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
2018 liban

6 maths questions

Q1 3 marks Exponential Distribution View
During the fifteen days preceding the start of the university term, the telephone switchboard of a student mutual aid organisation records a record number of calls. Callers are first placed on hold and hear background music and a pre-recorded message. During this first phase, the waiting time, expressed in seconds, is modelled by the random variable $X$ which follows the exponential distribution with parameter $\lambda = 0{,}02 \mathrm{~s}^{-1}$. Callers are then connected with a customer service representative who answers their questions. The exchange time, expressed in seconds, during this second phase is modelled by the random variable $Y$, expressed in seconds, which follows the normal distribution with mean $\mu = 96$ s and standard deviation $\sigma = 26 \mathrm{~s}$.
  1. What is the average total duration of a call to the telephone switchboard (waiting time and exchange time with the customer service representative)?
  2. A student is chosen at random from among the callers to the telephone switchboard. a. Calculate the probability that the student is placed on hold for more than 2 minutes. b. Calculate the probability that the exchange time with the adviser is less than 90 seconds.
  3. A female student, chosen at random from among the callers, has been waiting for more than one minute to be connected with the customer service. Tired, she hangs up and dials the number again. She hopes to wait less than thirty seconds this time. Does hanging up and calling back increase her chances of limiting the additional waiting time to 30 seconds, or would she have been better off staying on the line?
Q2 Complex numbers 2 Modulus and Argument Computation View
  1. Give the exponential and trigonometric forms of the complex numbers $1 + \mathrm{i}$ and $1 - \mathrm{i}$.
  2. For every natural number $n$, we define $$S_{n} = (1 + \mathrm{i})^{n} + (1 - \mathrm{i})^{n}.$$ a. Determine the trigonometric form of $S_{n}$. b. For each of the two following statements, say whether it is true or false by justifying your answer. An unjustified answer will not be taken into account and the absence of an answer is not penalised.
    Statement A: For every natural number $n$, the complex number $S_{n}$ is a real number. Statement B: There exist infinitely many natural numbers $n$ such that $S_{n} = 0$.
Q3 Vectors 3D & Lines Shortest Distance Between Two Lines View
The objective of this exercise is to study the trajectories of two submarines in the diving phase. We consider that these submarines move in a straight line, each at constant speed. At each instant $t$, expressed in minutes, the first submarine is located by the point $S_{1}(t)$ and the second submarine is located by the point $S_{2}(t)$ in an orthonormal reference frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$ with unit metre. The plane defined by $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$ represents the sea surface. The $z$ coordinate is zero at sea level, negative underwater.
  1. We admit that, for every real $t \geqslant 0$, the point $S_{1}(t)$ has coordinates: $$\left\{ \begin{array}{l} x(t) = 140 - 60t \\ y(t) = 105 - 90t \\ z(t) = -170 - 30t \end{array} \right.$$ a. Give the coordinates of the submarine at the beginning of the observation. b. What is the speed of the submarine? c. We place ourselves in the vertical plane containing the trajectory of the first submarine.
    Determine the angle $\alpha$ that the submarine's trajectory makes with the horizontal plane. Give the value of $\alpha$ rounded to 0.1 degree.
  2. At the beginning of the observation, the second submarine is located at the point $S_{2}(0)$ with coordinates $(68; 135; -68)$ and reaches after three minutes the point $S_{2}(3)$ with coordinates $(-202; -405; -248)$ at constant speed. At what instant $t$, expressed in minutes, are the two submarines at the same depth?
Q4 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
We consider, for every integer $n > 0$, the functions $f_{n}$ defined on the interval $[1; 5]$ by: $$f_{n}(x) = \frac{\ln x}{x^{n}}$$ For every integer $n > 0$, we denote by $\mathscr{C}_{n}$ the representative curve of the function $f_{n}$ in an orthogonal reference frame.
  1. Show that, for every integer $n > 0$ and every real $x$ in the interval $[1; 5]$: $$f_{n}^{\prime}(x) = \frac{1 - n\ln(x)}{x^{n+1}}$$
  2. For every integer $n > 0$, we admit that the function $f_{n}$ has a maximum on the interval $[1; 5]$. We denote by $A_{n}$ the point of the curve $\mathscr{C}_{n}$ having as ordinate this maximum. Show that all points $A_{n}$ belong to the same curve $\Gamma$ with equation $$y = \frac{1}{\mathrm{e}} \ln(x)$$
  3. a. Show that, for every integer $n > 1$ and every real $x$ in the interval $[1; 5]$: $$0 \leqslant \frac{\ln(x)}{x^{n}} \leqslant \frac{\ln(5)}{x^{n}}$$ b. Show that for every integer $n > 1$: $$\int_{1}^{5} \frac{1}{x^{n}} \mathrm{~d}x = \frac{1}{n-1}\left(1 - \frac{1}{5^{n-1}}\right)$$ c. For every integer $n > 0$, we are interested in the area, expressed in square units, of the surface under the curve $f_{n}$, that is the area of the region of the plane bounded by the lines with equations $x = 1$, $x = 5$, $y = 0$ and the curve $\mathscr{C}_{n}$. Determine the limiting value of this area as $n$ tends to $+\infty$.
Q5a 5 marks Geometric Sequences and Series Prove a Transformed Sequence is Geometric View
(Candidates who have not followed the specialisation course)
A computer game of chance is set up as follows:
  • If the player wins a game, the probability that he wins the next game is $\frac{1}{4}$;
  • If the player loses a game, the probability that he loses the next game is $\frac{1}{2}$;
  • The probability of winning the first game is $\frac{1}{4}$.
For every non-zero natural number $n$, we denote by $G_{n}$ the event ``the $n^{\mathrm{th}}$ game is won'' and we denote by $p_{n}$ the probability of this event. We thus have $p_{1} = \frac{1}{4}$.
  1. Show that $p_{2} = \frac{7}{16}$.
  2. Show that, for every non-zero natural number $n$, $p_{n+1} = -\frac{1}{4} p_{n} + \frac{1}{2}$.
  3. We thus obtain the first values of $p_{n}$:
    $n$1234567
    $p_{n}$0,250,43750,39060,40230,39940,40010,3999

    What conjecture can be made?
  4. We define, for every non-zero natural number $n$, the sequence $(u_{n})$ by $u_{n} = p_{n} - \frac{2}{5}$. a. Prove that the sequence $(u_{n})$ is a geometric sequence and specify its common ratio. b. Deduce that, for every non-zero natural number $n$, $p_{n} = \frac{2}{5} - \frac{3}{20}\left(-\frac{1}{4}\right)^{n-1}$. c. Does the sequence $(p_{n})$ converge? Interpret this result.
Q5b 5 marks Matrices Matrix Power Computation and Application View
(Candidates who have followed the specialisation course)
We define the sequence of real numbers $(a_{n})$ by: $$\begin{cases} a_{0} & = 0 \\ a_{1} & = 1 \\ a_{n+1} & = a_{n} + a_{n-1} \text{ for } n \geqslant 1 \end{cases}$$ This sequence is called the Fibonacci sequence.
  1. Copy and complete the algorithm below so that at the end of its execution the variable $A$ contains the term $a_{n}$. \begin{verbatim} $A \leftarrow 0$ $B \leftarrow 1$ For $i$ going from 1 to $n$ : $C \leftarrow A + B$ $A \leftarrow \ldots$ $B \leftarrow \ldots$ End For \end{verbatim} We thus obtain the first values of the sequence $a_{n}$:
    $n$012345678910
    $a_{n}$011235813213455

  2. Let the matrix $A = \left(\begin{array}{ll} 1 & 1 \\ 1 & 0 \end{array}\right)$.
    Calculate $A^{2}$, $A^{3}$ and $A^{4}$. Verify that $A^{5} = \left(\begin{array}{ll} 8 & 5 \\ 5 & 3 \end{array}\right)$.
  3. We can prove, and we will admit, that for every non-zero natural number $n$, $$A^{n} = \left(\begin{array}{cc} a_{n+1} & a_{n} \\ a_{n} & a_{n-1} \end{array}\right)$$ a. Let $p$ and $q$ be two non-zero natural numbers. Calculate the product $A^{p} \times A^{q}$ and deduce that $$a_{p+q} = a_{p} \times a_{q+1} + a_{p-1} \times a_{q}$$ b. Deduce that if an integer $r$ divides the integers $a_{p}$ and $a_{q}$, then $r$ also divides $a_{p+q}$.