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__europe_j2

4 maths questions

Q1 Differentiating Transcendental Functions Full function study with transcendental functions View
Consider the function $f$ defined on $\mathbb { R }$ by :
$$f ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - 3 x } }$$
We denote $\mathscr { C } _ { f }$ its representative curve in an orthogonal coordinate system of the plane. We name A the point with coordinates $\left( 0 ; \frac { 1 } { 2 } \right)$ and B the point with coordinates $\left( 1 ; \frac { 5 } { 4 } \right)$. Below we have drawn the curve $\mathscr { C } _ { f }$ and $\mathscr { T }$ the tangent line to the curve $\mathscr { C } _ { f }$ at the point with abscissa 0.
Part A: graphical readings
In this part, results will be obtained by graphical reading. No justification is required.
  1. Determine the reduced equation of the tangent line $\mathscr { T }$.
  2. Give the intervals on which the function $f$ appears to be convex or concave.

Part B : study of the function
  1. We admit that the function $f$ is differentiable on $\mathbb { R }$.

Determine the expression of its derivative function $f ^ { \prime }$.
2. Justify that the function $f$ is strictly increasing on $\mathbb { R }$.
3. a. Determine the limit at $+ \infty$ of the function $f$. b. Determine the limit at $- \infty$ of the function $f$.
4. Determine the exact value of the solution $\alpha$ of the equation $f ( x ) = 0,99$.
Part C : Tangent line and convexity
  1. Determine by calculation an equation of the tangent line $\mathscr { T }$ to the curve $\mathscr { C } _ { f }$ at the point with abscissa 0.

We admit that the function $f$ is twice differentiable on $\mathbb { R }$. We denote $f ^ { \prime \prime }$ the second derivative function of the function $f$. We admit that $f ^ { \prime \prime }$ is defined on $\mathbb { R }$ by:
$$f ^ { \prime \prime } ( x ) = \frac { 9 \mathrm { e } ^ { - 3 x } \left( \mathrm { e } ^ { - 3 x } - 1 \right) } { \left( 1 + \mathrm { e } ^ { - 3 x } \right) ^ { 3 } } .$$
    \setcounter{enumi}{1}
  1. Study the sign of the function $f ^ { \prime \prime }$ on $\mathbb { R }$.
  2. a. Indicate, by justifying, on which interval(s) the function $f$ is convex. b. What does point A represent for the curve $\mathscr { C } _ { f }$ ? c. Deduce the relative position of the tangent line $\mathscr { T }$ and the curve $\mathscr { C } _ { f }$. Justify the answer.
Q2 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Part A
Consider the function $f$ defined by :
$$f ( x ) = x - \ln ( 1 + x ) .$$
  1. Justify that the function $f$ is defined on the interval $] - 1 ; + \infty [$.
  2. We admit that the function $f$ is differentiable on $] - 1 ; + \infty [$.

Determine the expression of its derivative function $f ^ { \prime } ( x )$.
3. a. Deduce the direction of variation of the function $f$ on the interval $] - 1 ; + \infty [$. b. Deduce the sign of the function $f$ on the interval $] - 1 ; 0 [$.
4. a. Show that, for all $x$ belonging to the interval $] - 1 ; + \infty [$, we have :
$$f ( x ) = \ln \left( \frac { \mathrm { e } ^ { x } } { 1 + x } \right) .$$
b. Deduce the limit at $+ \infty$ of the function $f$.
Part B
Consider the sequence ( $u _ { n }$ ) defined by $u _ { 0 } = 10$ and, for all natural integer $n$,
$$u _ { n + 1 } = u _ { n } - \ln \left( 1 + u _ { n } \right) .$$
We admit that the sequence ( $u _ { n }$ ) is well defined.
  1. Give the value rounded to the nearest thousandth of $u _ { 1 }$.
  2. Using question 3. a. of Part A , prove by induction that, for all natural integer $n$, we have $u _ { n } \geqslant 0$.
  3. Prove that the sequence ( $u _ { n }$ ) is decreasing.
  4. Deduce from the previous questions that the sequence ( $u _ { n }$ ) converges.
  5. Determine the limit of the sequence $\left( u _ { n } \right)$.
Q3 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
$$\mathrm { A } ( 3 ; 0 ; 1 ) , \quad \mathrm { B } ( 2 ; 1 ; 2 ) \text { and } \quad \mathrm { C } ( - 2 ; - 5 ; 1 ) .$$
  1. Prove that the points $\mathrm { A } , \mathrm { B }$ and C are not collinear.
  2. Prove that the triangle ABC is right-angled at A .
  3. Verify that the plane $( \mathrm { ABC } )$ has the Cartesian equation :

$$- x + y - 2 z + 5 = 0$$
    \setcounter{enumi}{3}
  1. Consider the point $S ( 1 ; - 2 ; 4 )$.

Determine the parametric representation of the line ( $\Delta$ ), passing through S and orthogonal to the plane (ABC).
5. We call H the point of intersection of the line ( $\Delta$ ) and the plane (ABC).
Show that the coordinates of H are $( 0 ; - 1 ; 2 )$. 6. Calculate the exact value of the distance SH. 7. Consider the circle $\mathscr { C }$, included in the plane (ABC), with center H, passing through the point B. We call $\mathscr { D }$ the disk bounded by the circle $\mathscr { C }$.
Determine the exact value of the area of the disk $\mathscr { D }$. 8. Deduce the exact value of the volume of the cone with apex S and base the disk $\mathscr { D }$.
Q4 Binomial Distribution Compute Cumulative or Complement Binomial Probability View
This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required.
A wrong answer, multiple answers or the absence of an answer to a question neither awards nor deducts points. The five questions are independent.
A production line produces mechanical parts. It is estimated that $4 \%$ of the parts produced by this line are defective. We randomly choose $n$ parts produced by the production line. The number of parts produced is large enough that this choice can be treated as a draw with replacement. We denote $X$ the random variable equal to the number of defective parts drawn. In the following three questions, we take $n = 50$.
  1. What is the probability, rounded to the nearest thousandth, of drawing at least one defective part? a. 1 b. 0,870 c. 0,600 d. 0,599
  2. The probability $p ( 3 < X \leqslant 7 )$ is equal to : a. $p ( X \leqslant 7 ) - p ( X > 3 )$ b. $p ( X \leqslant 7 ) - p ( X \leqslant 3 )$ c. $p ( X < 7 ) - p ( X > 3 )$ d. $p ( X < 7 ) - p ( X \geqslant 3 )$
  3. What is the smallest natural integer $k$ such that the probability of drawing at most $k$ defective parts is greater than or equal to $95 \%$ ? a. 2 b. 3 c. 4 d. 5

In the following questions, $n$ no longer necessarily equals 50.
4. What is the probability of drawing only defective parts? a. $0,04 ^ { n }$ b. $0,96 ^ { n }$ c. $1 - 0,04 ^ { n }$ d. $1 - 0,96 ^ { n }$
5. Consider the Python function below. What does it return?
\begin{verbatim} def seuil (x) : n=1 while 1-0.96**n a. The smallest number $n$ such that the probability of drawing at least one defective part is greater than or equal to x . b. The smallest number $n$ such that the probability of drawing no defective parts is greater than or equal to x. c. The largest number $n$ such that the probability of drawing only defective parts is greater than or equal to x. d. The largest number $n$ such that the probability of drawing no defective parts is greater than or equal to x.