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
2015 centres-etrangers

9 maths questions

Q1A1 Modelling and Hypothesis Testing View
The supplier claims that, among the high-end padlocks, there are no more than $3\%$ of defective padlocks in his production. The manager of the hardware store wishes to verify the validity of this claim in his stock; for this purpose, he takes a random sample of 500 high-end padlocks, and finds 19 that are defective.
Does this check call into question the fact that the stock contains no more than $3\%$ of defective padlocks?
For this, you may use an asymptotic fluctuation interval at the $95\%$ threshold.
Q1A2 Confidence intervals Compute confidence interval for a proportion (estimation) View
The store manager wishes to estimate the proportion of defective padlocks in his stock of budget padlocks. For this, he takes a random sample of 500 budget padlocks, among which 39 prove to be defective.
Give a confidence interval for this proportion at the $95\%$ confidence level.
Q1B1 Normal Distribution Direct Probability Calculation from Given Normal Distribution View
According to a statistical study conducted over several months, it is assumed that the number $X$ of budget padlocks sold per month in the hardware store can be modeled by a random variable that follows a normal distribution with mean $\mu = 750$ and standard deviation $\sigma = 25$.
Calculate $P(725 \leqslant X \leqslant 775)$.
Q1B2 Normal Distribution Inverse Normal / Quantile Problem View
According to a statistical study conducted over several months, it is assumed that the number $X$ of budget padlocks sold per month in the hardware store can be modeled by a random variable that follows a normal distribution with mean $\mu = 750$ and standard deviation $\sigma = 25$.
The store manager wants to know the number $n$ of budget padlocks he must have in stock at the beginning of the month, so that the probability of running out of stock during the month is less than 0.05. The stock is not replenished during the month.
Determine the smallest integer value of $n$ satisfying this condition.
Q1C Conditional Probability Reverse Inference / Determining Unknown Quantities from Conditional Probability Constraints View
It is now assumed that, in the store:
  • $80\%$ of the padlocks offered for sale are budget models, the others are high-end;
  • $3\%$ of high-end padlocks are defective;
  • $7\%$ of padlocks are defective.

A padlock is randomly selected from the store. We denote:
  • $p$ the probability that a budget padlock is defective;
  • $H$ the event: ``the selected padlock is high-end'';
  • $D$ the event: ``the selected padlock is defective''.

  1. Represent the situation using a probability tree.
  2. Express $P(D)$ as a function of $p$. Deduce the value of the real number $p$.

Is the result obtained consistent with that of question A-2?
3. The selected padlock is in good condition. Determine the probability that it is a high-end padlock.
Q2 Complex Numbers Argand & Loci Intersection of Loci and Simultaneous Geometric Conditions View
For each of the four following statements, indicate whether it is true or false by justifying your answer.
One point is awarded for each correct answer with proper justification. An answer without justification is not taken into account. An absence of answer is not penalized.
  1. In the plane with an orthonormal coordinate system, let $S$ denote the set of points $M$ whose affix $z$ satisfies the two conditions: $$|z - 1| = |z - \mathrm{i}| \quad \text{and} \quad |z - 3 - 2\mathrm{i}| \leqslant 2.$$ In the figure below, we have represented the circle with center at the point with coordinates $(3;2)$ and radius 2, and the line with equation $y = x$. This line intersects the circle at two points A and B.
    Statement 1: the set $S$ is the segment $[AB]$.
  2. Statement 2: the complex number $(\sqrt{3} + \mathrm{i})^{1515}$ is a real number.
  3. For questions 3 and 4, consider the points $\mathrm{E}(2; 1; -3)$, $\mathrm{F}(1; -1; 2)$ and $\mathrm{G}(-1; 3; 1)$ whose coordinates are defined in an orthonormal coordinate system of space.
    Statement 3: a parametric representation of the line $(EF)$ is given by: $$\left\{\begin{array}{rlr} x & = & 2t \\ y & = & -3 + 4t, \quad t \in \mathbb{R} \\ z & = 7 - 10t \end{array}\right.$$
  4. Statement 4: a measure in degrees of the geometric angle $\widehat{\mathrm{FEG}}$, rounded to the nearest degree, is $50°$.
Q3 Sequences and series, recurrence and convergence Monotonicity and boundedness analysis View
Let $a$ be a fixed non-zero real number. The purpose of this exercise is to study the sequence $(u_n)$ defined by: $$u_0 = a \quad \text{and, for all } n \text{ in } \mathbb{N}, \quad u_{n+1} = \mathrm{e}^{2u_n} - \mathrm{e}^{u_n}.$$ Note that this equality can also be written: $u_{n+1} = e^{u_n}(\mathrm{e}^{u_n} - 1)$.
  1. Let $g$ be the function defined for all real $x$ by $$g(x) = \mathrm{e}^{2x} - \mathrm{e}^x - x.$$ a) Calculate $g'(x)$ and prove that, for all real $x$: $g'(x) = (\mathrm{e}^x - 1)(2\mathrm{e}^x + 1)$. b) Determine the variations of the function $g$ and give the value of its minimum. c) By noting that $u_{n+1} - u_n = g(u_n)$, study the direction of variation of the sequence $(u_n)$.
  2. In this question, we assume that $a \leqslant 0$. a) Prove by induction that, for all natural integer $n$, $u_n \leqslant 0$. b) Deduce from the previous questions that the sequence $(u_n)$ is convergent. c) In the case where $a$ equals 0, give the limit of the sequence $(u_n)$.
  3. In this question, we assume that $a > 0$.
    Since the sequence $(u_n)$ is increasing, question 1 allows us to assert that, for all natural integer $n$, $u_n \geqslant a$. a) Prove that, for all natural integer $n$, we have: $u_{n+1} - u_n \geqslant g(a)$. b) Prove by induction that, for all natural integer $n$, we have: $$u_n \geqslant a + n \times g(a).$$ c) Determine the limit of the sequence $(u_n)$.
  4. In this question, we take $a = 0.02$.
    The following algorithm is intended to determine the smallest integer $n$ such that $u_n > M$, where $M$ denotes a positive real number. This algorithm is incomplete.
    Variables$n$ is an integer, $u$ and $M$ are two real numbers
    Initialization\begin{tabular}{l} $u$ takes the value 0.02
    $n$ takes the value 0
    Enter the value of $M$
    \hline Processing & While $\cdots$ & $\ldots$ & $\ldots$ End while & \end{tabular}
    a) On your paper, rewrite the ``Processing'' part by completing it. b) Using a calculator, determine the value that this algorithm will display if $M = 60$.
Q4A Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View
The manufacturer of padlocks of the brand ``K'' wishes to print a logo for his company. This logo has the shape of a stylized capital letter K, inscribed in a square ABCD, with side length one unit of length, and respecting the following conditions C1 and C2:
  • Condition C1: the letter K must consist of three lines:
  • one of the lines is the segment $[AD]$;
  • a second line has endpoints at point A and a point E on segment $[DC]$;
  • the third line has endpoint at point B and a point G located on the second line.
  • Condition C2: the area of each of the three surfaces delimited by the three lines drawn in the square must be between 0.3 and 0.4, with the unit of area being that of the square. These areas are denoted $r$, $s$, $t$.

We place ourselves in the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD})$.
Part A: study of Proposal A
In this proposal, the three lines are segments and the three areas are equal: $r = s = t = \frac{1}{3}$. Determine the coordinates of points E and G.
Q4B Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The manufacturer of padlocks of the brand ``K'' wishes to print a logo for his company. This logo has the shape of a stylized capital letter K, inscribed in a square ABCD, with side length one unit of length. We place ourselves in the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD})$.
Part B: study of Proposal B
This proposal is characterized by the following two conditions:
  • the line with endpoints A and E is a portion of the graph of the function $f$ defined for all real $x \geqslant 0$ by: $f(x) = \ln(2x + 1)$;
  • the line with endpoints B and G is a portion of the graph of the function $g$ defined for all real $x > 0$ by: $g(x) = k\left(\frac{1 - x}{x}\right)$, where $k$ is a positive real number to be determined.

  1. a) Determine the abscissa of point E. b) Determine the value of the real number $k$, knowing that the abscissa of point G is equal to 0.5.
  2. a) Prove that the function $f$ has as a primitive the function $F$ defined for all real $x \geqslant 0$ by: $$F(x) = (x + 0.5) \times \ln(2x + 1) - x.$$ b) Prove that $r = \frac{\mathrm{e}}{2} - 1$.
  3. Determine a primitive $G$ of the function $g$ on the interval $]0; +\infty[$.
  4. It is admitted that the previous results allow us to establish that $s = [\ln(2)]^2 + \frac{\ln(2) - 1}{2}$. Does Proposal B satisfy the conditions imposed by the manufacturer?