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__polynesie-sept

4 maths questions

Q1 4 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
A car dealership sells vehicles with electric motors and vehicles with thermal engines. Some customers, before visiting the dealership website, consulted the dealership's digital platform. It was observed that:
  • $20\%$ of customers are interested in vehicles with electric motors and $80\%$ prefer to purchase a vehicle with a thermal engine;
  • when a customer wishes to buy a vehicle with an electric motor, the probability that the customer consulted the digital platform is 0.5;
  • when a customer wishes to buy a vehicle with a thermal engine, the probability that the customer consulted the digital platform is 0.375.
Consider the following events:
  • $C$: ``a customer consulted the digital platform'';
  • $E$: ``a customer wishes to acquire a vehicle with an electric motor'';
  • $T$: ``a customer wishes to acquire a vehicle with a thermal engine''.
Customers make choices independently of one another.
  1. a. Calculate the probability that a randomly chosen customer wishes to acquire a vehicle with an electric motor and consulted the digital platform.
    A weighted tree diagram may be used. b. Prove that $P(C) = 0.4$. c. Suppose that a customer consulted the digital platform. Calculate the probability that the customer wishes to buy a vehicle with an electric motor.
  2. The dealership welcomes an average of 17 clients daily. Let $X$ be the random variable giving the number of clients wishing to acquire a vehicle with an electric motor. a. Specify the nature and parameters of the probability distribution followed by $X$. b. Calculate the probability that at least three of the clients wish to buy a vehicle with an electric motor during a day. Give the result rounded to $10^{-2}$.
Q2 6 marks Applied differentiation Full function study (variation table, limits, asymptotes) View
Part A Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \left(x + \frac{1}{2}\right)e^{-x} + x.$$
  1. Determine the limits of $f$ at $-\infty$ and at $+\infty$.
  2. It is admitted that $f$ is twice differentiable on $\mathbb{R}$. a. Prove that, for all $x \in \mathbb{R}$, $$f''(x) = \left(x - \frac{3}{2}\right)e^{-x}.$$ b. Deduce the variations and the minimum of the function $f'$ on $\mathbb{R}$. c. Justify that for all $x \in \mathbb{R}$, $f'(x) > 0$. d. Deduce that the equation $f(x) = 0$ admits a unique solution on $\mathbb{R}$. e. Give a value rounded to $10^{-3}$ of this solution.

Part B Consider a function $h$, defined and differentiable on $\mathbb{R}$, having an expression of the form $$h(x) = (ax + b)e^{-x} + x \text{, where } a \text{ and } b \text{ are two real numbers.}$$ In an orthonormal coordinate system shown below are:
  • the representative curve $\mathscr{C}_h$ of the function $h$;
  • the points A and B with coordinates respectively $(-2; -2.5)$ and $(2; 3.5)$.
  1. Conjecture, with the precision allowed by the graph, the abscissae of any inflection points of the representative curve of the function $h$.
  2. Knowing that the function $h$ admits on $\mathbb{R}$ a second derivative with expression $$h''(x) = -\frac{3}{2}e^{-x} + xe^{-x}.$$ validate or not the previous conjecture.
  3. Determine an equation of the line (AB).
  4. Knowing that the line (AB) is tangent to the representative curve of the function $h$ at the point with abscissa 0, deduce the values of $a$ and $b$.
Q3 5 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \frac{3}{4}x^2 - 2x + 3$$
  1. Draw the table of variations of $f$ on $\mathbb{R}$.
  2. Deduce that for all $x$ belonging to the interval $\left[\frac{4}{3}; 2\right]$, $f(x)$ belongs to the interval $\left[\frac{4}{3}; 2\right]$.
  3. Prove that for all real $x$, $x \leq f(x)$. For this, one may prove that for all real $x$: $$f(x) - x = \frac{3}{4}(x - 2)^2.$$
Consider the sequence $(u_n)$ defined by a real $u_0$ and for all natural integer $n$: $$u_{n+1} = f(u_n).$$ We have therefore, for all natural integer $n$, $$u_{n+1} = \frac{3}{4}u_n^2 - 2u_n + 3.$$
    \setcounter{enumi}{3}
  1. Study of the case: $\frac{4}{3} \leq u_0 \leq 2$. a. Prove by induction that, for all natural integer $n$, $$u_n \leq u_{n+1} \leq 2.$$ b. Deduce that the sequence $(u_n)$ is convergent. c. Prove that the limit of the sequence is equal to 2.
  2. Study of the particular case: $u_0 = 3$. It is admitted that in this case the sequence $(u_n)$ tends to $+\infty$. Copy and complete the following ``threshold'' function written in Python, so that it returns the smallest value of $n$ such that $u_n$ is greater than or equal to 100. \begin{verbatim} def seuil() : u = 3 n = 0 while ... u = ... n = ... return n \end{verbatim}
  3. Study of the case: $u_0 > 2$. Using a proof by contradiction, show that $(u_n)$ is not convergent.
Q4 5 marks Vectors: Lines & Planes MCQ: Identify Correct Equation or Representation View
Space is equipped with an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$ in which we consider:
  • the points $A(6; -6; 6)$, $B(-6; 0; 6)$ and $C(-2; -2; 11)$.
  • the line $(d)$ orthogonal to the two secant lines $(AB)$ and $(BC)$ and passing through point A;
  • the line $(d')$ with parametric representation:
$$\left\{\begin{aligned} x &= -6 - 8t \\ y &= 4t, \text{ with } t \in \mathbb{R}. \\ z &= 6 + 5t \end{aligned}\right.$$
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers or absence of answer to a question neither awards nor deducts points. No justification is required.
Question 1 Among the following vectors, which is a direction vector of the line $(d)$? a. $\overrightarrow{u_1}\left(\begin{array}{c}-6 \\ 3 \\ 0\end{array}\right)$ b. $\overrightarrow{u_2}\left(\begin{array}{l}1 \\ 2 \\ 6\end{array}\right)$ c. $\overrightarrow{u_3}\left(\begin{array}{c}1 \\ 2 \\ 0.2\end{array}\right)$ d. $\overrightarrow{u_4}\left(\begin{array}{l}1 \\ 2 \\ 0\end{array}\right)$
Question 2 Among the following equations, which is a parametric representation of the line (AB)? a. $\left\{\begin{aligned}x &= 2t + 6 \\ y &= -6 \text{ with } t \in \mathbb{R} \\ z &= t + 6\end{aligned}\right.$ b. $\left\{\begin{aligned}x &= 2t - 6 \\ y &= -6 \text{ with } t \in \mathbb{R} \\ z &= -t - 6\end{aligned}\right.$ c. $\left\{\begin{aligned}x &= 2t + 6 \\ y &= -t - 6 \text{ with } t \in \mathbb{R} \\ z &= 6\end{aligned}\right.$ d. $\left\{\begin{aligned}x &= 2t + 6 \\ y &= t - 6 \text{ with } t \in \mathbb{R} \\ z &= 6\end{aligned}\right.$
Question 3
A direction vector of the line $(d')$ is: a. $\overrightarrow{v_1}\left(\begin{array}{c}-6 \\ 0 \\ 6\end{array}\right)$ b. $\overrightarrow{v_2}\left(\begin{array}{c}-14 \\ 4 \\ 11\end{array}\right)$ c. $\overrightarrow{v_3}\left(\begin{array}{c}8 \\ -4 \\ -5\end{array}\right)$ d. $\overrightarrow{v_4}\left(\begin{array}{l}8 \\ 4 \\ 5\end{array}\right)$
Question 4 Which of the following four points belongs to the line $(d')$? a. $M_1(50; -28; -29)$ b. $M_2(-14; -4; 1)$ c. $M_3(2; -4; -1)$ d. $M_4(-3; 0; 3)$
Question 5 The plane with equation $x = 1$ has as normal vector: a. $\overrightarrow{n_1}\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)$ b. $\overrightarrow{n_2}\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)$ c. $\overrightarrow{n_3}\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)$ d. $\overrightarrow{n_4}\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right)$