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
2013 centres-etrangers

8 maths questions

Q1A Exponential Distribution View
The lifetime of a valve, expressed in hours, is a random variable $T$ that follows the exponential distribution with parameter $\lambda = 0.0002$.
  1. What is the average lifetime of a valve?
  2. Calculate the probability, to 0.001, that the lifetime of a valve exceeds 6000 hours.
Q1B Conditional Probability Sequential/Multi-Stage Conditional Probability View
With three identical valves $V_1, V_2$ and $V_3$, we manufacture a hydraulic circuit. The circuit is operational if $V_1$ is operational or if $V_2$ and $V_3$ are simultaneously operational.
We treat as a random experiment the fact that each valve is or is not operational after 6000 hours. We denote:
  • $F_1$ the event: ``valve $V_1$ is operational after 6000 hours''.
  • $F_2$ the event: ``valve $V_2$ is operational after 6000 hours''.
  • $F_3$ the event: ``valve $V_3$ is operational after 6000 hours''.
  • $E$: the event: ``the circuit is operational after 6000 hours''.

We assume that the events $F_1, F_2$ and $F_3$ are pairwise independent and each have probability equal to 0.3.
  1. The probability tree shown represents part of the situation. Reproduce this tree and place the probabilities on the branches.
  2. Prove that $P(E) = 0.363$.
  3. Given that the circuit is operational after 6000 hours, calculate the probability that valve $V_1$ is operational at that time. Round to the nearest thousandth.
Q1C Modelling and Hypothesis Testing View
The industrialist claims that only $2\%$ of the valves he manufactures are defective. We assume this claim is true, and we denote $F$ the random variable equal to the frequency of defective valves in a random sample of 400 valves taken from total production.
  1. Determine the interval $I$ of asymptotic fluctuation at the $95\%$ threshold of the variable $F$.
  2. We choose 400 valves at random from production. We treat this choice as a random draw of 400 valves, with replacement, from production. Among these 400 valves, 10 are defective. In light of this result, can we question, at the $95\%$ threshold, the industrialist's claim?
Q1D Normal Distribution Direct Probability Calculation from Given Normal Distribution View
In this part, the calculated probabilities will be rounded to the nearest thousandth. The industrialist markets his valves to many customers. Monthly demand is a random variable $D$ that follows the normal distribution with mean $\mu = 800$ and standard deviation $\sigma = 40$.
  1. Determine $P(760 \leqslant D \leqslant 840)$.
  2. Determine $P(D \leqslant 880)$.
  3. The industrialist thinks that if he builds a monthly stock of 880 valves, he will have no more than a $1\%$ chance of running out of stock. Is he right?
Q2 4 marks Vectors: Lines & Planes True/False or Verify a Given Statement View
In space with an orthonormal coordinate system, we consider
  • the points $\mathrm{A}(12;0;0), \mathrm{B}(0;-15;0), \mathrm{C}(0;0;20), \mathrm{D}(2;7;-6), \mathrm{E}(7;3;-3)$;
  • the plane $\mathscr{P}$ with Cartesian equation: $2x + y - 2z - 5 = 0$

For each of the following statements, indicate whether it is true or false by justifying your answer. An unjustified answer will not be taken into account.
Statement 1
A Cartesian equation of the plane parallel to $\mathscr{P}$ and passing through point A is: $$2x + y + 2z - 24 = 0$$
Statement 2
A parametric representation of line (AC) is: $\left\{ \begin{array}{rl} x &= 9 - 3t \\ y &= 0 \\ z &= 5 + 5t \end{array}, t \in \mathbb{R} \right.$.
Statement 3 Line (DE) and plane $\mathscr{P}$ have at least one point in common.
Statement 4 Line (DE) is orthogonal to plane (ABC).
Q3 Indefinite & Definite Integrals Maximizing or Optimizing a Definite Integral View
We consider the function $g$ defined for all real $x$ in the interval $[0;1]$ by: $$g(x) = 1 + \mathrm{e}^{-x}$$ We admit that, for all real $x$ in the interval $[0;1], g(x) > 0$.
We denote $\mathscr{C}$ the representative curve of function $g$ in an orthogonal coordinate system, and $\mathscr{D}$ the plane region bounded on one hand between the $x$-axis and curve $\mathscr{C}$, on the other hand between the lines with equations $x = 0$ and $x = 1$.
The purpose of this exercise is to divide region $\mathscr{D}$ into two regions of equal area, first by a line parallel to the $y$-axis (part A), then by a line parallel to the $x$-axis (part B).
Part A
Let $a$ be a real number such that $0 \leqslant a \leqslant 1$. We denote $\mathscr{A}_1$ the area of the region between curve $\mathscr{C}$, the $x$-axis, the lines with equations $x = 0$ and $x = a$, and $\mathscr{A}_2$ that of the region between curve $\mathscr{C}$, the $x$-axis and the lines with equations $x = a$ and $x = 1$. $\mathscr{A}_1$ and $\mathscr{A}_2$ are expressed in square units.
  1. a. Prove that $\mathscr{A}_1 = a - \mathrm{e}^{-a} + 1$. b. Express $\mathscr{A}_2$ as a function of $a$.
  2. Let $f$ be the function defined for all real $x$ in the interval $[0;1]$ by: $$f(x) = 2x - 2\mathrm{e}^{-x} + \frac{1}{\mathrm{e}}$$ a. Draw the variation table of function $f$ on the interval $[0;1]$. The exact values of $f(0)$ and $f(1)$ will be specified. b. Prove that function $f$ vanishes once and only once on the interval $[0;1]$, at a real number $\alpha$. Give the value of $\alpha$ rounded to the nearest hundredth.
  3. Using the previous questions, determine an approximate value of the real $a$ for which the areas $\mathscr{A}_1$ and $\mathscr{A}_2$ are equal.

Part B
Let $b$ be a positive real number. In this part, we propose to divide region $\mathscr{D}$ into two regions of equal area by the line with equation $y = b$. We admit that there exists a unique positive real $b$ that is a solution.
  1. Justify the inequality $b < 1 + \frac{1}{\mathrm{e}}$. You may use a graphical argument.
  2. Determine the exact value of the real $b$.
Q4 (non-specialization) Sequences and series, recurrence and convergence Algorithm and programming for sequences View
The purpose of this exercise is the study of the sequence $(u_n)$ defined by its first term $u_1 = \frac{3}{2}$ and the recurrence relation: $u_{n+1} = \frac{n u_n + 1}{2(n+1)}$.
Part A - Algorithms and conjectures
To calculate and display the term $u_9$ of the sequence, a student proposes the algorithm below. He forgot to complete two lines.
Variables$n$ is a natural integer, $u$ is a real number
InitializationAssign to $n$ the value 1, Assign to $u$ the value 1.5
TreatmentWhile $n < 9$, Assign to $u$ the value $\cdots$, Assign to $n$ the value $\cdots$, End While
OutputDisplay the variable $u$

  1. Copy and complete the two lines of the algorithm where there are ellipses.
  2. How would this algorithm need to be modified so that it calculates and displays all terms of the sequence from $u_2$ to $u_9$?
  3. With this modified algorithm, the following results were obtained, rounded to the nearest ten-thousandth:
    n123456$\ldots$99100
    $u_n$1.50.6250.3750.26560.20630.1693$\ldots$0.01020.0101

    In light of these results, conjecture the direction of variation and convergence of the sequence $(u_n)$.

Part B - Mathematical study
We define an auxiliary sequence $(v_n)$ by: for all integer $n \geqslant 1, v_n = n u_n - 1$.
  1. Show that the sequence $(v_n)$ is geometric; specify its common ratio and its first term.
  2. Deduce that, for all natural integer $n \geqslant 1$, we have: $u_n = \frac{1 + (0.5)^n}{n}$.
  3. Determine the limit of the sequence $(u_n)$.
  4. Justify that, for all integer $n \geqslant 1$, we have: $u_{n+1} - u_n = -\frac{1 + (1 + 0.5n)(0.5)^n}{n(n+1)}$. Deduce the direction of variation of the sequence $(u_n)$.

Part C - Return to algorithms
Inspired by part A, write an algorithm to determine and display the smallest integer $n$ such that $u_n < 0.001$.
Q4 (specialization) Matrices Matrix Power Computation and Application View
A species of bird lives only on two islands A and B of an archipelago. At the beginning of 2013, 20 million birds of this species are present on island A and 10 million on island B.
Observations over several years have allowed ornithologists to estimate that, taking into account births, deaths, and migrations between the two islands, we find at the beginning of each year the following proportions:
  • on island A: $80\%$ of the number of birds present on island A at the beginning of the previous year and $30\%$ of the number of birds present on island B at the beginning of the previous year;
  • on island B: $20\%$ of the number of birds present on island A at the beginning of the previous year and $70\%$ of the number of birds present on island B at the beginning of the previous year.