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
2023 bac-spe-maths__amerique-nord_j1

4 maths questions

Q1 Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
In an effort to improve its sustainable development policy, a company conducted a statistical survey on its waste production.
In this survey, waste is classified into three categories:
  • $69 \%$ of waste is mineral and non-hazardous;
  • $28 \%$ of waste is non-mineral and non-hazardous;
  • the remaining waste is hazardous waste.

This statistical survey also tells us that:
  • $73 \%$ of mineral and non-hazardous waste is recyclable;
  • $49 \%$ of non-mineral and non-hazardous waste is recyclable;
  • $35 \%$ of hazardous waste is recyclable.

In this company, a piece of waste is randomly selected. We consider the following events:
  • $M$ : ``The selected waste is mineral and non-hazardous'';
  • N : ``The selected waste is non-mineral and non-hazardous'';
  • $D$ : ``The selected waste is hazardous'';
  • R: ``The selected waste is recyclable''.
We denote by $\bar{R}$ the complementary event of event $R$.
Part A
  1. Copy and complete the probability tree below representing the situation described in the problem.
  2. Justify that the probability that the waste is hazardous and recyclable is equal to 0.0105.
  3. Determine the probability $P(M \cap \bar{R})$ and interpret the answer obtained in the context of the exercise.
  4. Prove that the probability of event $R$ is $P(R) = 0.6514$.
  5. Suppose that the selected waste is recyclable. Determine the probability that this waste is non-mineral and non-hazardous. Give the answer rounded to the ten-thousandth.

Part B
We recall that the probability that a randomly selected piece of waste is recyclable is equal to 0.6514.
  1. In order to control the quality of collection in the company, a sample of 20 pieces of waste is randomly selected from production. We assume that the stock is sufficiently large to treat the sampling of this sample as drawing with replacement.
    We denote by $X$ the random variable equal to the number of recyclable pieces of waste in this sample. a. We admit that the random variable $X$ follows a binomial distribution. Specify its parameters. b. Give the probability that the sample contains exactly 14 recyclable pieces of waste. Give the answer rounded to the ten-thousandth.
  2. In this question, we now select $n$ pieces of waste, where $n$ denotes a strictly positive natural number. a. Give the expression as a function of $n$ of the probability $p_n$ that no piece of waste in this sample is recyclable. b. Determine the value of the natural number $n$ from which the probability that at least one piece of waste in the sample is recyclable is greater than or equal to 0.9999.
Q2 Differentiating Transcendental Functions Full function study with transcendental functions View
We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = \mathrm{e}^{3x} - (2x+1)\mathrm{e}^{x}$$
The purpose of this exercise is to study the function $f$ on $\mathbb{R}$.
Part A - Study of an auxiliary function
We define the function $g$ on $\mathbb{R}$ by: $$g(x) = 3\mathrm{e}^{2x} - 2x - 3$$
  1. a. Determine the limit of function $g$ at $-\infty$. b. Determine the limit of function $g$ at $+\infty$.
  2. a. We admit that function $g$ is differentiable on $\mathbb{R}$, and we denote by $g'$ its derivative function. Prove that for every real number $x$, we have $g'(x) = 6\mathrm{e}^{2x} - 2$. b. Study the sign of the derivative function $g'$ on $\mathbb{R}$. c. Deduce the table of variations of function $g$ on $\mathbb{R}$. Verify that function $g$ has a minimum equal to $\ln(3) - 2$.
  3. a. Show that $x = 0$ is a solution of the equation $g(x) = 0$. b. Show that the equation $g(x) = 0$ has a second non-zero solution, denoted $\alpha$, for which you will give an interval of amplitude $10^{-1}$.
  4. Deduce from the previous questions the sign of function $g$ on $\mathbb{R}$.

Part B - Study of function $f$
  1. Function $f$ is differentiable on $\mathbb{R}$, and we denote by $f'$ its derivative function. Prove that for every real number $x$, we have $f'(x) = \mathrm{e}^{x} g(x)$, where $g$ is the function defined in Part A.
  2. Deduce the sign of the derivative function $f'$ and then the variations of function $f$ on $\mathbb{R}$.
  3. Why is function $f$ not convex on $\mathbb{R}$? Explain.
Q3 Vectors: Lines & Planes MCQ: Identify Correct Equation or Representation View
This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required.
A wrong answer, multiple answers, or the absence of an answer to a question neither awards nor deducts points. The five questions are independent. Space is equipped with an orthonormal reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the points $\mathrm{A}(-1; 2; 5)$, $\mathrm{B}(3; 6; 3)$, $\mathrm{C}(3; 0; 9)$ and $\mathrm{D}(8; -3; -8)$. We admit that points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.
  1. Triangle ABC is: a. isosceles right-angled at A b. isosceles right-angled at B c. isosceles right-angled at C d. equilateral
  2. A Cartesian equation of plane (BCD) is: a. $2x + y + z - 15 = 0$ b. $9x - 5y + 3 = 0$ c. $4x + y + z - 21 = 0$ d. $11x + 5z - 73 = 0$
  3. We admit that plane $(\mathrm{ABC})$ has Cartesian equation $x - 2y - 2z + 15 = 0$. We call H the orthogonal projection of point D onto plane (ABC). We can affirm that: a. $\mathrm{H}(-2; 17; 12)$ b. $\mathrm{H}(3; 7; 2)$ c. $\mathrm{H}(3; 2; 7)$ d. $\mathrm{H}(-15; 1; -1)$
  4. Let the line $\Delta$ with parametric representation $\left\{\begin{array}{l} x = 5 + t \\ y = 3 - t \\ z = -1 + 3t \end{array}\right.$, with $t$ real. Lines (BC) and $\Delta$ are: a. coincident b. strictly parallel c. intersecting d. non-coplanar
  5. We consider the plane $\mathscr{P}$ with Cartesian equation $2x - y + 2z - 6 = 0$. We admit that plane (ABC) has Cartesian equation $x - 2y - 2z + 15 = 0$. We can affirm that: a. planes $\mathscr{P}$ and (ABC) are strictly parallel b. planes $\mathscr{P}$ and (ABC) are intersecting and their intersection is line (AB) c. planes $\mathscr{P}$ and (ABC) are intersecting and their intersection is line (AC) d. planes $\mathscr{P}$ and (ABC) are intersecting and their intersection is line (BC)
Q4 Sequences and series, recurrence and convergence Convergence proof and limit determination View
We consider the sequence $(u_n)$ defined by $u_0 = 5$ and for every natural number $n$, $$u_{n+1} = \frac{1}{2}\left(u_n + \frac{11}{u_n}\right)$$ We admit that the sequence $(u_n)$ is well defined.
Part A - Study of sequence $(u_n)$
  1. Give $u_1$ and $u_2$ in the form of irreducible fractions.
  2. We consider the function $f$ defined on the interval $]0; +\infty[$ by: $$f(x) = \frac{1}{2}\left(x + \frac{11}{x}\right)$$ Prove that function $f$ is increasing on the interval $[\sqrt{11}; +\infty[$.
  3. Prove by induction that for every natural number $n$, we have: $u_n \geqslant u_{n+1} \geqslant \sqrt{11}$.
  4. Deduce that the sequence $(u_n)$ converges to a real limit. We denote this limit by $a$.
  5. After determining and solving an equation of which $a$ is a solution, specify the exact value of $a$.

Part B - Geometric application
For every natural number $n$, we consider a rectangle $R_n$ with area 11 whose width is denoted $\ell_n$ and length $L_n$. The sequence $(L_n)$ is defined by $L_0 = 5$ and, for every natural number $n$, $$L_{n+1} = \frac{L_n + \ell_n}{2}$$
  1. a. Explain why $\ell_0 = 2.2$. b. Establish that for every natural number $n$, $$\ell_n = \frac{11}{L_n}.$$
  2. Verify that the sequence $(L_n)$ corresponds to the sequence $(u_n)$ from Part A.
  3. Show that for every natural number $n$, we have $\ell_n \leqslant \sqrt{11} \leqslant L_n$.
  4. We admit that the sequences $(L_n)$ and $(\ell_n)$ both converge to $\sqrt{11}$. Interpret this result geometrically in the context of Part B.
  5. Here is a script, written in Python language, relating to the sequences studied in this part: \begin{verbatim} def heron(n): L=5 l=2.2 for i in range(n): L = (L+l) / 2 l = 11 / L return round(l, 6), round(L, 6) \end{verbatim} We recall that the Python function round$(\mathrm{x}, \mathrm{k})$ returns a rounded version of the number x with k decimal places. a. If the user types heron(3) in a Python execution console, what output values does he obtain for $\ell$ and $L$? b. Give an interpretation of these two values.