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__centres-etrangers_j1

9 maths questions

QExercise 2 Part A Geometric Sequences and Series Prove a Transformed Sequence is Geometric View
In a large French city, electric scooters are made available to users. A company, responsible for maintaining the scooter fleet, checks their condition every Monday.
It is estimated that:
  • when a scooter is in good condition on a Monday, the probability that it is still in good condition the following Monday is 0.9;
  • when a scooter is in poor condition on a Monday, the probability that it is in good condition the following Monday is 0.4.

We are interested in the condition of a scooter during the inspection phases. Let $n$ be a natural integer. We denote $B_n$ the event ``the scooter is in good condition $n$ weeks after its commissioning'' and $p_n$ the probability of $B_n$. When commissioned, the scooter is in good condition. We therefore have $p_0 = 1$.
  1. Give $p_1$ and show that $p_2 = 0.85$. You may rely on a weighted tree.
  2. Copy and complete the weighted tree.
  3. Deduce that, for all natural integer $n$, $p_{n+1} = 0.5p_n + 0.4$.
  4. a. Prove by induction that for all natural integer $n$, $p_n \geqslant 0.8$. b. Based on this result, what communication can the company consider to highlight the reliability of the fleet?
  5. a. Consider the sequence $(u_n)$ defined for all natural integer $n$ by $u_n = p_n - 0.8$. Show that $(u_n)$ is a geometric sequence and give its first term and common ratio. b. Deduce the expression of $u_n$ then of $p_n$ as a function of $n$. c. Deduce the limit of the sequence $(p_n)$.
QExercise 2 Part B Binomial Distribution Justify Binomial Model and State Parameters View
In this part, we model the situation as follows:
  • the condition of a scooter is independent of that of the others;
  • the probability that a scooter is in good condition is equal to 0.8.

We denote $X$ the random variable which, to a batch of 15 scooters, associates the number of scooters in good condition. Since the number of scooters in the fleet is very large, the sampling of 15 scooters can be assimilated to a draw with replacement.
  1. Justify that $X$ follows a binomial distribution and specify the parameters of this distribution.
  2. Calculate the probability that all 15 scooters are in good condition.
  3. Calculate the probability that at least 10 scooters are in good condition in a batch of 15.
  4. We admit that $E(X) = 12$. Interpret the result.
QExercise 3 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Consider the right prism ABFEDCGH, with base ABFE, a right trapezoid at A. We associate with this prism the orthonormal coordinate system $(A; \vec{\imath}, \vec{\jmath}, \vec{k})$ such that:
$$\vec{\imath} = \frac{1}{4}\overrightarrow{AB}, \quad \vec{\jmath} = \frac{1}{4}\overrightarrow{AD}, \quad \vec{k} = \frac{1}{8}\overrightarrow{AE}$$
Moreover we have $\overrightarrow{BF} = \frac{1}{2}\overrightarrow{AE}$. We denote I the midpoint of segment $[EF]$. We denote J the midpoint of segment $[AE]$.
  1. Give the coordinates of points I and J.
  2. Let $\vec{n}$ be the vector with coordinates $\left(\begin{array}{c} -1 \\ 1 \\ 1 \end{array}\right)$. a. Show that the vector $\vec{n}$ is normal to the plane (IGJ). b. Determine a Cartesian equation of the plane (IGJ).
  3. Determine a parametric representation of the line $d$, perpendicular to the plane (IGJ) and passing through H.
  4. We denote L the orthogonal projection of point H onto the plane (IGJ). Show that the coordinates of L are $\left(\frac{8}{3}; \frac{4}{3}; \frac{16}{3}\right)$.
  5. Calculate the distance from point H to the plane (IGJ).
  6. Show that triangle IGJ is right-angled at I.
  7. Deduce the volume of tetrahedron IGJH.

We recall that the volume $V$ of a tetrahedron is given by the formula: $$V = \frac{1}{3} \times \text{(area of base) \times height.}$$
QExercise 4 3 marks Applied differentiation Applied modeling with differentiation View
A biologist has modeled the evolution of a bacterial population (in thousands of entities) by the function $f$ defined on $[0; +\infty[$ by
$$f(t) = e^3 - e^{-0.5t^2 + t + 2}$$
where $t$ denotes the time in hours since the beginning of the experiment. Based on this modeling, he proposes the three statements below. For each of them, indicate, by justifying, whether it is true or false.
  • Statement 1: ``The population increases permanently''.
  • Statement 2: ``In the long term, the population will exceed 21000 bacteria''.
  • Statement 3: ``The bacterial population will have a count of 10000 on two occasions over time''.
Q1 1 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
Consider the numerical sequence $(u_n)$ defined for all natural integer $n$ by
$$u_n = \frac{1 + 2^n}{3 + 5^n}$$
This sequence: a. diverges to $+\infty$ b. converges to $\frac{2}{5}$ c. converges to 0 d. converges to $\frac{1}{3}$.
Q2 1 marks Differentiating Transcendental Functions Compute derivative of transcendental function View
Let $f$ be the function defined on $]0; +\infty[$ by $f(x) = x^2 \ln x$. The expression of the derivative function of $f$ is: a. $f'(x) = 2x \ln x$. b. $f'(x) = x(2\ln x + 1)$. c. $f'(x) = 2$. d. $f'(x) = x$.
Q3 1 marks Indefinite & Definite Integrals Antiderivative Verification and Construction View
Consider a function $h$ defined and continuous on $\mathbb{R}$ whose variation table is given below:
$x$$-\infty$1$+\infty$
Variations of $h$0
$-\infty$

We denote $H$ the antiderivative of $h$ defined on $\mathbb{R}$ which vanishes at 0. It satisfies the property: a. $H$ is positive on $]-\infty; 0]$. b. $H$ is increasing on $]-\infty; 1]$. c. $H$ is negative on $]-\infty; 1]$. d. $H$ is increasing on $\mathbb{R}$.
Q4 1 marks Sign Change & Interval Methods View
Let two real numbers $a$ and $b$ with $a < b$. Consider a function $f$ defined, continuous, strictly increasing on the interval $[a; b]$ and which vanishes at a real number $\alpha$. Among the following propositions, the function in Python language that allows giving an approximate value of $\alpha$ to 0.001 is:
a. \begin{verbatim} def racine(a, b): while abs(b - a) >= 0.001: m = (a + b) / 2 if f(m) < 0: b = m else: a = m return m \end{verbatim}
c. \begin{verbatim} def racine(a, b): m = (a + b) / 2 while abs(b - a) <= 0.001: if f(m) < 0: a = m else: b = m return m \end{verbatim}
b. \begin{verbatim} def racine(a, b): m = (a + b) / 2 while abs(b - a) >= 0.001: if f(m) < 0: a = m else: b = m return m \end{verbatim}
d. \begin{verbatim} def racine(a, b): while abs(b - a) >= 0.001: m = (a + b) / 2 if f(m) < 0: a = m else: b = m return m \end{verbatim}
Q5 1 marks Binomial Distribution MCQ Selecting a Binomial Probability Expression or Value View
An urn contains 10 indistinguishable balls to the touch, of which 7 are blue and the others are green. Three successive draws are made with replacement. The probability of obtaining exactly two green balls is: a. $\left(\frac{7}{10}\right)^2 \times \frac{3}{10}$ b. $\left(\frac{3}{10}\right)^2$ c. $\binom{10}{2}\left(\frac{7}{10}\right)\left(\frac{3}{10}\right)^2$ d. $\binom{3}{2}\left(\frac{7}{10}\right)\left(\frac{3}{10}\right)^2$