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_j2

8 maths questions

QExercise 2 Fixed Point Iteration View
Consider the function $f$ defined on $]-1.5; +\infty[$ by $$f(x) = \ln(2x + 3) - 1$$ The purpose of this exercise is to study the convergence of the sequence $(u_{n})$ defined by: $$u_{0} = 0 \text{ and } u_{n+1} = f(u_{n}) \text{ for all natural integer } n.$$
Part A: Study of an auxiliary function Consider the function $g$ defined on $]-1.5; +\infty[$ by $g(x) = f(x) - x$.
  1. Determine the limit of the function $g$ at $-1.5$.

We admit that the limit of the function $g$ at $+\infty$ is $-\infty$.
    \setcounter{enumi}{1}
  1. Study the variations of the function $g$ on $]-1.5; +\infty[$.
  2. a. Prove that, in the interval $]-0.5; +\infty[$, the equation $g(x) = 0$ admits a unique solution $\alpha$. b. Determine an interval containing $\alpha$ with amplitude $10^{-2}$.

Part B: Study of the sequence $(u_{n})$ We admit that the function $f$ is strictly increasing on $]-1.5; +\infty[$.
  1. Let $x$ be a real number. Show that if $x \in [-1; \alpha]$ then $f(x) \in [-1; \alpha]$.
  2. a. Prove by induction that for all natural integer $n$: $$-1 \leqslant u_{n} \leqslant u_{n+1} \leqslant \alpha.$$ b. Deduce that the sequence $(u_{n})$ converges.
QExercise 3 6 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
The figure below corresponds to the model of an architectural project. It is a house with a cubic shape (ABCDEFGH) attached to a garage with a cubic shape (BIJKLMNO) where L is the midpoint of segment [BF] and K is the midpoint of segment [BC]. The garage is topped with a roof with a pyramidal shape (LMNOP) with square base LMNO and apex P positioned on the facade of the house.
We equip space with the orthonormal coordinate system $(A; \vec{\imath}, \vec{\jmath}, \vec{k})$, with $\vec{\imath} = \frac{1}{2}\overrightarrow{AB}$, $\vec{\jmath} = \frac{1}{2}\overrightarrow{AD}$ and $\vec{k} = \frac{1}{2}\overrightarrow{AE}$.
  1. a. By reading the graph, give the coordinates of points H, M and N. b. Determine a parametric representation of the line (HM).
  2. The architect places point P at the intersection of line (HM) and plane (BCF). Show that the coordinates of P are $\left(2; \frac{2}{3}; \frac{4}{3}\right)$.
  3. a. Calculate the dot product $\overrightarrow{PM} \cdot \overrightarrow{PN}$. b. Calculate the distance PM. We admit that the distance PN is equal to $\frac{\sqrt{11}}{3}$. c. To satisfy technical constraints, the roof can only be built if the angle $\widehat{MPN}$ does not exceed $55°$. Can the roof be built?
  4. Justify that the lines (HM) and (EN) are secant. What is their point of intersection?
QExercise 4 Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
A production company is considering whether to schedule a television game show. This game brings together four candidates and takes place in two phases:
  • The first phase is a qualification phase. This phase depends only on chance. For each candidate, the probability of qualifying is 0.6.
  • The second phase is a competition between the qualified candidates. It only takes place if at least two candidates are qualified. Its duration depends on the number of qualified candidates as indicated in the table below (when there is no second phase, its duration is considered to be zero).

\begin{tabular}{ l } Number of candidates qualified
for the second phase
& 0 & 1 & 2 & 3 & 4 \hline
Duration of the second phase in
minutes
& 0 & 0 & 5 & 9 & 11 \hline \end{tabular}
For the company to decide to retain this game, the following two conditions must be verified: Condition no. 1: The second phase must take place in at least 80\% of cases. Condition no. 2: The average duration of the second phase must not exceed 6 minutes. Can the game be retained?
Q1 1 marks Integration by Parts Multiple-Choice Primitive Identification View
Let $f$ be the function defined on $\mathbb{R}$ by $f(x) = x \mathrm{e}^{x}$.
A primitive $F$ on $\mathbb{R}$ of the function $f$ is defined by:
A. $F(x) = \frac{x^{2}}{2} \mathrm{e}^{x}$
B. $F(x) = (x - 1) \mathrm{e}^{x}$
C. $F(x) = (x + 1) \mathrm{e}^{x}$
D. $F(x) = \frac{2}{x} \mathrm{e}^{x^{2}}$.
Q2 1 marks Applied differentiation MCQ on derivative and graph interpretation View
The curve $\mathscr{C}$ below represents a function $f$ defined and twice differentiable on $]0; +\infty[$. We know that:
  • the maximum of the function $f$ is reached at the point with abscissa 3;
  • the point P with abscissa 5 is the unique inflection point of the curve $\mathscr{C}$.
We have:
A. for all $x \in ]0; 5[$, $f(x)$ and $f'(x)$ have the same sign;
C. for all $x \in ]0; 5[$, $f'(x)$ and $f''(x)$ have the same sign;
B. for all $x \in ]5; +\infty[$, $f(x)$ and $f'(x)$ have the same sign;
D. for all $x \in ]5; +\infty[$, $f(x)$ and $f''(x)$ have the same sign.
Q3 1 marks Exponential Functions Parameter Determination from Conditions View
Consider the function $g$ defined on $[0; +\infty[$ by $g(t) = \frac{a}{b + \mathrm{e}^{-t}}$ where $a$ and $b$ are two real numbers. We know that $g(0) = 2$ and $\lim_{t \rightarrow +\infty} g(t) = 3$. The values of $a$ and $b$ are:
A. $a = 2$ and $b = 3$
B. $a = 4$ and $b = \frac{4}{3}$
C. $a = 4$ and $b = 1$
D. $a = 6$ and $b = 2$
Q4 1 marks Conditional Probability Bayes' Theorem with Production/Source Identification View
Alice has two urns A and B each containing four indistinguishable balls. Urn A contains two green balls and two red balls. Urn B contains three green balls and one red ball. Alice randomly chooses an urn and then a ball from that urn. She obtains a green ball. The probability that she chose urn B is:
A. $\frac{3}{8}$
B. $\frac{1}{2}$
C. $\frac{3}{5}$
D. $\frac{5}{8}$
Q5 1 marks Arithmetic Sequences and Series Flowchart or Algorithm Tracing Involving Sequences View
Let $S = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots + \frac{1}{100}$. Among the Python scripts below, the one that allows calculating the sum $S$ is:
a. \begin{verbatim} def somme_a() : S = 0 for k in range(100) : S =1/( k+1) return S \end{verbatim}
b. \begin{verbatim} def somme_b() : S = 0 for k in range(100) : S = S + 1/(k + 1) return S \end{verbatim}
c. \begin{verbatim} def somme_c() : k = 0 while S < 100 : S = S+1 /(k+1) return S \end{verbatim}
d. \begin{verbatim} def somme_d() : k = 0 while k < 100: S = S + 1/(k + 1) return S \end{verbatim}