\mathrm{FG}}$. b. Justify that the points $\mathrm{E}$, $\mathrm{F}$ and $\mathrm{G}$ are not collinear.
  2. a. Determine a parametric representation of the line (FG). b. We call H the point with coordinates $(2; 2; -2)$. Verify that H is the orthogonal projection of E onto the line (FG). c. Show that the area of triangle EFG is equal to $12\text{ cm}^{2}$.
  3. a. Prove that the vector $\vec{n}\begin{pmatrix}2\\1\\2\end{pmatrix}$ is a normal vector to the plane (EFG). b. Determine a Cartesian equation of the plane (EFG). c. Determine a parametric representation of the line $(d)$ passing through point D and orthogonal to the plane (EFG). d. We denote K the orthogonal projection of point D onto the plane (EFG). Using the previous questions, calculate the coordinates of point K.
  4. a. Verify that the distance $DK$ is equal to 5 cm. b. Deduce the volume of the tetrahedron DEFG.
Q4 Discrete Probability Distributions Multiple Choice: Direct Probability or Distribution Calculation View
This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, a multiple answer, or the absence of an answer to a question neither awards nor deducts points. The five questions are independent.
  1. We consider the function $f$ defined on the interval $]1; +\infty[$ by $$f(x) = 0{,}05 - \frac{\ln x}{x-1}$$ The limit of the function $f$ at $+\infty$ is equal to: a. $+\infty$ b. 0.05 c. $-\infty$ d. 0
  2. We consider a function $h$ continuous on the interval $[-2;4]$ such that: $$h(-1) = 0, \quad h(1) = 4, \quad h(3) = -1.$$ We can affirm that: a. the function $h$ is increasing on the interval $[-1; 1]$. b. the function $h$ is positive on the interval $[-1; 1]$. c. there exists at least one real number $a$ in the interval $[1; 3]$ such that $h(a) = 1$. d. the equation $h(x) = 1$ has exactly two solutions in the interval $[-2; 4]$.
  3. We consider two sequences $(u_{n})$ and $(v_{n})$ with strictly positive terms such that $\lim_{n \rightarrow +\infty} u_{n} = +\infty$ and $(v_{n})$ converges to 0. We can affirm that: a. the sequence $\left(\dfrac{1}{v_{n}}\right)$ converges. b. the sequence $\left(\dfrac{v_{n}}{u_{n}}\right)$ converges. c. the sequence $(u_{n})$ is increasing. d. $\lim_{n \rightarrow +\infty} \left(-u_{n}\right)^{n} = -\infty$
  4. To participate in a game, a player must pay $4\,€$. They then roll a fair six-sided die:
    • if they get 1, they win $12\,€$;
    • if they get an even number, they win $3\,€$;
    • otherwise, they win nothing.
    On average, the player: a. wins $3.50\,€$ b. loses $3\,€$. c. loses $1.50\,€$ d. loses $0.50\,€$.
  5. We consider the random variable $X$ following the binomial distribution $\mathscr{B}(3; p)$. We know that $P(X = 0) = \dfrac{1}{125}$. We can affirm that: a. $p = \dfrac{1}{5}$ b. $P(X = 1) = \dfrac{124}{125}$ c. $p = \dfrac{4}{5}$ d. $P(X = 1) = \dfrac{4}{5}$