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
2017 amerique-nord

6 maths questions

Q1A Normal Distribution Inverse Normal / Quantile Problem View
Part A
As part of its activity, a company regularly receives quotation requests. The amounts of these quotations are calculated by its secretariat. A statistical study over the past year leads to modelling the amount of quotations by a random variable $X$ which follows the normal distribution with mean $\mu = 2900$ euros and standard deviation $\sigma = 1250$ euros.
  1. If a quotation request received by the company is chosen at random, what is the probability that the quotation amount exceeds 4000 euros?
  2. In order to improve the profitability of its activity, the entrepreneur decides not to follow up on $10\%$ of requests. He discards those with the lowest quotation amounts. What must be the minimum amount of a requested quotation for it to be taken into account? Give this amount to the nearest euro.
Q1B Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
Part B
This same entrepreneur decides to install anti-spam software. This software detects unwanted messages called spam (malicious messages, advertisements, etc.) and moves them to a file called the ``spam folder''. The manufacturer claims that $95\%$ of spam messages are moved. For his part, the entrepreneur knows that $60\%$ of the messages he receives are spam. After installing the software, he observes that $58.6\%$ of messages are moved to the spam folder. For a message chosen at random, we consider the following events:
  • $D$: ``the message is moved'';
  • $S$: ``the message is spam''.

  1. Calculate $P ( S \cap D )$.
  2. A message that is not spam is chosen at random. Show that the probability that it is moved equals 0.04.
  3. A message that is not moved is chosen at random. What is the probability that this message is spam?
  4. For the software chosen by the company, the manufacturer estimates that $2.7\%$ of messages moved to the spam folder are reliable messages. In order to test the software's effectiveness, the secretariat takes the trouble to count the number of reliable messages among the moved messages. It finds 13 reliable messages among the 231 messages moved during one week. Do these results call into question the manufacturer's claim?
Q2 5 marks Differentiating Transcendental Functions Determine parameters from function or curve conditions View
A manufacturer must create a solid wooden gate made to measure for a homeowner. The opening of the enclosure wall (not yet built) cannot exceed 4 meters wide. The gate consists of two panels of width $a$ such that $0 < a \leqslant 2$.
In the chosen model, the closed gate has the shape illustrated in the figure. The sides $[\mathrm{AD}]$ and $[\mathrm{BC}]$ are perpendicular to the threshold [CD] of the gate. Between points A and B, the top of the panels has the shape of a portion of curve. This portion of curve is part of the graph of the function $f$ defined on $[-2 ; 2]$ by:
$$f ( x ) = - \frac { b } { 8 } \left( \mathrm { e } ^ { \frac { x } { b } } + \mathrm { e } ^ { - \frac { x } { b } } \right) + \frac { 9 } { 4 } \quad \text { where } b > 0 .$$
The coordinate system is chosen so that the points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and D have coordinates respectively $(-a ; f(-a))$, $(a ; f(a))$, $(a ; 0)$ and $(-a ; 0)$ and we denote S the vertex of the curve of $f$.
Part A
  1. Show that, for all real $x$ belonging to the interval $[-2 ; 2], f(-x) = f(x)$. What can we deduce about the graph of the function $f$?
  2. Let $f^{\prime}$ denote the derivative function of $f$. Show that, for all real $x$ in the interval $[-2 ; 2]$: $$f^{\prime}(x) = -\frac{1}{8}\left(\mathrm{e}^{\frac{x}{b}} - \mathrm{e}^{-\frac{x}{b}}\right)$$
  3. Draw up the table of variations of the function $f$ on the interval $[-2 ; 2]$ and deduce the coordinates of point S as a function of $b$.

Part B
The height of the wall is $1.5\mathrm{~m}$. We want point S to be 2 m from the ground. We then seek the values of $a$ and $b$.
  1. Justify that $b = 1$.
  2. Show that the equation $f(x) = 1.5$ has a unique solution on the interval $[0 ; 2]$ and deduce an approximate value of $a$ to the nearest hundredth.
  3. In this question, we choose $a = 1.8$ and $b = 1$. The customer decides to automate his gate if the mass of a panel exceeds 60 kg. The density of the wooden planks used to manufacture the panels is equal to $20\mathrm{~kg\cdot m^{-2}}$. What does the customer decide?

Part C
We keep the values $a = 1.8$ and $b = 1$. To cut the panels, the manufacturer pre-cuts planks. He has a choice between two forms of pre-cut planks: either a rectangle OCES, or a trapezoid OCHG. In the second method, the line (GH) is tangent to the graph of the function $f$ at point F with abscissa 1.
Form 1 is the simplest, but visually form 2 seems more economical. Evaluate the savings achieved in terms of wood surface area by choosing form 2 rather than form 1. We recall the formula giving the area of a trapezoid. By denoting $b$ and $B$ respectively the lengths of the small base and the large base of the trapezoid (parallel sides) and $h$ the height of the trapezoid: $$\text{Area} = \frac{b + B}{2} \times h$$
Q3 Sequences and series, recurrence and convergence Summation of sequence terms View
The purpose of this exercise is to study sequences of positive terms whose first term $u_0$ is strictly greater than 1 and possessing the following property: for every natural number $n > 0$, the sum of the first $n$ consecutive terms equals the product of the first $n$ consecutive terms. We admit that such a sequence exists and we denote it $(u_n)$. It therefore satisfies three properties:
  • $u_0 > 1$,
  • for all $n \geqslant 0, u_n \geqslant 0$,
  • for all $n > 0, u_0 + u_1 + \cdots + u_{n-1} = u_0 \times u_1 \times \cdots \times u_{n-1}$.

  1. We choose $u_0 = 3$. Determine $u_1$ and $u_2$.
  2. For every integer $n > 0$, we denote $s_n = u_0 + u_1 + \cdots + u_{n-1} = u_0 \times u_1 \times \cdots \times u_{n-1}$.
    In particular, $s_1 = u_0$. a. Verify that for every integer $n > 0, s_{n+1} = s_n + u_n$ and $s_n > 1$. b. Deduce that for every integer $n > 0$, $$u_n = \frac{s_n}{s_n - 1}$$ c. Show that for all $n \geqslant 0, u_n > 1$.
  3. Using the algorithm opposite, we want to calculate the term $u_n$ for a given value of $n$. a. Copy and complete the processing part of the algorithm:
    Input:Enter $n$
    Enter $u$
    Processing:$s$ takes the value $u$
    For $i$ going from 1 to $n$:
    $u$ takes the value $\ldots$
    $s$ takes the value $\ldots$
    End For
    Output:Display $u$

    b. The table below gives values rounded to the nearest thousandth of $u_n$ for different values of the integer $n$:
    $n$0510203040
    $u_n$31.1401.0791.0431.0301.023

    What conjecture can be made about the convergence of the sequence $(u_n)$?
  4. a. Justify that for every integer $n > 0, s_n > n$. b. Deduce the limit of the sequence $(s_n)$ then that of the sequence $(u_n)$.
Q4 5 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
A homeowner is interested in the shadow cast on his future veranda by the roof of his house when the sun is at its zenith. This veranda is schematized in cavalier perspective in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. The roof of the veranda consists of two triangular faces SEF and SFG.
  • The planes (SOA) and (SOC) are perpendicular.
  • The planes (SOC) and (EAB) are parallel, as are the planes (SOA) and (GCB).
  • The edges [UV) and [EF] of the roofs are parallel.

The point K belongs to the segment [SE], the plane (UVK) separates the veranda into two zones, one illuminated and the other shaded. The plane (UVK) cuts the veranda along the polygonal line KMNP which is the shadow-sun boundary.
  1. Without calculation, justify that: a. the segment $[\mathrm{KM}]$ is parallel to the segment $[\mathrm{UV}]$; b. the segment [NP] is parallel to the segment [UK].
  2. In the rest of the exercise, we place ourselves in the orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. The coordinates of the different points are as follows: $\mathrm{A}(4 ; 0 ; 0)$, \ldots
Q5 Number Theory Modular Arithmetic Computation View
An association assigns each registered child a 6-digit number $c_1 c_2 c_3 c_4 c_5 k$ where:
  • $c_1 c_2$ represents the last two digits of the child's birth year;
  • $c_3 c_4 c_5$ are three digits chosen by the association;
  • $k$ is a check digit computed as follows:

  • we compute the sum $S = c_1 + c_3 + c_5 + a \times (c_2 + c_4)$ where $a$ is an integer between 1 and 9;
  • we perform the Euclidean division of $S$ by 10, the remainder obtained is the check digit $k$.

When an employee enters the 6-digit number of a child, an input error can be detected when the sixth digit is not equal to the check digit calculated from the first five digits.
  1. In this question only, we choose $a = 3$. a. Can the number 111383 be that of a child registered with the association? b. The employee, confusing a brother and sister, exchanges their birth years: 2008 and 2011. Thus, the number $08c_3c_4c_5k$ is transformed into $11c_3c_4c_5k$. Is this error detected thanks to the check digit?
  2. We denote $c_1c_2c_3c_4c_5k$ the number of a child. We seek the values of the integer $a$ for which the check digit systematically detects the typing error when the digits $c_3$ and $c_4$ are swapped. We therefore assume that the digits $c_3$ and $c_4$ are distinct. a. Show that the check digit does not detect the error of swapping the digits $c_3$ and $c_4$ if and only if $(a-1)(c_4 - c_3)$ is congruent to 0 modulo 10. b. Determine the integers $n$ between 0 and 9 for which there exists an integer $p$ between 1 and 9 such that $np \equiv 0 \pmod{10}$. c. Deduce the values of the integer $a$ which allow, thanks to the check digit, to systematically detect the swap of the digits $c_3$ and $c_4$.