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
2017 metropole

5 maths questions

Q1 Integration by Parts Integration by Parts within Function Analysis View
Exercise 1 -- Common to all candidates

Part A
We consider the function $h$ defined on the interval $[ 0 ; + \infty [$ by: $$h ( x ) = x \mathrm { e } ^ { - x }$$
  1. Determine the limit of the function $h$ at $+ \infty$.
  2. Study the variations of the function $h$ on the interval $[ 0 ; + \infty [$ and draw up its table of variations.
  3. The objective of this question is to determine a primitive of the function $h$. a. Verify that for every real number $x$ belonging to the interval $[ 0 ; + \infty [$, we have: $$h ( x ) = \mathrm { e } ^ { - x } - h ^ { \prime } ( x )$$ where $h ^ { \prime }$ denotes the derivative function of $h$. b. Determine a primitive on the interval $[ 0 ; + \infty [$ of the function $x \longmapsto \mathrm { e } ^ { - x }$. c. Deduce from the two previous questions a primitive of the function $h$ on the interval $[ 0 ; + \infty [$.

Part B
We define the functions $f$ and $g$ on the interval $[ 0 ; + \infty [$ by: $$f ( x ) = x \mathrm { e } ^ { - x } + \ln ( x + 1 ) \quad \text { and } \quad g ( x ) = \ln ( x + 1 )$$ We denote $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$ the respective graphical representations of the functions $f$ and $g$ in an orthonormal coordinate system.
  1. For a real number $x$ belonging to the interval $[ 0 ; + \infty [$, we call $M$ the point with coordinates $( x ; f ( x ) )$ and $N$ the point with coordinates $( x ; g ( x ) )$: $M$ and $N$ are therefore the points with abscissa $x$ belonging respectively to the curves $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$. a. Determine the value of $x$ for which the distance $MN$ is maximum and give this maximum distance. b. Place on the graph provided in the appendix the points $M$ and $N$ corresponding to the maximum value of $MN$.
  2. Let $\lambda$ be a real number belonging to the interval $[ 0 ; + \infty [$. We denote $D _ { \lambda }$ the region of the plane bounded by the curves $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$ and by the lines with equations $x = 0$ and $x = \lambda$. a. Shade the region $D _ { \lambda }$ corresponding to the value $\lambda$ proposed on the graph in the appendix. b. We denote $A _ { \lambda }$ the area of the region $D _ { \lambda }$, expressed in square units. Prove that: $$A _ { \lambda } = 1 - \frac { \lambda + 1 } { \mathrm { e } ^ { \lambda } } .$$ c. Calculate the limit of $A _ { \lambda }$ as $\lambda$ tends to $+ \infty$ and interpret the result.
  3. We consider the following algorithm: \begin{verbatim} Variables: $\lambda$ is a positive real number $S$ is a real number strictly between 0 and 1. Initialization: Input $S$ $\lambda$ takes the value 0 Processing: While $1 - \frac { \lambda + 1 } { \mathrm { e } ^ { \lambda } } < S$ do $\lambda$ takes the value $\lambda + 1$ End While Output: Display $\lambda$ \end{verbatim} a. What value does this algorithm display if we input the value $S = 0.8$? b. What is the role of this algorithm?
Q2 3 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 2 -- 3 points -- Common to all candidates
Space is equipped with a coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. Let $\mathcal{P}$ be the plane with Cartesian equation: $2x - z - 3 = 0$. We denote $A$ the point with coordinates $(1 ; a ; a^{2})$ where $a$ is a real number.
  1. Justify that, regardless of the value of $a$, the point $A$ does not belong to the plane $\mathcal{P}$.
  2. a. Determine a parametric representation of the line $\mathcal{D}$ (with parameter $t$) passing through the point $A$ and orthogonal to the plane $\mathcal{P}$. b. Let $M$ be a point belonging to the line $\mathcal{D}$, associated with the value $t$ of the parameter in the previous parametric representation. Express the distance $AM$ as a function of the real number $t$.
  3. We denote $H$ the point of intersection of the plane $\mathcal{P}$ and the line $\mathcal{D}$ orthogonal to $\mathcal{P}$ and passing through the point $A$. The point $H$ is called the orthogonal projection of the point $A$ onto the plane $\mathcal{P}$ and the distance $AH$ is called the distance from the point $A$ to the plane $\mathcal{P}$. Is there a value of $a$ for which the distance $AH$ from the point $A$ with coordinates $(1 ; a ; a^{2})$ to the plane $\mathcal{P}$ is minimal? Justify the answer.
Q3 Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View
Exercise 3 -- Common to all candidates
In a vast plain, a network of sensors makes it possible to detect lightning and to produce an image of storm phenomena. The radar screen is divided into forty sectors denoted by a letter and a number between 1 and 5. The lightning sensor is represented by the center of the screen; five concentric circles corresponding to the respective radii $20, 40, 60, 80$ and 100 kilometres delimit five zones numbered from 1 to 5, and eight segments starting from the sensor delimit eight portions of the same angular opening, named in the trigonometric direction from A to H.
We assimilate the radar screen to a part of the complex plane by defining an orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$:
  • the origin O marks the position of the sensor;
  • the abscissa axis is oriented from West to East;
  • the ordinate axis is oriented from South to North;
  • the chosen unit is the kilometre.
In the following, a point on the radar screen is associated with a point with affix $z$.
Part A
  1. We denote $z_{P}$ the affix of the point P located in sector B3 on the graph. We call $r$ the modulus of $z_{P}$ and $\theta$ its argument in the interval $]-\pi ; \pi]$. Among the four following propositions, determine the only one that proposes a correct bound for $r$ and for $\theta$ (no justification is required):
    Proposition AProposition BProposition CProposition D
    \begin{tabular}{ c } $40 < r < 60$
    and
    $0 < \theta < \frac{\pi}{4}$
    &
    $20 < r < 40$
    and
    $\frac{\pi}{2} < \theta < \frac{3\pi}{4}$
    &
    $40 < r < 60$
    and
    $\frac{\pi}{4} < \theta < \frac{\pi}{2}$
    &
    $0 < r < 60$
    and
    $-\frac{\pi}{2} < \theta < -\frac{\pi}{4}$
    \hline \end{tabular}
  2. A lightning impact is materialized on the screen at a point with affix $z$. In each of the two following cases, determine the sector to which this point belongs: a. $z = 70 \mathrm{e}^{-\mathrm{i}\frac{\pi}{3}}$; b. $z = -45\sqrt{3} + 45\mathrm{i}$.

Part B
We assume in this part that the sensor displays an impact at point P with affix $50\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$. When the sensor displays the impact point P with affix $50\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$, the affix $z$ of the actual lightning impact point admits:
  • a modulus that can be modeled by a random variable $M$ following a normal distribution with mean $\mu = 50$ and standard deviation $\sigma = 5$;
  • an argument that can be modeled by a random variable $T$ following a normal distribution with mean $\frac{\pi}{3}$ and standard deviation $\frac{\pi}{12}$.
We assume that the random variables $M$ and $T$ are independent. In the following, probabilities will be rounded to $10^{-3}$ near.
  1. Calculate the probability $P(M < 0)$ and interpret the result obtained.
  2. Calculate the probability $P(M \in ]40 ; 60[)$.
  3. We admit that $P\left(T \in \left]\frac{\pi}{4} ; \frac{\pi}{2}\right[\right) = 0.819$. Deduce the probability that the lightning actually struck sector B3 according to this modeling.
Q4 5 marks Discrete Probability Distributions Markov Chain and Transition Matrix Analysis View
Exercise 4 -- 5 points -- For candidates who have not followed the specialization course
We study a model of virus propagation in a population, week after week. Each individual in the population can be:
  • either susceptible to being affected by the virus (``of type S'');
  • either sick (affected by the virus);
  • either immunized (cannot be affected by the virus).

For any natural integer $n$, the model of virus propagation is defined by the following rules:
  • Among individuals of type S in week $n$, in week $n+1$: $85\%$ remain of type S, $5\%$ become sick and $10\%$ become immunized;
  • Among sick individuals in week $n$, in week $n+1$: $65\%$ remain sick, and $35\%$ are cured and become immunized.
  • Any individual immunized in week $n$ remains immunized in week $n+1$.

We randomly choose an individual from the population. We consider the following events: $S_{n}$: ``the individual is of type S in week $n$''; $M_{n}$: ``the individual is sick in week $n$''; $I_{n}$: ``the individual is immunized in week $n$''. In week 0, all individuals are considered ``of type S'', so: $$P(S_{0}) = 1 ; \quad P(M_{0}) = 0 \quad \text{and} \quad P(I_{0}) = 0.$$
Part A
We study the evolution of the epidemic during weeks 1 and 2.
  1. Reproduce and complete the probability tree.
  2. Show that $P(I_{2}) = 0.2025$.
  3. Given that an individual is immunized in week 2, what is the probability, rounded to the nearest thousandth, that he was sick in week 1?

Part B
We study the long-term evolution of the disease. For any natural integer $n$, we have: $u_{n} = P(S_{n})$, $v_{n} = P(M_{n})$ and $w_{n} = P(I_{n})$.
Q5 Number Theory Quadratic Diophantine Equations and Perfect Squares View
Exercise 5 -- For candidates who have followed the specialization course
A right-angled triangle with integer sides (TRPI) is a right-angled triangle whose three sides have lengths that are natural integers. If the triangle with sides $x$, $x+1$ and $y$, where $y$ is the length of the hypotenuse, is a TRPI, we will say that the couple $(x ; y)$ defines a TRPI.
Part A
  1. Prove that the couple of natural integers $(x ; y)$ defines a TRPI if, and only if, we have: $$y^{2} = 2x^{2} + 2x + 1$$
  2. Show that the TRPI having the smallest non-zero sides is defined by the couple $(3 ; 5)$.
  3. a. Let $n$ be a natural integer. Show that if $n^{2}$ is odd then $n$ is odd. b. Show that in a couple of integers $(x ; y)$ defining a TRPI, the number $y$ is necessarily odd.
  4. Show that if the couple of natural integers $(x ; y)$ defines a TRPI, then $x$ and $y$ are coprime.

Part B
We denote by $A$ the square matrix: $A = \left( \begin{array}{ll} 3 & 2 \\ 4 & 3 \end{array} \right)$, and $B$ the column matrix: $B = \binom{1}{2}$. Let $x$ and $y$ be two natural integers; we define the natural integers $x^{\prime}$ and $y^{\prime}$ by the relation: $$\binom{x^{\prime}}{y^{\prime}} = A\binom{x}{y} + B.$$
  1. Express $x^{\prime}$ and $y^{\prime}$ as functions of $x$ and $y$. a. Show that: $y^{\prime 2} - 2x^{\prime}(x^{\prime}+1) = y^{2} - 2x(x+1)$. b. Deduce that if the couple $(x ; y)$ defines a TRPI, then the couple $(x^{\prime} ; y^{\prime})$ also defines a TRPI.
  2. We consider the sequences $(x_{n})_{n \in \mathbb{N}}$ and $(y_{n})_{n \in \mathbb{N}}$ of natural integers, defined by $x_{0} = 3$, $y_{0} = 5$ and for every natural integer $n$: $$\binom{x_{n+1}}{y_{n+1}} = A\binom{x_{n}}{y_{n}} + B.$$ Show by induction that, for every natural integer $n$, the couple $(x_{n} ; y_{n})$ defines a TRPI.
  3. Determine, by the method of your choice which you will specify, a TRPI whose side lengths are greater than 2017.