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__centres-etrangers_j1

5 maths questions

Q1A Product & Quotient Rules View
The function $f$ is defined on the interval $[ 0 ; 1 ]$ by: $$f ( x ) = \frac { 0,96 x } { 0,93 x + 0,03 }$$
  1. Prove that, for all $x$ belonging to the interval $[ 0 ; 1 ]$, $$f ^ { \prime } ( x ) = \frac { 0,0288 } { ( 0,93 x + 0,03 ) ^ { 2 } }$$
  2. Determine the direction of variation of the function $f$ on the interval $[ 0 ; 1 ]$.
Q1B Conditional Probability Conditional Probability as a Function of a Parameter View
The function $f$ is defined on the interval $[ 0 ; 1 ]$ by: $$f ( x ) = \frac { 0,96 x } { 0,93 x + 0,03 }$$
The fight against doping involves carrying out anti-doping tests which aim to determine whether an athlete has used prohibited substances. During a competition bringing together 1000 athletes, a medical team tests all competitors. We propose to study the reliability of this test.
Let $x$ denote the real number between 0 and 1 which represents the proportion of doped athletes. During the development of this test, it was possible to determine that:
  • the probability that an athlete is declared positive given that they are doped is equal to 0.96;
  • the probability that an athlete is declared positive given that they are not doped is equal to 0.03.

We denote:
  • D the event: ``the athlete is doped''.
  • $T$ the event: ``the test is positive''.

  1. Copy and complete the probability tree.
  2. Determine, as a function of $x$, the probability that an athlete is doped and has a positive test.
  3. Prove that the probability of event $T$ is equal to $0,93 x + 0,03$.
  4. For this question only, assume that there are 50 doped athletes among the 1000 tested. Prove that the probability that an athlete is doped given that their test is positive is equal to $f ( 0,05 )$. Give an approximate value rounded to the nearest hundredth.
  5. The positive predictive value of a test is called the probability that the athlete is truly doped when the test result is positive.
    1. [a.] Determine from which value of $x$ the positive predictive value of the test studied will be greater than or equal to 0.9. Round the result to the nearest hundredth.
    2. [b.] A competition official decides to no longer test all athletes, but to target the most successful athletes who are assumed to be more frequently doped. What is the consequence of this decision on the positive predictive value of the test? Argue using a result from Part A.
Q2 5 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
Consider the function $f$ defined on the interval $[ 0 ; 1 ]$ by $$f ( x ) = 2 x \mathrm { e } ^ { - x } .$$ It is admitted that the function $f$ is differentiable on the interval $[ 0 ; 1 ]$.
    1. [a.] Solve on the interval $[ 0 ; 1 ]$ the equation $f ( x ) = x$.
    2. [b.] Prove that, for all $x$ belonging to the interval $[ 0 ; 1 ]$, $$f ^ { \prime } ( x ) = 2 ( 1 - x ) \mathrm { e } ^ { - x } .$$
    3. [c.] Give the table of variations of the function $f$ on the interval $[ 0 ; 1 ]$.

    Consider the sequence $(u_n)$ defined by $u_0 = 0,1$ and for all natural integer $n$, $$u_{n+1} = f(u_n).$$
    1. [a.] Prove by induction that, for all natural integer $n$, $$0 \leqslant u_n < u_{n+1} \leqslant 1.$$
    2. [b.] Deduce that the sequence $(u_n)$ is convergent.
  3. Prove that the limit of the sequence $(u_n)$ is $\ln(2)$.
    1. [a.] Justify that for all natural integer $n$, $\ln(2) - u_n$ is positive.
    2. [b.] It is desired to write a Python script that returns an approximate value of $\ln(2)$ by default to within $10^{-4}$, as well as the number of steps to achieve this. Copy and complete the script below so that it answers the problem posed. \begin{verbatim} def seuil() : n = 0 u=0.1 while ln (2) - u ...0.0001 : n=n+1 u=... return (u,n) \end{verbatim}
    3. [c.] Give the value of the variable $n$ returned by the function seuil().
Q3 Differential equations First-Order Linear DE: General Solution View
Consider the differential equation $$\left( E_0 \right) : \quad y^{\prime} = y$$ where $y$ is a differentiable function of the real variable $x$.
  1. Prove that the unique constant function solution of the differential equation $\left( E_0 \right)$ is the zero function.
  2. Determine all solutions of the differential equation $(E_0)$.

Consider the differential equation $$(E) : \quad y^{\prime} = y - \cos(x) - 3\sin(x)$$ where $y$ is a differentiable function of the real variable $x$.
    \setcounter{enumi}{2}
  1. The function $h$ is defined on $\mathbb{R}$ by $h(x) = 2\cos(x) + \sin(x)$. It is admitted that it is differentiable on $\mathbb{R}$. Prove that the function $h$ is a solution of the differential equation $(E)$.
  2. Consider a function $f$ defined and differentiable on $\mathbb{R}$. Prove that: ``$f$ is a solution of $(E)$'' is equivalent to ``$f - h$ is a solution of $\left(E_0\right)$''.
  3. Deduce all solutions of the differential equation $(E)$.
  4. Determine the unique solution $g$ of the differential equation $(E)$ such that $g(0) = 0$.
  5. Calculate: $$\int_{0}^{\frac{\pi}{2}} \left[ -2\mathrm{e}^{x} + \sin(x) + 2\cos(x) \right] \mathrm{d}x$$
Q4 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. Consider:
  • the points $\mathrm{A}(-2; 0; 2)$, $\mathrm{B}(-1; 3; 0)$, $\mathrm{C}(1; -1; 2)$ and $\mathrm{D}(0; 0; 3)$.
  • the line $\mathscr{D}_1$ whose parametric representation is $\left\{ \begin{aligned} x &= t \\ y &= 3t \\ z &= 3 + 5t \end{aligned} \right.$ with $t \in \mathbb{R}$.
  • the line $\mathscr{D}_2$ whose parametric representation is $\left\{ \begin{aligned} x &= 1 + 3s \\ y &= -1 - 5s \\ z &= 2 - 6s \end{aligned} \right.$ with $s \in \mathbb{R}$.

  1. Prove that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.
    1. [a.] Prove that the vector $\vec{n}\begin{pmatrix} 1 \\ 3 \\ 5 \end{pmatrix}$ is orthogonal to the plane (ABC).
    2. [b.] Justify that a Cartesian equation of the plane (ABC) is: $$x + 3y + 5z - 8 = 0$$
    3. [c.] Deduce that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$ are not coplanar.
    1. [a.] Justify that the line $\mathscr{D}_1$ is the altitude of the tetrahedron ABCD from D. It is admitted that the line $\mathscr{D}_2$ is the altitude of the tetrahedron ABCD from C.
    2. [b.] Prove that the lines $\mathscr{D}_1$ and $\mathscr{D}_2$ are secant and determine the coordinates of their point of intersection.
    1. [a.] Determine the coordinates of the orthogonal projection H of point D onto the plane (ABC).
    2. [b.] Calculate the distance from point D to the plane (ABC). Round the result to the nearest hundredth.