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__centres-etrangers_j2

4 maths questions

Q1 6 marks Geometric Sequences and Series Applied Geometric Model with Contextual Interpretation View
Exercise 1

We propose to compare the evolution of an animal population in two distinct environments A and B.
On January $1^{\text{st}}$ 2025, 6000 individuals are introduced into each of environments A and B.
Part A
In this part, we study the evolution of the population in environment A. We assume that in this environment, the evolution of the population is modelled by a geometric sequence $(u_n)$ with first term $u_0 = 6$ and common ratio 0.93. For every natural number $n$, $u_n$ represents the population on January $1^{\text{st}}$ of the year $2025 + n$, expressed in thousands of individuals.
  1. Give, according to this model, the population on January $1^{\text{st}}$ 2026.
  2. For every natural number $n$, express $u_n$ as a function of $n$.
  3. Determine the limit of the sequence $(u_n)$.

Interpret this result in the context of the exercise.
Part B
In this part, we study the evolution of the population in environment B. We assume that in this environment, the evolution of the population is modelled by the sequence $(v_n)$ defined by
$$v_0 = 6 \text{ and for every natural number } n, v_{n+1} = -0.05 v_n^2 + 1.1 v_n.$$
For every natural number $n$, $v_n$ represents the population on January $1^{\text{st}}$ of the year $2025 + n$, expressed in thousands of individuals.
  1. Give, according to this model, the population on January $1^{\text{st}}$ 2026.

Let $f$ be the function defined on the interval $[0; +\infty[$ by
$$f(x) = -0.05x^2 + 1.1x$$
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  1. Prove that the function $f$ is increasing on the interval $[0; 11]$.
  2. Prove by induction that for every natural number $n$, we have $$2 \leqslant v_{n+1} \leqslant v_n \leqslant 6$$
  3. Deduce that the sequence $(v_n)$ is convergent to a limit $\ell$.
  4. a. Justify that the limit $\ell$ satisfies $f(\ell) = \ell$ then deduce the value of $\ell$. b. Interpret this result in the context of the exercise.

Part C
This part aims to compare the evolution of the population in the two environments.
  1. By solving an inequality, determine the year from which the population of environment A will be strictly less than 3000 individuals.
  2. Using a calculator, determine the year from which the population of environment B will be strictly less than 3000 individuals.
  3. Justify that from a certain year onwards, the population of environment B will exceed the population of environment A.
  4. Consider the Python program opposite. a. Copy and complete this program so that after execution, it displays the year from which the population of environment B is strictly greater than the population of environment A. b. Determine the year displayed after execution of the programme.

\begin{verbatim} n=0 u=6 v = 6 while...: u = ... v=... n = n+1 print (2025 + n) \end{verbatim}
Q2 Differential equations Applied Modeling with Differential Equations View
Exercise 2

Part A
We consider the function $f$ defined on the interval $[0; +\infty[$ by:
$$f(x) = \frac{1}{a + \mathrm{e}^{-bx}}$$
where $a$ and $b$ are two strictly positive real constants. We admit that the function $f$ is differentiable on the interval $[0; +\infty[$. The function $f$ has for graphical representation the curve $\mathscr{C}_f$.
We consider the points $\mathrm{A}(0; 0.5)$ and $\mathrm{B}(10; 1)$. We admit that the line (AB) is tangent to the curve $\mathscr{C}_f$ at point A.
  1. By graphical reading, give an approximate value of $f(10)$.
  2. We admit that $\lim_{x \rightarrow +\infty} f(x) = 1$. Give a graphical interpretation of this result.
  3. Justify that $a = 1$.
  4. Determine the slope of the line (AB).
  5. a. Determine the expression of $f'(x)$ as a function of $x$ and the constant $b$. b. Deduce the value of $b$.

Part B
We admit, in the rest of the exercise, that the function $f$ is defined on the interval $[0; +\infty[$ by:
$$f(t) = \frac{1}{1 + \mathrm{e}^{-0.2x}}$$
  1. Determine $\lim_{x \rightarrow +\infty} f(x)$.
  2. Study the variations of the function $f$ on the interval $[0; +\infty[$.
  3. Show that there exists a unique positive real number $\alpha$ such that $f(\alpha) = 0.97$.
  4. Using a calculator, give a bound for the real number $\alpha$ by two consecutive integers. Interpret this result in the context of the statement.

Part C
  1. Show that, for all $x$ belonging to the interval $[0; +\infty[$, $f(x) = \dfrac{\mathrm{e}^{0.2x}}{1 + \mathrm{e}^{0.2x}}$.
  2. Deduce an antiderivative of the function $f$ on the interval $[0; +\infty[$.
  3. Calculate the average value of the function $f$ on the interval $[0; 40]$, that is: $$I = \frac{1}{40} \int_0^{40} \frac{1}{1 + \mathrm{e}^{-0.2x}} \,\mathrm{d}x$$ The exact value and an approximate value to the nearest thousandth will be given.
Q3 Binomial Distribution Derive or Prove a Binomial Distribution Identity View
Exercise 3
The ``base64'' encoding, used in computing, allows messages and other data such as images to be represented and transmitted using 64 characters: the 26 uppercase letters, the 26 lowercase letters, the digits 0 to 9 and two other special characters. Parts A, B and C are independent.
Part A
In this part, we are interested in sequences of 4 characters in base64. For example, ``gP3g'' is such a sequence. In a sequence, order must be taken into account: the sequences ``m5C2'' and ``5C2m'' are not identical.
  1. Determine the number of possible sequences.
  2. Determine the number of sequences if we require that the 4 characters are pairwise different.
  3. a. Determine the number of sequences containing no uppercase letter A. b. Deduce the number of sequences containing at least one uppercase letter A. c. Determine the number of sequences containing exactly one uppercase letter A. d. Determine the number of sequences containing exactly two uppercase letters A.

Part B
We are interested in the transmission of a sequence of 250 characters from one computer to another. We assume that the probability that a character is incorrectly transmitted is equal to 0.01 and that the transmissions of the different characters are independent of each other. We denote by $X$ the random variable equal to the number of characters incorrectly transmitted.
  1. We admit that the random variable $X$ follows the binomial distribution. Give its parameters.
  2. Determine the probability that all characters are correctly transmitted. The exact expression will be given, then an approximate value to $10^{-3}$ near.
  3. What do you think of the following statement: ``The probability that more than 16 characters are incorrectly transmitted is negligible''?

Part C
We are now interested in the transmission of 4 sequences of 250 characters. We denote by $X_1, X_2, X_3$ and $X_4$ the random variables corresponding to the numbers of characters incorrectly transmitted during the transmission of each of the 4 sequences. We admit that the random variables $X_1, X_2, X_3$ and $X_4$ are independent of each other and follow the same distribution as the random variable $X$ defined in part B. We denote by $S = X_1 + X_2 + X_3 + X_4$. Determine, by justifying, the expectation and the variance of the random variable $S$.
Q4 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 4
We place ourselves in an orthonormal frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ of space. We consider the points $\mathrm{A}(1; 0; 3)$, $\mathrm{B}(-2; 1; 2)$ and $\mathrm{C}(0; 3; 2)$.
  1. a. Show that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear. b. Let $\vec{n}$ be the vector with coordinates $\left(\begin{array}{c} -1 \\ 1 \\ 4 \end{array}\right)$. Verify that the vector $\vec{n}$ is orthogonal to the plane (ABC). c. Deduce that the plane $(\mathrm{ABC})$ has for Cartesian equation $-x + y + 4z - 11 = 0$.

We consider the plane $\mathscr{P}$ with Cartesian equation $3x - 3y + 2z - 9 = 0$ and the plane $\mathscr{P}'$ with Cartesian equation $x - y - z + 2 = 0$.
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  1. a. Prove that the planes $\mathscr{P}$ and $\mathscr{P}'$ are secant. We denote by (d) their line of intersection. b. Determine whether the planes $\mathscr{P}$ and $\mathscr{P}'$ are perpendicular.
  2. Show that the line (d) is directed by the vector $\vec{u}\left(\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right)$.
  3. Show that the point $\mathrm{M}(2; 1; 3)$ belongs to the planes $\mathscr{P}$ and $\mathscr{P}'$. Deduce a parametric representation of the line (d).
  4. Show that the line (d) is also included in the plane (ABC). What can we say about the three planes (ABC), $\mathscr{P}$ and $\mathscr{P}'$?