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
2022 bac-spe-maths__amerique-sud_j2

4 maths questions

Q1 7 marks Conditional Probability Bayes' Theorem with Production/Source Identification View
Exercise 1: Probability
A company manufactures components for the automotive industry. These components are designed on three assembly lines numbered 1 to 3.
  • Half of the components are designed on line $\mathrm{n}^{\circ}1$;
  • $30\%$ of the components are designed on line $\mathrm{n}^{\circ}2$;
  • the remaining components are designed on line $\mathrm{n}^{\circ}3$.
At the end of the manufacturing process, it appears that $1\%$ of the parts from line $\mathrm{n}^{\circ}1$ have a defect, as do $0.5\%$ of the parts from line $\mathrm{n}^{\circ}2$ and $4\%$ of the parts from line $\mathrm{n}^{\circ}3$. One of these components is randomly selected. We denote:
  • $C_1$ the event ``the component comes from line $\mathrm{n}^{\circ}1$'';
  • $C_2$ the event ``the component comes from line $\mathrm{n}^{\circ}2$'';
  • $C_3$ the event ``the component comes from line $\mathrm{n}^{\circ}3$'';
  • $D$ the event ``the component is defective'' and $\bar{D}$ its complementary event.
Throughout the exercise, probability calculations will be given as exact decimal values or rounded to $10^{-4}$ if necessary.
PART A
  1. Represent this situation with a probability tree.
  2. Calculate the probability that the selected component comes from line $\mathrm{n}^{\circ}3$ and is defective.
  3. Show that the probability of event $D$ is $P(D) = 0.0145$.
  4. Calculate the probability that a defective component comes from line $\mathrm{n}^{\circ}3$.

PART B
The company decides to package the produced components by forming batches of $n$ units. We denote $X$ the random variable which, to each batch of $n$ units, associates the number of defective components in this batch. Given the company's production and packaging methods, we can consider that $X$ follows the binomial distribution with parameters $n$ and $p = 0.0145$.
  1. In this question, the batches contain 20 units. We set $n = 20$. a. Calculate the probability that a batch contains exactly three defective components. b. Calculate the probability that a batch contains no defective components. Deduce the probability that a batch contains at least one defective component.
  2. The company director wishes the probability of having no defective components in a batch of $n$ components to be greater than 0.85. He proposes to form batches of at most 11 components. Is he correct? Justify your answer.

PART C
The manufacturing costs of the components of this company are 15 euros if they come from assembly line $\mathrm{n}^{\circ}1$, 12 euros if they come from assembly line $\mathrm{n}^{\circ}2$ and 9 euros if they come from assembly line $\mathrm{n}^{\circ}3$. Calculate the average manufacturing cost of a component for this company.
Q2 7 marks Differentiating Transcendental Functions Full function study with transcendental functions View
Exercise 2: Functions, logarithm function
The purpose of this exercise is to study the function $f$, defined on $]0; +\infty[$, by: $$f(x) = 3x - x\ln(x) - 2\ln(x).$$
PART A: Study of an auxiliary function $g$
Let $g$ be the function defined on $]0; +\infty[$ by $$g(x) = 2(x-1) - x\ln(x)$$ We denote $g'$ the derivative function of $g$. We admit that $\lim_{x \rightarrow +\infty} g(x) = -\infty$.
  1. Calculate $g(1)$ and $g(\mathrm{e})$.
  2. Determine $\lim_{x \rightarrow 0^+} g(x)$ by justifying your approach.
  3. Show that, for all $x > 0$, $g'(x) = 1 - \ln(x)$. Deduce the variation table of $g$ on $]0; +\infty[$.
  4. Show that the equation $g(x) = 0$ has exactly two distinct solutions on $]0; +\infty[$: 1 and $\alpha$ with $\alpha$ belonging to the interval $[\mathrm{e}; +\infty[$. Give an approximation of $\alpha$ to 0.01.
  5. Deduce the sign table of $g$ on $]0; +\infty[$.

PART B: Study of the function $f$
We consider in this part the function $f$, defined on $]0; +\infty[$, by $$f(x) = 3x - x\ln(x) - 2\ln(x).$$ We denote $f'$ the derivative function of $f$. We admit that: $\lim_{x \rightarrow 0^+} f(x) = +\infty$.
  1. Determine the limit of $f$ at $+\infty$ by justifying your approach.
  2. a. Justify that for all $x > 0$, $f'(x) = \dfrac{g(x)}{x}$. b. Deduce the variation table of $f$ on $]0; +\infty[$.
  3. We admit that, for all $x > 0$, the second derivative of $f$, denoted $f''$, is defined by $f''(x) = \dfrac{2-x}{x^2}$. Study the convexity of $f$ and specify the coordinates of the inflection point of $\mathscr{C}_f$.
Q3 7 marks Sequences and series, recurrence and convergence Applied/contextual sequence problem View
Exercise 3: Sequences
The population of an endangered species is closely monitored in a nature reserve. Climate conditions as well as poaching cause this population to decrease by $10\%$ each year. To compensate for these losses, 100 individuals are reintroduced into the reserve at the end of each year. We wish to study the evolution of the population size of this species over time. For this, we model the population size of the species by the sequence $(u_n)$ where $u_n$ represents the population size at the beginning of the year $2020 + n$. We admit that for all natural integer $n$, $u_n \geqslant 0$. At the beginning of the year 2020, the studied population has 2000 individuals, thus $u_0 = 2000$.
  1. Justify that the sequence $(u_n)$ satisfies the recurrence relation: $$u_{n+1} = 0.9u_n + 100.$$
  2. Calculate $u_1$ then $u_2$.
  3. Prove by induction that for all natural integer $n$: $1000 < u_{n+1} \leqslant u_n$.
  4. Is the sequence $(u_n)$ convergent? Justify your answer.
  5. We consider the sequence $(v_n)$ defined for all natural integer $n$ by $v_n = u_n - 1000$. a. Show that the sequence $(v_n)$ is geometric with common ratio 0.9. b. Deduce that, for all natural integer $n$, $u_n = 1000\left(1 + 0.9^n\right)$. c. Determine the limit of the sequence $(u_n)$. Give an interpretation of this in the context of this exercise.
  6. We wish to determine the number of years necessary for the population size to fall below a certain threshold $S$ (with $S > 1000$). a. Determine the smallest integer $n$ such that $u_n \leqslant 1020$. Justify your answer by a calculation. b. In the Python program opposite, the variable $n$ denotes the number of years elapsed since 2020, the variable $u$ denotes the population size. Copy and complete this program so that it returns the number of years necessary for the population size to fall below the threshold $S$. \begin{verbatim} def population(S) : n=0 u=2000 while ......: u= ... n = ... return ... \end{verbatim}
Q4 7 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 4: Geometry in Space
In space equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points $$\mathrm{A}(0; 8; 6), \quad \mathrm{B}(6; 4; 4) \quad \text{and} \quad \mathrm{C}(2; 4; 0).$$
  1. a. Justify that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear. b. Show that the vector $\vec{n}(1; 2; -1)$ is a normal vector to the plane (ABC). c. Determine a Cartesian equation of the plane (ABC).
  2. Let D and E be the points with respective coordinates $(0; 0; 6)$ and $(6; 6; 0)$. a. Determine a parametric representation of the line (DE). b. Show that the midpoint I of the segment [BC] belongs to the line (DE).
  3. We consider the triangle ABC. a. Determine the nature of triangle ABC. b. Calculate the area of triangle ABC in square units. c. Calculate $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AC}}$. d. Deduce a measure of the angle $\widehat{\mathrm{BAC}}$ rounded to 0.1 degree.
  4. We consider the point H with coordinates $\left(\dfrac{5}{3}; \dfrac{10}{3}; -\dfrac{5}{3}\right)$. Show that $H$ is the orthogonal projection of the point $O$ onto the plane (ABC). Deduce the distance from point O to the plane (ABC).