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
2015 amerique-nord

4 maths questions

Q1 5 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
In space, we consider a pyramid SABCE with square base ABCE with centre O. Let D be the point in space such that ( O ; $\overrightarrow { \mathrm { OA } } , \overrightarrow { \mathrm { OB } } , \overrightarrow { \mathrm { OD } }$ ) is an orthonormal frame. The point S has coordinates $( 0 ; 0 ; 3 )$ in this frame.
Part A
  1. Let U be the point on the line $( \mathrm { SB } )$ with height 1. Construct the point U on the attached figure in appendix 1.
  2. Let V be the intersection point of the plane (AEU) and the line (SC). Show that the lines (UV) and (BC) are parallel. Construct the point V on the attached figure in appendix 1.
  3. Let K be the point with coordinates $\left( \frac { 5 } { 6 } ; - \frac { 1 } { 6 } ; 0 \right)$. Show that K is the foot of the altitude from U in the trapezoid AUVE.

Part B
In this part, we admit that the area of the quadrilateral AUVE is $\frac { 5 \sqrt { 43 } } { 18 }$.
  1. We admit that the point $U$ has coordinates $\left( 0 ; \frac { 2 } { 3 } ; 1 \right)$. Verify that the plane (EAU) has equation $3 x - 3 y + 5 z - 3 = 0$.
  2. Give a parametric representation of the line (d) perpendicular to the plane (EAU) passing through the point $S$.
  3. Determine the coordinates of H, the intersection point of the line (d) and the plane (EAU).
  4. The plane (EAU) divides the pyramid (SABCE) into two solids. Do these two solids have the same volume?
Q2A 5 marks Complex numbers 2 Complex Recurrence Sequences View
Exercise 2 — Candidates who have not followed the specialization course
We place ourselves in an orthonormal frame and, for every natural integer $n$, we define the points $\left( A _ { n } \right)$ by their coordinates $\left( x _ { n } ; y _ { n } \right)$ in the following way:
$$\left\{ \begin{array} { l } x _ { 0 } = - 3 \\ y _ { 0 } = 4 \end{array} \text { and for every natural integer } n : \left\{ \begin{array} { l } x _ { n + 1 } = 0,8 x _ { n } - 0,6 y _ { n } \\ y _ { n + 1 } = 0,6 x _ { n } + 0,8 y _ { n } \end{array} \right. \right.$$
  1. a. Determine the coordinates of the points $A _ { 0 } , A _ { 1 }$ and $A _ { 2 }$. b. Copy and complete the following algorithm so that it constructs the points $A _ { 0 }$ to $A _ { 20 }$: \begin{verbatim} Variables : i,x,y,t: real numbers Initialization : x takes the value -3 y takes the value 4 Processing : For i ranging from 0 to 20 Construct the point with coordinates (x;y) t takes the value x x takes the value .... y takes the value .... End For \end{verbatim} c. Identify the points $A _ { 0 } , A _ { 1 }$ and $A _ { 2 }$ on the point cloud figure. What appears to be the set to which the points $A _ { n }$ belong for every natural integer $n$?
  2. The purpose of this question is to construct geometrically the points $A _ { n }$ for every natural integer $n$. In the complex plane, we denote, for every natural integer $n$, $z _ { n } = x _ { n } + \mathrm { i } y _ { n }$ the affix of the point $A _ { n }$. a. Let $u _ { n } = \left| z _ { n } \right|$. Show that, for every natural integer $n , u _ { n } = 5$. What geometric interpretation can be made of this result? b. We admit that there exists a real number $\theta$ such that $\cos ( \theta ) = 0,8$ and $\sin ( \theta ) = 0,6$. Show that, for every natural integer $n , \mathrm { e } ^ { \mathrm { i } \theta } z _ { n } = z _ { n + 1 }$. c. Prove that, for every natural integer $n , z _ { n } = \mathrm { e } ^ { \mathrm { i } n \theta } z _ { 0 }$. d. Show that $\theta + \frac { \pi } { 2 }$ is an argument of the complex number $z _ { 0 }$. e. For every natural integer $n$, determine, as a function of $n$ and $\theta$, an argument of the complex number $z _ { n }$. Explain, for every natural integer $n$, how to construct the point $A _ { n + 1 }$ from the point $A _ { n }$.
Q2B 5 marks 3x3 Matrices Solving a 3×3 Linear System Explicitly View
Exercise 2 — Candidates who have followed the specialization course
We are given the matrices $M = \left( \begin{array} { c c c } 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 4 & 2 & 1 \end{array} \right)$ and $I = \left( \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$.
Part A
  1. Determine the matrix $M ^ { 2 }$. We are given $M ^ { 3 } = \left( \begin{array} { c c c } 20 & 10 & 11 \\ 12 & 2 & 9 \\ 42 & 20 & 21 \end{array} \right)$.
  2. Verify that $M ^ { 3 } = M ^ { 2 } + 8 M + 6 I$.
  3. Deduce that $M$ is invertible and that $M ^ { - 1 } = \frac { 1 } { 6 } \left( M ^ { 2 } - M - 8 I \right)$.

Part B Study of a particular case
We seek to determine three integers $a , b$ and $c$ such that the parabola with equation $y = a x ^ { 2 } + b x + c$ passes through the points $\mathrm { A } ( 1 ; 1 ) , \mathrm { B } ( - 1 ; - 1 )$ and $\mathrm { C } ( 2 ; 5 )$.
  1. Prove that the problem amounts to finding three integers $a , b$ and $c$ such that $$M \left( \begin{array} { l } a \\ b \\ c \end{array} \right) = \left( \begin{array} { c } 1 \\ - 1 \\ 5 \end{array} \right)$$
  2. Calculate the numbers $a$, $b$ and $c$ and verify that these numbers are integers.

Part C Return to the general case
The numbers $a , b , c , p , q , r$ are integers. In a frame ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ), we consider the points $\mathrm { A } ( 1 ; p ) , \mathrm { B } ( - 1 ; q )$ and $\mathrm { C } ( 2 ; r )$. We seek values of $p , q$ and $r$ for which there exists a parabola with equation $y = a x ^ { 2 } + b x + c$ passing through A, B and C.
  1. Prove that if $\left( \begin{array} { l } a \\ b \\ c \end{array} \right) = M ^ { - 1 } \left( \begin{array} { c } p \\ q \\ r \end{array} \right)$ with $a , b$ and $c$ integers, then $$\begin{cases} - 3 p + q + 2 r & \equiv 0 [ 6 ] \\ 3 p - 3 q & \equiv 0 [ 6 ] \\ 6 p + 2 q - 2 r & \equiv 0 [ 6 ] \end{cases}$$
  2. Deduce that $\left\{ \begin{array} { l l l } q - r & \equiv & 0 [ 3 ] \\ p - q & \equiv & 0 [ 2 ] \end{array} \right.$.
  3. Conversely, we admit that if $\left\{ \begin{array} { l } q - r \equiv 0 [ 3 ] \\ p - q \equiv 0 [ 2 ] \\ \mathrm { A } , \mathrm { B } , \mathrm { C } \text{ are not collinear} \end{array} \right.$ then there exist three integers $a , b$ and $c$ such that the parabola with equation $y = a x ^ { 2 } + b x + c$ passes through the points $\mathrm { A } , \mathrm { B }$ and C. a. Show that the points $\mathrm { A } , \mathrm { B }$ and C are collinear if and only if $2 r + q - 3 p = 0$. b. We choose $p = 7$. Determine integers $q , r , a , b$ and $c$ such that the parabola with equation $y = a x ^ { 2 } + b x + c$ passes through the points $\mathrm { A } , \mathrm { B }$ and C.
Q3 Conditional Probability Conditional Probability with Normal Distribution View
A company manufactures chocolate tablets of 100 grams. The quality control department performs several types of control.
Part A Control before market release
A chocolate tablet must weigh 100 grams with a tolerance of two grams more or less. It is therefore put on the market if its mass is between 98 and 102 grams. The mass (expressed in grams) of a chocolate tablet can be modelled by a random variable $X$ following the normal distribution with mean $\mu = 100$ and standard deviation $\sigma = 1$. The adjustment of the manufacturing chain machines allows us to modify the value of $\sigma$.
  1. Calculate the probability of the event $M$ : ``the tablet is put on the market''.
  2. We wish to modify the adjustment of the machines so that the probability of this event reaches 0.97. Determine the value of $\sigma$ so that the probability of the event ``the tablet is put on the market'' equals 0.97.

Part B Control upon reception
The department controls the quality of cocoa beans delivered by producers. One of the quality criteria is the moisture content which must be $7\%$. The bean is then said to be compliant. The company has three different suppliers: the first supplier provides half of the bean stock, the second $30\%$ and the last provides $20\%$ of the stock. For the first, $98\%$ of its production respects the moisture content; for the second, which is somewhat cheaper, $90\%$ of its production is compliant, and the third supplies $20\%$ of non-compliant beans. We randomly choose a bean from the received stock. We denote $F _ { i }$ the event ``the bean comes from supplier $i$'', for $i$ taking the values 1, 2 or 3, and $C$ the event ``the bean is compliant''.
  1. Determine the probability that the bean comes from supplier 1, given that it is compliant.