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
2022 bac-spe-maths__amerique-nord_j2

4 maths questions

Q1 7 marks Conditional Probability Markov Chain / Day-to-Day Transition Probabilities View
Exercise 1 (7 points) Theme: probabilities, sequences In a tourist region, a company offers a bicycle rental service for the day. The company has two distinct rental points, point A and point B. Bicycles can be borrowed and returned indifferently at either of the two rental points. It is assumed that the total number of bicycles is constant and that every morning, when the service opens, each bicycle is at point A or point B. According to a statistical study:
  • If a bicycle is at point A one morning, the probability that it is at point A the next morning is equal to 0.84;
  • If a bicycle is at point B one morning, the probability that it is at point B the next morning is equal to 0.76.
When the service opens on the first morning, the company has placed half of its bicycles at point A and the other half at point B. We consider a bicycle from the company chosen at random. For every positive integer $n$, the following events are defined:
  • $A _ { n }$ : ``the bicycle is at point A on the $n$-th morning''
  • $B _ { n }$ : ``the bicycle is at point B on the $n$-th morning''.
For every positive integer $n$, we denote by $a _ { n }$ the probability of event $A _ { n }$ and by $b _ { n }$ the probability of event $B _ { n }$. Thus $a _ { 1 } = 0.5$ and $b _ { 1 } = 0.5$.
  1. Copy and complete the weighted tree that models the situation for the first two mornings.
  2. a. Calculate $a _ { 2 }$. b. The bicycle is at point A on the second morning. Calculate the probability that it was at point B on the first morning. The probability will be rounded to the nearest thousandth.
  3. a. Copy and complete the weighted tree that models the situation for the $n$-th and $(n + 1)$-th mornings. b. Justify that for every positive integer $n$, $a _ { n + 1 } = 0.6 a _ { n } + 0.24$.
  4. Show by induction that, for every positive integer $n$, $a _ { n } = 0.6 - 0.1 \times 0.6 ^ { n - 1 }$.
  5. Determine the limit of the sequence $(a _ { n })$ and interpret this limit in the context of the exercise.
  6. Determine the smallest positive integer $n$ such that $a _ { n } \geqslant 0.599$ and interpret the result obtained in the context of the exercise.
Q2 7 marks Differentiating Transcendental Functions Full function study with transcendental functions View
Exercise 2 (7 points) Theme: functions, exponential function
Part A
Let $p$ be the function defined on the interval $[ - 3 ; 4 ]$ by: $$p ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 5 x + 1$$
  1. Determine the variations of the function $p$ on the interval $[ - 3 ; 4 ]$.
  2. Justify that the equation $p ( x ) = 0$ admits in the interval $[-3;4]$ a unique solution which will be denoted $\alpha$.
  3. Determine an approximate value of the real number $\alpha$ to the nearest tenth.
  4. Give the sign table of the function $p$ on the interval $[ - 3 ; 4 ]$.

Part B
Let $f$ be the function defined on the interval $[ - 3 ; 4 ]$ by: $$f ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + x ^ { 2 } }$$ We denote by $\mathscr { C } _ { f }$ its representative curve in an orthogonal coordinate system.
  1. a. Determine the derivative of the function $f$ on the interval $[ - 3 ; 4 ]$. b. Justify that the curve $\mathscr { C } _ { f }$ admits a horizontal tangent at the point with abscissa 1.
  2. The designers of a water slide use the curve $\mathscr { C } _ { f }$ as the profile of a water slide. They estimate that the water slide provides good sensations if the profile has at least two inflection points. a. Based on the graph, does the water slide seem to provide good sensations? Argue. b. It is admitted that the function $f ^ { \prime \prime }$, the second derivative of the function $f$, has the following expression for every real $x$ in the interval $[ - 3 ; 4 ]$: $$f ^ { \prime \prime } ( x ) = \frac { p ( x ) ( x - 1 ) \mathrm { e } ^ { x } } { \left( 1 + x ^ { 2 } \right) ^ { 3 } }$$ where $p$ is the function defined in Part A. Using the above expression for $f ^ { \prime \prime }$, answer the question: ``does the water slide provide good sensations?''. Justify.
Q3 7 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 3 (7 points) Theme: geometry in space An exhibition of contemporary art takes place in a room in the shape of a rectangular parallelepiped with width 6 m, length 8 m and height 4 m. It is represented by the rectangular parallelepiped OBCDEFGH where $\mathrm { OB } = 6 \mathrm {~m} , \mathrm { OD } = 8 \mathrm {~m}$ and $\mathrm { OE } = 4 \mathrm {~m}$. We use the orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$ such that $\vec { \imath } = \frac { 1 } { 6 } \overrightarrow { \mathrm { OB } } , \vec { \jmath } = \frac { 1 } { 8 } \overrightarrow { \mathrm { OD } }$ and $\vec { k } = \frac { 1 } { 4 } \overrightarrow { \mathrm { OE } }$. In this coordinate system we have, in particular $\mathrm { C } ( 6 ; 8 ; 0 ) , \mathrm { F } ( 6 ; 0 ; 4 )$ and $\mathrm { G } ( 6 ; 8 ; 4 )$. One of the exhibited works is a glass triangle represented by triangle ART which has vertices $\mathrm { A } ( 6 ; 0 ; 2 )$, $\mathrm { R } ( 6 ; 3 ; 4 )$ and $\mathrm { T } ( 3 ; 0 ; 4 )$. Finally, S is the point with coordinates $\left( 3 ; \frac { 5 } { 2 } ; 0 \right)$.
  1. a. Verify that triangle ART is isosceles with apex A. b. Calculate the dot product $\overrightarrow { \mathrm { AR } } \cdot \overrightarrow { \mathrm { AT } }$. c. Deduce an approximate value to 0.1 degree of the angle $\widehat { \mathrm { RAT } }$.
  2. a. Justify that the vector $\vec { n } \left( \begin{array} { c } 2 \\ - 2 \\ 3 \end{array} \right)$ is a normal vector to the plane (ART). b. Deduce a Cartesian equation of the plane (ART).
  3. A laser beam directed towards triangle ART is emitted from the floor from point S. It is admitted that this beam is perpendicular to the plane (ART). a. Let $\Delta$ be the line perpendicular to the plane (ART) and passing through point S. Justify that the system below is a parametric representation of the line $\Delta$: $$\left\{ \begin{aligned} x & = 3 + 2 k \\ y & = \frac { 5 } { 2 } - 2 k , \text { with } k \in \mathbb { R } . \\ z & = 3 k \end{aligned} \right.$$ b. Let L be the point of intersection of the line $\Delta$ with the plane (ART). Prove that L has coordinates $\left( 5 ; \frac { 1 } { 2 } ; 3 \right)$.
  4. The artist installs a rail represented by the segment [DK] where K is the midpoint of segment [EH]. On this rail, he positions a laser light source at a point N of segment [DK] and directs this second laser beam towards point S. a. Show that, for every real $t$ in the interval $[ 0 ; 1 ]$, the point N with coordinates $( 0 ; 8 - 4 t ; 4 t )$ is a point of segment [DK]. b. Calculate the exact coordinates of point N such that the two laser beams represented by segments [SL] and [SN] are perpendicular.
Q4 7 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
Exercise 4 (7 points) Theme: natural logarithm function, probabilities This exercise is a multiple choice questionnaire (MCQ) comprising six questions. The six questions are independent. For each question, only one of the four answers is correct. The candidate will indicate on his answer sheet the number of the question followed by the letter corresponding to the correct answer. No justification is required.
A wrong answer, a multiple answer or no answer gives neither points nor deducts any points.
Question 1 The real number $a$ defined by $a = \ln ( 9 ) + \ln \left( \frac { \sqrt { 3 } } { 3 } \right) + \ln \left( \frac { 1 } { 9 } \right)$ is equal to: a. $1 - \frac { 1 } { 2 } \ln ( 3 )$ b. $\frac { 1 } { 2 } \ln ( 3 )$ c. $3 \ln ( 3 ) + \frac { 1 } { 2 }$ d. $- \frac { 1 } { 2 } \ln ( 3 )$
Question 2 We denote by $(E)$ the following equation $\ln x + \ln ( x - 10 ) = \ln 3 + \ln 7$ with unknown real $x$. a. 3 is a solution of $(E)$. b. $5 - \sqrt { 46 }$ is a solution of $(E)$. c. The equation $(E)$ admits a unique real solution. d. The equation $(E)$ admits two real solutions.
Question 3 The function $f$ is defined on the interval $] 0 ; + \infty [$ by the expression $f ( x ) = x ^ { 2 } ( - 1 + \ln x )$. We denote by $\mathscr { C } _ { f }$ its representative curve in the plane with a coordinate system. a. For every real $x$ in the interval $] 0 ; + \infty [$ , $f ^ { \prime } ( x ) = 2 x + \frac { 1 } { x }$. b. The function $f$ is increasing on the interval $] 0 ; + \infty [$. c. $f ^ { \prime } ( \sqrt { \mathrm { e } } )$ is different from 0. d. The line with equation $y = - \frac { 1 } { 2 } e$ is tangent to the curve $\mathscr { C } _ { f }$ at the point with abscissa $\sqrt { e }$.
Question 4
A bag contains 20 yellow tokens and 30 blue tokens. We draw successively and with replacement 5 tokens from the bag. The probability of drawing exactly 2 yellow tokens, rounded to the nearest thousandth, is: a. 0.683 b. 0.346 c. 0.230 d. 0.165
Question 5
A bag contains 20 yellow tokens and 30 blue tokens. We draw successively and with replacement 5 tokens from the bag. The probability of drawing at least one yellow token, rounded to the nearest thousandth, is: a. 0.078 b. 0.259 c. 0.337 d. 0.922
Question 6
A bag contains 20 yellow tokens and 30 blue tokens. We perform the following random experiment: we draw successively and with replacement five tokens from the bag. We note the number of yellow tokens obtained after these five draws. If we repeat this random experiment a very large number of times then, on average, the number of yellow tokens is equal to: a. 0.4 b. 1.2 c. 2 d. 2.5