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
2024 bac-spe-maths__polynesie-sept

4 maths questions

Q1 Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
A car dealership sells two types of vehicles:
  • $60\%$ are fully electric vehicles;
  • $40\%$ are rechargeable hybrid vehicles.
$75\%$ of buyers of fully electric vehicles and $52\%$ of buyers of rechargeable hybrid vehicles have the material possibility of installing a charging station at home.
A buyer is chosen at random and the following events are considered:
  • $E$: ``the buyer chooses a fully electric vehicle'';
  • $B$: ``the buyer has the possibility of installing a charging station at home''.
Throughout the exercise, probabilities should be rounded to the nearest thousandth if necessary.
  1. Calculate the probability that the buyer chooses a fully electric vehicle and has the possibility of installing a charging station at home.
    A weighted tree diagram may be used.
  2. Prove that $P(B) = 0.658$.
  3. A buyer has the possibility of installing a charging station at home. What is the probability that he chooses a fully electric vehicle?
  4. A sample of 20 buyers is chosen. This sampling is treated as drawing with replacement. Let $X$ be the random variable that gives the total number of buyers able to install a charging station at home among the sample of 20 buyers. a. Determine the nature and parameters of the probability distribution followed by $X$. b. Calculate $P(X = 8)$. c. Calculate the probability that at least 10 buyers can install a charging station. d. Calculate the expected value of $X$. e. The dealership manager decides to offer the installation of the charging station to buyers who have the possibility of installing one at home. This installation costs $1200$~\euro. On average, what amount should she plan to spend on this offer when selling 20 vehicles?
Q2 6 marks Differential equations Verification that a Function Satisfies a DE View
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer receives no points.
  1. Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \mathrm{e}^{x} + x.$$ Statement A: The function $f$ has the following variation table:
    $x$$-\infty$
    variations of $f$$+\infty$

    Statement B: The equation $f(x) = -2$ has two solutions in $\mathbb{R}$.
  2. Statement C: $$\lim_{x \rightarrow +\infty} \frac{\ln(x) - x^{2} + 2}{3x^{2}} = -\frac{1}{3}.$$
  3. Consider the function $k$ defined and continuous on $\mathbb{R}$ by $$k(x) = 1 + 2\mathrm{e}^{-x^{2}+1}$$ Statement D: There exists a primitive of the function $k$ that is decreasing on $\mathbb{R}$.
  4. Consider the differential equation $$(E): \quad 3y' + y = 1.$$ Statement E: The function $g$ defined on $\mathbb{R}$ by $$g(x) = 4\mathrm{e}^{-\frac{1}{3}x} + 1$$ is a solution of the differential equation $(E)$ with $g(0) = 5$.
  5. Statement F: Integration by parts allows us to obtain: $$\int_{0}^{1} x\mathrm{e}^{-x}\,\mathrm{d}x = 1 - 2\mathrm{e}^{-1}$$
Q3 4 marks Proof by induction Prove a summation identity by induction View
Consider a pyramid with a square base formed of identical balls stacked on top of each other:
  • the $1^{\text{st}}$ level, located at the highest level, is composed of 1 ball;
  • the $2^{\mathrm{nd}}$ level, just below, is composed of 4 balls;
  • the $3^{\mathrm{rd}}$ level has 9 balls;
  • the $n$-th level has $n^{2}$ balls.
For any integer $n \geqslant 1$, we denote by $u_{n}$ the number of balls that make up the $n$-th level from the top of the pyramid. Thus, $u_{n} = n^{2}$.
  1. Calculate the total number of balls in a pyramid with 4 levels.
  2. Consider the sequence $(S_{n})$ defined for any integer $n \geqslant 1$ by $$S_{n} = u_{1} + u_{2} + \ldots + u_{n}.$$ a. Calculate $S_{5}$ and interpret this result. b. Consider the pyramid function below written incompletely in Python. Copy and complete on your paper the box below so that, for any non-zero natural integer $n$, the instruction \texttt{pyramide(n)} returns the number of balls making up a pyramid with $n$ levels. \begin{verbatim} def pyramide(n) : S = 0 for i in range(1, n+1): S = ... return... \end{verbatim} c. Verify that for any natural integer $n$: $$\frac{n(n+1)(2n+1)}{6} + (n+1)^{2} = \frac{(n+1)(n+2)[2(n+1)+1]}{6}$$ d. Prove by induction that for any integer $n \geqslant 1$: $$S_{n} = \frac{n(n+1)(2n+1)}{6}.$$
  3. A merchant wishes to arrange oranges in a pyramid with a square base. He has 200 oranges. How many oranges does he use to build the largest possible pyramid?
Q4 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Consider a cube ABCDEFGH and the space is referred to the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. For any real $m$ belonging to the interval $[0; 1]$, we consider the points $K$ and $L$ with coordinates: $$K(m; 0; 0) \text{ and } L(1-m; 1; 1).$$
  1. Give the coordinates of points E and C in this coordinate system.
  2. In this question, $m = 0$. Thus, the point $\mathrm{L}(1; 1; 1)$ coincides with point G, the point $\mathrm{K}(0; 0; 0)$ coincides with point A and the plane (LEK) is therefore the plane (GEA). a. Justify that the vector $\overrightarrow{\mathrm{DB}} \left(\begin{array}{c} 1 \\ -1 \\ 0 \end{array}\right)$ is normal to the plane (GEA). b. Determine a Cartesian equation of the plane (GEA).
  3. In this question, $m$ is any real number in the interval $[0; 1]$. a. Prove that $\mathrm{CKEL}$ is a parallelogram. b. Justify that $\overrightarrow{KC} \cdot \overrightarrow{KE} = m(m-1)$. c. Prove that $\mathrm{CKEL}$ is a rectangle if, and only if, $m = 0$ or $m = 1$.
  4. In this question, $m = \frac{1}{2}$. Thus, L has coordinates $\left(\frac{1}{2}; 1; 1\right)$ and K has coordinates $\left(\frac{1}{2}; 0; 0\right)$. a. Prove that the parallelogram CKEL is then a rhombus. b. Using question 3.b., determine an approximate value to the nearest degree of the measure of the angle $\widehat{\mathrm{CKE}}$.