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 metropole-sept

5 maths questions

Q1 Indefinite & Definite Integrals Antiderivative Verification and Construction View
Let $f$ be a function defined and differentiable on $\mathbb{R}$. We denote by $\mathscr{C}$ its representative curve in the plane equipped with a coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$.
Part A
In the graphs below, we have represented the curve $\mathscr{C}$ and three other curves $\mathscr{C}_1, \mathscr{C}_2, \mathscr{C}_3$ with the tangent line at their point with abscissa 0.
  1. Determine by reading the graph, the sign of $f(x)$ according to the values of $x$.
  2. We denote by $F$ a primitive of the function $f$ on $\mathbb{R}$. a. Using the curve $\mathscr{C}$, determine $F'(0)$ and $F'(-2)$. b. One of the curves $\mathscr{C}_1, \mathscr{C}_2, \mathscr{C}_3$ is the representative curve of the function $F$. Determine which one by justifying the elimination of the other two.

Part B
In this part, we assume that the function $f$ mentioned in Part A is the function defined on $\mathbb{R}$ by $$f(x) = (x+2)\mathrm{e}^{\frac{1}{2}x}$$
  1. Observation of the curve $\mathscr{C}$ allows us to conjecture that the function $f$ admits a minimum. a. Prove that for all real $x$, $f'(x) = \frac{1}{2}(x+4)\mathrm{e}^{\frac{1}{2}x}$. b. Deduce a validation of the previous conjecture.
  2. Let $I = \int_0^1 f(x)\,\mathrm{d}x$. a. Give a geometric interpretation of the real number $I$. b. Let $u$ and $v$ be the functions defined on $\mathbb{R}$ by $u(x) = x$ and $v(x) = \mathrm{e}^{\frac{1}{2}x}$. Verify that $f = 2(u'v + uv')$. c. Deduce the exact value of the integral $I$.
  3. The algorithm below is given.
    Variables :$k$ and $n$ are natural integers. $s$ is a real number.
    Input :Assign to $s$ the value 0.
    Processing :For $k$ ranging from 0 to $n-1$
    End of loop.
    Output :Display $s$.

    We denote by $s_n$ the number displayed by this algorithm when the user enters a strictly positive natural integer as the value of $n$. a. Justify that $s_3$ represents the area, expressed in square units, of the shaded region in the graph below where the three rectangles have the same width. b. What can be said about the value of $s_n$ provided by the proposed algorithm when $n$ becomes large?
Q2 4 marks Complex Numbers Argand & Loci True/False or Multiple-Statement Verification View
This exercise is a multiple choice questionnaire. For each question, three answers are proposed and only one of them is correct. The candidate will write on the answer sheet the number of the question followed by the chosen answer and will justify their choice. One point is awarded for each correct and properly justified answer. An unjustified answer will not be taken into account. No points are deducted in the absence of an answer or in case of an incorrect answer.
For questions 1 and 2, space is equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. The line $\mathscr{D}$ is defined by the parametric representation $\left\{\begin{array}{rl} x &= 5-2t \\ y &= 1+3t \\ z &= 4 \end{array},\, t \in \mathbb{R}\right.$.
  1. We denote by $\mathscr{P}$ the plane with Cartesian equation $3x + 2y + z - 6 = 0$. a. The line $\mathscr{D}$ is perpendicular to the plane $\mathscr{P}$. b. The line $\mathscr{D}$ is parallel to the plane $\mathscr{P}$. c. The line $\mathscr{D}$ is contained in the plane $\mathscr{P}$.
  2. We denote by $\mathscr{D}'$ the line that passes through point A with coordinates $(3;1;1)$ and has direction vector $\vec{u} = 2\vec{i} - \vec{j} + 2\vec{k}$. a. The lines $\mathscr{D}$ and $\mathscr{D}'$ are parallel. b. The lines $\mathscr{D}$ and $\mathscr{D}'$ are secant. c. The lines $\mathscr{D}$ and $\mathscr{D}'$ are not coplanar.

For questions 3 and 4, the plane is equipped with a direct orthonormal coordinate system with origin O.
    \setcounter{enumi}{2}
  1. Let $\mathscr{E}$ be the set of points $M$ with affix $z$ satisfying $|z + \mathrm{i}| = |z - \mathrm{i}|$. a. $\mathscr{E}$ is the $x$-axis. b. $\mathscr{E}$ is the $y$-axis. c. $\mathscr{E}$ is the circle with center O and radius 1.
  2. We denote by B and C two points in the plane whose respective affixes $b$ and $c$ satisfy the equality $\dfrac{c}{b} = \sqrt{2}\,\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$. a. The triangle OBC is isosceles with apex O. b. The points O, B, C are collinear. c. The triangle OBC is isosceles and right-angled at B.
Q3 Conditional Probability Bayes' Theorem with Production/Source Identification View
In a factory, two machines A and B are used to manufacture parts.
Machine A ensures $40\%$ of production and machine B ensures $60\%$. It is estimated that $10\%$ of parts from machine A have a defect and that $9\%$ of parts from machine B have a defect.
A part is chosen at random and we consider the following events:
  • $A$: ``The part is produced by machine A''
  • $B$: ``The part is produced by machine B''
  • $D$: ``The part has a defect''
  • $\bar{D}$: the opposite event of event $D$.

  1. a. Translate the situation using a probability tree. b. Calculate the probability that the chosen part has a defect and was manufactured by machine A. c. Prove that the probability $P(D)$ of event $D$ is equal to 0.094. d. It is observed that the chosen part has a defect. What is the probability that this part comes from machine A?
  2. It is estimated that machine A is properly adjusted if $90\%$ of the parts it manufactures are conforming. It is decided to check this machine by examining $n$ parts chosen at random ($n$ natural integer) from the production of machine A. These $n$ draws are treated as successive independent draws with replacement. We denote by $X_n$ the number of parts that are conforming in the sample of $n$ parts, and $F_n = \dfrac{X_n}{n}$ the corresponding proportion. a. Justify that the random variable $X_n$ follows a binomial distribution and specify its parameters. b. In this question, we take $n = 150$. Determine the asymptotic fluctuation interval $I$ at the $95\%$ threshold of the random variable $F_{150}$. c. A quality test counts 21 non-conforming parts in a sample of 150 parts produced. Does this call into question the adjustment of the machine? Justify the answer.
Q4a Sequences and series, recurrence and convergence Auxiliary sequence transformation View
Exercise 4 — Candidates who have NOT followed the specialization course
We consider the sequence $(u_n)$ defined on $\mathbb{N}$ by: $$u_0 = 2 \text{ and for all natural integer } n,\quad u_{n+1} = \frac{u_n + 2}{2u_n + 1}.$$ We admit that for all natural integer $n$, $u_n > 0$.
  1. a. Calculate $u_1, u_2, u_3, u_4$. An approximate value to $10^{-2}$ may be given. b. Verify that if $n$ is one of the integers $0,1,2,3,4$ then $u_n - 1$ has the same sign as $(-1)^n$. c. Establish that for all natural integer $n$, $u_{n+1} - 1 = \dfrac{-u_n + 1}{2u_n + 1}$. d. Prove by induction that for all natural integer $n$, $u_n - 1$ has the same sign as $(-1)^n$.
  2. For all natural integer $n$, we set $v_n = \dfrac{u_n - 1}{u_n + 1}$. a. Establish that for all natural integer $n$, $v_{n+1} = \dfrac{-u_n + 1}{3u_n + 3}$. b. Prove that the sequence $(v_n)$ is a geometric sequence with ratio $-\dfrac{1}{3}$. Deduce the expression of $v_n$ as a function of $n$. c. We admit that for all natural integer $n$, $u_n = \dfrac{1 + v_n}{1 - v_n}$. Express $u_n$ as a function of $n$ and determine the limit of the sequence $(u_n)$.
Q4b Matrices Matrix Power Computation and Application View
Exercise 4 — Candidates who have followed the specialization course
In an imaginary isolated village, a new contagious but non-fatal disease has appeared. Scientists discovered that an individual could be in one of three following states:
  • $S$: ``the individual is healthy, that is, not sick and not infected'',
  • $I$: ``the individual is a healthy carrier, that is, not sick but infected'',
  • $M$: ``the individual is sick and infected''.

Part A
Scientists estimate that a single individual is at the origin of the disease among the 100 people in the population and that, from one week to the next, an individual changes state according to the following process:
  • among healthy individuals, the proportion of those who become healthy carriers is equal to $\dfrac{1}{3}$ and the proportion of those who become sick is equal to $\dfrac{1}{3}$,
  • among healthy carriers, the proportion of those who become sick is equal to $\dfrac{1}{2}$.

We denote by $P_n = \begin{pmatrix} s_n & i_n & m_n \end{pmatrix}$ the row matrix giving the probabilistic state after $n$ weeks where $s_n, i_n$ and $m_n$ denote respectively the probability that the individual is healthy, a healthy carrier, or sick in the $n$-th week.
We have $P_0 = (0.99 \quad 0 \quad 0.01)$ and for all natural integer $n$, $$\left\{\begin{aligned} s_{n+1} &= \frac{1}{3}s_n \\ i_{n+1} &= \frac{1}{3}s_n + \frac{1}{2}i_n \\ m_{n+1} &= \frac{1}{3}s_n + \frac{1}{2}i_n + m_n \end{aligned}\right.$$
  1. Write the matrix $A$ called the transition matrix, such that for all natural integer $n$, $P_{n+1} = P_n \cdot A$.