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_j2

4 maths questions

Q1 4 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
A survey conducted in France provides the following information:
  • $60\%$ of people over 15 years old intend to watch the Paris 2024 Olympic and Paralympic Games (OPG) on television;
  • among those who intend to watch the OPG, 8 out of 9 people declare that they regularly practice a sport.
A person over 15 years old is chosen at random. The following events are considered:
  • $J$: ``the person intends to watch the Paris 2024 OPG on television'';
  • $S$: ``the chosen person declares that they regularly practice a sport''.
We denote by $\bar{J}$ and $\bar{S}$ their complementary events.
In questions 1. and 2., probabilities will be given in the form of an irreducible fraction.
  1. Demonstrate that the probability that the chosen person intends to watch the Paris 2024 OPG on television and declares that they regularly practice a sport is $\frac{8}{15}$. A weighted tree diagram may be used.
    According to this survey, two out of three people over 15 years old declare that they regularly practice a sport.
  2. [2.] a. Calculate the probability that the chosen person does not intend to watch the Paris 2024 OPG on television and declares that they regularly practice a sport. b. Deduce the probability of $S$ given $\bar{J}$ denoted $P_{\bar{J}}(S)$.
    In the rest of the exercise, results will be rounded to the nearest thousandth.
  3. [3.] As part of a promotional operation, 30 people over 15 years old are chosen at random. This choice is treated as sampling with replacement. Let $X$ be the random variable that gives the number of people declaring that they regularly practice a sport among the 30 people. a. Determine the nature and parameters of the probability distribution followed by $X$. b. Calculate the probability that exactly 16 people declare that they regularly practice a sport among the 30 people. c. The French judo federation wishes to offer a ticket for the final of the mixed team judo event at the Arena Champ-de-Mars for each person declaring that they regularly practice a sport among these 30 people. The price of a ticket is $380\,€$ and a budget of 10000 euros is available for this operation. What is the probability that this budget is insufficient?
Q2 5 marks Differential equations First-Order Linear DE: General Solution View
This exercise is a multiple choice questionnaire (MCQ) comprising five questions. The five questions are independent. For each question, only one of the four answers is correct.
  1. The solution $f$ of the differential equation $y' = -3y + 7$ such that $f(0) = 1$ is the function defined on $\mathbb{R}$ by:
    A. $f(x) = \mathrm{e}^{-3x}$
    B. $f(x) = -\frac{4}{3}\mathrm{e}^{-3x} + \frac{7}{3}$
    C. $f(x) = \mathrm{e}^{-3x} + \frac{7}{3}$
    D. $f(x) = -\frac{10}{3}\mathrm{e}^{-3x} - \frac{7}{3}$
  2. The curve of a function $f$ defined on $[0; +\infty[$ is given below.
    A bound for the integral $I = \int_{1}^{5} f(x)\,\mathrm{d}x$ is:
    A. $0 \leqslant I \leqslant 4$
    B. $1 \leqslant I \leqslant 5$
    C. $5 \leqslant I \leqslant 10$
    D. $10 \leqslant I \leqslant 15$
  3. Consider the function $g$ defined on $\mathbb{R}$ by $g(x) = x^2 \ln\left(x^2 + 4\right)$.
    Then $\int_{0}^{2} g'(x)\,\mathrm{d}x$ equals, to $10^{-1}$ near:
    A. 4.9
    B. 8.3
    C. 1.7
    D. 7.5
  4. A teacher teaches the mathematics specialization in a class of 31 final year students. She wants to form a group of 5 students. In how many different ways can she form such a group of 5 students?
    A. $31^5$
    B. $31 \times 30 \times 29 \times 28 \times 27$
    C. $31 + 30 + 29 + 28 + 27$
    D. $\binom{31}{5}$
  5. The teacher is now interested in the other specialization of the 31 students in her group:
    • 10 students chose the physics-chemistry specialization;
    • 20 students chose the SES specialization;
    • 1 student chose the Spanish LLCE specialization.
    She wants to form a group of 5 students containing exactly 3 students who chose the SES specialization. In how many different ways can she form such a group?
    A. $\binom{20}{3} \times \binom{11}{2}$
    B. $\binom{20}{3} + \binom{11}{2}$
    C. $\binom{20}{3}$
    D. $20^3 \times 11^2$
Q3 6 marks Sequences and series, recurrence and convergence Algorithm and programming for sequences View
Consider the sequence $(u_n)$ defined by: $$u_0 = 8 \text{ and for every natural number } n,\quad u_{n+1} = u_n - \ln\left(\frac{u_n}{4}\right).$$
  1. a. Give the values rounded to the nearest hundredth of $u_1$ and $u_2$. b. Consider the mystery function defined below in Python. We admit that, for every strictly positive real number $a$, $\log(a)$ returns the value of the natural logarithm of $a$. \begin{verbatim} def mystery(k) : u = 8 S = 0 for i in range(k) : S = S + u u = u - log( u / 4) return S \end{verbatim} The execution of \texttt{mystery(10)} returns 58.44045206721732. What does this result represent? c. Modify the previous function so that it returns the average of the first $k$ terms of the sequence $(u_n)$.
  2. Consider the function $f$ defined and differentiable on $]0; +\infty[$ by: $$f(x) = x - \ln\left(\frac{x}{4}\right).$$ Study the variations of $f$ on $]0; +\infty[$ and draw its variation table. The exact value of the minimum of $f$ on $]0; +\infty[$ will be specified. Limits are not required.
    In the rest of the exercise, it will be noted that for every natural number $n$, $u_{n+1} = f(u_n)$.
  3. a. Prove, by induction, that for every natural number $n$, we have: $$1 \leqslant u_{n+1} \leqslant u_n.$$ b. Deduce that the sequence $(u_n)$ converges to a real limit.
    Let $\ell$ denote the value of this limit. c. Solve the equation $f(x) = x$. d. Deduce the value of $\ell$.
Q4 5 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
A municipality decides to replace the traditional July 14 fireworks with a luminous drone show. For drone piloting, space is equipped with an orthonormal reference frame $(\mathrm{O};\vec{\imath},\vec{\jmath},\vec{k})$ whose unit is one hundred meters.
The position of each drone is modeled by a point and each drone is sent from a starting point D with coordinates $(2;5;1)$. It is desired to form figures with drones by positioning them in the same plane $\mathscr{P}$. Three drones are positioned at points $\mathrm{A}(-1;-1;17)$, $\mathrm{B}(4;-2;4)$ and $\mathrm{C}(1;-3;7)$.
  1. Justify that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.
    In the following, we denote by $\mathscr{P}$ the plane (ABC) and we consider the vector $\vec{n}\begin{pmatrix}2\\-3\\1\end{pmatrix}$.
  2. a. Justify that $\vec{n}$ is normal to the plane (ABC). b. Prove that a Cartesian equation of the plane $\mathscr{P}$ is $2x - 3y + z - 18 = 0$.
  3. The drone pilot decides to send a fourth drone taking as trajectory the line $d$ whose parametric representation is given by $$d : \left\{\begin{array}{rl} x &= 3t + 2 \\ y &= t + 5 \\ z &= 4t + 1 \end{array},\text{ with } t \in \mathbb{R}.\right.$$ a. Determine a direction vector of the line $d$. b. So that this new drone is also placed in the plane $\mathscr{P}$, determine by calculation the coordinates of point E, the intersection of the line $d$ with the plane $\mathscr{P}$.
  4. The drone pilot decides to send a fifth drone along the line $\Delta$ which passes through point $\mathrm{D}$ and which is perpendicular to the plane $\mathscr{P}$. This fifth drone is also placed in the plane $\mathscr{P}$, at the intersection between the line $\Delta$ and the plane $\mathscr{P}$. We admit that the point $\mathrm{F}(6;-1;3)$ corresponds to this location. Prove that the distance between the starting point D and the plane $\mathscr{P}$ equals $2\sqrt{14}$ hundreds of meters.
  5. The show organizer asks the pilot to send a new drone in the plane (no matter its position in the plane), always starting from point D. Knowing that there are 40 seconds left before the start of the show and that the drone flies in a straight trajectory at $18.6\,\mathrm{m.s}^{-1}$, can the new drone arrive on time?