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
2013 asie

5 maths questions

Q1 5 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
Exercise 1 (5 points) -- Common to all candidates
In this exercise, probabilities should be rounded to the nearest hundredth.
Part A
A wholesaler buys boxes of green tea from two suppliers. He buys 80\% of his boxes from supplier A and 20\% from supplier B. 10\% of the boxes from supplier A show traces of pesticides and 20\% of those from supplier B also show traces of pesticides.
A box is randomly selected from the wholesaler's stock and the following events are considered: --- event $A$: ``the box comes from supplier A''; --- event $B$: ``the box comes from supplier B''; --- event $S$: ``the box shows traces of pesticides''.
  1. Translate the statement in the form of a weighted tree diagram.
  2. a. What is the probability of event $B \cap \bar{S}$? b. Justify that the probability that the selected box shows no traces of pesticides is equal to 0.88.
  3. It is observed that the selected box shows traces of pesticides. What is the probability that this box comes from supplier B?

Part B
The manager of a tea salon buys 10 boxes from the above wholesaler. It is assumed that the latter's stock is sufficiently large to model this situation by random selection of 10 boxes with replacement. Consider the random variable $X$ which associates with this sample of 10 boxes the number of boxes without traces of pesticides.
  1. Justify that the random variable $X$ follows a binomial distribution and specify its parameters.
  2. Calculate the probability that all 10 boxes are free of pesticide traces.
  3. Calculate the probability that at least 8 boxes show no traces of pesticides.

Part C
For advertising purposes, the wholesaler displays on his leaflets: ``88\% of our tea is guaranteed free of pesticide traces''.
An inspector from the fraud prevention unit wishes to study the validity of this claim. To this end, he randomly selects 50 boxes from the wholesaler's stock and finds 12 with traces of pesticides.
It is assumed that in the wholesaler's stock, the proportion of boxes without traces of pesticides is indeed equal to 0.88. Let $F$ be the random variable which, for any sample of 50 boxes, associates the frequency of boxes containing no traces of pesticides.
  1. Give the asymptotic confidence interval for the random variable $F$ at the 95\% confidence level.
  2. Can the fraud prevention inspector decide, at the 95\% confidence level, that the advertisement is misleading?
Q2 Differentiating Transcendental Functions Determine parameters from function or curve conditions View
Exercise 2 -- Common to all candidates
Consider the functions $f$ and $g$ defined for all real $x$ by: $$f(x) = \mathrm{e}^{x} \quad \text{and} \quad g(x) = 1 - \mathrm{e}^{-x}.$$ The representative curves of these functions in an orthogonal coordinate system of the plane, denoted respectively $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$, are provided in the appendix.
Part A
These curves appear to admit two common tangent lines. Draw these tangent lines as accurately as possible on the figure in the appendix.
Part B
In this part, the existence of these common tangent lines is admitted. Let $\mathscr{D}$ denote one of them. This line is tangent to the curve $\mathscr{C}_{f}$ at point A with abscissa $a$ and tangent to the curve $\mathscr{C}_{g}$ at point B with abscissa $b$.
  1. a. Express in terms of $a$ the slope of the tangent line to the curve $\mathscr{C}_{f}$ at point A. b. Express in terms of $b$ the slope of the tangent line to the curve $\mathscr{C}_{g}$ at point B. c. Deduce that $b = -a$.
  2. Prove that the real number $a$ is a solution of the equation $$2(x - 1)\mathrm{e}^{x} + 1 = 0.$$

Part C
Consider the function $\varphi$ defined on $\mathbb{R}$ by $$\varphi(x) = 2(x - 1)\mathrm{e}^{x} + 1$$
  1. a. Calculate the limits of the function $\varphi$ at $-\infty$ and $+\infty$. b. Calculate the derivative of the function $\varphi$, then study its sign. c. Draw the variation table of the function $\varphi$ on $\mathbb{R}$. Specify the value of $\varphi(0)$.
  2. a. Prove that the equation $\varphi(x) = 0$ has exactly two solutions in $\mathbb{R}$. b. Let $\alpha$ denote the negative solution of the equation $\varphi(x) = 0$ and $\beta$ the positive solution of this equation. Using a calculator, give the values of $\alpha$ and $\beta$ rounded to the nearest hundredth.

Part D
In this part, we prove the existence of these common tangent lines, which was admitted in Part B. Let E be the point on the curve $\mathscr{C}_{f}$ with abscissa $\alpha$ and F the point on the curve $\mathscr{C}_{g}$ with abscissa $-\alpha$ ($\alpha$ is the real number defined in Part C).
  1. Prove that the line $(EF)$ is tangent to the curve $\mathscr{C}_{f}$ at point E.
  2. Prove that $(EF)$ is tangent to $\mathscr{C}_{g}$ at point F.
Q3 Complex Numbers Argand & Loci True/False or Multiple-Statement Verification View
Exercise 3 -- Common to all candidates
The four questions in this exercise are independent. For each question, a statement is proposed. Indicate whether each of them is true or false, by justifying your answer. An unjustified answer earns no points. In questions 1. and 2., the plane is referred to the direct orthonormal coordinate system $(O, \vec{u}, \vec{v})$. Consider the points A, B, C, D and E with complex numbers respectively: $$a = 2 + 2\mathrm{i}, \quad b = -\sqrt{3} + \mathrm{i}, \quad c = 1 + \mathrm{i}\sqrt{3}, \quad d = -1 + \frac{\sqrt{3}}{2}\mathrm{i} \quad \text{and} \quad e = -1 + (2 + \sqrt{3})\mathrm{i}.$$
  1. Statement 1: the points A, B and C are collinear.
  2. Statement 2: the points B, C and D belong to the same circle with center E.
  3. In this question, space is equipped with a coordinate system $(O, \vec{\imath}, \vec{\jmath}, \vec{k})$. Consider the points $I(1; 0; 0)$, $J(0; 1; 0)$ and $K(0; 0; 1)$. Statement 3: the line $\mathscr{D}$ with parametric representation $\left\{\begin{aligned} x &= 2 - t \\ y &= 6 - 2t \\ z &= -2 + t \end{aligned}\right.$ where $t \in \mathbb{R}$, intersects the plane (IJK) at point $E\left(-\frac{1}{2}; 1; \frac{1}{2}\right)$.
  4. In the cube ABCDEFGH, the point T is the midpoint of segment $[HF]$. Statement 4: the lines (AT) and (EC) are orthogonal.
Q4A 5 marks Sequences and series, recurrence and convergence Monotonicity and boundedness analysis View
Exercise 4 (5 points) -- Candidates who have NOT chosen the specialization option
Part A
Consider the sequence $(u_{n})$ defined by: $u_{0} = 2$ and, for all natural integers $n$: $$u_{n+1} = \frac{1 + 3u_{n}}{3 + u_{n}}$$ It is admitted that all terms of this sequence are defined and strictly positive.
  1. Prove by induction that, for all natural integers $n$, we have: $u_{n} > 1$.
  2. a. Establish that, for all natural integers $n$, we have: $u_{n+1} - u_{n} = \frac{(1 - u_{n})(1 + u_{n})}{3 + u_{n}}$. b. Determine the direction of variation of the sequence $(u_{n})$. Deduce that the sequence $(u_{n})$ converges.

Part B
Consider the sequence $(u_{n})$ defined by: $u_{0} = 2$ and, for all natural integers $n$: $$u_{n+1} = \frac{1 + 0.5u_{n}}{0.5 + u_{n}}$$ It is admitted that all terms of this sequence are defined and strictly positive.
  1. Consider the following algorithm:
    InputLet $n$ be a non-zero natural integer
    InitializationAssign to $u$ the value 2
    ProcessingFOR $i$ going from 1 to $n$
    andAssign to $u$ the value $\frac{1 + 0.5u}{0.5 + u}$
    outputDisplay $u$
    END FOR

    Reproduce and complete the following table by running this algorithm for $n = 3$. The values of $u$ should be rounded to the nearest thousandth.
    $i$123
    $u$

  2. For $n = 12$, the previous table was extended and we obtained:
    $i$456789101112
    $u$1.00830.99731.00090.99971.00010.999971.000010.9999961.000001

    Conjecture the behavior of the sequence $(u_{n})$ at infinity.
  3. Consider the sequence $(v_{n})$ defined, for all natural integers $n$, by: $v_{n} = \frac{u_{n} - 1}{u_{n} + 1}$. a. Prove that the sequence $(v_{n})$ is geometric with common ratio $-\frac{1}{3}$. b. Calculate $v_{0}$ then write $v_{n}$ as a function of $n$.
  4. a. Show that, for all natural integers $n$, we have: $v_{n} \neq 1$. b. Show that, for all natural integers $n$, we have: $u_{n} = \frac{1 + v_{n}}{1 - v_{n}}$. c. Determine the limit of the sequence $(u_{n})$.
Q4B 5 marks Linear transformations View
Exercise 4 (5 points) -- Candidates who have chosen the specialization option
A software allows transforming a rectangular element of a photograph. Thus, the initial rectangle OEFG is transformed into a rectangle $OE'F'G'$, called the image of OEFG.
The purpose of this exercise is to study the rectangle obtained after several successive transformations.
Part A
The plane is referred to an orthonormal coordinate system $(O, \vec{\imath}, \vec{\jmath})$. The points E, F and G have coordinates respectively $(2; 2)$, $(-1; 5)$ and $(-3; 3)$. The software transformation associates with any point $M(x; y)$ of the plane the point $M'(x'; y')$, image of point $M$ such that: $$\left\{\begin{aligned} x' &= \frac{5}{4}x + \frac{3}{4}y \\ y' &= \frac{3}{4}x + \frac{5}{4}y \end{aligned}\right.$$
  1. a. Calculate the coordinates of points $E'$, $F'$ and $G'$, images of points E, F and G by this transformation. b. Compare the lengths OE and $OE'$ on one hand, OG and $OG'$ on the other hand.

Give the square matrix of order 2, denoted $A$, such that: $\binom{x'}{y'} = A\binom{x}{y}$.
Part B
In this part, we study the coordinates of the successive images of vertex F of rectangle OEFG when the software transformation is applied multiple times.
  1. Consider the algorithm intended to display the coordinates of these successive images.
  2. a. Prove that, for every natural integer $n$, the point $E_{n}$ is located on the line with equation $y = x$. One may use the fact that, for every natural integer $n$, the coordinates $(x_{n}; y_{n})$ of point $E_{n}$ satisfy: $$\binom{x_{n}}{y_{n}} = A^{n}\binom{2}{2}$$ b. Prove that the length $\mathrm{O}E_{n}$ tends towards $+\infty$ when $n$ tends towards $+\infty$.