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
2019 antilles-guyane

5 maths questions

Q1 Differentiating Transcendental Functions Determine parameters from function or curve conditions View
Exercise 1 (6 points) -- Part A
Let $a$ and $b$ be real numbers. We consider a function $f$ defined on $[ 0 ; + \infty [$ by $$f ( x ) = \frac { a } { 1 + \mathrm { e } ^ { - b x } }$$ The curve $\mathscr { C } _ { f }$ representing the function $f$ in an orthogonal coordinate system is given. The curve $\mathscr { C } _ { f }$ passes through the point $\mathrm { A } ( 0 ; 0.5 )$. The tangent line to the curve $\mathscr { C } _ { f }$ at point A passes through the point B(10; 1).
  1. Justify that $a = 1$.

We then obtain, for all real $x \geqslant 0$, $$f ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - b x } }$$
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  1. It is admitted that the function $f$ is differentiable on $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $f ^ { \prime }$ its derivative function. Verify that, for all real $x \geqslant 0$ $$f ^ { \prime } ( x ) = \frac { b \mathrm { e } ^ { - b x } } { \left( 1 + \mathrm { e } ^ { - b x } \right) ^ { 2 } }$$
  2. Using the data from the problem statement, determine $b$.
Q1B 6 marks Standard Integrals and Reverse Chain Rule Qualitative Properties of Antiderivatives View
Exercise 1 -- Part B
The proportion of individuals who possess a certain type of equipment in a population is modelled by the function $p$ defined on $[ 0 ; + \infty [$ by $$p ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - 0,2 x } } .$$ The real number $x$ represents the time elapsed, in years, since January 1st, 2000. The number $p ( x )$ models the proportion of equipped individuals after $x$ years. Thus, for this model, $p ( 0 )$ is the proportion of equipped individuals on January 1st, 2000 and $p ( 3.5 )$ is the proportion of equipped individuals in the middle of 2003.
  1. What is, for this model, the proportion of equipped individuals on January 1st, 2010? Give a value rounded to the nearest hundredth.
    1. [a.] Determine the direction of variation of the function $p$ on $[ 0 ; + \infty [$.
    2. [b.] Calculate the limit of the function $p$ as $x \to + \infty$.
    3. [c.] Interpret this limit in the context of the exercise.
  3. It is considered that, when the proportion of equipped individuals exceeds $95\%$, the market is saturated. Determine, by explaining the approach, the year in which this occurs.
  4. The average proportion of equipped individuals between 2008 and 2010 is defined by $$m = \frac { 1 } { 2 } \int _ { 8 } ^ { 10 } p ( x ) \mathrm { d } x$$
    1. [a.] Verify that, for all real $x \geqslant 0$, $$p ( x ) = \frac { \mathrm { e } ^ { 0,2 x } } { 1 + \mathrm { e } ^ { 0,2 x } }$$
    2. [b.] Deduce an antiderivative of the function $p$ on $[ 0 ; + \infty [$.
    3. [c.] Determine the exact value of $m$ and its approximation to the nearest hundredth.
Q2 Vectors 3D & Lines Multi-Part 3D Geometry Problem View
Exercise 2 -- Common to all candidates
Alex and Élisa, two drone pilots, are training on a terrain consisting of a flat part bordered by an obstacle. We consider an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ), with one unit corresponding to ten metres. Six points are defined by their coordinates: $$\mathrm { O } ( 0 ; 0 ; 0 ) , \mathrm { P } ( 0 ; 10 ; 0 ) , \mathrm { Q } ( 0 ; 11 ; 1 ) , \mathrm { T } ( 10 ; 11 ; 1 ) , \mathrm { U } ( 10 ; 10 ; 0 ) \text { and } \mathrm { V } ( 10 ; 0 ; 0 )$$ The flat part is delimited by the rectangle OPUV and the obstacle by the rectangle PQTU.
The two drones are assimilable to two points and follow rectilinear trajectories:
  • Alex's drone follows the trajectory carried by the line $( \mathrm { AB } )$ with $\mathrm { A } ( 2 ; 4 ; 0.25 )$ and $\mathrm { B } ( 2 ; 6 ; 0.75 )$;
  • Élisa's drone follows the trajectory carried by the line (CD) with C(4; 6; 0.25) and D(2; 6; 0.25).

Part A: Study of Alex's drone trajectory
  1. Determine a parametric representation of the line ( AB ).
    1. [a.] Justify that the vector $\vec { n } ( 0 ; 1 ; - 1 )$ is a normal vector to the plane (PQU).
    2. [b.] Deduce a Cartesian equation of the plane (PQU).
  3. Prove that the line (AB) and the plane (PQU) intersect at the point I with coordinates $\left( 2 ; \frac { 37 } { 3 } ; \frac { 7 } { 3 } \right)$.
  4. Explain why, following this trajectory, Alex's drone does not encounter the obstacle.

Part B: Minimum distance between the two trajectories
To avoid a collision between their two devices, Alex and Élisa impose a minimum distance of 4 metres between the trajectories of their drones. For this, we consider a point $M$ on the line (AB) and a point $N$ on the line (CD). There then exist two real numbers $a$ and $b$ such that $\overrightarrow { \mathrm { A } M } = a \overrightarrow { \mathrm { AB } }$ and $\overrightarrow { \mathrm { C } N } = b \overrightarrow { \mathrm { CD } }$.
  1. Prove that the coordinates of the vector $\overrightarrow { M N }$ are $( 2 - 2 b ; 2 - 2 a ; - 0.5 a )$.
  2. It is admitted that the lines (AB) and (CD) are not coplanar. It is also admitted that the distance $MN$ is minimal when the line ( $MN$ ) is perpendicular to both the line ( AB ) and the line (CD). Prove then that the distance $MN$ is minimal when $a = \frac { 16 } { 17 }$ and $b = 1$.
  3. Deduce the minimum value of the distance $MN$ and conclude.
Q3 4 marks Complex Numbers Arithmetic True/False or Property Verification Statements View
Exercise 3 (4 points) -- Common to all candidates
For each of the four following statements, indicate whether it is true or false, by justifying the answer. One point is awarded for each correct answer that is properly justified. An unjustified answer earns no points. An absence of an answer is not penalised.
The complex plane is equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). We consider the complex number $c = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } }$ and the points S and T with affixes respectively $c ^ { 2 }$ and $\frac { 1 } { c }$.
  1. Statement 1: The number $c$ can be written as $c = \frac { 1 } { 4 } ( 1 - \mathrm { i } \sqrt { 3 } )$.
  2. Statement 2: For all natural integer $n$, $c ^ { 3 n }$ is a real number.
  3. Statement 3: The points $\mathrm { O }$, $\mathrm { S }$ and T are collinear.
  4. Statement 4: For all non-zero natural integer $n$, $$| c | + \left| c ^ { 2 } \right| + \ldots + \left| c ^ { n } \right| = 1 - \left( \frac { 1 } { 2 } \right) ^ { n } .$$
Q4 6 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
Exercise 4 -- Candidates who have not followed the specialisation course
Part A
During an evening, a television channel broadcast a match. This channel then offered a programme analysing this match. We have the following information:
  • $56\%$ of viewers watched the match;
  • one quarter of viewers who watched the match also watched the programme;
  • $16.2\%$ of viewers watched the programme.
We randomly interview a viewer. We denote the events:
  • $M$: ``the viewer watched the match'';
  • $E$: ``the viewer watched the programme''.
We denote by $x$ the probability that a viewer watched the programme given that they did not watch the match.
  1. Construct a probability tree illustrating the situation.
  2. Determine the probability of $M \cap E$.
    1. [a.] Verify that $p ( E ) = 0.44 x + 0.14$.
    2. [b.] Deduce the value of $x$.
  4. The interviewed viewer did not watch the programme. What is the probability, rounded to $10 ^ { - 2 }$, that they watched the match?

Part B
This institute decides to model the time spent, in hours, by a viewer watching television on the evening of the match, by a random variable $T$ following the normal distribution with mean $\mu = 1.5$ and standard deviation $\sigma = 0.5$.
  1. What is the probability, rounded to $10 ^ { - 3 }$, that a viewer spent between one hour and two hours watching television on the evening of the match?
  2. Determine the approximation to $10 ^ { - 2 }$ of the real number $t$ such that $P ( T \geqslant t ) = 0.066$. Interpret the result.

Part C
The lifetime of an individual set-top box, expressed in years, is modelled by a random variable denoted $S$ which follows an exponential distribution with parameter $\lambda$ strictly positive. The probability density function of $S$ is the function $f$ defined on $[ 0 ; + \infty [$ by $$f ( x ) = \lambda \mathrm { e } ^ { - \lambda x }$$ The polling institute has observed that one quarter of the set-top boxes have a lifetime between one and two years. The factory that manufactures the set-top boxes claims that their average lifetime is greater than three years. Is the factory's claim correct? The answer must be justified.