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__caledonie_j2

4 maths questions

Q1 5 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
We consider the cube ABCDEFGH with edge length 1 represented opposite. We denote K the midpoint of segment [HG]. We place ourselves in the orthonormal coordinate system $( \mathrm { A } ; \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AE } } )$.
  1. Justify that the points $\mathrm { C } , \mathrm { F }$ and K define a plane.
  2. a. Give, without justification, the lengths KG, GF and GC. b. Calculate the area of triangle FGC. c. Calculate the volume of tetrahedron FGCK.

We recall that the volume $V$ of a tetrahedron 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.
3. a. We denote $\vec { n }$ the vector with coordinates $\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)$.
Prove that $\vec { n }$ is normal to the plane (CFK). b. Deduce that a Cartesian equation of the plane (CFK) is: $$x + 2 y + z - 3 = 0 .$$
    \setcounter{enumi}{3}
  1. We denote $\Delta$ the line passing through point G and perpendicular to the plane (CFK). Prove that a parametric representation of the line $\Delta$ is:

$$\left\{ \begin{aligned} x & = 1 + t \\ y & = 1 + 2 t \\ z & = 1 + t \end{aligned} \quad ( t \in \mathbb { R } ) \right)$$
    \setcounter{enumi}{4}
  1. Let L be the point of intersection between the line $\Delta$ and the plane (CFK). a. Determine the coordinates of point L . b. Deduce that $\mathrm { LG } = \frac { \sqrt { 6 } } { 6 }$.
  2. Using question 2., determine the exact value of the area of triangle CFK.
Q2 5 marks Applied differentiation Full function study (variation table, limits, asymptotes) View
We consider the function $f$ defined on $[ 0 ; + \infty [$ by: $$f ( x ) = x \mathrm { e } ^ { - x }$$ We denote $\mathscr { C } _ { f }$ its representative curve in an orthonormal coordinate system of the plane. We admit that $f$ is twice differentiable on $[ 0 ; + \infty [$. We denote $f ^ { \prime }$ its derivative and $f ^ { \prime \prime }$ its second derivative.
  1. By noting that for all $x$ in $[ 0 ; + \infty [$, we have
$$f ( x ) = \frac { x } { \mathrm { e } ^ { x } }$$ prove that the curve $\mathscr { C } _ { f }$ has an asymptote at $+ \infty$ for which you will give an equation.
2. Prove that for all real $x$ belonging to $[ 0 ; + \infty [$ : $$f ^ { \prime } ( x ) = ( 1 - x ) \mathrm { e } ^ { - x }$$
    \setcounter{enumi}{2}
  1. Draw up the table of variations of $f$ on $[ 0 ; + \infty [$, on which you will show the values at the boundaries as well as the exact value of the extremum.
  2. Determine, on the interval $[ 0 ; + \infty [$, the number of solutions of the equation
$$f ( x ) = \frac { 367 } { 1000 }$$
    \setcounter{enumi}{4}
  1. We admit that for all $x$ belonging to $[ 0 ; + \infty [$ :
$$f ^ { \prime \prime } ( x ) = \mathrm { e } ^ { - x } ( x - 2 )$$ Study the convexity of the function $f$ on the interval $[ 0 ; + \infty [$. 6. Let $a$ be a real number belonging to $[ 0 ; + \infty [$ and A the point of the curve $\mathscr { C } _ { f }$ with abscissa $a$. We denote $T _ { a }$ the tangent to $\mathscr { C } _ { f }$ at A. We denote $\mathrm { H } _ { a }$ the point of intersection of the line $T _ { a }$ and the ordinate axis. We denote $g ( a )$ the ordinate of $\mathrm { H } _ { a }$. a. Prove that a reduced equation of the tangent $T _ { a }$ is: $$y = \left[ ( 1 - a ) \mathrm { e } ^ { - a } \right] x + a ^ { 2 } \mathrm { e } ^ { - a }$$ b. Deduce the expression of $g ( a )$. c. Prove that $g ( a )$ is maximum when A is an inflection point of the curve $\mathscr { C } _ { f }$.
Q3 5 marks Sequences and series, recurrence and convergence Algorithm and programming for sequences View
We consider the sequence ( $u _ { n }$ ) such that $u _ { 0 } = 0$ and for all natural integer $n$ : $$u _ { n + 1 } = \frac { - u _ { n } - 4 } { u _ { n } + 3 } .$$ We admit that $u _ { n }$ is defined for all natural integer $n$.
  1. Calculate the exact values of $u _ { 1 }$ and $u _ { 2 }$.
  2. We consider the function term below written incompletely in Python language:

\begin{verbatim} def terme (n) : u = ... for i in range(n): u = ... return(u) \end{verbatim}
We recall that in Python language, «i in range (n) » means that $i$ varies from 0 to $n - 1$.
Rewrite and complete the box above so that, for all natural integer $n$, the instruction terme (n) returns the value of $u _ { n }$.
3. Let the function $f$ defined on $] - 3 ; + \infty [$ by: $$f ( x ) = \frac { - x - 4 } { x + 3 }$$ Thus, for all natural integer $n$, we have $u _ { n + 1 } = f \left( u _ { n } \right)$. Prove that the function $f$ is strictly increasing on $] - 3 ; + \infty [$.
4. Prove by induction that for all natural integer $n$ : $$- 2 < u _ { n + 1 } \leqslant u _ { n } .$$
    \setcounter{enumi}{4}
  1. Deduce that the sequence ( $u _ { n }$ ) is convergent.
  2. Let the sequence $\left( v _ { n } \right)$ defined for all natural integer $n$ by:
$$v _ { n } = \frac { 1 } { u _ { n } + 2 }$$ a. Give $v _ { 0 }$. b. Prove that the sequence ( $v _ { n }$ ) is arithmetic with common difference 1 . c. Deduce that for all natural integer $n \geqslant 1$ : $$u _ { n } = \frac { 1 } { n + 0,5 } - 2 .$$ d. Determine the limit of the sequence $\left( u _ { n } \right)$.
Q4 5 marks Conditional Probability Combinatorial Conditional Probability (Counting-Based) View
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. To answer, indicate on your paper the number of the question and the letter of the chosen answer. No justification is required.
A wrong answer, an absence of answer, or a multiple answer, neither gives nor removes points.
The 200 members of a club are girls or boys. These members practice rowing or basketball according to the distribution shown in the table below.
\cline { 2 - 4 } \multicolumn{1}{c|}{}RowingBasketballTotal
Girls2580105
Boys504595
Total75125200

We choose a member at random and consider the following events: $F$ : the member is a girl; $A$ : the member practices rowing.
  1. The probability of $F$ given $A$ is equal to : a. $\frac { 25 } { 100 }$ b. $\frac { 25 } { 75 }$ c. $\frac { 25 } { 105 }$ d. $\frac { 75 } { 105 }$
  2. The probability of the event $A \cup F$ is equal to : a. $\frac { 9 } { 10 }$ b. $\frac { 1 } { 8 }$ c. $\frac { 31 } { 40 }$ d. $\frac { 5 } { 36 }$

To get to work, Albert can use either the bus or the train. The probability that the bus breaks down is equal to $b$. The probability that the train breaks down is equal to $t$. Bus and train breakdowns occur independently.
3. The probability $p _ { 1 }$ that the bus or the train breaks down is equal to : a. $p _ { 1 } = b t$ b. $p _ { 1 } = 1 - b t$ c. $p _ { 1 } = b + t$ d. $p _ { 1 } = b + t - b t$
4. The probability $p _ { 2 }$ that Albert can get to work is equal to : a. $p _ { 2 } = b t$ b. $p _ { 2 } = 1 - b t$ c. $p _ { 2 } = b + t$ d. $p _ { 2 } = b + t - b t$
5. We consider a coin for which the probability of obtaining HEADS is equal to $x$. We flip the coin $n$ times. The flips are independent. The probability $p$ of obtaining at least one HEADS in the $n$ flips is equal to a. $p = x ^ { n }$ b. $p = ( 1 - x ) ^ { n }$ c. $p = 1 - x ^ { n }$ d. $p = 1 - ( 1 - x ) ^ { n }$