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
2025 bac-spe-maths__centres-etrangers_j1

6 maths questions

Q1A Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
A store is equipped with self-service automatic checkouts where the customer scans their own items. The checkout software regularly triggers verification requests.
The check can be either ``complete'': the store employee then scans all of the customer's items again; or ``partial'': the employee then selects one or more of the customer's items to verify that they have been scanned correctly.
If a check is triggered, it is a complete check one time out of ten. When a complete check is triggered, a customer error is detected in $30\%$ of cases. When a partial check is performed, in $85\%$ of cases, there is no error.
A check is triggered at an automatic checkout. We consider the following events:
  • T: ``The check is a complete check'';
  • E: ``An error is detected during the check''.
We denote $\bar{T}$ and $\bar{E}$ the complementary events of $T$ and $E$.
  1. Construct a probability tree representing the situation and then determine $P(\bar{T} \cap E)$.
  2. Calculate the probability that an error is detected during a check.
  3. Determine the probability that a complete check was performed, given that an error was detected. The answer will be given rounded to the nearest hundredth.
Q1B Binomial Distribution Find Minimum n for a Probability Threshold View
On a given day, an automatic checkout triggers 15 checks. The probability that a check reveals an error is $p = 0.165$. The detection of an error during a check is independent of other checks. We denote $X$ the random variable equal to the number of errors detected during the checks on this day.
  1. We admit that the random variable $X$ follows a binomial distribution. Specify its parameters.
  2. Determine the probability that exactly 5 errors are detected. The answer will be given rounded to the nearest hundredth.
  3. Determine the probability that at least one error is detected. The answer will be given rounded to the nearest hundredth.
  4. We wish to modify the number of checks triggered by the checkout so that the probability that at least one error is detected each day is greater than $99\%$. Determine the number of checks that the checkout must trigger each day for this constraint to be satisfied.
Q1C Central limit theorem View
The store has three identical automatic checkouts which, during a day, each triggered 20 checks. We denote $X_1, X_2$ and $X_3$ the random variables associating to each checkout the number of errors detected during this day. We admit that the random variables $X_1, X_2$ and $X_3$ are independent of each other and each follow a binomial distribution $\mathscr{B}(20; 0.165)$.
  1. Determine the exact values of the expectation and variance of the random variable $X_1$.
  2. We define the random variable $S$ by $S = X_1 + X_2 + X_3$.
    Justify that $E(S) = 9.9$ and that $V(S) = 8.2665$. For this question, we will use 10 as the value of $E(S)$. Using the Bienaymé-Chebyshev inequality, show that the probability that the total number of errors on the day is strictly between 6 and 14 is greater than 0.48.
Q2 Vectors: Lines & Planes MCQ: Identify Correct Equation or Representation View
This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, multiple answers, or the absence of an answer earns neither points nor deducts points.
Throughout the exercise, we consider that space is equipped with an orthonormal reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider:
  • the points $\mathrm{A}(-3; 1; 4)$ and $\mathrm{B}(1; 5; 2)$
  • the plane $\mathscr{P}$ with Cartesian equation $4x + 4y - 2z + 3 = 0$
  • the line $(d)$ with parametric representation $\left\{\begin{aligned} x &= -6 + 3t \\ y &= 1 \\ z &= 9 - 5t \end{aligned}\right.$, where $t \in \mathbb{R}$.

  1. The lines $(\mathrm{AB})$ and $(d)$ are: a. secant and non-perpendicular. b. perpendicular. c. non-coplanar. d. parallel.
  2. The line $(\mathrm{AB})$ is: a. included in the plane $\mathscr{P}$. b. strictly parallel to the plane $\mathscr{P}$. c. secant and non-orthogonal to the plane $\mathscr{P}$. d. orthogonal to the plane $\mathscr{P}$.
  3. We consider the plane $\mathscr{P}'$ with Cartesian equation $2x + y + 6z + 5 = 0$. The planes $\mathscr{P}$ and $\mathscr{P}'$ are: a. secant and non-perpendicular. b. perpendicular. c. identical. d. strictly parallel.
  4. We consider the point $\mathrm{C}(0; 1; -1)$. The value of the angle $\widehat{\mathrm{BAC}}$ rounded to the nearest degree is: a. $90^\circ$ b. $51^\circ$ c. $39^\circ$ d. $0^\circ$
Q3 Differential equations Qualitative Analysis of DE Solutions View
Part A
We consider the function $f$ defined on the interval $]-1; +\infty[$ by $$f(x) = 4\ln(x+1) - \frac{x^2}{25}$$ We admit that the function $f$ is differentiable on the interval $]-1; +\infty[$.
  1. Determine the limit of the function $f$ at $-1$.
  2. Show that, for all $x$ belonging to the interval $]-1; +\infty[$, we have: $$f'(x) = \frac{100 - 2x - 2x^2}{25(x+1)}$$
  3. Study the variations of the function $f$ on the interval $]-1; +\infty[$ and then deduce that the function $f$ is strictly increasing on the interval $[2; 6.5]$.
  4. We consider $h$ the function defined on the interval $[2; 6.5]$ by $h(x) = f(x) - x$.
    The table of variations of the function $h$ is given (showing $h$ increases then decreases on $[2;6.5]$ with $h(2) < 0$ and $h(6.5) < 0$ and a positive maximum in between).
    Show that the equation $h(x) = 0$ admits a unique solution $\alpha \in [2; 6.5]$.
  5. Consider the following script, written in Python language: \begin{verbatim} from math import * def f(x): return 4*log(1+x)-(x**2)/25 def bornes(n) : p = 1/10**n x = 6 while f(x)-x > 0 : x = x + p return (x-p,x) \end{verbatim} We recall that in Python language:
    • the command $\log(x)$ returns the value $\ln x$;
    • the command $\mathrm{c}**\mathrm{d}$ returns the value of $c^d$.
    a. Give the values returned by the command \texttt{bornes(2)}. The values will be given rounded to the nearest hundredth. b. Interpret these values in the context of the exercise.

Part B
In this part, we may use the results obtained in Part A. We consider the sequence $(u_n)$ defined by $u_0 = 2$, and, for all natural integer $n$, $u_{n+1} = f(u_n)$.
  1. Show by induction that for all natural integer $n$, $$2 \leqslant u_n \leqslant u_{n+1} < 6.5.$$
  2. Deduce that the sequence $(u_n)$ converges to a limit $\ell$.
  3. We recall that the real number $\alpha$, defined in Part A, is the solution of the equation $h(x) = 0$ on the interval $[2; 6.5]$. Justify that $\ell = \alpha$.
Q4 Second order differential equations Solving non-homogeneous second-order linear ODE View
Part A
We consider the differential equation $$\left(E_1\right): \quad y' + 0.48y = \frac{1}{250}$$ where $y$ is a function of the variable $t$ belonging to the interval $[0; +\infty[$.
  1. We consider the constant function $h$ defined on the interval $[0; +\infty[$ by $h(t) = \frac{1}{120}$. Show that the function $h$ is a solution of the differential equation $(E_1)$.
  2. Give the general form of the solutions of the differential equation $y' + 0.48y = 0$.
  3. Deduce the set of solutions of the differential equation $(E_1)$.

Part B
We are now interested in the evolution of a population of bacteria in a culture medium. At an instant $t = 0$, an initial population of 30000 bacteria is introduced into the medium. We denote $p(t)$ the quantity of bacteria, expressed in thousands of individuals, present in the medium after a time $t$, expressed in hours. We therefore have $p(0) = 30$. We admit that the function $p$ defined on the interval $[0; +\infty[$ is differentiable, strictly positive on this interval and that it is a solution of the differential equation $(E_2)$: $$p' = \frac{1}{250} p \times (120 - p)$$ Let $y$ be the function strictly positive on the interval $[0; +\infty[$ such that, for all $t$ belonging to the interval $[0; +\infty[$, we have $p(t) = \frac{1}{y(t)}$.
  1. Show that if $p$ is a solution of the differential equation $(E_2)$, then $y$ is a solution of the differential equation $\left(E_1\right): \quad y' + 0.48y = \frac{1}{250}$.
  2. We admit conversely that, if $y$ is a strictly positive solution of the differential equation $(E_1)$, then $p = \frac{1}{y}$ is a solution of the differential equation $(E_2)$. Show that, for all $t$ belonging to the interval $[0; +\infty[$, we have: $$p(t) = \frac{120}{1 + K\mathrm{e}^{-0.48t}} \text{ with } K \text{ a real constant.}$$
  3. Using the initial condition, determine the value of $K$.
  4. Determine $\lim_{t \rightarrow +\infty} p(t)$. Give an interpretation of this in the context of the exercise.
  5. Determine the time required for the bacterial population to exceed 60000 individuals. The result will be given in the form of a rounded value expressed in hours and minutes.