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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
2025 bac-spe-maths__polynesie_j1

4 maths questions

Q1 Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
An American team mapped food allergies in children in the United States for the first time in 2020. It is known that in 2020, $17\%$ of the population of the United States lives in rural areas and $83\%$ in urban areas. Among children in the United States living in rural areas, $6.2\%$ are affected by food allergies. Also, $9\%$ of children in the United States are affected by food allergies.
For any event $E$, we denote $P(E)$ its probability and $\bar{E}$ its complementary event. Unless otherwise stated, probabilities will be given in exact form.
A child is randomly selected from the population of the United States and we denote:
  • R The event: ``the child interviewed lives in a rural area'';
  • A The event: ``the child interviewed is affected by food allergies''.

Part A
  1. Translate this situation using a probability tree. This tree may be completed later.
  2. a. Calculate the probability that the child interviewed lives in a rural area and is affected by food allergies. b. Deduce the probability that the child interviewed lives in an urban area and is affected by food allergies. c. The child interviewed lives in an urban area. What is the probability that he/she is affected by food allergies? Round the result to $10^{-4}$.

Part B
A study is conducted by randomly interviewing 100 children in the United States. We assume that this choice amounts to successive independent draws with replacement. We denote $X$ the random variable giving the number of children affected by food allergies in the sample considered.
  1. Justify that the random variable $X$ follows a binomial distribution and specify its parameters.
  2. What is the probability that at least 10 children among the 100 interviewed are affected by food allergies? Round the result to $10^{-4}$.

Part C
We are interested in a sample of 20 children affected by food allergies chosen at random. The age of onset of the first allergic symptoms of these 20 children is modeled by the random variables $A_1, A_2, \ldots, A_{20}$. We assume that these random variables are independent and follow the same distribution with expectation 4 and variance 2.25. We consider the random variable: $$M_{20} = \frac{A_1 + A_2 + \ldots + A_{20}}{20}.$$
  1. What does the random variable $M_{20}$ represent in the context of the exercise?
  2. Determine the expectation and variance of $M_{20}$.
  3. Justify, using the concentration inequality, that $$P\left(2 < M_{20} < 6\right) > 0.97.$$ Interpret this result in the context of the exercise.
Q2 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Two aircraft are approaching an airport. We equip space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ whose origin O is the base of the control tower, and the ground is the plane $P_0$ with equation $z = 0$. The unit of the axes corresponds to 1 km. We model the aircraft as points.
Aircraft Alpha transmits to the tower its position at $\mathrm{A}(-7; 1; 7)$ and its trajectory is directed by the vector $\vec{u}\begin{pmatrix} 2 \\ -1 \\ -3 \end{pmatrix}$.
Aircraft Beta transmits a trajectory defined by the line $d_{\mathrm{B}}$ passing through point B with a parametric representation: $$\left\{\begin{aligned} x &= -11 + 5t \\ y &= -5 + t \\ z &= 11 - 4t \end{aligned}\right. \text{ where } t \text{ describes } \mathbb{R}.$$
  1. If it does not deviate from its trajectory, determine the coordinates of point S where aircraft Beta will touch the ground.
  2. a. Determine a parametric representation of the line $d_{\mathrm{A}}$ characterizing the trajectory of aircraft Alpha. b. Can the two aircraft collide?
  3. a. Prove that aircraft Alpha passes through position $\mathrm{E}(-3; -1; 1)$. b. Justify that a Cartesian equation of the plane $P_{\mathrm{E}}$ passing through E and perpendicular to the line $d_{\mathrm{A}}$ is: $$2x - y - 3z + 8 = 0.$$ c. Verify that the point $\mathrm{F}(-1; -3; 3)$ is the intersection point of the plane $P_{\mathrm{E}}$ and the line $d_{\mathrm{B}}$. d. Calculate the exact value of the distance EF, then verify that this corresponds to a distance of 3464 m, to the nearest 1 m.
  4. Air traffic regulations stipulate that two aircraft on approach must be at least 3 nautical miles apart at all times (1 nautical mile equals 1852 m). If aircraft Alpha and Beta are respectively at E and F at the same instant, is their safety distance respected?
Q3 Integration by Parts Reduction Formula or Recurrence via Integration by Parts View
We equip the plane with an orthonormal coordinate system. For every natural integer $n$, we consider the function $f_n$ defined on $[0; +\infty[$ by: $$f_0(x) = \mathrm{e}^{-x} \text{ and, for } n \geqslant 1,\ f_n(x) = x^n \mathrm{e}^{-x}.$$ For every natural integer $n$, we denote $\mathscr{C}_n$ the representative curve of the function $f_n$.
Parts A and B are independent.
Part A: Study of the functions $f_n$ for $n \geqslant 1$
We consider a natural integer $n \geqslant 1$.
  1. a. We admit that the function $f_n$ is differentiable on $[0; +\infty[$. Show that for all $x \geqslant 0$, $$f_n'(x) = (n - x)x^{n-1}\mathrm{e}^{-x}.$$ b. Justify all elements of the table below:
    $x$0$n$$+\infty$
    $f_n'(x)$+0-
    $\left(\frac{n}{\mathrm{e}}\right)^n$
    $f_n$
    00

  2. Justify by calculation that the point $\mathrm{A}\left(1; \mathrm{e}^{-1}\right)$ belongs to the curve $\mathscr{C}_n$.

Part B: Study of the integrals $\int_0^1 f_n(x)\,\mathrm{d}x$ for $n \geqslant 0$
In this part, we study the functions $f_n$ on $[0; 1]$ and we consider the sequence $(I_n)$ defined for every natural integer $n$ by: $$I_n = \int_0^1 f_n(x)\,\mathrm{d}x = \int_0^1 x^n \mathrm{e}^{-x}\,\mathrm{d}x.$$
  1. On the graph in APPENDIX, the curves $\mathscr{C}_0, \mathscr{C}_1, \mathscr{C}_2, \mathscr{C}_{10}$ and $\mathscr{C}_{100}$ are represented. a. Give a graphical interpretation of $I_n$. b. By reading this graph, what conjecture can be made about the limit of the sequence $(I_n)$?
  2. Calculate $I_0$.
  3. a. Let $n$ be a natural integer. Prove that for all $x \in [0; 1]$, $$0 \leqslant x^{n+1} \leqslant x^n.$$ b. Deduce that for every natural integer $n$, we have: $$0 \leqslant I_{n+1} \leqslant I_n.$$
  4. Prove that the sequence $(I_n)$ is convergent, towards a limit greater than or equal to zero that we will denote $\ell$.
  5. Using integration by parts, prove that for every natural integer $n$ we have: $$I_{n+1} = (n+1)I_n - \frac{1}{\mathrm{e}}.$$
  6. a. Prove that if $\ell > 0$, the equality from question 5 leads to a contradiction. b. Prove that $\ell = 0$. You may use question 6.a.
  7. The script of the \texttt{mystere} function is given below, written in Python language. The constant \texttt{e} has been imported. \begin{verbatim} def mystere(n): I = 1 - 1/e L = [I] for i in range(n): I = (i + 1)*I - 1/e L.append(I) return L \end{verbatim} What does \texttt{mystere(100)} return in the context of the exercise?
Q4 5 marks Second order differential equations Solving non-homogeneous second-order linear ODE View
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. In this exercise, the questions are independent of one another.
  1. We consider the differential equation: $$(E) \quad y' = \frac{1}{2}y + 4.$$ Statement 1: The solutions of $(E)$ are the functions $f$ defined on $\mathbb{R}$ by: $$f(x) = k\mathrm{e}^{\frac{1}{2}x} - 8, \quad \text{with } k \in \mathbb{R}.$$
  2. In a final year class, there are 18 girls and 14 boys. A volleyball team is formed by randomly choosing 3 girls and 3 boys. Statement 2: There are 297024 possibilities for forming such a team.
  3. Let $(v_n)$ be the sequence defined for every natural integer $n$ by: $$v_n = \frac{n}{2 + \cos(n)}.$$ Statement 3: The sequence $(v_n)$ diverges to $+\infty$.
  4. In space with respect to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points $\mathrm{A}(1; 1; 2)$, $\mathrm{B}(5; -1; 8)$ and $\mathrm{C}(2; 1; 3)$. Statement 4: $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AC}} = 10$ and a measure of the angle $\widehat{\mathrm{BAC}}$ is $30^\circ$.
  5. We consider a function $h$ defined on $]0; +\infty[$ whose second derivative is defined on $]0; +\infty[$ by: $$h''(x) = x\ln x - 3x.$$ Statement 5: The function $h$ is convex on $[\mathrm{e}^3; +\infty[$.