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
2016 asie

5 maths questions

Q1 Normal Distribution Normal Distribution Combined with Total Probability or Bayes' Theorem View
A market gardener specializes in strawberry production.
Part A: strawberry production
The market gardener produces strawberries in two greenhouses denoted A and B; $55\%$ of strawberry flowers are in greenhouse A, and $45\%$ in greenhouse B. In greenhouse A, the probability that each flower produces fruit is equal to 0.88; in greenhouse B, it is equal to 0.84.
For each of the following propositions, indicate whether it is true or false by justifying the answer. An unjustified answer will not be taken into account.
Proposition 1: The probability that a strawberry flower, chosen at random from this farm, produces fruit is equal to 0.862.
Proposition 2: It is observed that a flower, chosen at random from this farm, produces fruit. The probability that it is located in greenhouse A, rounded to the nearest thousandth, is equal to 0.439.
Part B: strawberry packaging
Strawberries are packaged in trays. The mass (expressed in grams) of a tray can be modeled by a random variable $X$ which follows the normal distribution with mean $\mu = 250$ and standard deviation $\sigma$.
  1. We are given $P ( X \leqslant 237 ) = 0.14$. Calculate the probability of the event ``the mass of the tray is between 237 and 263 grams''.
  2. Let $Y$ be the random variable defined by: $Y = \frac { X - 250 } { \sigma }$. a. What is the distribution of the random variable $Y$? b. Prove that $P \left( Y \leqslant - \frac { 13 } { \sigma } \right) = 0.14$. c. Deduce the value of $\sigma$ rounded to the nearest integer.
  3. In this question, we assume that $\sigma$ equals 12. We denote by $n$ and $m$ two integers. a. A tray is compliant if its mass, expressed in grams, lies in the interval $[ 250 - n ; 250 + n ]$. Determine the smallest value of $n$ for a tray to be compliant with a probability greater than or equal to $95\%$. b. In this question, we consider that a tray is compliant if its mass, expressed in grams, lies in the interval $[230; m ]$. Determine the smallest value of $m$ for a tray to be compliant with a probability greater than or equal to $95\%$.
Q2 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Let $a$ be a real number between 0 and 1. We denote by $f _ { a }$ the function defined on $\mathbf { R }$ by:
$$f _ { a } ( x ) = a \mathrm { e } ^ { a x } + a .$$
We denote by $I ( a )$ the integral of the function $f _ { a }$ between 0 and 1:
$$I ( a ) = \int _ { 0 } ^ { 1 } f _ { a } ( x ) \mathrm { d } x$$
  1. In this question, we set $a = 0$. Determine $I ( 0 )$.
  2. In this question, we set $a = 1$. We therefore study the function $f _ { 1 }$ defined on $\mathbf { R }$ by: $$f _ { 1 } ( x ) = \mathrm { e } ^ { x } + 1$$ a. Without detailed study, sketch the graph of the function $f _ { 1 }$ on your paper in an orthogonal coordinate system and show the number $I ( 1 )$. b. Calculate the exact value of $I ( 1 )$, then round to the nearest tenth.
  3. Does there exist a value of $a$ for which $I ( a )$ equals 2? If so, give an interval of width $10 ^ { - 2 }$ containing this value.
Q3 Geometric Sequences and Series Applied Geometric Model with Contextual Interpretation View
A company produces bacteria for industry. In the laboratory, it was measured that, in an appropriate nutrient medium, the mass of these bacteria, measured in grams, increases by $20\%$ in one day. The company implements the following industrial process. In a vat of nutrient medium, 1 kg of bacteria is initially introduced. Then, each day, at a fixed time, the nutrient medium in the vat is replaced. During this operation, 100 g of bacteria are lost. The company's objective is to produce 30 kg of bacteria.
Part A: first model - with a sequence
The evolution of the bacterial population in the vat is modeled by the sequence $(u _ { n })$ defined as follows:
$$u _ { 0 } = 1000 \text{ and, for all natural integers } n , u _ { n + 1 } = 1.2 u _ { n } - 100 .$$
  1. a. Explain how this model corresponds to the situation described in the problem. You will specify in particular what $u _ { n }$ represents. b. The company wants to know after how many days the mass of bacteria will exceed 30 kg. Using a calculator, give the answer to this problem. c. We can also use the following algorithm to answer the problem posed in the previous question. Copy and complete this algorithm.
    Variables$u$ and $n$ are numbers
    Processing\begin{tabular}{l} $u$ takes the value 1000
    $n$ takes the value 0
    While $\_\_\_\_$ do
    $u$ takes the value $\_\_\_\_$ $n$ takes the value $n + 1$
    End While
    \hline Output & Display .......... \hline \end{tabular}
  2. a. Prove by induction that, for all natural integers $n$, $u _ { n } \geqslant 1000$. b. Prove that the sequence $( u _ { n } )$ is increasing.
  3. We define the sequence $( v _ { n } )$ by: for all natural integers $n$, $v _ { n } = u _ { n } - 500$. a. Prove that the sequence $( v _ { n } )$ is a geometric sequence. b. Express $v _ { n }$, then $u _ { n }$, as a function of $n$. c. Determine the limit of the sequence $( u _ { n } )$.

Part B: second model - with a function
It is observed that in practice, the mass of bacteria in the vat will never exceed 50 kg. This leads to studying a second model in which the mass of bacteria is modeled by the function $f$ defined on $[ 0 ; +\infty[$ by:
$$f ( t ) = \frac { 50 } { 1 + 49 \mathrm { e } ^ { - 0.2 t } }$$
where $t$ represents time expressed in days and where $f ( t )$ represents the mass, expressed in kg, of bacteria at time $t$.
  1. a. Calculate $f ( 0 )$. b. Prove that, for all real $t \geqslant 0$, $f ( t ) < 50$. c. Study the monotonicity of the function $f$. d. Determine the limit of the function $f$ as $t \to + \infty$.
  2. Interpret the results of question 1 in the context of the problem.
  3. Using this model, we seek to determine after how many days the mass of bacteria will exceed 30 kg. Solve the inequality with unknown $t$: $f ( t ) > 30$. Deduce the answer to the problem.

Part C: quality control
Bacteria can be of two types: type A, which effectively produces a protein useful to industry, and type B, which does not produce it and is therefore commercially useless. The company claims that $80\%$ of the bacteria produced are of type A. To verify this claim, a laboratory analyzes a random sample of 200 bacteria at the end of production. The analysis shows that 146 of them are of type A. Should the company's claim be questioned?
Q4 4 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
The points O, A, B and C are vertices of a cube, such that the coordinate system $(\mathrm{O} ; \overrightarrow{\mathrm{OA}}, \overrightarrow{\mathrm{OB}}, \overrightarrow{\mathrm{OC}})$ is an orthonormal coordinate system. This coordinate system will be used throughout the exercise. The three mirrors of the retroreflector are represented by the planes (OAB), (OBC) and (OAC). Light rays are modeled by lines.
Rules for reflection of a light ray (admitted):
  • when a light ray with direction vector $\vec{v}(a ; b ; c)$ is reflected by the plane (OAB), a direction vector of the reflected ray is $\vec{v}(a ; b ; -c)$;
  • when a light ray with direction vector $\vec{v}(a ; b ; c)$ is reflected by the plane (OBC), a direction vector of the reflected ray is $\vec{v}(-a ; b ; c)$;
  • when a light ray with direction vector $\vec{v}(a ; b ; c)$ is reflected by the plane (OAC), a direction vector of the reflected ray is $\vec{v}(a ; -b ; c)$.

1. Property of retroreflectors
Using the above rules, prove that if a light ray with direction vector $\vec{v}(a ; b ; c)$ is reflected successively by the planes (OAB), (OBC) and (OAC), the final ray is parallel to the initial ray.
For the rest, we consider a light ray modeled by a line $d _ { 1 }$ with direction vector $\overrightarrow{v _ { 1 }}(-2 ; -1 ; -1)$ which strikes the plane (OAB) at the point $\mathrm{I} _ { 1 }(2 ; 3 ; 0)$. The reflected ray is modeled by the line $d _ { 2 }$ with direction vector $\overrightarrow{v _ { 2 }}(-2 ; -1 ; 1)$ and passing through the point $\mathrm{I} _ { 1 }$.
2. Reflection of $d_2$ on the plane (OBC)
a. Give a parametric representation of the line $d _ { 2 }$. b. Give, without justification, a normal vector to the plane (OBC) and a Cartesian equation of this plane. c. Let $\mathrm{I} _ { 2 }$ be the point with coordinates $(0 ; 2 ; 1)$. Verify that the plane (OBC) and the line $d _ { 2 }$ intersect at $\mathrm{I} _ { 2 }$.
We denote by $d _ { 3 }$ the line representing the light ray after reflection on the plane (OBC). $d _ { 3 }$ is therefore the line with direction vector $\overrightarrow{v _ { 3 }}(2 ; -1 ; 1)$ passing through the point $\mathrm{I} _ { 2 }(0 ; 2 ; 1)$.
3. Reflection of $d_3$ on the plane (OAC)
Calculate the coordinates of the intersection point $\mathrm{I} _ { 3 }$ of the line $d _ { 3 }$ with the plane (OAC).
We denote by $d _ { 4 }$ the line representing the light ray after reflection on the plane (OAC). It is therefore parallel to the line $d _ { 1 }$.
4. Study of the light path
We are given the vector $\vec{u}(1 ; -2 ; 0)$, and we denote by $\mathscr{P}$ the plane defined by the lines $d _ { 1 }$ and $d _ { 2 }$. a. Prove that the vector $\vec{u}$ is a normal vector to the plane $\mathscr{P}$. b. Are the lines $d _ { 1 }$, $d _ { 2 }$ and $d_3$ coplanar?
Q5 Matrices Linear System and Inverse Existence View
Encryption Method (Hill cipher)
The following table gives a correspondence between letters and numbers:
Encryption proceeds as follows:
  1. We replace the letters with the values associated using the table above, and we place the pairs of numbers obtained in column matrices: $C _ { 1 } = \binom { 12 } { 0 }$, $C _ { 2 } = \binom { 19 } { 7 }$
  2. We multiply the column matrices on the left by the matrix $A = \left( \begin{array} { l l } 9 & 4 \\ 7 & 3 \end{array} \right)$: $A C _ { 1 } = \binom { 108 } { 84 }$, $A C _ { 2 } = \binom { 199 } { 154 }$
  3. We replace each coefficient of the column matrices obtained by its remainder in the Euclidean division by 26: $108 = 4 \times 26 + 4$, $84 = 3 \times 26 + 6$, we obtain $\binom { 4 } { 6 }$
  4. We use the correspondence table between letters and numbers to obtain the encrypted word: EGRY

Question 1: By encrypting the word ``PION'' using this method, we obtain ``LZWH''. By detailing the steps for the letters ``ES'', encrypt the word ``ESPION''.
2. Decryption Method
Let $a, b, x, y, x^{\prime}$ and $y^{\prime}$ be relative integers. We know that if $x \equiv x^{\prime}$ modulo 26 and $y \equiv y^{\prime}$ modulo 26 then $ax + by \equiv ax^{\prime} + by^{\prime}$ modulo 26. This result allows us to write that, if $A$ is a $2 \times 2$ matrix, and $B$ and $C$ are two column matrices $2 \times 1$, then $B \equiv C$ modulo 26 implies $AB \equiv AC$ modulo 26.
a. Establish that the matrix $A = \left( \begin{array} { l l } 9 & 4 \\ 7 & 3 \end{array} \right)$ is invertible, and determine its inverse. b. Decrypt the word: XQGY.