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 pondichery

5 maths questions

Q1 Indefinite & Definite Integrals Antiderivative Verification and Construction View
Exercise 1 -- Common to all candidates

Part A
Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = \frac{3}{1 + \mathrm{e}^{-2x}}$$ In the graph below, we have drawn, in an orthogonal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, the representative curve $\mathscr{C}$ of the function $f$ and the line $\Delta$ with equation $y = 3$.
  1. Prove that the function $f$ is strictly increasing on $\mathbb{R}$.
  2. Justify that the line $\Delta$ is an asymptote to the curve $\mathscr{C}$.
  3. Prove that the equation $f(x) = 2.999$ has a unique solution $\alpha$ on $\mathbb{R}$.
Determine an interval containing $\alpha$ with amplitude $10^{-2}$.
Part B
Let $h$ be the function defined on $\mathbb{R}$ by $h(x) = 3 - f(x)$.
  1. Justify that the function $h$ is positive on $\mathbb{R}$.
  2. We denote by $H$ the function defined on $\mathbb{R}$ by $H(x) = -\frac{3}{2}\ln\left(1 + \mathrm{e}^{-2x}\right)$.
    Prove that $H$ is an antiderivative of $h$ on $\mathbb{R}$.
  3. Let $a$ be a strictly positive real number. a. Give a graphical interpretation of the integral $\int_{0}^{a} h(x)\,\mathrm{d}x$. b. Prove that $\int_{0}^{a} h(x)\,\mathrm{d}x = \frac{3}{2}\ln\left(\frac{2}{1 + \mathrm{e}^{-2a}}\right)$. c. We denote by $\mathscr{D}$ the set of points $M(x\,;\,y)$ in the plane defined by $$\left\{\begin{array}{l} x \geqslant 0 \\ f(x) \leqslant y \leqslant 3 \end{array}\right.$$ Determine the area, in square units, of the region $\mathscr{D}$.
Q2 5 marks Sequences and series, recurrence and convergence Auxiliary sequence transformation View
Exercise 2 (5 points) -- Common to all candidates

Part A
Let $(u_n)$ be the sequence defined by its first term $u_0$ and, for every natural number $n$, by the relation $$u_{n+1} = a u_n + b \quad (a \text{ and } b \text{ non-zero real numbers such that } a \neq 1).$$ We set, for every natural number $n$, $\quad v_n = u_n - \dfrac{b}{1-a}$.
  1. Prove that the sequence $(v_n)$ is geometric with common ratio $a$.
  2. Deduce that if $a$ belongs to the interval $]-1\,;\,1[$, then the sequence $(u_n)$ has limit $\dfrac{b}{1-a}$.

Part B
In March 2015, Max buys a green plant measuring 80 cm. He is advised to prune it every year, in March, by cutting a quarter of its height. The plant will then grow 30 cm over the following twelve months. As soon as he gets home, Max prunes his plant.
  1. What will be the height of the plant in March 2016 before Max prunes it?
  2. For every natural number $n$, we denote by $h_n$ the height of the plant, before pruning, in March of the year $(2015 + n)$. a. Justify that, for every natural number $n$, $\quad h_{n+1} = 0.75\,h_n + 30$. b. Conjecture using a calculator the direction of variation of the sequence $(h_n)$. Prove this conjecture (you may use a proof by induction). c. Is the sequence $(h_n)$ convergent? Justify your answer.
Q3 Normal Distribution Algebraic Relationship Between Normal Parameters and Probability View
Exercise 3 -- Common to all candidates

Part A: Study of the lifespan of a household appliance
Statistical studies have made it possible to model the lifespan, in months, of a type of dishwasher by a random variable $X$ following a normal distribution $\mathscr{N}(\mu, \sigma^2)$ with mean $\mu = 84$ and standard deviation $\sigma$. Furthermore, we have $P(X \leqslant 64) = 0.16$.
  1. a. By exploiting the graph, determine $P(64 \leqslant X \leqslant 104)$. b. What approximate integer value of $\sigma$ can we propose?
  2. We denote by $Z$ the random variable defined by $Z = \dfrac{X - 84}{\sigma}$. a. What is the probability distribution followed by $Z$? b. Justify that $P(X \leqslant 64) = P\!\left(Z \leqslant \dfrac{-20}{\sigma}\right)$. c. Deduce the value of $\sigma$, rounded to $10^{-3}$.
  3. In this question, we consider that $\sigma = 20.1$.
    The probabilities requested will be rounded to $10^{-3}$. a. Calculate the probability that the lifespan of the dishwasher is between 2 and 5 years. b. Calculate the probability that the dishwasher has a lifespan greater than 10 years.

Part B: Study of the warranty extension offered by El'Ectro
The dishwasher is guaranteed free of charge for the first two years. The company El'Ectro offers its customers a warranty extension of 3 additional years. Statistical studies conducted on customers who take the warranty extension show that $11.5\%$ of them use the warranty extension.
  1. We randomly choose 12 customers among those who have taken the warranty extension (this choice can be treated as random sampling with replacement given the large number of customers). a. What is the probability that exactly 3 of these customers use this warranty extension? Detail the approach by specifying the probability distribution used. Round to $10^{-3}$. b. What is the probability that at least 6 of these customers use this warranty extension? Round to $10^{-3}$.
  2. The warranty extension offer is as follows: for 65 euros additional, El'Ectro will reimburse the customer the initial value of the dishwasher, namely 399 euros, if an irreparable breakdown occurs between the beginning of the third year and the end of the fifth year. The customer cannot use this warranty extension if the breakdown is repairable.
    We randomly choose a customer among those who have subscribed to the warranty extension, and we denote by $Y$ the random variable representing the algebraic gain in euros realized on this customer by the company El'Ectro, thanks to the warranty extension. a. Justify that $Y$ takes the values 65 and $-334$ then give the probability distribution of $Y$. b. Is this warranty extension offer financially advantageous for the company? Justify.
Q4A 5 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 4 (5 points) -- Candidate who has NOT followed the specialization course
Let a cube ABCDEFGH with edge length 1. In the coordinate system $(A;\,\overrightarrow{AB},\,\overrightarrow{AD},\,\overrightarrow{AE})$, we consider the points $M$, $N$ and $P$ with respective coordinates $\mathrm{M}\!\left(1\,;\,1\,;\,\tfrac{3}{4}\right)$, $\mathrm{N}\!\left(0\,;\,\tfrac{1}{2}\,;\,1\right)$, $\mathrm{P}\!\left(1\,;\,0\,;\,-\tfrac{5}{4}\right)$.
  1. Plot $\mathrm{M}$, $\mathrm{N}$ and $\mathrm{P}$ on the figure provided in the appendix.
  2. Determine the coordinates of the vectors $\overrightarrow{\mathrm{MN}}$ and $\overrightarrow{\mathrm{MP}}$.
    Deduce that the points $\mathrm{M}$, $\mathrm{N}$ and $\mathrm{P}$ are not collinear.
  3. We consider algorithm 1 given in the appendix. a. Execute this algorithm by hand with the coordinates of the points $\mathrm{M}$, $\mathrm{N}$ and $\mathrm{P}$ given above. b. What does the result displayed by the algorithm correspond to? What can we deduce about triangle MNP?
  4. We consider algorithm 2 given in the appendix. Complete it so that it tests and displays whether a triangle MNP is right-angled and isosceles at M.
  5. We consider the vector $\vec{n}(5\,;\,-8\,;\,4)$ normal to the plane (MNP). a. Determine a Cartesian equation of the plane (MNP). b. We consider the line $\Delta$ passing through F and with direction vector $\vec{n}$.
    Determine a parametric representation of the line $\Delta$.
  6. Let K be the point of intersection of the plane (MNP) and the line $\Delta$. a. Prove that the coordinates of point K are $\left(\dfrac{4}{7}\,;\,\dfrac{24}{35}\,;\,\dfrac{23}{35}\right)$. b. We are given $FK = \sqrt{\dfrac{27}{35}}$.
    Calculate the volume of the tetrahedron MNPF.
Q4B 5 marks Number Theory Divisibility and Divisor Analysis View
Exercise 4 (5 points) -- Candidate who has followed the specialization course
Numbers of the form $2^n - 1$ where $n$ is a non-zero natural number are called Mersenne numbers.
  1. We denote by $a$, $b$ and $c$ three non-zero natural numbers such that $\operatorname{GCD}(b\,;\,c) = 1$.
    Prove, using Gauss's theorem, that: if $b$ divides $a$ and $c$ divides $a$ then the product $bc$ divides $a$.
  2. We consider the Mersenne number $2^{33} - 1$.
    A student uses his calculator and obtains the results below:
    $\left(2^{33}-1\right) \div 3$2863311530
    $\left(2^{33}-1\right) \div 4$2147483648
    $\left(2^{33}-1\right) \div 12$715827882.6

    He claims that 3 divides $\left(2^{33}-1\right)$ and 4 divides $\left(2^{33}-1\right)$ and 12 does not divide $\left(2^{33}-1\right)$. a. How does this claim contradict the result proved in question 1? b. Justify that, in reality, 4 does not divide $\left(2^{33}-1\right)$. c. By noting that $2 \equiv -1\ [3]$, show that, in reality, 3 does not divide $2^{33}-1$. d. Calculate the sum $S = 1 + 2^3 + \left(2^3\right)^2 + \left(2^3\right)^3 + \cdots + \left(2^3\right)^{10}$. e. Deduce that 7 divides $2^{33}-1$.
  3. We consider the Mersenne number $2^7 - 1$. Is it prime? Justify.
  4. We are given the following algorithm where $\operatorname{MOD}(N,k)$ represents the remainder of the Euclidean division of $N$ by $k$:
    Variables:
    $n$ natural number greater than or equal to 3
    $k$ natural number greater than or equal to 2
    Initialization:
    Ask the user for the value of $n$.
    Assign to $k$ the value 2.
    Processing:
    While $\operatorname{MOD}\left(2^n - 1,\,k\right) \neq 0$ and $k \leqslant \sqrt{2^n - 1}$
    \quad Assign to $k$ the value $k+1$
    End While.
    Output:
    Otherwise
    Display ``CASE 2''
    End If

    a. What does this algorithm display if we enter $n = 33$? And if we enter $n = 7$? b. What does CASE 2 correspond to?