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
2023 bac-spe-maths__metropole_j2

4 maths questions

Q1 5 marks Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
A video game has a large community of online players. Before starting a game, the player must choose between two ``worlds'': either world A or world B. An individual is chosen at random from the community of players. When playing a game, we assume that:
  • the probability that the player chooses world A is equal to $\frac{2}{5}$;
  • if the player chooses world A, the probability that they win the game is $\frac{7}{10}$;
  • the probability that the player wins the game is $\frac{12}{25}$.
We consider the following events:
  • A: ``The player chooses world A'';
  • B: ``The player chooses world B'';
  • G: ``The player wins the game''.

This exercise is a multiple choice questionnaire (5 questions). For each question, only one of the four proposed answers is correct.
  1. The probability that the player chooses world A and wins the game is equal to: a. $\frac{7}{10}$ b. $\frac{3}{25}$ c. $\frac{7}{25}$ d. $\frac{24}{125}$
  2. The probability $P_{B}(G)$ of event $G$ given that $B$ is realized is equal to: a. $\frac{1}{5}$ b. $\frac{1}{3}$ c. $\frac{7}{15}$ d. $\frac{5}{12}$

In the rest of the exercise, a player plays 10 successive games. This situation is treated as a random draw with replacement. We recall that the probability of winning a game is $\frac{12}{25}$.
3. The probability, rounded to the nearest thousandth, that the player wins exactly 6 games is equal to: a. 0.859 b. 0.671 c. 0.188 d. 0.187
4. We consider a natural number $n$ for which the probability, rounded to the nearest thousandth, that the player wins at most $n$ games is 0.207. Then: a. $n = 2$ b. $n = 3$ c. $n = 4$ d. $n = 5$
5. The probability that the player wins at least one game is equal to: a. $1 - \left(\frac{12}{25}\right)^{10}$ b. $\left(\frac{13}{25}\right)^{10}$ c. $\left(\frac{12}{25}\right)^{10}$ d. $1 - \left(\frac{13}{25}\right)^{10}$
Q2 Geometric Sequences and Series Applied Geometric Model with Contextual Interpretation View
Biologists are studying the evolution of an insect population in a botanical garden. At the beginning of the study, the population is 100,000 insects. To preserve the balance of the natural environment, the number of insects must not exceed 400,000.
Part A: Study of a first model in the laboratory
Observation of the evolution of these insect populations in the laboratory, in the absence of any predator, shows that the number of insects increases by $60\%$ each month. Taking this observation into account, biologists model the evolution of the insect population using a sequence $(u_n)$ where, for every natural number $n$, $u_n$ models the number of insects, expressed in millions, after $n$ months. We therefore have $u_0 = 0.1$.
  1. Justify that for every natural number $n$: $u_n = 0.1 \times 1.6^n$.
  2. Determine the limit of the sequence $(u_n)$.
  3. By solving an inequality, determine the smallest natural number $n$ from which $u_n > 0.4$.
  4. According to this model, would the balance of the natural environment be preserved? Justify your answer.

Part B: Study of a second model
Taking into account the constraints of the natural environment in which the insects evolve, biologists choose a new model. They model the number of insects using the sequence $(v_n)$, defined by: $$v_0 = 0.1 \text{ and, for every natural number } n, v_{n+1} = 1.6v_n - 1.6v_n^2,$$ where, for every natural number $n$, $v_n$ is the number of insects, expressed in millions, after $n$ months.
  1. Determine the number of insects after one month.
  2. We consider the function $f$ defined on the interval $\left[0; \frac{1}{2}\right]$ by $$f(x) = 1.6x - 1.6x^2.$$ a. Solve the equation $f(x) = x$. b. Show that the function $f$ is increasing on the interval $\left[0; \frac{1}{2}\right]$.
  3. a. Show by induction that, for every natural number $n$, $0 \leqslant v_n \leqslant v_{n+1} \leqslant \frac{1}{2}$. b. Show that the sequence $(v_n)$ is convergent. We denote by $\ell$ the value of its limit. We admit that $\ell$ is a solution of the equation $f(x) = x$. c. Determine the value of $\ell$. According to this model, will the balance of the natural environment be preserved? Justify your answer.
  4. The threshold function is given below, written in Python language. a. What do we observe if we enter \texttt{seuil(0.4)}? b. Determine the value returned by entering \texttt{seuil(0.35)}. Interpret this value in the context of the exercise. \begin{verbatim} def seuil(a) : v=0.1 n=0 while v
Q3 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
In space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider:
  • the plane $\mathscr{P}_1$ whose Cartesian equation is $2x + y - z + 2 = 0$,
  • the plane $\mathscr{P}_2$ passing through point $\mathrm{B}(1; 1; 2)$ and whose normal vector is $\overrightarrow{n_2}\left(\begin{array}{c}1\\-1\\1\end{array}\right)$.

  1. a. Give the coordinates of a vector $\overrightarrow{n_1}$ normal to the plane $\mathscr{P}_1$. b. We recall that two planes are perpendicular if a normal vector to one of the planes is orthogonal to a normal vector to the other plane. Show that the planes $\mathscr{P}_1$ and $\mathscr{P}_2$ are perpendicular.
  2. a. Determine a Cartesian equation of the plane $\mathscr{P}_2$. b. We denote by $\Delta$ the line whose parametric representation is: $$\left\{\begin{array}{rl} x &= 0 \\ y &= -2 + t \\ z &= t \end{array},\quad t \in \mathbb{R}\right.$$ Show that the line $\Delta$ is the intersection of the planes $\mathscr{P}_1$ and $\mathscr{P}_2$.
  3. We consider the point $\mathrm{A}(1; 1; 1)$ and we admit that point A belongs to neither $\mathscr{P}_1$ nor $\mathscr{P}_2$. We denote by H the orthogonal projection of point A onto the line $\Delta$. We recall that, from question 2.b, the line $\Delta$ is the set of points $M_t$ with coordinates $(0; -2+t; t)$, where $t$ denotes any real number. a. Show that, for every real $t$, $\mathrm{A}M_t = \sqrt{2t^2 - 8t + 11}$. b. Deduce that $\mathrm{AH} = \sqrt{3}$.
  4. We denote by $\mathscr{D}_1$ the line perpendicular to the plane $\mathscr{P}_1$ passing through point A and $\mathrm{H}_1$ the orthogonal projection of point A onto the plane $\mathscr{P}_1$. a. Determine a parametric representation of the line $\mathscr{D}_1$. b. Deduce that the point $\mathrm{H}_1$ has coordinates $\left(-\frac{1}{3}; \frac{1}{3}; \frac{5}{3}\right)$.
  5. Let $\mathrm{H}_2$ be the orthogonal projection of A onto the plane $\mathscr{P}_2$. We admit that $\mathrm{H}_2$ has coordinates $\left(\frac{4}{3}; \frac{2}{3}; \frac{4}{3}\right)$ and that H has coordinates $(0; 0; 2)$. Show that $\mathrm{AH}_1\mathrm{HH}_2$ is a rectangle.
Q4 Differentiating Transcendental Functions Full function study with transcendental functions View
We consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \ln\left(1 + \mathrm{e}^{-x}\right),$$ where $\ln$ denotes the natural logarithm function. We denote by $\mathscr{C}$ its representative curve in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.
  1. a. Determine the limit of the function $f$ at $-\infty$. b. Determine the limit of the function $f$ at $+\infty$. Interpret this result graphically. c. We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote by $f'$ its derivative function. Calculate $f'(x)$ then show that, for every real number $x$, $f'(x) = \frac{-1}{1 + \mathrm{e}^x}$. d. Draw the complete table of variations of the function $f$ on $\mathbb{R}$.
  2. We denote by $T_0$ the tangent line to the curve $\mathscr{C}$ at its point with abscissa 0. a. Determine an equation of the tangent line $T_0$. b. Show that the function $f$ is convex on $\mathbb{R}$. c. Deduce that, for every real number $x$, we have: $$f(x) \geqslant -\frac{1}{2}x + \ln(2)$$
  3. For every real number $a$ different from 0, we denote by $M_a$ and $N_a$ the points of the curve $\mathscr{C}$ with abscissas $-a$ and $a$ respectively. We therefore have: $M_a(-a; f(-a))$ and $N_a(a; f(a))$. a. Show that, for every real number $x$, we have: $f(x) - f(-x) = -x$. b. Deduce that the lines $T_0$ and $(M_a N_a)$ are parallel.