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 centres-etrangers

8 maths questions

QI.1 1 marks Normal Distribution Direct Probability Calculation from Given Normal Distribution View
In an industrial bakery, a baguette is randomly selected from production. It is admitted that the random variable expressing its mass, in grams, follows the normal distribution with mean 200 and standard deviation 10.
Statement 1: The probability that the mass of the baguette is greater than 187 g is greater than 0.9.
Indicate whether this statement is true or false, justifying your answer.
QI.2 1 marks Sign Change & Interval Methods View
Statement 2: The equation $x - \cos x = 0$ has a unique solution in the interval $\left[ 0 ; \frac{\pi}{2} \right]$.
Indicate whether this statement is true or false, justifying your answer.
QI.3 1 marks Vectors 3D & Lines MCQ: Relationship Between Two Lines View
In space referred to an orthonormal coordinate system, consider the lines $\mathscr{D}_1$ and $\mathscr{D}_2$ which have the following parametric representations respectively: $$\left\{ \begin{array}{l} x = 1 + 2t \\ y = 2 - 3t \\ z = 4t \end{array} \right., t \in \mathbb{R} \quad \text{and} \quad \left\{ \begin{array}{l} x = -5t' + 3 \\ y = 2t' \\ z = t' + 4 \end{array} \right., t' \in \mathbb{R}$$
Statement 3: The lines $\mathscr{D}_1$ and $\mathscr{D}_2$ are secant.
Indicate whether this statement is true or false, justifying your answer.
QI.4 1 marks Vectors: Lines & Planes True/False or Verify a Given Statement View
In space referred to an orthonormal coordinate system, consider the line $\mathscr{D}_1$ with parametric representation: $$\left\{ \begin{array}{l} x = 1 + 2t \\ y = 2 - 3t \\ z = 4t \end{array} \right., t \in \mathbb{R}$$
Statement 4: The line $\mathscr{D}_1$ is parallel to the plane with equation $x + 2y + z - 3 = 0$.
Indicate whether this statement is true or false, justifying your answer.
QII 6 marks Indefinite & Definite Integrals Maximizing or Optimizing a Definite Integral View
Let $f$ be a function defined on the interval $[0;1]$, continuous and positive on this interval, and $a$ a real number such that $0 < a < 1$.
We denote:
  • $\mathscr{C}$ the representative curve of the function $f$ in an orthogonal coordinate system;
  • $\mathscr{A}_1$ the area of the plane region bounded by the $x$-axis and the curve $\mathscr{C}$ on one hand, the lines with equations $x = 0$ and $x = a$ on the other hand.
  • $\mathscr{A}_2$ the area of the plane region bounded by the $x$-axis and the curve $\mathscr{C}$ on one hand, the lines with equations $x = a$ and $x = 1$ on the other hand.

The purpose of this exercise is to determine, for different functions $f$, a value of the real number $a$ satisfying condition (E): ``the areas $\mathscr{A}_1$ and $\mathscr{A}_2$ are equal''. We admit the existence of such a real number $a$ for each of the functions considered.
Part A: Study of some examples
  1. Verify that in the following cases, condition (E) is satisfied for a unique real number $a$ and determine its value. a. $f$ is a strictly positive constant function. b. $f$ is defined on $[0;1]$ by $f(x) = x$.
  2. a. Using integrals, express, in units of area, the areas $\mathscr{A}_1$ and $\mathscr{A}_2$. b. Let $F$ be a primitive of the function $f$ on the interval $[0;1]$. Prove that if the real number $a$ satisfies condition (E), then $F(a) = \dfrac{F(0) + F(1)}{2}$. Is the converse true?
  3. In this question, we consider two other particular functions. a. The function $f$ is defined for all real $x$ in $[0;1]$ by $f(x) = \mathrm{e}^x$. Verify that condition (E) is satisfied for a unique real number $a$ and give its value. b. The function $f$ defined for all real $x$ in $[0;1]$ by $f(x) = \dfrac{1}{(x+2)^2}$. Verify that the value $a = \dfrac{2}{5}$ works.

Part B: Using a sequence to determine an approximate value of $a$
In this part, we consider the function $f$ defined for all real $x$ in $[0;1]$ by $f(x) = 4 - 3x^2$.
  1. Prove that if $a$ is a real number satisfying condition (E), then $a$ is a solution of the equation: $$x = \frac{x^3}{4} + \frac{3}{8}$$ In the rest of the exercise, we will admit that this equation has a unique solution in the interval $[0;1]$. We denote this solution by $a$.
  2. We consider the function $g$ defined for all real $x$ in $[0;1]$ by $g(x) = \dfrac{x^3}{4} + \dfrac{3}{8}$ and the sequence $(u_n)$ defined by: $u_0 = 0$ and, for all natural number $n$, $u_{n+1} = g(u_n)$. a. Calculate $u_1$. b. Prove that the function $g$ is increasing on the interval $[0;1]$. c. Prove by induction that, for all natural number $n$, we have $0 \leqslant u_n \leqslant u_{n+1} \leqslant 1$. d. Prove that the sequence $(u_n)$ is convergent. Using operations on limits, prove that the limit is $a$. e. We admit that the real number $a$ satisfies the inequality $0 < a - u_{10} < 10^{-9}$. Calculate $u_{10}$ to $10^{-8}$ precision.
QIII Confidence intervals Determine minimum sample size for a desired interval width View
An institute conducts a survey to determine, in a given population, the proportion of people who are in favour of a territorial development project. A random sample of people from this population is interviewed, and one question is asked to each person.
Part A: Number of people who agree to answer the survey
We admit in this part that the probability that a person interviewed agrees to answer the question is equal to 0.6.
  1. The survey institute interviews 700 people. We denote by $X$ the random variable corresponding to the number of people interviewed who agree to answer the question asked. a. What is the distribution of the random variable $X$? Justify the answer. b. What is the best approximation of $P(X \geqslant 400)$ among the following numbers? 0.92 0.93 0.94 0.95.
  2. How many people must the institute interview at minimum to guarantee, with a probability greater than 0.9, that the number of people answering the survey is greater than or equal to 400?

Part B: Proportion of people in favour of the project in the population
In this part, we assume that $n$ people have answered the question, and we admit that these people constitute a random sample of size $n$ (where $n$ is a natural number greater than 50). Among these people, 29\% are in favour of the development project.
  1. Give a confidence interval, at the 95\% confidence level, for the proportion of people who are in favour of the project in the total population.
  2. Determine the minimum value of the integer $n$ so that the confidence interval, at the 95\% confidence level, has an amplitude less than or equal to 0.04.

Part C: Correction due to insincere responses
In this part, we assume that, among the surveyed people who agreed to answer the question asked, 29\% claim that they are in favour of the project. The survey institute also knows that some people are not sincere and answer the opposite of their true opinion. Based on experience, the institute estimates at 15\% the rate of insincere responses among the people who responded, and admits that this rate is the same regardless of the opinion of the person interviewed.
A file of a person who responded is randomly selected, and we define:
  • $F$ the event ``the person is actually in favour of the project'';
  • $\bar{F}$ the event ``the person is actually opposed to the project'';
  • $A$ the event ``the person claims that they are in favour of the project'';
  • $\bar{A}$ the event ``the person claims that they are opposed to the project''.

Thus, according to the data, we have $p(A) = 0.29$.
  1. By interpreting the data in the statement, indicate the values of $P_F(A)$ and $P_{\bar{F}}(A)$.
  2. We set $x = P(F)$. a. Reproduce on your paper and complete the probability tree. b. Deduce an equality satisfied by $x$.
  3. Determine, among the people who responded to the survey, the proportion of those who are actually in favour of the project.
QIV Complex Numbers Arithmetic Trigonometric/Polar Form and De Moivre's Theorem View
We want to model in the plane the shell of a nautilus using a broken line in the form of a spiral. We are interested in the area delimited by this line.
We equip the plane with a direct orthonormal coordinate system $(O; \vec{u}; \vec{v})$. Let $n$ be an integer greater than or equal to 2. For all integer $k$ ranging from 0 to $n$, we define the complex numbers $z_k = \left(1 + \dfrac{k}{n}\right) \mathrm{e}^{\mathrm{i}\frac{2k\pi}{n}}$ and we denote by $M_k$ the point with affix $z_k$. In this model, the perimeter of the nautilus is the broken line connecting all the points $M_k$ with $0 \leqslant k \leqslant n$.
Part A: Broken line formed from seven points
In this part, we assume that $n = 6$. Thus, for $0 \leqslant k \leqslant 6$, we have $z_k = \left(1 + \dfrac{k}{6}\right) \mathrm{e}^{\mathrm{i}\frac{2k\pi}{6}}$.
  1. Determine the algebraic form of $z_1$.
  2. Verify that $z_0$ and $z_6$ are integers that you will determine.
  3. Calculate the length of the altitude from $M_1$ in the triangle $OM_0M_1$ then establish that the area of this triangle is equal to $\dfrac{7\sqrt{3}}{24}$.

Part B: Broken line formed from $n+1$ points
In this part, $n$ is an integer greater than or equal to 2.
  1. For all integer $k$ such that $0 \leqslant k \leqslant n$, determine the length $OM_k$.
  2. For $k$ an integer such that $0 \leqslant k \leqslant n-1$, determine a measure of the angles $(\vec{u}; \overrightarrow{OM_k})$ and $(\vec{u}; \overrightarrow{OM_{k+1}})$. Deduce a measure of the angle $(\overrightarrow{OM_k}; \overrightarrow{OM_{k+1}})$.
  3. For $k$ an integer such that $0 \leqslant k \leqslant n-1$, calculate the area of the triangle $OM_kM_{k+1}$ as a function of $n$ and $k$.
QV Matrices Linear System and Inverse Existence View
Part A - Hill Cipher
Here are the different encryption steps for a word with an even number of letters:
Step 1The word is divided into blocks of two consecutive letters, then for each block, each of the following steps is performed.
Step 2To the two letters of the block are associated two integers $x_1$ and $x_2$ both between 0 and 25, which correspond to the two letters in the same order, in the following table: \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
A		BCDEFGHIJKLM
0123456789101112
NOPQRSTUVWXYZ
13141516171819202122232425
\hline Step 3 & The matrix $X = \binom{x_1}{x_2}$ is transformed into the matrix $Y = \binom{y_1}{y_2}$ satisfying $Y = AX$, where $A = \left(\begin{array}{ll} 5 & 2 \\ 7 & 7 \end{array}\right)$. \hline Step 4 & The matrix $Y = \binom{y_1}{y_2}$ is transformed into the matrix $R = \binom{r_1}{r_2}$, where $r_1$ is the remainder of the Euclidean division of $y_1$ by 26 and $r_2$ is the remainder of the Euclidean division of $y_2$ by 26. \hline Step 5 & To the integers $r_1$ and $r_2$ are associated the two corresponding letters from the table in step 2. The encrypted block is the block obtained by concatenating these two letters. \hline \end{tabular}
Use the encryption method presented to encrypt the word ``HILL''.
Part B - Some mathematical tools necessary for decryption
  1. Let $a$ be an integer relatively prime to 26. Prove that there exists an integer $u$ such that $u \times a \equiv 1$ modulo 26.
  2. Consider the following algorithm:
    VARIABLES : PROCESSING :\begin{tabular}{l} $a, u$, and $r$ are numbers ($a$ is a natural number and relatively prime to 26)
    Read $a$
    $u$ takes the value 0, and $r$ takes the value 0
    While $r \neq 1$
    $u$ takes the value $u + 1$
    $r$ takes the value of the remainder of the Euclidean division of $u \times a$ by 26
    End While
    Display $u$
    \hline \end{tabular}
    The value $a = 21$ is entered into this algorithm. a. Reproduce on your paper and complete the following table, until the algorithm stops.
    $u$012$\ldots$
    $r$021$\ldots$$\ldots$

    b. Deduce that $5 \times 21 \equiv 1$ modulo 26.
  3. Recall that $A$ is the matrix $A = \left(\begin{array}{ll} 5 & 2 \\ 7 & 7 \end{array}\right)$ and denote by $I$ the matrix: $I = \left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)$. a. Calculate the matrix $12A - A^2$. b. Deduce the matrix $B$ such that $BA = 21I$. c. Prove that if $AX = Y$, then $21X = BY$.

Part C - Decryption
We want to decrypt the word VLUP. We denote by $X = \binom{x_1}{x_2}$ the matrix associated, according to the correspondence table, to a block of two letters before encryption, and $Y = \binom{y_1}{y_2}$ the matrix defined by the equality: $Y = AX = \left(\begin{array}{ll} 5 & 2 \\ 7 & 7 \end{array}\right) X$. If $r_1$ and $r_2$ are the respective remainders of $y_1$ and $y_2$ in the Euclidean division by 26, the block of two letters after encryption is associated with the matrix $R = \binom{r_1}{r_2}$.
  1. Prove that: $\left\{ \begin{aligned} 21x_1 &= 7y_1 - 2y_2 \\ 21x_2 &= -7y_1 + 5y_2 \end{aligned} \right.$
  2. Using question B.2., establish that: $\begin{cases} x_1 \equiv 9r_1 + 16r_2 & \text{modulo } 26 \\ x_2 \equiv 17r_1 + 25r_2 & \text{modulo } 26 \end{cases}$
  3. Decrypt the word VLUP, associated with the matrices $\binom{21}{11}$ and $\binom{20}{15}$.