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
2024 bac-spe-maths__amerique-nord_j2

4 maths questions

Q1 5 marks Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
Data published on March 1st, 2023 by the Ministry of Ecological Transition on the registration of private vehicles in France in 2022 contain the following information:
  • $22.86\%$ of vehicles were new vehicles;
  • $8.08\%$ of new vehicles were rechargeable hybrids;
  • $1.27\%$ of used vehicles (that is, those that are not new) were rechargeable hybrids.

Throughout the exercise, probabilities will be rounded to the ten-thousandth.
Part I
In this part, we consider a private vehicle registered in France in 2022. We denote:
  • $N$ the event ``the vehicle is new'';
  • $R$ the event ``the vehicle is a rechargeable hybrid'';
  • $\bar{N}$ and $\bar{R}$ the complementary events of $N$ and $R$.

  1. Represent the situation with a probability tree.
  2. Calculate the probability that this vehicle is new and a rechargeable hybrid.
  3. Prove that the value rounded to the ten-thousandth of the probability that this vehicle is a rechargeable hybrid is 0.0283.
  4. Calculate the probability that this vehicle is new given that it is a rechargeable hybrid.

Part II
In this part, we choose 500 private rechargeable hybrid vehicles registered in France in 2022. In what follows, we will assume that the probability that such a vehicle is new is equal to 0.65. We treat the choice of these 500 vehicles as a random draw with replacement. We call $X$ the random variable representing the number of new vehicles among the 500 vehicles chosen.
  1. We assume that the random variable $X$ follows a binomial distribution. Specify the values of its parameters.
  2. Determine the probability that exactly 325 of these vehicles are new.
  3. Determine the probability $p(X \geq 325)$ then interpret the result in the context of the exercise.

Part III
We now choose $n$ private rechargeable hybrid vehicles registered in France in 2022, where $n$ denotes a strictly positive natural number. We recall that the probability that such a vehicle is new is equal to 0.65. We treat the choice of these $n$ vehicles as a random draw with replacement.
  1. Give the expression as a function of $n$ of the probability $p_n$ that all these vehicles are used.
  2. We denote $q_n$ the probability that at least one of these vehicles is new. By solving an inequality, determine the smallest value of $n$ such that $q_n \geqslant 0.9999$.
Q2 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
We consider the rectangular prism ABCDEFGH such that $\mathrm{AB} = 3$ and $\mathrm{AD} = \mathrm{AE} = 1$.
We consider the point I on the segment $[\mathrm{AB}]$ such that $\overrightarrow{\mathrm{AB}} = 3\overrightarrow{\mathrm{AI}}$ and we call $M$ the midpoint of the segment [CD]. We place ourselves in the orthonormal coordinate system ($A$; $\overrightarrow{AI}, \overrightarrow{AD}, \overrightarrow{AE}$).
  1. Without justification, give the coordinates of the points $\mathrm{F}$, $\mathrm{H}$ and $M$.
    1. [a.] Show that the vector $\vec{n}\begin{pmatrix} 2 \\ 6 \\ 3 \end{pmatrix}$ is a normal vector to the plane (HMF).
    2. [b.] Deduce that a Cartesian equation of the plane (HMF) is: $$2x + 6y + 3z - 9 = 0$$
    3. [c.] Is the plane $\mathscr{P}$ whose Cartesian equation is $5x + 15y - 3z + 7 = 0$ parallel to the plane (HMF)? Justify your answer.
  3. Determine a parametric representation of the line (DG).
  4. We call $N$ the point of intersection of the line (DG) with the plane (HMF). Determine the coordinates of point N.
  5. Is the point R with coordinates $\left(3; \frac{1}{4}; \frac{1}{2}\right)$ the orthogonal projection of point G onto the plane (HMF)? Justify your answer.
Q3 6 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
We consider the function $g$ defined on the interval $[0; 1]$ by $$g(x) = 2x - x^2$$
  1. Show that the function $g$ is strictly increasing on the interval $[0; 1]$ and specify the values of $g(0)$ and $g(1)$.

We consider the sequence $\left(u_n\right)$ defined by $\left\{\begin{array}{l} u_0 = \dfrac{1}{2} \\ u_{n+1} = g\left(u_n\right) \end{array}\right.$ for every natural number $n$.
    \setcounter{enumi}{1}
  1. Calculate $u_1$ and $u_2$.
  2. Prove by induction that, for every natural number $n$, we have: $0 < u_n < u_{n+1} < 1$.
  3. Deduce that the sequence $(u_n)$ is convergent.
  4. Determine the limit $\ell$ of the sequence $(u_n)$.

We consider the sequence $(\nu_n)$ defined for every natural number $n$ by $\nu_n = \ln\left(1 - u_n\right)$.
    \setcounter{enumi}{5}
  1. Prove that the sequence $\left(v_n\right)$ is a geometric sequence with common ratio 2 and specify its first term.
  2. Deduce an expression for $v_n$ as a function of $n$.
  3. Deduce an expression for $u_n$ as a function of $n$ and find again the limit determined in question 5.
  4. Copy and complete the Python script below so that it returns the rank $n$ from which the sequence exceeds 0.95. \begin{verbatim} def seuil() : n=0 u=0.5 while u < 0.95: n=... u=... return n \end{verbatim}
Q4 4 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Let $a$ be a strictly positive real number. We consider the function $f$ defined on the interval $]0; +\infty[$ by $$f(x) = a\ln(x)$$ We denote $\mathscr{C}$ its representative curve in an orthonormal coordinate system. Let $x_0$ be a real number strictly greater than 1.
  1. Determine the abscissa of the point of intersection of the curve $\mathscr{C}$ and the x-axis.
  2. Verify that the function $F$ defined by $F(x) = a[x\ln(x) - x]$ is a primitive of the function $f$ on the interval $]0; +\infty[$.
  3. Deduce the area of the blue region as a function of $a$ and $x_0$.

We denote $T$ the tangent line to the curve $\mathscr{C}$ at the point $M$ with abscissa $x_0$. We call $A$ the point of intersection of the tangent line $T$ with the y-axis and $B$ the orthogonal projection of $M$ onto the y-axis.
    \setcounter{enumi}{3}
  1. Prove that the length AB is equal to a constant (that is, to a number that does not depend on $x_0$) which we will determine. The candidate will take care to make their approach explicit.