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
2018 caledonie

5 maths questions

Q1 Applied differentiation Full function study (variation table, limits, asymptotes) View
Let $f$ and $g$ be the functions defined on $] 0 ; + \infty [$ by
$$f ( x ) = \mathrm { e } ^ { - x } \quad \text { and } \quad g ( x ) = \frac { 1 } { x ^ { 2 } } \mathrm { e } ^ { - \frac { 1 } { x } } .$$
We admit that $f$ and $g$ are differentiable on $] 0 ; + \infty [$. We denote $f ^ { \prime }$ and $g ^ { \prime }$ their respective derivative functions.
Part A - Graphical conjectures
In each of the questions in this part, no explanation is required.
  1. Conjecture graphically a solution to the equation $f ( x ) = g ( x )$ on $] 0 ; + \infty [$.
  2. Conjecture graphically a solution to the equation $g ^ { \prime } ( x ) = 0$ on $] 0 ; + \infty [$.

Part B - Study of the function $g$
  1. Calculate the limit of $g ( x )$ as $x$ tends to $+ \infty$.
  2. We admit that the function $g$ is strictly positive on $] 0 ; + \infty [$.

Let $h$ be the function defined on $] 0 ; + \infty [$ by $h ( x ) = \ln ( g ( x ) )$. a. Prove that, for every strictly positive real number $x$,
$$h ( x ) = \frac { - 1 - 2 x \ln x } { x } .$$
b. Calculate the limit of $h ( x )$ as $x$ tends to 0. c. Deduce the limit of $g ( x )$ as $x$ tends to 0.
3. Prove that, for every strictly positive real number $x$,
$$g ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { - \frac { 1 } { x } } ( 1 - 2 x ) } { x ^ { 4 } } .$$
    \setcounter{enumi}{3}
  1. Deduce the variations of the function $g$ on $] 0 ; + \infty [$.

Part C - Area of the two regions between the curves $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$
  1. Prove that point A with coordinates $( 1 ; \mathrm { e } ^ { - 1 } )$ is an intersection point of $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$. We admit that this point is the unique intersection point of $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$, and that $\mathscr { C } _ { f }$ is above $\mathscr { C } _ { g }$ on the interval $] 0 ; 1 [$ and below on the interval $] 1 ; + \infty [$.
  2. Let $a$ and $b$ be two strictly positive real numbers. Prove that

$$\int _ { a } ^ { b } ( f ( x ) - g ( x ) ) \mathrm { d } x = \mathrm { e } ^ { - a } + \mathrm { e } ^ { - \frac { 1 } { a } } - \mathrm { e } ^ { - b } - \mathrm { e } ^ { - \frac { 1 } { b } } .$$
    \setcounter{enumi}{2}
  1. Prove that

$$\lim _ { a \rightarrow 0 } \int _ { a } ^ { 1 } ( f ( x ) - g ( x ) ) \mathrm { d } x = 1 - 2 \mathrm { e } ^ { - 1 } .$$
    \setcounter{enumi}{3}
  1. We admit that

$$\lim _ { a \rightarrow 0 } \int _ { a } ^ { 1 } ( f ( x ) - g ( x ) ) \mathrm { d } x = \lim _ { b \rightarrow + \infty } \int _ { 1 } ^ { b } ( g ( x ) - f ( x ) ) \mathrm { d } x .$$
Give a graphical interpretation of this equality.
Q2 Binomial Distribution Compute Expectation, Variance, or Standard Deviation View
A general knowledge test consists of a multiple choice questionnaire (MCQ) with twenty questions. For each one, the subject proposes four possible answers, of which only one is correct. For each question, the candidate must necessarily choose a single answer. This person earns one point for each correct answer and loses no points if their answer is wrong.
We consider three candidates:
  • Anselme answers completely at random to each of the twenty questions. In other words, for each of the questions, the probability that he answers correctly is equal to $\frac { 1 } { 4 }$;
  • Barbara is somewhat better prepared. We consider that for each of the twenty questions, the probability that she answers correctly is $\frac { 1 } { 2 }$;
  • Camille does even better: for each of the questions, the probability that she answers correctly is $\frac { 2 } { 3 }$.

  1. We denote $X , Y$ and $Z$ the random variables equal to the scores respectively obtained by Anselme, Barbara and Camille. a. What is the probability distribution followed by the random variable $X$? Justify. b. Using a calculator, give the answer rounded to the nearest thousandth of the probability $P ( X \geqslant 10 )$. In the following, we will admit that $P ( Y \geqslant 10 ) \approx 0.588$ and $P ( Z \geqslant 10 ) \approx 0.962$.
  2. We randomly choose the copy of one of these three candidates.

We denote $A , B , C$ and $M$ the events:
  • $A$: ``the chosen copy is Anselme's'';
  • $B$: ``the chosen copy is Barbara's'';
  • $C$: ``the chosen copy is Camille's'';
  • $M$: ``the chosen copy obtains a score greater than or equal to 10''.

We observe, after correcting it, that the chosen copy obtains a score greater than or equal to 10 out of 20.
What is the probability that it is Barbara's copy? Give the answer rounded to the nearest thousandth of this probability.
Q3 Vectors 3D & Lines Multi-Part 3D Geometry Problem View
Let ABCDEFGH be a cube. We consider:
  • I and J the midpoints respectively of segments [AD] and [BC];
  • P the center of face ABFE, that is, the intersection of diagonals (AF) and (BE);
  • Q the midpoint of segment [FG].

We place ourselves in the orthonormal coordinate system $( \mathrm { A } ; \frac { 1 } { 2 } \overrightarrow { \mathrm { AB } } , \frac { 1 } { 2 } \overrightarrow { \mathrm { AD } } , \frac { 1 } { 2 } \overrightarrow { \mathrm { AE } } )$. Throughout the exercise, we may use the coordinates of the points in the figure without justifying them. We admit that a parametric representation of the line (IJ) is
$$\left\{ \begin{array} { l } x = r \\ y = 1 , \quad r \in \mathbf { R } \\ z = 0 \end{array} \right.$$
  1. Verify that a parametric representation of the line (PQ) is

$$\left\{ \begin{array} { l } x = 1 + t \\ y = t , \quad t \in \mathbf { R } \\ z = 1 + t \end{array} \right.$$
Let $t$ be a real number and $\mathrm { M } ( 1 + t ; t ; 1 + t )$ be the point on the line (PQ) with parameter $t$.
2. a. We admit that there exists a unique point K belonging to the line (IJ) such that (MK) is orthogonal to (IJ). Prove that the coordinates of this point $\mathrm { K }$ are $( 1 + t ; 1 ; 0 )$. b. Deduce that $\mathrm { MK } = \sqrt { 2 + 2 t ^ { 2 } }$.
3. a. Verify that $y - z = 0$ is a Cartesian equation of the plane (HGB). b. We admit that there exists a unique point L belonging to the plane (HGB) such that (ML) is orthogonal to (HGB). Verify that the coordinates of this point L are $\left( 1 + t ; \frac { 1 } { 2 } + t ; \frac { 1 } { 2 } + t \right)$. c. Deduce that the distance ML is independent of $t$.
4. Does there exist a value of $t$ for which the distance MK is equal to the distance ML?
Q4a Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
Exercise 4 — Candidates who have not followed the specialization course
We define the sequence of complex numbers $( z _ { n } )$ in the following way: $z _ { 0 } = 1$ and, for every natural integer $n$,
$$z _ { n + 1 } = \frac { 1 } { 3 } z _ { n } + \frac { 2 } { 3 } \mathrm { i } .$$
We place ourselves in a plane with an orthonormal direct coordinate system $( \mathrm { O } ; \vec { u } , \vec { v } )$. For every natural integer $n$, we denote $\mathrm { A } _ { n }$ the point in the plane with affix $z _ { n }$. For every natural integer $n$, we set $u _ { n } = z _ { n } - \mathrm { i }$ and we denote $\mathrm { B } _ { n }$ the point with affix $u _ { n }$. We denote C the point with affix i.
  1. Express $u _ { n + 1 }$ as a function of $u _ { n }$, for every natural integer $n$.
  2. Prove that, for every natural integer $n$,

$$u _ { n } = \left( \frac { 1 } { 3 } \right) ^ { n } ( 1 - \mathrm { i } ) .$$
    \setcounter{enumi}{2}
  1. a. For every natural integer $n$, calculate, as a function of $n$, the modulus of $u _ { n }$. b. Prove that

$$\lim _ { n \rightarrow + \infty } \left| z _ { n } - \mathrm { i } \right| = 0$$
c. What geometric interpretation can be given of this result?
4. a. Let $n$ be a natural integer. Determine an argument of $u _ { n }$. b. Prove that, as $n$ ranges over the set of natural integers, the points $\mathrm { B } _ { n }$ are collinear. c. Prove that, for every natural integer $n$, the point $\mathrm { A } _ { n }$ belongs to the line with reduced equation:
$$y = - x + 1 .$$
Q4b Proof by induction Prove a sequence bound or inequality by induction View
Exercise 4 — Candidates who have followed the specialization course
We call Fibonacci sequence the sequence $( u _ { n } )$ defined by $u _ { 0 } = 0 , u _ { 1 } = 1$ and, for every natural integer $n$,
$$u _ { n + 2 } = u _ { n + 1 } + u _ { n }$$
We admit that, for every natural integer $n$, $u _ { n }$ is a natural integer. Parts A and B can be treated independently.
Part A
  1. a. Calculate the terms of the Fibonacci sequence up to $u _ { 10 }$. b. What can be conjectured about the GCD of $u _ { n }$ and $u _ { n + 1 }$ for every natural integer $n$?
  2. We define the sequence $\left( v _ { n } \right)$ by $v _ { n } = u _ { n } ^ { 2 } - u _ { n + 1 } \times u _ { n - 1 }$ for every non-zero natural integer $n$. a. Prove that, for every non-zero natural integer $n$, $v _ { n + 1 } = - v _ { n }$. b. Deduce that, for every non-zero natural integer $n$,

$$u _ { n } ^ { 2 } - u _ { n + 1 } \times u _ { n - 1 } = ( - 1 ) ^ { n - 1 }$$
c. Then prove the conjecture made in question 1.b.
Part B
We consider the matrix $F = \left( \begin{array} { l l } 1 & 1 \\ 1 & 0 \end{array} \right)$.
  1. Calculate $F ^ { 2 }$ and $F ^ { 3 }$. You may use a calculator.
  2. Prove by induction that, for every non-zero natural integer $n$,

$$F ^ { n } = \left( \begin{array} { c c } u_{n+1} & u_n \\ u_n & u_{n-1} \end{array} \right)$$