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__reunion_j2

4 maths questions

Q1 Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
A merchant sells two types of mattresses: SPRING mattresses and FOAM mattresses. We assume that each customer buys only one mattress.
We have the following information:
  • $20\%$ of customers buy a SPRING mattress. Among them, $90\%$ are satisfied with their purchase.
  • $82\%$ of customers are satisfied with their purchase.

The two parts can be treated independently.
Part A
We randomly select a customer and note the events:
  • R: ``the customer buys a SPRING mattress'',
  • S: ``the customer is satisfied with their purchase''.

We denote $x = P_{\bar{R}}(S)$, where $P_{\bar{R}}(S)$ denotes the probability of $S$ given that $R$ is not realized.
  1. Copy and complete the probability tree below describing the situation.
  2. Prove that $x = 0.8$.
  3. A customer satisfied with their purchase is selected. What is the probability that they bought a SPRING mattress? Round the result to $10^{-2}$.

Part B
  1. We randomly select 5 customers. We consider the random variable $X$ which gives the number of customers satisfied with their purchase among these 5 customers.
    a. We admit that $X$ follows a binomial distribution. Give its parameters.
    b. Determine the probability that at most three customers are satisfied with their purchase. Round the result to $10^{-3}$.
  2. Let $n$ be a non-zero natural number. We now randomly select $n$ customers. This selection can be treated as a random draw with replacement.
    a. We denote $p_n$ the probability that all $n$ customers are satisfied with their purchase. Prove that $p_n = 0.82^n$.
    b. Determine the natural numbers $n$ such that $p_n < 0.01$. Interpret in the context of the exercise.
Q2 Sequences and series, recurrence and convergence Convergence proof and limit determination View
We consider the sequence $(u_n)$ defined by $u_0 = 8$ and, for every natural number $n$, $$u_{n+1} = \frac{6u_n + 2}{u_n + 5}.$$
  1. Calculate $u_1$.
  2. Let $f$ be the function defined on the interval $[0; +\infty[$ by: $$f(x) = \frac{6x + 2}{x + 5}.$$ Thus, for every natural number $n$, we have: $u_{n+1} = f(u_n)$.
    a. Prove that the function $f$ is strictly increasing on the interval $[0; +\infty[$. Deduce that for every real number $x > 2$, we have $f(x) > 2$.
    b. Prove by induction that, for every natural number $n$, we have $u_n > 2$.
  3. We admit that, for every natural number $n$, we have: $$u_{n+1} - u_n = \frac{(2 - u_n)(u_n + 1)}{u_n + 5}.$$
    a. Prove that the sequence $(u_n)$ is decreasing.
    b. Deduce that the sequence $(u_n)$ is convergent.
  4. We define the sequence $(v_n)$ for every natural number by: $$v_n = \frac{u_n - 2}{u_n + 1}.$$
    a. Calculate $v_0$.
    b. Prove that $(v_n)$ is a geometric sequence with common ratio $\dfrac{4}{7}$.
    c. Determine, by justifying, the limit of $(v_n)$. Deduce the limit of $(u_n)$.
  5. We consider the Python function threshold below, where A is a real number strictly greater than 2. \begin{verbatim} def seuil(A): n = 0 u = 8 while u > A: u = (6*u + 2)/(u + 5) n = n + 1 return n \end{verbatim} Give, without justification, the value returned by the command \texttt{seuil(2.001)} then interpret this value in the context of the exercise.
Q3 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
We place ourselves in space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.
We consider the point $\mathrm{A}(1; 1; 0)$ and the vector $\vec{u}\begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix}$.
We consider the plane $\mathscr{P}$ with equation: $x + 4y + 2z + 1 = 0$.
  1. We denote (d) the line passing through A and directed by the vector $\vec{u}$. Determine a parametric representation of (d).
  2. Justify that the line (d) and the plane $\mathscr{P}$ intersect at a point B whose coordinates are $(1; -1; 1)$.
  3. We consider the point $\mathrm{C}(1; -1; -1)$.
    a. Verify that the points $\mathrm{A}$, $\mathrm{B}$ and C do indeed define a plane.
    b. Show that the vector $\vec{n}\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$ is a normal vector to the plane (ABC).
    c. Determine a Cartesian equation of the plane (ABC).
  4. a. Justify that the triangle ABC is isosceles at A.
    b. Let H be the midpoint of segment [BC]. Calculate the length AH then the area of triangle ABC.
  5. Let D be the point with coordinates $(0; -1; 1)$.
    a. Show that the line (BD) is a height of the pyramid ABCD.
    b. Deduce from the previous questions the volume of the pyramid ABCD.

We recall that the volume $V$ of a pyramid is given by: $$V = \frac{1}{3}\mathscr{B} \times h,$$ where $\mathscr{B}$ is the area of a base and $h$ the corresponding height.
Q4 5 marks Differentiating Transcendental Functions Full function study with transcendental functions View
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. No justification is required. A wrong answer, no answer, or multiple answers, neither gives nor takes away points.
  1. We consider the function $f$ defined on $\mathbb{R}$ by $f(x) = 2x\mathrm{e}^x$.
    The number of solutions on $\mathbb{R}$ of the equation $f(x) = -\dfrac{73}{100}$ is equal to:
    a. 0
    b. 1
    c. 2
    d. infinitely many.
  2. We consider the function $g$ defined on $\mathbb{R}$ by: $$g(x) = \frac{x+1}{\mathrm{e}^x}.$$ The limit of the function $g$ at $-\infty$ is equal to:
    a. $-\infty$
    b. $+\infty$
    c. $0$
    d. it does not exist.
  3. We consider the function $h$ defined on $\mathbb{R}$ by: $$h(x) = (4x - 16)\mathrm{e}^{2x}.$$ We denote $\mathscr{C}_h$ the representative curve of $h$ in an orthogonal coordinate system. We can affirm that:
    a. $h$ is convex on $\mathbb{R}$.
    b. $\mathscr{C}_h$ has an inflection point at $x = 3$.
    c. $h$ is concave on $\mathbb{R}$.
    d. $\mathscr{C}_h$ has an inflection point at $x = 3.5$.
  4. We consider the function $k$ defined on the interval $]0; +\infty[$ by: $$k(x) = 3\ln(x) - x.$$ We denote $\mathscr{C}$ the representative curve of the function $k$ in an orthonormal coordinate system. We denote $T$ the tangent line to the curve $\mathscr{C}$ at the point with abscissa $x = \mathrm{e}$. An equation of $T$ is:
    a. $y = (3 - \mathrm{e})x$
    b. $y = \left(\dfrac{3 - \mathrm{e}}{\mathrm{e}}\right)x$
    c. $y = \left(\dfrac{3}{\mathrm{e}} - 1\right)x + 1$
    d. $y = (\mathrm{e} - 1)x + 1$
  5. We consider the equation $[\ln(x)]^2 + 10\ln(x) + 21 = 0$, with $x \in ]0; +\infty[$.
    The number of solutions of this equation is equal to:
    a. 0
    b. 1
    c. 2
    d. infinitely many.