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
2016 antilles-guyane

10 maths questions

Q1A Tree Diagrams Construct a Tree Diagram View
A light bulb manufacturer has two machines, denoted A and B. Machine A provides $65\%$ of production, and machine B provides the rest. Some light bulbs have a manufacturing defect:
  • at the output of machine $\mathrm{A}$, $8\%$ of light bulbs have a defect;
  • at the output of machine B, $5\%$ of light bulbs have a defect.
The following events are defined:
  • A: ``the light bulb comes from machine A'';
  • B: ``the light bulb comes from machine B'';
  • $D$: ``the light bulb has a defect''.

  1. A light bulb is randomly selected from the total production of one day. a. Construct a probability tree representing the situation. b. Show that the probability of drawing a light bulb without a defect is equal to 0.9305. c. The light bulb drawn has no defect. Calculate the probability that it comes from machine A.
  2. 10 light bulbs are randomly selected from the production of one day at the output of machine A. The size of the stock allows us to consider the trials as independent and to assimilate the draws to draws with replacement. Calculate the probability of obtaining at least 9 light bulbs without a defect.
Q1B Exponential Distribution View
  1. Recall that if $T$ follows an exponential distribution with parameter $\lambda$ ($\lambda$ being a strictly positive real number) then for any positive real $a$, $P(T \leqslant a) = \int_{0}^{a} \lambda \mathrm{e}^{-\lambda x} \mathrm{~d}x$. a. Show that $P(T \geqslant a) = \mathrm{e}^{-\lambda a}$. b. Show that if $T$ follows an exponential distribution then for all positive real numbers $t$ and $a$ we have $$P_{T \geqslant t}(T \geqslant t + a) = P(T \geqslant a).$$
  2. In this part, the lifetime in hours of a light bulb without a defect is a random variable $T$ that follows the exponential distribution with expectation 10000. a. Determine the exact value of the parameter $\lambda$ of this distribution. b. Calculate the probability $P(T \geqslant 5000)$. c. Given that a light bulb without a defect has already operated for 7000 hours, calculate the probability that its total lifetime exceeds 12000 hours.
Q1C Confidence intervals Hypothesis test via confidence interval for a proportion View
The company sought to improve the quality of its production and claims that there are no more than $6\%$ defective light bulbs in its production. A consumer association conducts a test on a sample and obtains 71 defective light bulbs out of 1000.
  1. In the case where there would be exactly $6\%$ defective light bulbs, determine an asymptotic confidence interval at the $95\%$ level for the frequency of defective light bulbs in a random sample of size 1000.
  2. Do we have reasons to question the company's claim?
Q2 3 marks Circles Circle-Line Intersection and Point Conditions View
The complex plane is equipped with a direct orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). We denote by $\mathscr{C}$ the set of points $M$ in the plane with affix $z$ such that $|z - 2| = 1$.
  1. Justify that $\mathscr{C}$ is a circle, and specify its center and radius.
  2. Let $a$ be a real number. We call $\mathscr{D}$ the line with equation $y = ax$. Determine the number of intersection points between $\mathscr{C}$ and $\mathscr{D}$ as a function of the values of the real number $a$.
Q3A Differentiating Transcendental Functions Full function study with transcendental functions View
We consider the function $f$ defined for all real $x$ by $f(x) = x\mathrm{e}^{1-x^{2}}$.
  1. Calculate the limit of the function $f$ at $+\infty$. Hint: you may use the fact that for all real $x$ different from 0, $f(x) = \frac{\mathrm{e}}{x} \times \frac{x^{2}}{\mathrm{e}^{x^{2}}}$. It is admitted that the limit of the function $f$ at $-\infty$ is equal to 0.
  2. a. It is admitted that $f$ is differentiable on $\mathbb{R}$ and we denote by $f'$ its derivative. Prove that for all real $x$, $$f'(x) = \left(1 - 2x^{2}\right)\mathrm{e}^{1-x^{2}}$$ b. Deduce the table of variations of the function $f$.
Q3B Applied differentiation Inequality proof via function study View
We consider the function $g$ defined for all real $x$ by $g(x) = \mathrm{e}^{1-x}$. The representative curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ of the functions $f$ and $g$ respectively are drawn in a coordinate system. The purpose of this part is to study the relative position of these two curves.
  1. After observing the graph, what conjecture can be made?
  2. Justify that, for all real $x$ belonging to $]-\infty; 0]$, $f(x) < g(x)$.
  3. In this question, we consider the interval $]0; +\infty[$. We set, for all strictly positive real $x$, $\Phi(x) = \ln x - x^{2} + x$. a. Show that, for all strictly positive real $x$, $$f(x) \leqslant g(x) \text{ is equivalent to } \Phi(x) \leqslant 0$$ It is admitted for the rest that $f(x) = g(x)$ is equivalent to $\Phi(x) = 0$. b. It is admitted that the function $\Phi$ is differentiable on $]0; +\infty[$. Draw the table of variations of the function $\Phi$. (The limits at 0 and $+\infty$ are not required.) c. Deduce that, for all strictly positive real $x$, $\Phi(x) \leqslant 0$.
  4. a. Is the conjecture made in question 1 of Part B valid? b. Show that $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ have a unique common point, denoted $A$. c. Show that at this point $A$, these two curves have the same tangent line.
Q3C Standard Integrals and Reverse Chain Rule Reverse Chain Rule Antiderivative (MCQ) View
We consider the functions $f(x) = x\mathrm{e}^{1-x^{2}}$ and $g(x) = \mathrm{e}^{1-x}$.
  1. Find a primitive $F$ of the function $f$ on $\mathbb{R}$.
  2. Deduce the value of $\int_{0}^{1} \left(\mathrm{e}^{1-x} - x\mathrm{e}^{1-x^{2}}\right) \mathrm{d}x$.
  3. Give a graphical interpretation of this result.
Q4 (non-specialization) Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
$ABCDEFGH$ is a cube with edge length equal to 1. The space is equipped with the orthonormal coordinate system ($D; \overrightarrow{DC}, \overrightarrow{DA}, \overrightarrow{DH}$). In this coordinate system, we have: $D(0;0;0)$, $C(1;0;0)$, $A(0;1;0)$, $H(0;0;1)$ and $E(0;1;1)$. Let $I$ be the midpoint of $[AB]$. Let $\mathscr{P}$ be the plane parallel to the plane $(BGE)$ and passing through the point $I$. It is admitted that the section of the cube by the plane $\mathscr{P}$ is a hexagon whose vertices $I, J, K, L, M$, and $N$ belong respectively to the edges $[AB], [BC], [CG], [GH], [HE]$ and $[AE]$.
  1. a. Show that the vector $\overrightarrow{DF}$ is normal to the plane $(BGE)$. b. Deduce a Cartesian equation of the plane $\mathscr{P}$.
  2. Show that the point $N$ is the midpoint of the segment $[AE]$.
  3. a. Determine a parametric representation of the line $(HB)$. b. Deduce that the line $(HB)$ and the plane $\mathscr{P}$ intersect at a point $T$ whose coordinates you will specify.
  4. Calculate, in units of volume, the volume of the tetrahedron $FBGE$.
Q4 (specialization) Part A Number Theory Linear Diophantine Equations View
We consider the following equation with unknowns $x$ and $y$ integers: $$7x - 3y = 1 \tag{E}$$
  1. An incomplete algorithm is given below. Copy it and complete it, writing its missing lines (1) and (2) so that it gives the integer solutions $(x; y)$ of the equation (E) satisfying $-5 \leqslant x \leqslant 10$ and $-5 \leqslant y \leqslant 10$.
    Variables:
    X is an integer
    Y is an integer
    Start:
    For X varying from $-5$ to 10
    (1) \ldots
    (2) \ldots
    Then Display X and Y
    End If
    End For
    End

  2. a. Give a particular solution of equation (E). b. Determine the set of pairs of integers solutions of equation (E). c. Determine the set of pairs $(x; y)$ of integers solutions of equation (E) such that $-5 \leqslant x \leqslant 10$ and $-5 \leqslant y \leqslant 10$.
Q4 (specialization) Part B Invariant lines and eigenvalues and vectors Recurrence relations via matrix eigenvalues View
The complex plane is equipped with an orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). We consider the line $\mathscr{D}$ with equation $$7x - 3y - 1 = 0$$ We define the sequence $(A_{n})$ of points in the plane with coordinates $(x_{n}; y_{n})$ satisfying for all natural integer $n$: $$\left\{ \begin{array}{l} x_{0} = 1 \\ y_{0} = 2 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} x_{n+1} = -\frac{13}{2} x_{n} + 3 y_{n} \\ y_{n+1} = -\frac{35}{2} x_{n} + 8 y_{n} \end{array} \right. \right.$$
We denote by $M$ the matrix $\left( \begin{array}{cc} \frac{-13}{2} & 3 \\ \frac{-35}{2} & 8 \end{array} \right)$.