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
2023 bac-spe-maths__europe_j1

4 maths questions

Q1 Differentiating Transcendental Functions Full function study with transcendental functions View
Exercise 1
Part A
We consider the function $g$ defined on the interval $] 0 ; + \infty [$ by
$$g ( x ) = \ln \left( x ^ { 2 } \right) + x - 2$$
  1. Determine the limits of the function $g$ at the boundaries of its domain.
  2. It is admitted that the function $g$ is differentiable on the interval $] 0 ; + \infty [$. Study the variations of the function $g$ on the interval $] 0 ; + \infty [$.
  3. a. Prove that there exists a unique strictly positive real number $\alpha$ such that $g ( \alpha ) = 0$. b. Determine an interval containing $\alpha$ with amplitude $10 ^ { - 2 }$.
  4. Deduce the sign table of the function $g$ on the interval $] 0 ; + \infty [$.

Part B
We consider the function $f$ defined on the interval $] 0 ; + \infty [$ by :
$$f ( x ) = \frac { ( x - 2 ) } { x } \ln ( x ) .$$
We denote $\mathscr { C } _ { f }$ its representative curve in an orthonormal coordinate system.
  1. a. Determine the limit of the function $f$ at 0. b. Interpret the result graphically.
  2. Determine the limit of the function $f$ at $+ \infty$.
  3. It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$.

Show that for every strictly positive real number $x$, we have $f ^ { \prime } ( x ) = \frac { g ( x ) } { x ^ { 2 } }$.
4. Deduce the variations of the function $f$ on the interval $] 0 ; + \infty [$.
Part C
Study the relative position of the curve $\mathscr { C } _ { f }$ and the representative curve of the natural logarithm function on the interval $] 0 ; + \infty [$.
Q2 5 marks Conditional Probability Markov Chain / Day-to-Day Transition Probabilities View
Exercise 2
5 points With a concern for environmental preservation, Mr. Durand decides to go to work each morning using his bicycle or public transport. If he chooses to take public transport one morning, he takes public transport again the next day with a probability equal to 0.8. If he uses his bicycle one morning, he uses his bicycle again the next day with a probability equal to 0.4. For every non-zero natural number $n$, we denote:
  • $T _ { n }$ the event ``Mr. Durand uses public transport on the $n$-th day''
  • $V _ { n }$ the event ``Mr. Durand uses his bicycle on the $n$-th day''
  • We denote $p _ { n }$ the probability of the event $T _ { n }$,

On the first morning, he decides to use public transport. Thus, the probability of the event $T _ { 1 }$ is $p _ { 1 } = 1$.
  1. Copy and complete the probability tree below representing the situation for the $2 ^ { \mathrm { nd } }$ and $3 ^ { \mathrm { rd } }$ days.
  2. Calculate $p _ { 3 }$
  3. On the $3 ^ { \mathrm { rd } }$ day, Mr. Durand uses his bicycle. Calculate the probability that he took public transport the day before.
  4. Copy and complete the probability tree below representing the situation for the $n$-th and ( $n + 1$ )-th days.
  5. Show that, for every non-zero natural number $n$, $p _ { n + 1 } = 0,2 p _ { n } + 0,6$.
  6. Show by induction that, for every non-zero natural number $n$, we have $$p _ { n } = 0,75 + 0,25 \times 0,2 ^ { n - 1 } .$$
  7. Determine the limit of the sequence ( $p _ { n }$ ) and interpret the result in the context of the exercise.
Q3 Applied differentiation MCQ on derivative and graph interpretation View
Exercise 3
This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. The candidate will indicate on their answer sheet the number of the question and the chosen answer. No justification is required.
A wrong answer, multiple answers, or the absence of an answer to a question neither awards nor deducts points. The five questions are independent. Throughout the exercise, $\mathbb { R }$ denotes the set of real numbers.
  1. A primitive of the function $f$, defined on $\mathbb { R }$ by $f ( x ) = x \mathrm { e } ^ { x }$, is the function $F$, defined on $\mathbb { R }$, by: a. $F ( x ) = \frac { x ^ { 2 } } { 2 } \mathrm { e } ^ { x }$ b. $F ( x ) = ( x - 1 ) \mathrm { e } ^ { x }$ c. $F ( x ) = ( x + 1 ) \mathrm { e } ^ { x }$ d. $F ( x ) = x ^ { 2 } \mathrm { e } ^ { x ^ { 2 } }$
  2. We consider the function $g$ defined by $g ( x ) = \ln \left( \frac { x - 1 } { 2 x + 4 } \right)$. The function $g$ is defined on: a. $\mathbb { R }$ b. $] - \infty ; - 2 [ \cup ] 1 ; + \infty [$ c. $] - 2 ; + \infty [$ d. $] - 2 ; 1 [$
  3. The function $h$ defined on $\mathbb { R }$ by $h ( x ) = ( x + 1 ) \mathrm { e } ^ { x }$ is: a. concave on $\mathbb { R }$ b. convex on $\mathbb { R }$ c. convex on $] - \infty ; - 3 ]$ and concave on $[-3; + \infty [$ d. concave on $] - \infty ; - 3 ]$ and convex on $[-3; + \infty [$
  4. A sequence ( $u _ { n }$ ) is bounded below by 3 and converges to a real number $\ell$. We can affirm that: a. $\ell = 3$ b. $\ell \geqslant 3$ c. The sequence ( $u _ { n }$ ) is decreasing. d. The sequence ( $u _ { n }$ ) is constant from a certain rank onwards.
  5. The sequence ( $w _ { n }$ ) is defined by $w _ { 1 } = 2$ and for every strictly positive natural number $n$, $w _ { n + 1 } = \frac { 1 } { n } w _ { n }$. a. The sequence ( $w _ { n }$ ) is geometric b. The sequence ( $w _ { n }$ ) does not have a limit c. $w _ { 5 } = \frac { 1 } { 15 }$ d. The sequence ( $w _ { n }$ ) converges to 0.
Q4 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 4
In space equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ), we consider the points
$$\mathrm { A } ( - 1 ; - 3 ; 2 ) , \quad \mathrm { B } ( 3 ; - 2 ; 6 ) \quad \text { and } \quad \mathrm { C } ( 1 ; 2 ; - 4 ) .$$
  1. Prove that the points $\mathrm { A } , \mathrm { B }$ and C define a plane which we will denote $\mathscr { P }$.
  2. a. Show that the vector $\vec { n } \left( \begin{array} { c } 13 \\ - 16 \\ - 9 \end{array} \right)$ is normal to the plane $\mathscr { P }$. b. Prove that a Cartesian equation of the plane $\mathscr{P}$ is $13 x - 16 y - 9 z - 17 = 0$.

We denote $\mathscr { D }$ the line passing through the point $\mathrm { F } ( 15 ; - 16 ; - 8 )$ and perpendicular to the plane $\mathscr { P }$.
3. Give a parametric representation of the line $\mathscr { D }$.
4. We call E the point of intersection of the line $\mathscr { D }$ and the plane $\mathscr { P }$. Prove that the point E has coordinates $( 2 ; 0 ; 1 )$.
5. Determine the exact value of the distance from point F to the plane $\mathscr { P }$. 6. Determine the coordinates of the point(s) on the line $\mathscr { D }$ whose distance to the plane $\mathscr { P }$ is equal to half the distance from point F to the plane $\mathscr { P }$.