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_j1

4 maths questions

Q1 4 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
Exercise 1 — 4 points Theme: probability Parts A and B can be treated independently Bicycle users in a city are classified into two disjoint categories:
  • those who use bicycles for professional travel;
  • those who use bicycles only for leisure.
A survey gives the following results:
  • $21\%$ of users are under 35 years old. Among them, $68\%$ use their bicycle only for leisure while the others use it for professional travel;
  • among those 35 years or older, only $20\%$ use their bicycle for professional travel, the others use it only for leisure.
A bicycle user from this city is randomly interviewed. Throughout the exercise, the following events are considered:
  • $J$: ``the person interviewed is under 35 years old'';
  • $T$: ``the person interviewed uses the bicycle for professional travel'';
  • $\bar{J}$ and $\bar{T}$ are the complementary events of $J$ and $T$.

Part A
  1. Calculate the probability that the person interviewed is under 35 years old and uses their bicycle for professional travel. You may use a probability tree.
  2. Calculate the exact value of the probability of $T$.
  3. Now consider a resident who uses their bicycle for professional travel. Prove that the probability that they are under 35 years old is 0.30 to within $10^{-2}$.

Part B In this part, we are interested only in people using their bicycle for professional travel. We assume that $30\%$ of them are under 35 years old.
A sample of 120 people is randomly selected from among them to complete an additional questionnaire. The selection of this sample is treated as random sampling with replacement. Each individual in this sample is asked their age. $X$ represents the number of people in the sample who are under 35 years old. In this part, results should be rounded to $10^{-3}$.
  1. Determine the nature and parameters of the probability distribution followed by $X$.
  2. Calculate the probability that at least 50 bicycle users among the 120 are under 35 years old.
Q2 5 marks Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
Exercise 2 — 5 points Theme: geometry in space Space is equipped with an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider:
  • $d_1$ the line passing through point $H(2; 3; 0)$ with direction vector $\vec{u}\left(\begin{array}{c}1\\-1\\1\end{array}\right)$;
  • $d_2$ the line with parametric representation:
$$\left\{\begin{aligned} x &= 2k - 3\\ y &= k\\ z &= 5 \end{aligned}\quad\text{where }k\text{ describes }\mathbb{R}.\right.$$ The purpose of this exercise is to determine a parametric representation of a line $\Delta$ that is perpendicular to both lines $d_1$ and $d_2$.
  1. a. Determine a direction vector $\vec{v}$ of line $d_2$. b. Prove that lines $d_1$ and $d_2$ are not parallel. c. Prove that lines $d_1$ and $d_2$ are not intersecting. d. What is the relative position of lines $d_1$ and $d_2$?
  2. a. Verify that the vector $\vec{w}\left(\begin{array}{c}-1\\2\\3\end{array}\right)$ is orthogonal to both $\vec{u}$ and $\vec{v}$. b. We consider the plane $P$ passing through point $H$ and directed by vectors $\vec{u}$ and $\vec{w}$. We admit that a Cartesian equation of this plane is: $$5x + 4y - z - 22 = 0.$$ Prove that the intersection of plane $P$ and line $d_2$ is the point $M(3; 3; 5)$.
  3. Let $\Delta$ be the line with direction vector $\vec{w}$ passing through point $M$.
    A parametric representation of $\Delta$ is therefore given by: $$\left\{\begin{array}{l} x = -r + 3\\ y = 2r + 3\\ z = 3r + 5 \end{array}\text{ where }r\text{ describes }\mathbb{R}.\right.$$ a. Justify that lines $\Delta$ and $d_1$ are perpendicular at a point $L$ whose coordinates you will determine. b. Explain why line $\Delta$ is a solution to the problem posed.
Q3 5 marks Exponential Functions True/False or Multiple-Statement Verification View
Exercise 3 — 5 points Theme: exponential function, algorithms 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. Statement: The function $f$ defined on $\mathbb{R}$ by $f(x) = \mathrm{e}^x - x$ is convex.
  2. Statement: The equation $(2\mathrm{e}^x - 6)(\mathrm{e}^x + 2) = 0$ has $\ln(3)$ as its unique solution in $\mathbb{R}$.
  3. Statement: $$\lim_{x \to +\infty} \frac{\mathrm{e}^{2x} - 1}{\mathrm{e}^x - x} = 0.$$
  4. Let $f$ be the function defined on $\mathbb{R}$ by $f(x) = (6x + 5)\mathrm{e}^{3x}$ and $F$ the function defined on $\mathbb{R}$ by: $F(x) = (2x + 1)\mathrm{e}^{3x} + 4$. Statement: $F$ is the antiderivative of $f$ on $\mathbb{R}$ that takes the value 5 when $x = 0$.
  5. We consider the function \texttt{mystere} defined below which takes a list $L$ of numbers as a parameter. We recall that \texttt{len(L)} represents the length of list $L$. \begin{verbatim} def mystere(L) : S = 0 for i in range(len(L)) : S = S + L[i] return S / len(L) \end{verbatim} Statement: The execution of \texttt{mystere([1, 9, 9, 5, 0, 3, 6, 12, 0, 5])} returns 50.
Q4 6 marks Sequences and series, recurrence and convergence Closed-form expression derivation View
Exercise 4 — 6 points Theme: sequences, functions Let $(u_n)$ be the sequence defined by $u_0 = -1$ and, for every natural number $n$: $$u_{n+1} = 0.9u_n - 0.3.$$
  1. a. Prove by induction that, for all $n \in \mathbb{N}$, $u_n = 2 \times 0.9^n - 3$. b. Deduce that for all $n \in \mathbb{N}$, $-3 < u_n \leq -1$. c. Prove that the sequence $(u_n)$ is strictly decreasing. d. Prove that the sequence $(u_n)$ converges and specify its limit.
  2. We propose to study the function $g$ defined on $]-3; -1]$ by: $$g(x) = \ln(0.5x + 1.5) - x.$$ a. Justify all the information given by the variations table of function $g$ (limits, variations, image of $-1$). b. Deduce that the equation $g(x) = 0$ has exactly one solution which we will denote $\alpha$ and for which we will give an interval of amplitude $10^{-3}$.
  3. In the rest of the exercise, we consider the sequence $(v_n)$ defined for all $n \in \mathbb{N}$ by: $$v_n = \ln(0.5u_n + 1.5).$$ a. Using the formula given in question 1.a., prove that the sequence $v$ is arithmetic with common difference $\ln(0.9)$. b. Let $n$ be a natural number. Prove that $u_n = v_n$ if and only if $g(u_n) = 0$. c. Prove that there is no rank $k \in \mathbb{N}$ for which $u_k = \alpha$. d. Deduce that there is no rank $k \in \mathbb{N}$ for which $v_k = u_k$.