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
2019 asie

6 maths questions

Q1A 6 marks Geometric Sequences and Series Prove a Transformed Sequence is Geometric View
Newton's law of cooling states that the rate of change of the temperature of a body is proportional to the difference between the temperature of this body and that of the surrounding environment.
A cup of coffee is served at an initial temperature of $80^{\circ}\mathrm{C}$ in an environment whose temperature, expressed in degrees Celsius, assumed to be constant, is denoted $M$.
In this part, for any natural number $n$, we denote $T_n$ the temperature of the coffee at instant $n$, with $T_n$ expressed in degrees Celsius and $n$ in minutes. Thus $T_0 = 80$.
We model Newton's law between any two consecutive minutes $n$ and $n+1$ by the equality: $$T_{n+1} - T_n = k(T_n - M)$$ where $k$ is a real constant.
In the rest of part A, we choose $M = 10$ and $k = -0{,}2$. Thus, for any natural number $n$, we have: $T_{n+1} - T_n = -0{,}2(T_n - 10)$.
  1. Based on the context, can we conjecture the direction of variation of the sequence $(T_n)$?
  2. Show that for any natural number $n$: $T_{n+1} = 0{,}8T_n + 2$.
  3. We set, for any natural number $n$: $u_n = T_n - 10$. a. Show that $(u_n)$ is a geometric sequence. Specify its common ratio and its first term $u_0$. b. Show that, for any natural number $n$, we have: $T_n = 70 \times 0{,}8^n + 10$. c. Determine the limit of the sequence $(T_n)$.
  4. Consider the following algorithm: \begin{verbatim} While $T \geqslant 40$ $T \leftarrow 0,8T + 2$ $n \leftarrow n + 1$ End While \end{verbatim} a. Initially, we assign the value 80 to the variable $T$ and the value 0 to the variable $n$. What numerical value does the variable $n$ contain at the end of the algorithm's execution? b. Interpret this value in the context of the exercise.
Q1B 6 marks Differential equations Applied Modeling with Differential Equations View
Newton's law of cooling states that the rate of change of the temperature of a body is proportional to the difference between the temperature of this body and that of the surrounding environment.
A cup of coffee is served at an initial temperature of $80^{\circ}\mathrm{C}$ in an environment whose temperature, expressed in degrees Celsius, assumed to be constant, is denoted $M$.
In this part, for any non-negative real $t$, we denote $\theta(t)$ the temperature of the coffee at instant $t$, with $\theta(t)$ expressed in degrees Celsius and $t$ in minutes. Thus $\theta(0) = 80$.
In this model, more precise than that of part A, we assume that $\theta$ is a function differentiable on the interval $[0; +\infty[$ and that, for any real $t$ in this interval, Newton's law is modeled by the equality: $$\theta'(t) = -0{,}2(\theta(t) - M).$$
  1. In this question, we choose $M = 0$. We then seek a function $\theta$ differentiable on the interval $[0; +\infty[$ satisfying $\theta(0) = 80$ and, for any real $t$ in this interval: $\theta'(t) = -0{,}2\theta(t)$. a. If $\theta$ is such a function, we set for any $t$ in the interval $[0; +\infty[$, $f(t) = \frac{\theta(t)}{\mathrm{e}^{-0{,}2t}}$. Show that the function $f$ is differentiable on $[0; +\infty[$ and that, for any real $t$ in this interval, $f'(t) = 0$. b. Keeping the hypothesis from a., calculate $f(0)$. Deduce, for any $t$ in the interval $[0; +\infty[$, an expression for $f(t)$, then for $\theta(t)$. c. Verify that the function $\theta$ found in b. is a solution to the problem.
  2. In this question, we choose $M = 10$. We admit that there exists a unique function $g$ differentiable on $[0; +\infty[$, modeling the temperature of the coffee at any non-negative instant $t$, and that, for any $t$ in the interval $[0; +\infty[$: $$g(t) = 10 + 70\mathrm{e}^{-0{,}2t},$$ where $t$ is expressed in minutes and $g(t)$ in degrees Celsius.
    A person likes to drink their coffee at $40^{\circ}\mathrm{C}$. Show that there exists a unique real $t_0$ in $[0; +\infty[$ such that $g(t_0) = 40$. Give the value of $t_0$ rounded to the nearest second.
Q2 4 marks Vectors 3D & Lines MCQ: Perpendicularity or Parallelism of Lines and Planes View
For each of the following questions, only one of the four statements is correct. Indicate on your answer sheet the question number and copy the letter corresponding to the correct statement. One point is awarded if the letter corresponds to the correct statement, 0 otherwise.
Throughout the exercise, we work in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ in space. The four questions are independent. No justification is required.
  1. Consider the plane $P$ with Cartesian equation $3x + 2y + 9z - 5 = 0$ and the line $d$ with parametric representation: $\left\{\begin{array}{l} x = 4t + 3 \\ y = -t + 2 \\ z = -t + 9 \end{array}, t \in \mathbb{R}\right.$. Statement A: the intersection of plane $P$ and line $d$ is reduced to the point with coordinates $(3;2;9)$. Statement B: plane $P$ and line $d$ are orthogonal. Statement C: plane $P$ and line $d$ are parallel. Statement D: the intersection of plane $P$ and line $d$ is reduced to the point with coordinates $(-353; 91; 98)$.
  2. Consider the cube ABCDEFGH and the points I, J and K defined by the vector equalities: $$\overrightarrow{\mathrm{AI}} = \frac{3}{4}\overrightarrow{\mathrm{AB}}, \quad \overrightarrow{\mathrm{DJ}} = \frac{1}{4}\overrightarrow{\mathrm{DC}}, \quad \overrightarrow{\mathrm{HK}} = \frac{3}{4}\overrightarrow{\mathrm{HG}}$$ Statement A: the cross-section of cube ABCDEFGH by plane (IJK) is a triangle. Statement B: the cross-section of cube ABCDEFGH by plane (IJK) is a quadrilateral. Statement C: the cross-section of cube ABCDEFGH by plane (IJK) is a pentagon. Statement D: the cross-section of cube ABCDEFGH by plane (IJK) is a hexagon.
  3. Consider the line $d$ with parametric representation $\left\{\begin{aligned} x &= t + 2 \\ y &= 2 \\ z &= 5t - 6 \end{aligned}\right.$, with $t \in \mathbb{R}$, and the point $\mathrm{A}(-2; 1; 0)$. Let $M$ be a variable point on line $d$. Statement A: the smallest length $AM$ is equal to $\sqrt{53}$. Statement B: the smallest length $AM$ is equal to $\sqrt{27}$. Statement C: the smallest length $AM$ is attained when point $M$ has coordinates $(-2; 1; 0)$. Statement D: the smallest length $AM$ is attained when point $M$ has coordinates $(2; 2; -6)$.
  4. Consider the plane $P$ with Cartesian equation $x + 2y - 3z + 1 = 0$ and the plane $P'$ with Cartesian equation $2x - y + 2 = 0$. Statement A: planes $P$ and $P'$ are parallel. Statement B: the intersection of planes $P$ and $P'$ is a line passing through points $\mathrm{A}(5; 12; 10)$ and $\mathrm{B}(3; 1; 2)$. Statement C: the intersection of planes $P$ and $P'$ is a line passing through point $\mathrm{C}(2; 6; 5)$ and having a direction vector $\vec{u}(1; 2; 2)$. Statement D: the intersection of planes $P$ and $P'$ is a line passing through point $\mathrm{D}(-1; 0; 0)$ and having a direction vector $\vec{v}(3; 6; 5)$.
Q3A Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
In France, the consumption of organic products has been growing for several years.
In 2017, the country had $52\%$ women. That same year, $92\%$ of French people had already consumed organic products. Furthermore, among consumers of organic products, $55\%$ were women.
We randomly choose a person from the file of French people in 2017. We denote:
  • $F$ the event ``the chosen person is a woman'';
  • $H$ the event ``the chosen person is a man'';
  • $B$ the event ``the chosen person has already consumed organic products''.

  1. Translate the numerical data from the statement using events $F$ and $B$.
  2. a. Show that $P(F \cap B) = 0{,}506$. b. Deduce the probability that a person consumed organic products in 2017, given that they are a woman.
  3. Calculate $P_H(\bar{B})$. Interpret this result in the context of the exercise.
Q3B Modelling and Hypothesis Testing View
In a supermarket, a department manager wishes to develop the supply of organic products. To justify his approach, he claims to his supervisor that $75\%$ of customers buy organic products at least once a month.
The supervisor wishes to verify his claims. To do this, he organizes a survey at the store exit. Of 2000 people interviewed, 1421 respond that they consume organic products at least once a month.
At the $95\%$ confidence level, what can we think of the department manager's claim?
Q3C Continuous Probability Distributions and Random Variables PDF Graph Interpretation and Probability Computation View
To promote the organic products of his store, a store manager decides to organize a game that consists, for a customer, of filling a basket with a certain mass of apricots from organic farming. It is announced that the customer wins the contents of the basket if the mass of apricots deposited is between 3.2 and 3.5 kilograms.
The mass of fruit in kg, placed in the basket by customers, can be modeled by a random variable $X$ following the probability distribution with density $f$ defined on the interval $[3; 4]$ by: $$f(x) = \frac{2}{(x-2)^2}$$
Reminder: a probability density function on the interval $[a; b]$ is any function $f$ defined, continuous and positive on $[a; b]$, such that the integral of $f$ over $[a; b]$ is equal to 1.
  1. Verify that the function $f$ previously defined is indeed a probability density function on the interval $[3;4]$.
  2. The store announces: ``One customer in three wins the basket!''. Is this announcement accurate?
  3. The purpose of this question is to calculate the mathematical expectation $\mathrm{E}(X)$.