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
2024 bac-spe-maths__metropole-sept_j1

4 maths questions

Q1 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Consider a cube ABCDEFGH with side length 1.
The point I is the midpoint of segment [BD]. We define the point L such that $\overrightarrow { \mathrm { IL } } = \frac { 3 } { 4 } \overrightarrow { \mathrm { IG } }$. We use the orthonormal coordinate system ( $A ; \overrightarrow { A B } , \overrightarrow { A D } , \overrightarrow { A E }$ ).
  1. a. Specify the coordinates of points $\mathrm { D } , \mathrm { B } , \mathrm { I }$ and G.
    No justification is required. b. Show that point L has coordinates $\left( \frac { 7 } { 8 } ; \frac { 7 } { 8 } ; \frac { 3 } { 4 } \right)$.
  2. Verify that a Cartesian equation of plane (BDG) is $x + y - z - 1 = 0$.
  3. Consider the line $\Delta$ perpendicular to plane (BDG) passing through L. a. Justify that a parametric representation of line $\Delta$ is: $$\left\{ \begin{aligned} x & = \frac { 7 } { 8 } + t \\ y & = \frac { 7 } { 8 } + t \text { where } t \in \mathbb { R } . \\ z & = \frac { 3 } { 4 } - t \end{aligned} \right.$$ b. Show that lines $\Delta$ and (AE) intersect at point K with coordinates $\left( 0 ; 0 ; \frac { 13 } { 8 } \right)$. c. What does point L represent for point K? Justify your answer.
  4. a. Calculate the distance KL. b. We admit that triangle DBG is equilateral. Show that its area equals $\frac { \sqrt { 3 } } { 2 }$. c. Deduce the volume of tetrahedron KDBG.
    We recall that:
    • the volume of a pyramid is given by the formula $V = \frac { 1 } { 3 } \times \mathscr { B } \times h$ where $\mathscr { B }$ is the area of a base and $h$ is the length of the height relative to this base;
    • a tetrahedron is a pyramid with a triangular base.
  5. We denote by $a$ a real number belonging to the interval $] 0 ; + \infty \left[ \right.$ and we note $K _ { a }$ the point with coordinates ( $0 ; 0 ; a$ ). a. Express the volume $V _ { a }$ of pyramid $\mathrm { ABCD } K _ { a }$ as a function of $a$. b. We denote $\Delta _ { a }$ the line with parametric representation $$\left\{ \begin{aligned} x & = t ^ { \prime } \\ y & = t ^ { \prime } \\ z & = - t ^ { \prime } + a \end{aligned} \quad \text { where } t ^ { \prime } \in \mathbb { R } . \right.$$ We call $L _ { a }$ the point of intersection of line $\Delta _ { a }$ with plane (BDG). Show that the coordinates of point $L _ { a }$ are $\left( \frac { a + 1 } { 3 } ; \frac { a + 1 } { 3 } ; \frac { 2 a - 1 } { 3 } \right)$. c. Determine, if it exists, a strictly positive real number $a$ such that tetrahedron $\mathrm { GDB } L _ { a }$ and pyramid $\mathrm { ABCD } K _ { a }$ have the same volume.
Q2 5 marks Integration by Parts Area or Volume Computation Requiring Integration by Parts View
Exercise 2 (5 points)
Part A
A craftsman creates chocolate candies whose shape recalls the profile of the local mountain. The base of such a candy is modeled by the shaded surface defined in an orthonormal coordinate system with unit 1 cm.
This surface is bounded by the x-axis and the graph denoted $\mathscr { C } _ { f }$ of the function $f$ defined on $[ - 1 ; 1 ]$ by: $$f ( x ) = \left( 1 - x ^ { 2 } \right) \mathrm { e } ^ { x } .$$
The objective of this part is to calculate the volume of chocolate needed to manufacture a chocolate candy.
  1. a. Justify that for all $x$ belonging to the interval $[ - 1 ; 1 ]$ we have $f ( x ) \geqslant 0$. b. Show using integration by parts that: $$\int _ { - 1 } ^ { 1 } f ( x ) \mathrm { d } x = 2 \int _ { - 1 } ^ { 1 } x \mathrm { e } ^ { x } \mathrm {~d} x .$$
  2. The volume $V$ of chocolate, in $\mathrm { cm } ^ { 3 }$, needed to manufacture a candy is given by: $$V = 3 \times S$$ where $S$ is the area, in $\mathrm { cm } ^ { 2 }$, of the colored surface. Deduce that this volume, rounded to $0.1 \mathrm {~cm} ^ { 3 }$, equals $4.4 \mathrm {~cm} ^ { 3 }$.

Part B
We now consider the profit realized by the craftsman on the sale of these chocolate candies as a function of the weekly sales volume.
This profit can be modeled by the function $B$ defined on the interval $[ 0.01 ; + \infty [$ by: $$B ( q ) = 8 q ^ { 2 } [ 2 - 3 \ln ( q ) ] - 3 .$$
The profit is expressed in tens of euros and the quantity $q$ in hundreds of candies. We admit that the function $B$ is differentiable on $\left[ 0.01 ; + \infty \left[ \right. \right.$. We denote $B ^ { \prime }$ its derivative function.
  1. a. Determine $\lim _ { q \rightarrow + \infty } B ( q )$. b. Show that, for all $q \geqslant 0.01 , B ^ { \prime } ( q ) = 8 q ( 1 - 6 \ln ( q ) )$. c. Study the sign of $B ^ { \prime } ( q )$, and deduce the direction of variation of $B$ on $[ 0.01 ; + \infty [$. Draw the complete variation table of function $B$. d. What is the maximum profit, to the nearest euro, that the craftsman can expect?
  2. a. Show that the equation $B ( q ) = 10$ has a unique solution $\beta$ on the interval $[1.2; + \infty [$. Give an approximate value of $\beta$ to $10 ^ { - 3 }$ near. b. We admit that the equation $B ( q ) = 10$ has a unique solution $\alpha$ on $[ 0.01 ; 1.2 [$. We are given $\alpha \approx 0.757$. Deduce the minimum and maximum number of chocolate candies to sell to achieve a profit greater than 100 euros.
Q3 5 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
Exercise 3 (5 points)
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. The five questions in this exercise are independent.
  1. Consider a sequence ( $t _ { n }$ ) satisfying the recurrence relation: $$\text { for all natural integer } n , t _ { n + 1 } = - 0.8 t _ { n } + 18 .$$ Statement 1: The sequence ( $w _ { n }$ ) defined for all natural integer $n$ by $w _ { n } = t _ { n } - 10$ is geometric.
  2. Consider a sequence ( $S _ { n }$ ) that satisfies for all non-zero natural integer $n$: $$3 n - 4 \leqslant S _ { n } \leqslant 3 n + 4 .$$ The sequence ( $u _ { n }$ ) is defined, for all non-zero natural integer $n$, by: $u _ { n } = \frac { S _ { n } } { n }$. Statement 2: The sequence ( $u _ { n }$ ) converges.
  3. Consider the sequence $\left( v _ { n } \right)$ defined by: $$v _ { 1 } = 2 \text { and for all natural integer } n \geqslant 1 , v _ { n + 1 } = 2 - \frac { 1 } { v _ { n } } .$$ Statement 3: For all natural integer $n \geqslant 1 , v _ { n } = \frac { n + 1 } { n }$.
  4. Consider the sequence ( $u _ { n }$ ) defined for all natural integer $n$ by $u _ { n } = \mathrm { e } ^ { n } - n$. Statement 4: The sequence $\left( u _ { n } \right)$ converges.
  5. Consider the sequence ( $u _ { n }$ ) defined using the script written below in Python language, which returns the value of $u _ { n }$. \begin{verbatim} def u(n) : valeur = 2 for k in range(n) : valeur = 0.5 * (valeur + 2/valeur) return valeur \end{verbatim} We admit that ( $u _ { n }$ ) is decreasing and satisfies for all natural integer $n$: $$\sqrt { 2 } \leqslant u _ { n } \leqslant 2 .$$ Statement 5: The sequence $\left( u _ { n } \right)$ converges to $\sqrt { 2 }$.
Q4 4 marks Binomial Distribution Justify Binomial Model and State Parameters View
Exercise 4 (4 points)
The two parts are independent.
A laboratory manufactures a medicine packaged in the form of tablets.
Part A
A quality control, concerning the mass of tablets, showed that $2 \%$ of tablets have non-conforming mass. These tablets are packaged in boxes of 100 chosen at random from the production line. We admit that the conformity of a tablet is independent of that of the others.
We denote by $N$ the random variable that associates to each box of 100 tablets the number of non-conforming tablets in this box.
  1. Justify that the random variable $N$ follows a binomial distribution whose parameters you will specify.
  2. Calculate the expectation of $N$ and give an interpretation in the context of the exercise.
  3. Results will be rounded to $10 ^ { - 3 }$ near. a. Calculate the probability that a box contains exactly three non-conforming tablets. b. Calculate the probability that a box contains at least 95 conforming tablets.
  4. The laboratory director wants to modify the number of tablets per box to be able to state: ``The probability that a box contains only conforming tablets is greater than 0.5''. What is the maximum number of tablets a box should contain to meet this criterion? Justify.

Part B
We admit that the masses of tablets are independent of one another. We take a sample of 100 tablets and we denote $M _ { i }$, for $i$ natural integer between 1 and 100, the random variable that gives the mass in grams of the $i$-th tablet sampled. We consider the random variable $S$ defined by: $$S = M _ { 1 } + M _ { 2 } + \ldots + M _ { 100 } .$$ We admit that the random variables $M _ { 1 } , M _ { 2 } , \ldots , M _ { 100 }$ follow the same probability distribution with expectation $\mu = 2$ and standard deviation $\sigma$.
  1. Determine $E ( S )$ and interpret the result in the context of the exercise.
  2. We denote by $s$ the standard deviation of the random variable $S$. Show that: $s = 10 \sigma$.
  3. We wish that the total mass, in grams, of the tablets contained in a box be strictly between 199 and 201 with a probability at least equal to 0.9. a. Show that this condition is equivalent to: $$P ( | S - 200 | \geqslant 1 ) \leqslant 0.1 .$$ b. Deduce the maximum value of $\sigma$ which allows, using the Bienaymé--Chebyshev inequality, to ensure this condition.