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
2013 pondichery

4 maths questions

Q1 5 marks Exponential Equations & Modelling Exponential Growth/Decay Modelling with Contextual Interpretation View
We are interested in the evolution of the height of a corn plant as a function of time.
We decide to model this growth by a logistic function of the type: $$h ( t ) = \frac { a } { 1 + b \mathrm { e } ^ { - 0,04 t } }$$ where $a$ and $b$ are positive real constants, $t$ is the time variable expressed in days and $h ( t )$ denotes the height of the plant, expressed in metres.
We know that initially, for $t = 0$, the plant measures $0,1 \mathrm{~m}$ and that its height tends towards a limiting height of 2 m.
Part 1. Determine the constants $a$ and $b$ so that the function $h$ corresponds to the growth of the corn plant studied.
Part 2. We now consider that the growth of the corn plant is given by the function $f$ defined on $[0;250]$ by $$f ( t ) = \frac { 2 } { 1 + 19 \mathrm { e } ^ { - 0,04 t } }$$
  1. Determine $f ^ { \prime } ( t )$ as a function of $t$ ($f ^ { \prime }$ denoting the derivative function of the function $f$). Deduce the variations of the function $f$ on the interval $[ 0 ; 250 ]$.
  2. Calculate the time required for the corn plant to reach a height greater than $1,5 \mathrm{~m}$.
  3. a. Verify that for all real $t$ belonging to the interval $[ 0 ; 250 ]$ we have $f ( t ) = \frac { 2 \mathrm { e } ^ { 0,04 t } } { \mathrm { e } ^ { 0,04 t } + 19 }$.
    Show that the function $F$ defined on the interval $[ 0 ; 250]$ by $F ( t ) = 50 \ln \left( \mathrm { e } ^ { 0,04 t } + 19 \right)$ is an antiderivative of the function $f$.
    b. Determine the average value of $f$ on the interval $[ 50 ; 100 ]$. Give an approximate value to $10 ^ { - 2 }$ and interpret this result.
  4. We are interested in the growth rate of the corn plant; it is given by the derivative function of the function $f$. The growth rate is maximum for a value of $t$. Using the graph given in the appendix, determine an approximate value of this. Then estimate the height of the plant.
Q2 4 marks Vectors: Lines & Planes MCQ: Identify Correct Equation or Representation View
For each question, four answer options are given, of which only one is correct. For each question, indicate, without justification, the correct answer on your paper. A correct answer is worth 1 point. An incorrect answer or the absence of an answer gives neither points nor deducts any points.
Space is referred to an orthonormal coordinate system. $t$ and $t ^ { \prime }$ denote real parameters. The plane (P) has equation $x - 2 y + 3 z + 5 = 0$. The plane (S) has parametric representation $\left\{ \begin{aligned} x & = - 2 + t + 2 t ^ { \prime } \\ y & = - t - 2 t ^ { \prime } \\ z & = - 1 - t + 3 t ^ { \prime } \end{aligned} \right.$ The line (D) has parametric representation $\left\{ \begin{aligned} x & = - 2 + t \\ y & = - t \\ z & = - 1 - t \end{aligned} \right.$ We are given the points in space $\mathrm { M } ( - 1 ; 2 ; 3 )$ and $\mathrm { N } ( 1 ; - 2 ; 9 )$.
  1. A parametric representation of the plane (P) is: a. $\left\{ \begin{array} { r l r } x & = & t \\ y & = & 1 - 2 t \\ z & = & - 1 + 3 t \end{array} \right.$ b. $\left\{ \begin{array} { r l r } x & = t + 2 t ^ { \prime } \\ y & = 1 - t + t ^ { \prime } \\ z & = - 1 - t \end{array} \right.$ c. $\left\{ \begin{aligned} x & = t + t ^ { \prime } \\ y & = 1 - t - 2 t ^ { \prime } \\ z & = 1 - t - 3 t ^ { \prime } \end{aligned} \right.$ d. $\left\{ \begin{array} { l } x = 1 + 2 t + t ^ { \prime } \\ y = 1 - 2 t + 2 t ^ { \prime } \\ z = - 1 - t ^ { \prime } \end{array} \right.$
  2. a. The line (D) and the plane (P) are secant at point A(-8;3;2). b. The line (D) and the plane (P) are perpendicular. c. The line (D) is a line of the plane (P). d. The line (D) and the plane (P) are strictly parallel.
  3. a. The line (MN) and the line (D) are orthogonal. b. The line (MN) and the line (D) are parallel. c. The line (MN) and the line (D) are secant. d. The line (MN) and the line (D) are coincident.
  4. a. The planes $( \mathrm { P } )$ and $( \mathrm { S } )$ are parallel. b. The line $( \Delta )$ with parametric representation $\left\{ \begin{aligned} x & = t \\ y & = - 2 - t \\ z & = - 3 - t \end{aligned} \right.$ is the line of intersection of the planes (P) and (S). c. The point M belongs to the intersection of the planes (P) and (S). d. The planes $( \mathrm { P } )$ and $( \mathrm { S } )$ are perpendicular.
Q3 Complex Numbers Argand & Loci Similarity, Rotation, and Geometric Transformations in the Complex Plane View
The complex plane is equipped with a direct orthonormal coordinate system $( \mathrm { O } , \vec { u } , \vec { v } )$. We denote by i the complex number such that $\mathrm { i } ^ { 2 } = - 1$. We consider the point A with affixe $z _ { \mathrm { A } } = 1$ and the point B with affixe $z _ { \mathrm { B } } = \mathrm { i }$. To any point $M$ with affixe $z _ { M } = x + \mathrm { i } y$, with $x$ and $y$ two real numbers such that $y \neq 0$, we associate the point $M ^ { \prime }$ with affixe $z _ { M ^ { \prime } } = - \mathrm { i } z _ { M }$. We denote by $I$ the midpoint of the segment [AM]. The purpose of the exercise is to show that for any point $M$ not belonging to (OA), the median $( \mathrm { O } I )$ of the triangle $\mathrm { OA} M$ is also a height of the triangle $\mathrm { OB } M ^ { \prime }$ (property 1) and that $\mathrm { B } M ^ { \prime } = 2 \mathrm { O } I$ (property 2).
  1. In this question and only in this question, we take $z _ { M } = 2 \mathrm { e } ^ { - \mathrm { i } \frac { \pi } { 3 } }$. a. Determine the algebraic form of $z _ { M }$. b. Show that $z _ { M ^ { \prime } } = - \sqrt { 3 } - \mathrm { i }$. Determine the modulus and an argument of $z _ { M ^ { \prime } }$. c. Place the points $\mathrm { A } , \mathrm { B } , M , M ^ { \prime }$ and $I$ in the coordinate system $( \mathrm { O } , \vec { u } , \vec { v } )$ using 2 cm as the graphical unit. Draw the line $( \mathrm { O } I )$ and quickly verify properties 1 and 2 using the graph.
  2. We return to the general case by taking $z _ { M } = x + \mathrm { i } y$ with $y \neq 0$. a. Determine the affixe of point $I$ as a function of $x$ and $y$. b. Determine the affixe of point $M ^ { \prime }$ as a function of $x$ and $y$. c. Write the coordinates of points $I$, B and $M ^ { \prime }$. d. Show that the line $( \mathrm { O } I )$ is a height of the triangle $\mathrm { OB } M ^ { \prime }$. e. Show that $\mathrm { B } M ^ { \prime } = 2 \mathrm { O } I$.
Q3 (specialization) 5 marks Matrices Matrix Power Computation and Application View
We study the evolution over time of the number of young and adult animals in a population. For any natural integer $n$, we denote by $j _ { n }$ the number of young animals after $n$ years of observation and $a _ { n }$ the number of adult animals after $n$ years of observation. At the beginning of the first year of the study, there are 200 young animals and 500 adult animals. Thus $j _ { 0 } = 200$ and $a _ { 0 } = 500$. We admit that for any natural integer $n$ we have: $$\left\{ \begin{array} { l } j _ { n + 1 } = 0,125 j _ { n } + 0,525 a _ { n } \\ a _ { n + 1 } = 0,625 j _ { n } + 0,625 a _ { n } \end{array} \right.$$ We introduce the following matrices: $$A = \left( \begin{array} { l l } 0,125 & 0,525 \\ 0,625 & 0,625 \end{array} \right) \text { and, for any natural integer } n , U _ { n } = \binom { j _ { n } } { a _ { n } } .$$
  1. a. Show that for any natural integer $n , U _ { n + 1 } = A \times U _ { n }$. b. Calculate the number of young animals and adult animals after one year of observation and then after two years of observation (results rounded down to the nearest unit). c. For any non-zero natural integer $n$, express $U _ { n }$ as a function of $A ^ { n }$ and $U _ { 0 }$.
  2. We introduce the following matrices $Q = \left( \begin{array} { c c } 7 & 3 \\ - 5 & 5 \end{array} \right)$ and $D = \left( \begin{array} { c c } - 0,25 & 0 \\ 0 & 1 \end{array} \right)$. a. We admit that the matrix $Q$ is invertible and that $Q ^ { - 1 } = \left( \begin{array} { c c } 0,1 & - 0,06 \\ 0,1 & 0,14 \end{array} \right)$. Show that $Q \times D \times Q ^ { - 1 } = A$. b. Show by induction on $n$ that for any non-zero natural integer $n$: $A ^ { n } = Q \times D ^ { n } \times Q ^ { - 1 }$. c. For any non-zero natural integer $n$, determine $D ^ { n }$ as a function of $n$.
  3. We admit that for any non-zero natural integer $n$, $$A ^ { n } = \left( \begin{array} { l l } 0,3 + 0,7 \times ( - 0,25 ) ^ { n } & 0,42 - 0,42 \times ( - 0,25 ) ^ { n } \\ 0,5 - 0,5 \times ( - 0,25 ) ^ { n } & 0,7 + 0,3 \times ( - 0,25 ) ^ { n } \end{array} \right)$$ a. Deduce the expressions of $j _ { n }$ and $a _ { n }$ as functions of $n$ and determine the limits of these two sequences. b. What can we conclude about the population of animals studied?