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
2021 bac-spe-maths__asie_j1

5 maths questions

QA Differentiating Transcendental Functions Full function study with transcendental functions View
EXERCISE-A
Main topics covered: convexity, logarithm function
Part I: graphical readings
$f$ denotes a function defined and differentiable on $\mathbb{R}$. We give below the representative curve of the derivative function $f'$.
With the precision allowed by the graph, answer the following questions
  1. Determine the slope of the tangent line to the curve of function $f$ at 0.
  2. a. Give the variations of the derivative function $f'$. b. Deduce an interval on which $f$ is convex.

Part II: function study
The function $f$ is defined on $\mathbb{R}$ by $$f(x) = \ln\left(x^{2} + x + \frac{5}{2}\right)$$
  1. Calculate the limits of function $f$ at $+\infty$ and at $-\infty$.
  2. Determine an expression $f'(x)$ of the derivative function of $f$ for all $x \in \mathbb{R}$.
  3. Deduce the table of variations of $f$. Be sure to place the limits in this table.
  4. a. Justify that the equation $f(x) = 2$ has a unique solution $\alpha$ in the interval $\left[-\frac{1}{2}; +\infty\right[$. b. Give an approximate value of $\alpha$ to $10^{-1}$ near.
  5. The function $f'$ is differentiable on $\mathbb{R}$. We admit that, for all $x \in \mathbb{R}$, $f''(x) = \frac{-2x^{2} - 2x + 4}{\left(x^{2} + x + \frac{5}{2}\right)^{2}}$. Determine the number of inflection points of the representative curve of $f$.
QB Differential equations First-Order Linear DE: General Solution View
EXERCISE - B
Main topics covered: Function study, exponential function; Differential equations
Part I
Let us consider the differential equation $$y' = -0.4y + 0.4$$ where $y$ denotes a function of the variable $t$, defined and differentiable on $[0; +\infty[$.
  1. a. Determine a particular constant solution of this differential equation. b. Deduce the set of solutions of this differential equation. c. Determine the function $g$, solution of this differential equation, which satisfies $g(0) = 10$.

Part II
Let $p$ be the function defined and differentiable on the interval $[0; +\infty[$ by $$p(t) = \frac{1}{g(t)} = \frac{1}{1 + 9\mathrm{e}^{-0.4t}}$$
  1. Determine the limit of $p$ at $+\infty$.
  2. Show that $p'(t) = \frac{3.6\mathrm{e}^{-0.4t}}{\left(1 + 9\mathrm{e}^{-0.4t}\right)^{2}}$ for all $t \in [0; +\infty[$.
  3. a. Show that the equation $p(t) = \frac{1}{2}$ has a unique solution $\alpha$ on $[0; +\infty[$. b. Determine an approximate value of $\alpha$ to $10^{-1}$ near using a calculator.

Part III
  1. $p$ denotes the function from Part II. Verify that $p$ is a solution of the differential equation $y' = 0.4y(1 - y)$ with the initial condition $y(0) = \frac{1}{10}$ where $y$ denotes a function defined and differentiable on $[0; +\infty[$.
  2. In a developing country, in the year 2020, 10\% of schools have access to the internet.
    A voluntary equipment policy is implemented and we are interested in the evolution of the proportion of schools with access to the internet. We denote $t$ the time elapsed, expressed in years, since the year 2020. The proportion of schools with access to the internet at time $t$ is modelled by $p(t)$. Interpret in this context the limit from question II 1 then the approximate value of $\alpha$ from question II 3. b. as well as the value $p(0)$.
Q1 5 marks Geometric Sequences and Series Applied Geometric Model with Contextual Interpretation View
In 2020, an influencer on social media has 1000 followers on her profile. The number of followers is modelled as follows: each year, she loses $10\%$ of her followers to which 250 new followers are added. For any natural integer $n$, we denote $u_{n}$ the number of followers on her profile in the year $(2020 + n)$, following this model. Thus $u_{0} = 1000$.
  1. Calculate $u_{1}$.
  2. Justify that for any natural integer $n$, $u_{n+1} = 0.9 u_{n} + 250$.
  3. The Python function named ``suite'' is defined below. In the context of the exercise, interpret the value returned by suite(10).

\begin{verbatim} def suite(n) : u=1000 for i in range(n) : u=0.9*u+250 return u \end{verbatim}
    \setcounter{enumi}{3}
  1. a. Show, using a proof by induction, that for any natural integer $n$, $u_{n} \leqslant 2500$. b. Prove that the sequence $(u_{n})$ is increasing. c. Deduce from the previous questions that the sequence $(u_{n})$ is convergent.
  2. Let $(v_{n})$ be the sequence defined by $v_{n} = u_{n} - 2500$ for any natural integer $n$. a. Show that the sequence $(v_{n})$ is a geometric sequence with common ratio 0.9 and initial term $v_{0} = -1500$. b. For any natural integer $n$, express $v_{n}$ as a function of $n$ and show that: $$u_{n} = -1500 \times 0.9^{n} + 2500$$ c. Determine the limit of the sequence $(u_{n})$ and interpret it in the context of the exercise.
  3. Write a program that determines in which year the number of followers will exceed 2200. Determine this year.
Q2 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
We consider a cube ABCDEFGH with edge 8 cm and centre $\Omega$.
The points P, Q and R are defined by $\overrightarrow{AP} = \frac{3}{4}\overrightarrow{AB}$, $\overrightarrow{AQ} = \frac{3}{4}\overrightarrow{AE}$ and $\overrightarrow{FR} = \frac{1}{4}\overrightarrow{FG}$. We use the orthonormal coordinate system $(A; \vec{\imath}, \vec{\jmath}, \vec{k})$ with: $\vec{\imath} = \frac{1}{8}\overrightarrow{AB}$, $\vec{\jmath} = \frac{1}{8}\overrightarrow{AD}$ and $\vec{k} = \frac{1}{8}\overrightarrow{AE}$.
Part I
  1. In this coordinate system, we admit that the coordinates of point R are $(8; 2; 8)$. Give the coordinates of points P and Q.
  2. Show that the vector $\vec{n}(1; -5; 1)$ is a normal vector to the plane (PQR).
  3. Justify that a Cartesian equation of the plane (PQR) is $x - 5y + z - 6 = 0$.

Part II
We denote L the orthogonal projection of point $\Omega$ onto the plane (PQR).
  1. Justify that the coordinates of point $\Omega$ are $(4; 4; 4)$.
  2. Give a parametric representation of the line $d$ perpendicular to the plane (PQR) and passing through $\Omega$.
  3. Show that the coordinates of point L are $\left(\frac{14}{3}; \frac{2}{3}; \frac{14}{3}\right)$.
  4. Calculate the distance from point $\Omega$ to the plane (PQR).
Q3 Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
A bag contains the following eight letters: A B C D E F G H (2 vowels and 6 consonants).
A game consists of drawing simultaneously at random two letters from this bag. You win if the draw consists of one vowel and one consonant.
  1. A player draws simultaneously two letters from the bag. a. Determine the number of possible draws. b. Determine the probability that the player wins this game.

Questions 2 and 3 of this exercise are independent.
For the rest of the exercise, we admit that the probability that the player wins is equal to $\frac{3}{7}$.
    \setcounter{enumi}{1}
  1. To play, the player must pay $k$ euros, where $k$ is a non-zero natural integer. If the player wins, he receives 10 euros, otherwise he receives nothing. We denote $G$ the random variable equal to the algebraic gain of a player (that is, the sum received minus the sum paid). a. Determine the probability distribution of $G$. b. What must be the maximum value of the sum paid at the start for the game to remain favourable to the player?
  2. Ten players each play one game. The letters drawn are returned to the bag after each game. We denote $X$ the random variable equal to the number of winning players. a. Justify that $X$ follows a binomial distribution and give its parameters. b. Calculate the probability, rounded to $10^{-3}$, that there are exactly four winning players. c. Calculate $P(X \geqslant 5)$ by rounding to $10^{-3}$. Give an interpretation of the result obtained. d. Determine the smallest natural integer $n$ such that $P(X \leqslant n) \geqslant 0.9$.