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
2025 bac-spe-maths__polynesie-sept_j1

4 maths questions

Q1 5 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
In France there are two formulas for obtaining a driving license:
  • Follow supervised driving training from age 15 for 2 years;
  • Follow classical training (without supervised driving) from age 17.
In France currently, among young people who follow driving license training, $16\%$ choose supervised driving training, and among them, $74.7\%$ pass the driving test on their first attempt. Following classical training, the success rate on the first attempt is only $56.8\%$.
A young French person who has already taken the driving test is chosen at random, and we consider the following events $A$ and $R$:
  • $A$: ``the young person followed supervised driving training'';
  • $R$: ``the young person obtained their license on their first attempt''.

Results should be rounded to $10^{-3}$ if necessary.
Part A
  1. Draw a probability tree modeling this situation.
  2. a. Prove that $P(R) = 0.59664$.
    In the following, we will keep the value 0.597 rounded to $10^{-3}$. b. Give this result as a percentage and interpret it in the context of the exercise.
  3. A young person who obtained their license on their first attempt is chosen. What is the probability that they followed supervised driving training?
  4. What should be the proportion of young people following supervised driving training if we wanted the overall success rate (regardless of the training chosen) on the first attempt at the driving test to exceed $70\%$?

Part B
A driving school presents 10 candidates for the driving test for the first time, all of whom have followed supervised driving training. The fact of taking driving tests is modeled by independent random trials.
Let $X$ be the random variable giving the number of these 10 candidates who will obtain their license on their first attempt.
  1. Justify that $X$ follows a binomial distribution with parameters $n = 10$ and $p = 0.747$.
  2. Calculate $P(X \geqslant 6)$. Interpret this result.
  3. Determine $E(X)$ and $V(X)$.
  4. There are also 40 candidates who have not followed supervised driving training and who are taking the driving test for the first time. In the same way, let $Y$ be the random variable giving the number of these candidates who will obtain their license on their first attempt. We admit that $Y$ is independent of the variable $X$ and that in fact $E(Y) = 22.53$ and $V(Y) = 9.81$.
    Let $Z$ be the random variable counting the total number of candidates (among the 50) who will obtain their driving license on their first attempt at this driving school. a. Express $Z$ as a function of $X$ and $Y$. Deduce $E(Z)$ and $V(Z)$. b. Using Bienaymé-Chebyshev's inequality, show that the probability that there are fewer than 20 or more than 40 candidates who obtain their license on their first attempt is less than 0.12.
Q2 Differential equations Applied Modeling with Differential Equations View
We study the evolution of the population of an animal species within a nature reserve. The numbers of this population were recorded in different years. The collected data are presented in the following table:
Year2000200520102015
Number of individuals506480100

To anticipate the evolution of this population, the reserve management chose to model the number of individuals as a function of time. For this, it uses a function, defined on the interval $[0; +\infty[$, where the variable $x$ represents the elapsed time, in years, from the year 2000. In its model, the image of 0 by this function equals 50, which corresponds to the number of individuals in the year 2000.
Part A. Model 1
In this part, the reserve management makes the hypothesis that the function sought satisfies the following differential equation: $$y' = 0.05y - 0.5 \quad (E_1)$$
  1. Solve the differential equation $(E_1)$ with the initial condition $y(0) = 50$.
  2. Compare the results in the table with those that would be obtained with this model.

Part B. Model 2
In this part, the reserve management makes the hypothesis that the function sought satisfies the following differential equation: $$y' = 0.05y(1 - 0.00125y)$$
Let $f$ be the function defined on $[0; +\infty[$ by: $$f(x) = \frac{800}{1 + 15\mathrm{e}^{-0.05x}}$$ and $C$ its representative curve in an orthonormal coordinate system.
Using computer algebra software, the following results were obtained. For the rest of the exercise, these results may be used without proof, except for question 5.
InstructionResult
1$f(x) := \frac{800}{1 + 15\mathrm{e}^{-0.05x}}$$f(x) = \frac{800}{1 + 15\mathrm{e}^{-0.05x}}$
2$f'(x) :=$ Derivative$(f(x))$$f'(x) = \frac{600\mathrm{e}^{-0.05x}}{\left(1 + 15\mathrm{e}^{-0.05x}\right)^2}$
3$f''(x) :=$ Derivative$(f'(x))$$f''(x) = \frac{30\mathrm{e}^{-0.05x}}{\left(1 + 15\mathrm{e}^{-0.05x}\right)^3}\left(15\mathrm{e}^{-0.05x} - 1\right)$
4Solve$(15\mathrm{e}^{-0.05x} - 1 \geqslant 0)$$x \leqslant 20\ln(15)$

  1. Prove that the function $f$ satisfies $f(0) = 50$ and that for all $x \in \mathbb{R}$: $$f'(x) = 0.05f(x)(1 - 0.00125f(x))$$ We admit that this function $f$ is the unique solution of $(E_2)$ taking the initial value of 50 at 0.
  2. With this new model $f$, estimate the population size in 2050. Round the result to the nearest integer.
  3. Calculate the limit of $f$ as $x \to +\infty$. What can be deduced about the curve $C$? Interpret this limit in the context of this concrete problem.
  4. Justify that the function $f$ is increasing on $[0; +\infty[$.
  5. Prove the result obtained in line 4 of the software.
  6. We admit that the growth rate of the population of this species, expressed in number of individuals per year, is modeled by the function $f'$. a. Study the convexity of the function $f$ on the interval $[0; +\infty[$ and determine the coordinates of any inflection points of the curve $C$. b. The reserve management claims: ``According to this model, the growth rate of the population of this species will increase for a little more than fifty years, then will decrease''. Is the management correct? Justify.
Q3 Differential equations Qualitative Analysis of DE Solutions View
We consider the sequence $(u_n)$ defined by $u_0 = 5$ and, for all natural integers $n$: $$u_{n+1} = 2 + \ln\left(u_n^2 - 3\right)$$ We admit that this sequence is well defined.
Part A: Exploitation of Python programs
  1. Copy and complete the Python script below so that \texttt{suite(k)} which takes a natural integer $k$ as parameter returns the list of the first $k$ values of the sequence $(u_n)$.
    Remark: We specify that, for any strictly positive real number $a$, $\log(a)$ returns the value of the natural logarithm of $a$.
    \begin{verbatim} def suite(k): L = [] u = 5 for i in range(......): L.append(u) u=............ return(......) \end{verbatim}
  2. We executed \texttt{suite(9)} below. Make two conjectures: one on the direction of variation of the sequence $(u_n)$ and another on its possible convergence.
    \begin{verbatim} >>> suite(9) [ 5, 5.091042453358316, 5.131953749864703, 5.150037910978289, 5.157974010229213, 5.1614456706362954, 5.162962248594583, 5.163624356938671, 5.163913344065642] \end{verbatim}
  3. We then created the function \texttt{mystere(n)} given below and executed \texttt{mystere(10000)}, which returned 1. Does this output contradict the conjecture made about the direction of variation of the sequence $(u_n)$? Justify.
    \begin{verbatim} def mystere(n): L = suite(n) c = 1 for i in range(n - 1): if L[i] > L[i + 1]: c = 0 return c
    >>> mystere(10000) 1 \end{verbatim}

Part B: Study of the convergence of the sequence $(u_n)$
We consider the function $g$ defined on $[2; +\infty[$ by: $$g(x) = 2 + \ln\left(x^2 - 3\right)$$ We admit that $g$ is differentiable on $[2; +\infty[$ and we denote $g'$ its derivative function.
  1. Prove that the function $g$ is increasing on $[2; +\infty[$.
  2. a. Prove by induction that, for all natural integers $n$: $$4 \leqslant u_n \leqslant u_{n+1} \leqslant 6$$ b. Deduce that the sequence $(u_n)$ converges.

Part C: Study of the limit value
We consider the function $f$ defined on $[2; +\infty[$ by: $$f(x) = 2 + \ln\left(x^2 - 3\right) - x$$ We admit that $f$ is differentiable on $[2; +\infty[$ and we denote $f'$ its derivative function. We give the following variation table of $f$. No justification is requested.
$x$23$+\infty$
$\ln(6) - 1$
$f(x)$
0$-\infty$

  1. a. Show that the equation $f(x) = 0$ has exactly two solutions on $[2; +\infty[$ which we will denote $\alpha$ and $\beta$ with $\alpha < \beta$. b. Give the exact value of $\alpha$ and an approximate value to $10^{-3}$ of $\beta$.
  2. Let $\ell$ be the limit of the sequence $(u_n)$. Justify that $f(\ell) = 0$ and determine $\ell$.
Q4 Vectors: Lines & Planes True/False or Verify a Given Statement 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.
  1. We consider the function $f$ defined on $]0; +\infty[$ by: $f(x) = x\ln(x)$.
    Statement 1: $$\int_1^{\mathrm{e}} f(x)\,\mathrm{d}x = \frac{\mathrm{e}^2 + 1}{4}$$
  2. Let $n$ and $k$ be two non-zero natural integers such that $k \leqslant n$.
    Statement 2: $$n \times \binom{n-1}{k-1} = k \times \binom{n}{k}$$
  3. For the three following statements, we consider that space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.
    Let $d$ be the line with parametric representation: $\left\{\begin{array}{rl} x &= t + 1 \\ y &= 2t + 1 \\ z &= -t \end{array}\right., t \in \mathbb{R}$. Let $d'$ be the line with parametric representation: $\left\{\begin{array}{rl} x &= 2t' - 1 \\ y &= -t' + 2 \\ z &= t' + 1 \end{array}\right., t' \in \mathbb{R}$. Let $P$ be the plane with Cartesian equation: $2x + y - 2z + 18 = 0$. Let A be the point with coordinates $(-1; -3; 2)$ and B be the point with coordinates $(-5; -5; 6)$. We call the perpendicular bisector plane of segment $[\mathrm{AB}]$ the plane passing through the midpoint of segment $[\mathrm{AB}]$ and perpendicular to the line $(\mathrm{AB})$.
    Statement 3: Point A belongs to line $d$. Statement 4: Lines $d$ and $d'$ are secant. Statement 5: Plane $P$ is the perpendicular bisector plane of segment $[\mathrm{AB}]$.