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
2021 bac-spe-maths__metropole_j1

5 maths questions

QA 5 marks Sequences and series, recurrence and convergence Conjecture from numerical data or computation View
Main topics covered: Numerical sequences; proof by induction; geometric sequences.
The sequence $(u_{n})$ is defined on $\mathbb{N}$ by $u_{0} = 1$ and for every natural number $n$, $$u_{n+1} = \frac{3}{4}u_{n} + \frac{1}{4}n + 1.$$
  1. Calculate, showing the calculations in detail, $u_{1}$ and $u_{2}$ in the form of irreducible fractions.

The extract, reproduced below, from a spreadsheet created with a spreadsheet application presents the values of the first terms of the sequence $(u_{n})$.
AB
1$n$$u_{n}$
201
311.75
422.5625
533.421875
644.31640625

    \setcounter{enumi}{1}
  1. a. What formula, then extended downward, can be written in cell B3 of the spreadsheet to obtain the successive terms of $(u_{n})$ in column B? b. Conjecture the direction of variation of the sequence $(u_{n})$.
  2. a. Prove by induction that, for every natural number $n$, we have: $n \leqslant u_{n} \leqslant n+1$. b. Deduce from this, justifying the answer, the direction of variation and the limit of the sequence $(u_{n})$. c. Prove that: $$\lim_{n \rightarrow +\infty} \frac{u_{n}}{n} = 1$$
  3. We denote by $(v_{n})$ the sequence defined on $\mathbb{N}$ by $v_{n} = u_{n} - n$ a. Prove that the sequence $(v_{n})$ is geometric with common ratio $\frac{3}{4}$. b. Deduce from this that, for every natural number $n$, we have: $u_{n} = \left(\frac{3}{4}\right)^{n} + n$.
QB 5 marks Differentiating Transcendental Functions Full function study with transcendental functions View
Main topics covered: Natural logarithm function; convexity
We consider the function $f$ defined on the interval $]0;+\infty[$ by: $$f(x) = x + 4 - 4\ln(x) - \frac{3}{x}$$ where ln denotes the natural logarithm function. We denote $\mathscr{C}$ the graphical representation of $f$ in an orthonormal coordinate system.
  1. Determine the limit of the function $f$ at $+\infty$.
  2. We assume that the function $f$ is differentiable on $]0;+\infty[$ and we denote $f^{\prime}$ its derivative function.
    Prove that, for every real number $x > 0$, we have: $$f^{\prime}(x) = \frac{x^{2} - 4x + 3}{x^{2}}$$
  3. a. Give the variation table of the function $f$ on the interval $]0;+\infty[$.
    The exact values of the extrema and the limits of $f$ at 0 and at $+\infty$ will be shown. We will assume that $\lim_{x \rightarrow 0} f(x) = -\infty$. b. By simply reading the variation table, specify the number of solutions of the equation $f(x) = \frac{5}{3}$.
  4. Study the convexity of the function $f$, that is, specify the parts of the interval $]0;+\infty[$ on which $f$ is convex, and those on which $f$ is concave. We will justify that the curve $\mathscr{C}$ admits a unique inflection point, whose coordinates we will specify.
Q1 5 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
In a statistics school, after reviewing candidate files, recruitment is done in two ways:
  • $10\%$ of candidates are selected based on their file. These candidates must then take an oral examination after which $60\%$ of them are finally admitted to the school.
  • Candidates who were not selected based on their file take a written examination after which $20\%$ of them are admitted to the school.

Part 1
A candidate for this recruitment competition is chosen at random. We denote:
  • $D$ the event ``the candidate was selected based on their file'';
  • $A$ the event ``the candidate was admitted to the school'';
  • $\bar{D}$ and $\bar{A}$ the complementary events of events $D$ and $A$ respectively.

  1. Represent the situation with a probability tree.
  2. Calculate the probability that the candidate is selected based on their file and admitted to the school.
  3. Show that the probability of event $A$ is equal to 0.24.
  4. A candidate admitted to the school is chosen at random. What is the probability that their file was not selected?

Part 2
  1. We assume that the probability for a candidate to be admitted to the school is equal to 0.24.
    We consider a sample of seven candidates chosen at random, treating this choice as a random draw with replacement. We denote by $X$ the random variable counting the candidates admitted to the school among the seven drawn. a. We assume that the random variable $X$ follows a binomial distribution. What are the parameters of this distribution? b. Calculate the probability that only one of the seven candidates drawn is admitted to the school. Give an answer rounded to the nearest hundredth. c. Calculate the probability that at least two of the seven candidates drawn are admitted to this school. Give an answer rounded to the nearest hundredth.
  2. A secondary school presents $n$ candidates for recruitment in this school, where $n$ is a non-zero natural number. We assume that the probability for any candidate from the secondary school to be admitted to the school is equal to 0.24 and that the results of the candidates are independent of each other. a. Give the expression, as a function of $n$, of the probability that no candidate from this secondary school is admitted to the school. b. From what value of the integer $n$ is the probability that at least one student from this secondary school is admitted to the school greater than or equal to 0.99?
Q2 Stationary points and optimisation Construct or complete a full variation table View
Let $f$ be the function defined on the interval $]0;+\infty[$ by: $$f(x) = \frac{\mathrm{e}^{x}}{x}.$$ We denote $\mathscr{C}_{f}$ the representative curve of the function $f$ in an orthonormal coordinate system.
  1. a. Specify the limit of the function $f$ at $+\infty$. b. Justify that the $y$-axis is an asymptote to the curve $\mathscr{C}_{f}$.
  2. Show that, for every real number $x$ in the interval $]0;+\infty[$, we have: $$f^{\prime}(x) = \frac{\mathrm{e}^{x}(x-1)}{x^{2}}$$ where $f^{\prime}$ denotes the derivative function of the function $f$.
  3. Determine the variations of the function $f$ on the interval $]0;+\infty[$. A variation table of the function $f$ will be established in which the limits appear.
  4. Let $m$ be a real number. Specify, depending on the values of the real number $m$, the number of solutions of the equation $f(x) = m$.
  5. We denote $\Delta$ the line with equation $y = -x$.
    We denote A a possible point of $\mathscr{C}_{f}$ with abscissa $a$ at which the tangent to the curve $\mathscr{C}_{f}$ is parallel to the line $\Delta$. a. Show that $a$ is a solution of the equation $\mathrm{e}^{x}(x-1) + x^{2} = 0$.
    We denote $g$ the function defined on $[0;+\infty[$ by $g(x) = \mathrm{e}^{x}(x-1) + x^{2}$. We assume that the function $g$ is differentiable and we denote $g^{\prime}$ its derivative function. b. Calculate $g^{\prime}(x)$ for every real number $x$ in the interval $[0;+\infty[$, then establish the variation table of $g$ on $[0;+\infty[$. c. Show that there exists a unique point $A$ at which the tangent to $\mathscr{C}_{f}$ is parallel to the line $\Delta$.
Q3 Vectors: Lines & Planes MCQ: Identify Correct Equation or Representation View
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct.
A correct answer earns one point. An incorrect answer, a multiple answer, or the absence of an answer to a question earns neither points nor deducts points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
SABCD is a regular pyramid with square base ABCD in which all edges have the same length. The point I is the center of the square ABCD. We assume that: $\mathrm{IC} = \mathrm{IB} = \mathrm{IS} = 1$. The points K, L and M are the midpoints of edges [SD], [SC] and [SB] respectively.
  1. The following lines are not coplanar: a. (DK) and (SD) b. (AS) and (IC) c. (AC) and (SB) d. (LM) and (AD)

For the following questions, we place ourselves in the orthonormal coordinate system of space $(\mathrm{I}; \overrightarrow{\mathrm{IC}}, \overrightarrow{\mathrm{IB}}, \overrightarrow{\mathrm{IS}})$. In this coordinate system, we are given the coordinates of the following points: $$\mathrm{I}(0;0;0) \quad ; \quad \mathrm{A}(-1;0;0) \quad ; \quad \mathrm{B}(0;1;0) \quad ; \quad \mathrm{C}(1;0;0) \quad ; \quad \mathrm{D}(0;-1;0) \quad ; \quad \mathrm{S}(0;0;1).$$
    \setcounter{enumi}{1}
  1. The coordinates of the midpoint N of [KL] are: a. $\left(\frac{1}{4};\frac{1}{4};\frac{1}{4}\right)$ b. $\left(\frac{1}{4};-\frac{1}{4};\frac{1}{2}\right)$ c. $\left(-\frac{1}{4};\frac{1}{4};\frac{1}{2}\right)$ d. $\left(-\frac{1}{2};\frac{1}{2};1\right)$
  2. The coordinates of the vector $\overrightarrow{\mathrm{AS}}$ are: a. $\left(\begin{array}{l}1\\1\\0\end{array}\right)$ b. $\left(\begin{array}{l}1\\0\\1\end{array}\right)$ c. $\left(\begin{array}{c}2\\1\\-1\end{array}\right)$ d. $\left(\begin{array}{l}1\\1\\1\end{array}\right)$
  3. A parametric representation of the line (AS) is: a. $\left\{\begin{array}{rl}x &= -1-t\\y &= t\\z &= -t\end{array}\right.$ b. $\left\{\begin{aligned}x =& -1+2t\\y =& 0\\z =& 1+2t\end{aligned}\right.$ c. $\left\{\begin{aligned}x &= t\\y &= 0\\z &= 1+t\end{aligned}\right.$ d. $\left\{\begin{aligned}x &= -1-t\\y &= 1+t\\z &= 1-t\end{aligned}\right. (t \in \mathbb{R})$
  4. A Cartesian equation of the plane (SCB) is: a. $y+z-1=0$ b. $x+y+z-1=0$ c. $x-y+z=0$ d. $x+z-1=0$