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
2015 polynesie

6 maths questions

Q1 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Consider the rectangular prism ABCDEFGH below, for which $\mathrm { AB } = 6 , \mathrm { AD } = 4$ and $\mathrm { AE } = 2$. I, J and K are points such that $\overrightarrow { A I } = \frac { 1 } { 6 } \overrightarrow { A B } , \overrightarrow { A J } = \frac { 1 } { 4 } \overrightarrow { A D } , \overrightarrow { A K } = \frac { 1 } { 2 } \overrightarrow { A E }$. We use the orthonormal coordinate system ( $A$; $\overrightarrow { A I } , \overrightarrow { A J } , \overrightarrow { A K }$ ).
  1. Verify that the vector $\vec { n }$ with coordinates $\left( \begin{array} { c } 2 \\ 2 \\ - 9 \end{array} \right)$ is normal to the plane (IJG).
  2. Determine an equation of the plane (IJG).
  3. Determine the coordinates of the intersection point L of the plane (IJG) and the line (BF).
  4. Draw the cross-section of the rectangular prism ABCDEFGH by the plane (IJG). This drawing should be done on the figure provided in the appendix to be returned with your work). No justification is required.
Q2 Complex numbers 2 Complex Mappings and Transformations View
The complex plane is given an orthonormal coordinate system ( $\mathrm { O } , \vec { u } , \vec { v }$ ). To every point $M$ with affixe $z$ in the plane, we associate the point $M ^ { \prime }$ with affixe $z ^ { \prime }$ defined by:
$$z ^ { \prime } = z ^ { 2 } + 4 z + 3 .$$
  1. A point $M$ is called invariant when it coincides with the associated point $M ^ { \prime }$.
    Prove that there exist two invariant points. Give the affixe of each of these points in algebraic form, then in exponential form.
  2. Let A be the point with affixe $\frac { - 3 - \mathrm { i } \sqrt { 3 } } { 2 }$ and B the point with affixe $\frac { - 3 + \mathrm { i } \sqrt { 3 } } { 2 }$.
    Show that OAB is an equilateral triangle.
  3. Determine the set $\mathcal { E }$ of points $M$ with affixe $z = x + \mathrm { i } y$ where $x$ and $y$ are real, such that the associated point $M ^ { \prime }$ lies on the real axis.
  4. In the complex plane, represent the points A and B as well as the set $\mathcal { E }$.
Q3 Normal Distribution Normal Distribution Combined with Total Probability or Bayes' Theorem View
In a country, the height in centimetres of women aged 18 to 65 can be modelled by a random variable $X _ { 1 }$ following a normal distribution with mean $\mu _ { 1 } = 165 \mathrm {~cm}$ and standard deviation $\sigma _ { 1 } = 6 \mathrm {~cm}$, and that of men aged 18 to 65 by a random variable $X _ { 2 }$ following a normal distribution with mean $\mu _ { 2 } = 175 \mathrm {~cm}$ and standard deviation $\sigma _ { 2 } = 11 \mathrm {~cm}$. In this exercise all results should be rounded to $10 ^ { - 2 }$.
  1. What is the probability that a woman chosen at random in this country measures between 1.53 metres and 1.77 metres?
  2. a. Determine the probability that a man chosen at random in this country measures more than 1.70 metres. b. Furthermore, it is known that in this country women represent $52 \%$ of the population of people aged between 18 and 65. A person aged between 18 and 65 is chosen at random. They measure more than $1.70 \mathrm {~m}$. What is the probability that this person is a woman?
Q4 5 marks Differentiating Transcendental Functions Determine parameters from function or curve conditions View
The profile of a slide is modelled by the curve $\mathcal { C }$ representing the function $f$ defined on the interval [1;8] by
$$f ( x ) = ( a x + b ) \mathrm { e } ^ { - x } \text { where } a \text { and } b \text { are two natural integers. }$$
The curve $\mathcal { C }$ is drawn in an orthonormal coordinate system with unit of one metre.
Part A Modelling
  1. We want the tangent to the curve $\mathcal { C }$ at its point with abscissa 1 to be horizontal. Determine the value of the integer $b$.
  2. We want the top of the slide to be located between 3.5 and 4 metres high. Determine the value of the integer $a$.

Part B An amenity for visitors
We assume in the following that the function $f$ introduced in Part A is defined for all real $x \in [ 1 ; 8 ]$ by
$$f ( x ) = 10 x \mathrm { e } ^ { - x }$$
The retaining wall of the slide will be painted by an artist on a single face. In the quote he proposes, he asks for a flat fee of 300 euros plus 50 euros per square metre painted.
  1. Let $g$ be the function defined on [ $1 ; 8$ ] by
    $$g ( x ) = 10 ( - x - 1 ) \mathrm { e } ^ { - x }$$
    Determine the derivative of the function $g$.
  2. What is the amount of the artist's quote?

Part C A constraint to verify
Safety considerations require limiting the maximum slope of the slide. Consider a point $M$ on the curve $\mathcal { C }$, with abscissa different from 1. We call $\alpha$ the acute angle formed by the tangent to $\mathcal { C }$ at $M$ and the horizontal axis. The constraints require that the angle $\alpha$ be less than 55 degrees.
  1. We denote $f ^ { \prime }$ the derivative of the function $f$ on the interval $[ 1 ; 8 ]$. We admit that, for all $x$ in the interval $[ 1 ; 8 ] , f ^ { \prime } ( x ) = 10 ( 1 - x ) \mathrm { e } ^ { - x }$. Study the variations of the function $f ^ { \prime }$ on the interval [ $1 ; 8$ ].
  2. Let $x$ be a real number in the interval ] 1; 8] and let $M$ be the point with abscissa $x$ on the curve $\mathcal { C }$. Justify that $\tan \alpha = \left| f ^ { \prime } ( x ) \right|$.
  3. Is the slide compliant with the imposed constraints?
Q5a Sequences and series, recurrence and convergence Algorithm and programming for sequences View
Exercise 5 — Candidates who have not chosen the specialisation option
Let ( $v _ { n }$ ) be the sequence defined by
$$v _ { 1 } = \ln ( 2 ) \text { and, for all natural integer } n \text { non-zero, } v _ { n + 1 } = \ln \left( 2 - \mathrm { e } ^ { - v _ { n } } \right) .$$
We admit that this sequence is defined for all non-zero natural integer $n$. We then define the sequence ( $S _ { n }$ ) for all non-zero natural integer $n$ by :
$$S _ { n } = \sum _ { k = 1 } ^ { n } v _ { k } = v _ { 1 } + v _ { 2 } + \cdots + v _ { n } .$$
The purpose of this exercise is to determine the limit of ( $S _ { n }$ ).
Part A - Conjectures using an algorithm
  1. Copy and complete the following algorithm which calculates and displays the value of $S _ { n }$ for a value of $n$ chosen by the user :
    Variables :
    $n , k$ integers
    $S , v$ real numbers
    Initialisation :
    Input the value of $n$
    $v$ takes the value $\ldots$
    $S$ takes the value $\ldots$
    Processing:
    For $k$ varying from \ldots to \ldots do
    \ldots takes the value \ldots
    \ldots takes the value \ldots
    End For
    Output :
    Display $S$

  2. Using this algorithm, we obtain some values of $S _ { n }$. The values rounded to the nearest tenth are given in the table below :
    $n$101001000100001000001000000
    $S _ { n }$2.44.66.99.211.513.8

    By explaining your approach, make a conjecture about the behaviour of the sequence $\left( S _ { n } \right)$.

Part B - Study of an auxiliary sequence
For all non-zero natural integer $n$, we define the sequence ( $u _ { n }$ ) by $u _ { n } = \mathrm { e } ^ { v _ { n } }$.
  1. Verify that $u _ { 1 } = 2$ and that, for all non-zero natural integer $n$, $u _ { n + 1 } = 2 - \frac { 1 } { u _ { n } }$.
  2. Calculate $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$. Results should be given in fractional form.
  3. Prove that, for all non-zero natural integer $n$, $u _ { n } = \frac { n + 1 } { n }$.

Part C - Study of ( $S _ { n }$ )
  1. For all non-zero natural integer $n$, express $v _ { n }$ as a function of $u _ { n }$, then $v _ { n }$ as a function of $n$.
  2. Verify that $S _ { 3 } = \ln ( 4 )$.
  3. For all non-zero natural integer $n$, express $S _ { n }$ as a function of $n$. Deduce the limit of the sequence $\left( S _ { n } \right)$.
Q5b Matrices Matrix Power Computation and Application View
Exercise 5 — Candidates who have chosen the specialisation option
Consider the matrix $A = \left( \begin{array} { l l } - 4 & 6 \\ - 3 & 5 \end{array} \right)$.
  1. We call $I$ the identity matrix of order 2.
    Verify that $A ^ { 2 } = A + 2 I$.
  2. Deduce an expression for $A ^ { 3 }$ and an expression for $A ^ { 4 }$ in the form $\alpha A + \beta I$ where $\alpha$ and $\beta$ are real numbers.
  3. Consider the sequences $\left( r _ { n } \right)$ and $\left( s _ { n } \right)$ defined by $r _ { 0 } = 0$ and $s _ { 0 } = 1$ and, for all natural integer $n$,
    $$\left\{ \begin{array} { l } r _ { n + 1 } = r _ { n } + s _ { n } \\ s _ { n + 1 } = 2 r _ { n } \end{array} \right.$$
    Prove that, for all natural integer $n , A ^ { n } = r _ { n } A + s _ { n } I$.
  4. Prove that the sequence ( $k _ { n }$ ) defined for all natural integer $n$ by $k _ { n } = r _ { n } - s _ { n }$ is geometric with common ratio $- 1$. Deduce, for all natural integer $n$, an explicit expression for $k _ { n }$ as a function of $n$.
  5. We admit that the sequence ( $t _ { n }$ ) defined for all natural integer $n$ by $t _ { n } = r _ { n } + \frac { ( - 1 ) ^ { n } } { 3 }$ is geometric with common ratio 2. Deduce, for all natural integer $n$, an explicit expression for $t _ { n }$ as a function of $n$.
  6. From the previous questions, deduce, for all natural integer $n$, an explicit expression for $r _ { n }$ and $s _ { n }$ as a function of $n$.
  7. Deduce then, for all natural integer $n$, an expression for $A ^ { n }$.