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
2022 bac-spe-maths__metropole_j1

4 maths questions

Q1 7 marks Differentiating Transcendental Functions Full function study with transcendental functions View
Exercise 1 (7 points) Themes: exponential function, sequences In the context of a clinical trial, two treatment protocols for a disease are being considered. The objective of this exercise is to study, for these two protocols, the evolution of the quantity of medication present in a patient's blood as a function of time.
Parts $A$ and $B$ are independent
Part A: Study of the first protocol
The first protocol consists of having the patient take a medication in tablet form. The quantity of medication present in the patient's blood, expressed in mg, is modelled by the function $f$ defined on the interval $[0; 10]$ by $$f(t) = 3t \mathrm{e}^{-0.5t + 1},$$ where $t$ denotes the time, expressed in hours, elapsed since taking the tablet.
  1. a. It is admitted that the function $f$ is differentiable on the interval $[0; 10]$ and we denote $f'$ its derivative function. Show that, for every real number $t$ in $[0; 10]$, we have: $f'(t) = 3(-0.5t + 1)\mathrm{e}^{-0.5t + 1}$. b. Deduce the table of variations of the function $f$ on the interval $[0; 10]$. c. According to this model, after how much time will the quantity of medication present in the patient's blood be maximum? What is this maximum quantity?
  2. a. Show that the equation $f(t) = 5$ admits a unique solution on the interval $[0; 2]$ denoted $\alpha$, of which you will give an approximate value to $10^{-2}$ near. It is admitted that the equation $f(t) = 5$ admits a unique solution on the interval $[2; 10]$, denoted $\beta$, and that an approximate value of $\beta$ to $10^{-2}$ near is 3.46. b. It is considered that this treatment is effective when the quantity of medication present in the patient's blood is greater than or equal to 5 mg. Determine, to the nearest minute, the duration of effectiveness of the medication in the case of this protocol.

Part B: Study of the second protocol
The second protocol consists of initially injecting the patient, by intravenous injection, a dose of 2 mg of medication and then re-injecting every hour a dose of $1.8$ mg. It is assumed that the medication diffuses instantaneously into the blood and is then progressively eliminated. It is estimated that when one hour has elapsed after an injection, the quantity of medication in the blood has decreased by $30\%$ compared to the quantity present immediately after this injection. This situation is modelled using the sequence $(u_n)$ where, for every natural number $n$, $u_n$ denotes the quantity of medication, expressed in mg, present in the patient's blood immediately after the injection at the $n$-th hour. We therefore have $u_0 = 2$.
  1. Calculate, according to this model, the quantity $u_1$, of medication (in mg) present in the patient's blood immediately after the injection at the first hour.
  2. Justify that, for every natural number $n$, we have: $u_{n+1} = 0.7u_n + 1.8$.
  3. a. Show by induction that, for every natural number $n$, we have: $u_n \leqslant u_{n+1} < 6$. b. Deduce that the sequence $(u_n)$ is convergent. We denote its limit by $\ell$. c. Determine the value of $\ell$. Interpret this value in the context of the exercise.
  4. Consider the sequence $(v_n)$ defined, for every natural number $n$, by $v_n = 6 - u_n$. a. Show that the sequence $(v_n)$ is a geometric sequence with ratio 0.7 and specify its first term. b. Determine the expression of $v_n$ as a function of $n$, then of $u_n$ as a function of $n$. c. With this protocol, injections are stopped when the quantity of medication present in the patient's blood is greater than or equal to $5.5$ mg. Determine, by detailing the calculations, the number of injections carried out when applying this protocol.
Q2 7 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 2 (7 points) Theme: geometry in space In space with respect to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider:
  • the point A with coordinates $(-1; 1; 3)$,
  • the line $\mathscr{D}$ whose parametric representation is: $\left\{\begin{aligned} x &= 1 + 2t \\ y &= 2 - t \\ z &= 2 + 2t \end{aligned} \quad t \in \mathbb{R}\right.$.
It is admitted that point A does not belong to line $\mathscr{D}$.
  1. a. Give the coordinates of a direction vector $\vec{u}$ of line $\mathscr{D}$. b. Show that point $B(-1; 3; 0)$ belongs to line $\mathscr{D}$. c. Calculate the dot product $\overrightarrow{AB} \cdot \vec{u}$.
  2. We denote by $\mathscr{P}$ the plane passing through point A and perpendicular to line $\mathscr{D}$, and we call H the point of intersection of plane $\mathscr{P}$ and line $\mathscr{D}$. Thus, H is the orthogonal projection of A onto line $\mathscr{D}$. a. Show that plane $\mathscr{P}$ has the Cartesian equation: $2x - y + 2z - 3 = 0$. b. Deduce that point H has coordinates $\left(\frac{7}{9}; \frac{19}{9}; \frac{16}{9}\right)$. c. Calculate the length AH. An exact value will be given.
  3. In this question, we propose to find the coordinates of point H, the orthogonal projection of point A onto line $\mathscr{D}$, by another method. We recall that point $B(-1; 3; 0)$ belongs to line $\mathscr{D}$ and that vector $\vec{u}$ is a direction vector of line $\mathscr{D}$. a. Justify that there exists a real number $k$ such that $\overrightarrow{HB} = k\vec{u}$. b. Show that $k = \frac{\overrightarrow{AB} \cdot \vec{u}}{\|\vec{u}\|^2}$. c. Calculate the value of the real number $k$ and find the coordinates of point H.
  4. We consider a point C belonging to plane $\mathscr{P}$ such that the volume of tetrahedron ABCH is equal to $\frac{8}{9}$. Calculate the area of triangle ACH. We recall that the volume of a tetrahedron is given by: $V = \frac{1}{3} \times \mathscr{B} \times h$ where $\mathscr{B}$ denotes the area of a base and $h$ the height relative to this base.
Q3 7 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
Exercise 3 (7 points) The director of a large company proposed a training course to all its employees on the use of new software. This course was followed by $25\%$ of employees.
  1. In this company, $52\%$ of employees are women, of whom $40\%$ followed the course.

A random employee of the company is questioned and we consider the events:
  • $F$: ``the employee questioned is a woman'',
  • $S$: ``the employee questioned followed the course''. $\bar{F}$ and $\bar{S}$ denote respectively the complementary events of events $F$ and $S$. a. Give the probability of event $S$. b. Copy and complete the blanks of the probability tree below on the four indicated branches. c. Demonstrate that the probability that the person questioned is a woman who followed the course is equal to 0.208. d. Given that the person questioned followed the course, what is the probability that it is a woman? e. The director claims that, among the male employees of the company, fewer than $10\%$ followed the course. Justify the director's claim.

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  1. We denote by $X$ the random variable that associates to a sample of 20 employees of this company chosen at random the number of employees in this sample who followed the course. We assume that the number of employees in the company is sufficiently large to assimilate this choice to sampling with replacement. a. Determine, by justifying, the probability distribution followed by the random variable $X$. b. Determine, to $10^{-3}$ near, the probability that 5 employees in a sample of 20 followed the course. c. The program below, written in Python language, uses the function binomial$(i, n, p)$ created for this purpose which returns the value of the probability $P(X = i)$ in the case where the random variable $X$ follows a binomial distribution with parameters $n$ and $p$. \begin{verbatim} def proba(k) : P=0 for i in range(0,k+1) : P=P+binomiale(i,20,0.25) return P \end{verbatim} Determine, to $10^{-3}$ near, the value returned by this program when proba(5) is entered in the Python console. Interpret this value in the context of the exercise. d. Determine, to $10^{-3}$ near, the probability that at least 6 employees in a sample of 20 followed the course.
  2. This question is independent of questions 1 and 2. To encourage employees to follow the course, the company had decided to increase the salaries of employees who followed the course by $5\%$, compared to $2\%$ increase for employees who did not follow the course. What is the average percentage increase in salaries for this company under these conditions?
Q4 7 marks Applied differentiation MCQ on derivative and graph interpretation View
Exercise 4 (7 points) Theme: numerical functions 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 to a question neither awards nor deducts points. The six questions are independent.
  1. The representative curve of the function $f$ defined on $\mathbb{R}$ by $f(x) = \frac{-2x^2 + 3x - 1}{x^2 + 1}$ admits as an asymptote the line with equation: a. $x = -2$; b. $y = -1$; c. $y = -2$; d. $y = 0$
  2. Let $f$ be the function defined on $\mathbb{R}$ by $f(x) = x\mathrm{e}^{x^2}$. The antiderivative $F$ of $f$ on $\mathbb{R}$ which satisfies $F(0) = 1$ is defined by: a. $F(x) = \frac{x^2}{2}\mathrm{e}^{x^2}$; b. $F(x) = \frac{1}{2}\mathrm{e}^{x^2}$ c. $F(x) = \left(1 + 2x^2\right)\mathrm{e}^{x^2}$; d. $F(x) = \frac{1}{2}\mathrm{e}^{x^2} + \frac{1}{2}$
  3. The representative graph $\mathscr{C}_{f'}$ of the derivative function $f'$ of a function $f$ defined on $\mathbb{R}$ is given below. We can affirm that the function $f$ is: a. concave on $]0; +\infty[$; b. convex on $]0; +\infty[$; c. convex on $[0; 2]$; d. convex on $[2; +\infty[$.
  4. Among the antiderivatives of the function $f$ defined on $\mathbb{R}$ by $f(x) = 3\mathrm{e}^{-x^2} + 2$: a. all are increasing on $\mathbb{R}$; b. all are decreasing on $\mathbb{R}$; c. some are increasing on $\mathbb{R}$ and others decreasing on $\mathbb{R}$; d. all are increasing on $]-\infty; 0]$ and decreasing on $[0; +\infty[$.
  5. The limit at $+\infty$ of the function $f$ defined on the interval $]0; +\infty[$ by $f(x) = \frac{2\ln x}{3x^2 + 1}$ is equal to: a. $\frac{2}{3}$; b. $+\infty$; c. $-\infty$; d. 0.
  6. The equation $\mathrm{e}^{2x} + \mathrm{e}^x - 12 = 0$ admits in $\mathbb{R}$: a. three solutions; b. two solutions; c. only one solution; d. no solution.