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 amerique-nord

5 maths questions

Q1 6 marks Normal Distribution Direct Probability Calculation from Given Normal Distribution View
Exercise 1 (6 points)

We study certain characteristics of a supermarket in a small town.
Part A - Preliminary Demonstration
Let $X$ be a random variable that follows the exponential distribution with parameter 0.2. Recall that the expectation of the random variable $X$, denoted $E(X)$, is equal to: $$\lim_{x \rightarrow +\infty} \int_{0}^{x} 0.2t\, \mathrm{e}^{-0.2t} \mathrm{~d}t$$ The purpose of this part is to demonstrate that $E(X) = 5$.
  1. Let $g$ be the function defined on the interval $[0; +\infty[$ by $g(t) = 0.2t\,\mathrm{e}^{-0.2t}$.
    We define the function $G$ on the interval $[0; +\infty[$ by $G(t) = (-t-5)\mathrm{e}^{-0.2t}$. Verify that $G$ is a primitive of $g$ on the interval $[0; +\infty[$.
  2. Deduce that the exact value of $E(X)$ is 5.
    Hint: you may use, without proving it, the following result: $$\lim_{x \rightarrow +\infty} x\,\mathrm{e}^{-0.2x} = 0$$

Part B - Study of the duration of a customer's presence in the supermarket
A study commissioned by the supermarket manager makes it possible to model the duration, expressed in minutes, spent in the supermarket by a randomly chosen customer using a random variable $T$. This variable $T$ follows a normal distribution with expectation 40 minutes and standard deviation a positive real number denoted $\sigma$. Thanks to this study, it is estimated that $P(T < 10) = 0.067$.
  1. Determine an approximate value of the real number $\sigma$ to the nearest second.
  2. In this question, we take $\sigma = 20$ minutes. What is then the proportion of customers who spend more than one hour in the supermarket?

Part C - Waiting time for payment
This supermarket gives customers the choice to use automatic payment terminals alone or to go through a checkout managed by an operator.
  1. The waiting time at an automatic terminal, expressed in minutes, is modeled by a random variable that follows the exponential distribution with parameter $0.2\,\mathrm{min}^{-1}$. a. Give the average waiting time for a customer at an automatic payment terminal. b. Calculate the probability, rounded to $10^{-3}$, that the waiting time for a customer at an automatic payment terminal is greater than 10 minutes.
  2. The study commissioned by the manager leads to the following modeling:
    • among customers who chose to use an automatic terminal, 86\% wait less than 10 minutes;
    • among customers using a checkout, 63\% wait less than 10 minutes.
    We randomly choose a customer from the store and define the following events: $B$: ``the customer pays at an automatic terminal''; $\bar{B}$: ``the customer pays at a checkout with an operator''; $S$: ``the customer's waiting time during payment is less than 10 minutes''.
    A waiting time greater than ten minutes at a checkout with an operator or at an automatic terminal creates a negative perception of the store in the customer. The manager wants more than 75\% of customers to wait less than 10 minutes. What is the minimum proportion of customers who must choose an automatic payment terminal for this objective to be achieved?

Part D - Gift vouchers
During payment, scratch cards, winning or losing, are distributed to customers. The number of cards distributed depends on the amount of purchases. Each customer receives one scratch card per 10~\euro{} of purchases. For example, if the purchase amount is 58.64~\euro{}, then the customer receives 5 cards; if the amount is 124.31~\euro{}, the customer receives 12 cards. Winning cards represent 0.5\% of the entire stock of cards. Furthermore, this stock is large enough to treat the distribution of a card as a draw with replacement.
  1. A customer makes purchases for an amount of 158.02~\euro{}.
    What is the probability, rounded to $10^{-2}$, that they obtain at least one winning card?
  2. From what purchase amount, rounded to 10~\euro{}, is the probability of obtaining at least one winning card greater than 50\%?
Q2 Stationary points and optimisation Geometric or applied optimisation problem View
Exercise 2

During a laboratory experiment, a projectile is launched into a fluid medium. The objective is to determine for which firing angle $\theta$ with respect to the horizontal the height of the projectile does not exceed 1.6 meters. Since the projectile does not move through air but through a fluid, the usual parabolic model is not adopted. Here we model the projectile as a point that moves, in a vertical plane, on the curve representing the function $f$ defined on the interval $[0; 1[$ by: $$f(x) = bx + 2\ln(1-x)$$ where $b$ is a real parameter greater than or equal to 2, $x$ is the abscissa of the projectile, $f(x)$ its ordinate, both expressed in meters.
  1. The function $f$ is differentiable on the interval $[0; 1[$. We denote $f'$ its derivative function.
    We admit that the function $f$ has a maximum on the interval $[0; 1[$ and that, for every real $x$ in the interval $[0; 1[$: $$f'(x) = \frac{-bx + b - 2}{1 - x}$$ Show that the maximum of the function $f$ is equal to $b - 2 + 2\ln\left(\frac{2}{b}\right)$.
  2. Determine for which values of the parameter $b$ the maximum height of the projectile does not exceed 1.6 meters.
  3. In this question, we choose $b = 5.69$.
    The firing angle $\theta$ corresponds to the angle between the abscissa axis and the tangent to the curve of the function $f$ at the point with abscissa 0. Determine an approximate value of the angle $\theta$ to the nearest tenth of a degree.
Q3 5 marks Vectors 3D & Lines Shortest Distance Between Two Lines View
Exercise 3 (5 points)

We place ourselves in space equipped with an orthonormal coordinate system whose origin is point A. We consider the points $\mathrm{B}(10; -8; 2)$, $\mathrm{C}(-1; -8; 5)$ and $\mathrm{D}(14; 4; 8)$.
  1. a. Determine a system of parametric equations for each of the lines (AB) and (CD). b. Verify that the lines (AB) and (CD) are not coplanar.
  2. We consider the point I on the line (AB) with abscissa 5 and the point J on the line (CD) with abscissa 4. a. Determine the coordinates of points I and J and deduce the distance IJ. b. Demonstrate that the line (IJ) is perpendicular to the lines (AB) and (CD). The line (IJ) is called the common perpendicular to the lines (AB) and (CD).
  3. The purpose of this question is to verify that the distance IJ is the minimum distance between the lines (AB) and (CD). We consider a point $M$ on the line (AB) distinct from point I. We consider a point $M'$ on the line (CD) distinct from point J. a. Justify that the parallel to the line (IJ) passing through point $M'$ intersects the line $\Delta$ (the line parallel to (CD) passing through I) at a point that we will denote $P$. b. Demonstrate that the triangle $MPM'$ is right-angled at $P$. c. Justify that $MM' > IJ$ and conclude.
Q4 5 marks Vectors Introduction & 2D Vector Word Problem / Physical Application View
Exercise 4 (5 points)
Candidates who have not followed the specialization course

A radio-controlled scooter moves in a straight line at the constant speed of $1\,\mathrm{m.s}^{-1}$. It is pursued by a dog that moves at the same speed. We represent the situation from above in an orthonormal coordinate system of the plane with unit 1 meter. The origin of this coordinate system is the initial position of the dog. The scooter is represented by a point belonging to the line with equation $x = 5$. It moves on this line in the direction of increasing ordinates.
Part A - Modeling using a sequence
At the initial instant, the scooter is represented by the point $S_0$. The dog pursuing it is represented by the point $M_0$. We consider that at each second, the dog instantly orients itself in the direction of the scooter and moves in a straight line over a distance of 1 meter. We then model the trajectories of the dog and the scooter by two sequences of points denoted $(M_n)$ and $(S_n)$. After $n$ seconds, the coordinates of point $S_n$ are $(5; n)$. We denote $(x_n; y_n)$ the coordinates of point $M_n$.
  1. Construct on graph $\mathrm{n}^\circ 1$ given in the appendix the points $M_2$ and $M_3$.
  2. We denote $d_n$ the distance between the dog and the scooter $n$ seconds after the start of the pursuit, $d_n = M_nS_n$. Calculate $d_0$ and $d_1$.
  3. Justify that the point $M_2$ has coordinates $\left(1 + \frac{4}{\sqrt{17}}; \frac{1}{\sqrt{17}}\right)$.
  4. We admit that, for every natural integer $n$: $$\left\{\begin{array}{l} x_{n+1} = x_n + \dfrac{5 - x_n}{d_n} \\[6pt] y_{n+1} = y_n + \dfrac{n - y_n}{d_n} \end{array}\right.$$ a. The table below, obtained using a spreadsheet, gives the coordinates of points $M_n$ and $S_n$ as well as the distance $d_n$ as a function of $n$. What formulas should be written in cells C5 and F5 and copied downward to fill columns C and F?
    ABCDEF
    1$n$\multicolumn{2}{|c|}{$M_n$}\multicolumn{2}{|c|}{$S_n$}$d_n$
    2$x_n$$y_n$5n
    3000505
    4110514.12310563
    521.9701425$\cdots$$\cdots$$\cdots$$\cdots$

Q5 5 marks Matrices Matrix Power Computation and Application View
Exercise 4 (5 points)
Candidates who followed the specialization course

We denote $u_n$ as the number of voles and $v_n$ as the number of foxes on July $1^{\text{st}}$ of the year $2012 + n$.
Part A - A simple model
We model the evolution of populations using the following relations: $$\left\{\begin{array}{l} u_{n+1} = 1{,}1\, u_n - 2000\, v_n \\ v_{n+1} = 2 \times 10^{-5}\, u_n + 0{,}6\, v_n \end{array}\right. \quad \text{for all integers } n \geqslant 0, \text{ with } u_0 = 2000000 \text{ and } v_0 = 120.$$
  1. a. We consider the column matrix $U_n = \binom{u_n}{v_n}$ for all integers $n \geqslant 0$.
    Determine the matrix $A$ such that $U_{n+1} = A \times U_n$ for all integers $n$ and give the matrix $U_0$. b. Calculate the number of voles and foxes estimated using this model on July $1^{\text{st}}$ 2018.
  2. Let the matrices $P = \left(\begin{array}{cc} 20000 & 5000 \\ 1 & 1 \end{array}\right)$, $D = \left(\begin{array}{cc} 1 & 0 \\ 0 & 0{,}7 \end{array}\right)$ and $P^{-1} = \dfrac{1}{15000}\left(\begin{array}{cc} 1 & -5000 \\ -1 & 20000 \end{array}\right)$.
    We admit that $P^{-1}$ is the inverse matrix of matrix $P$ and that $A = P \times D \times P^{-1}$. a. Show that for all natural integers $n$, $U_n = P \times D^n \times P^{-1} \times U_0$. b. Give without justification the expression of matrix $D^n$ as a function of $n$. c. We admit that, for all natural integers $n$: $$\left\{\begin{array}{lcl} u_n & = & \dfrac{2{,}8 \times 10^7 + 2 \times 10^6 \times 0{,}7^n}{15} \\[6pt] v_n & = & \dfrac{1400 + 400 \times 0{,}7^n}{15} \end{array}\right.$$ Describe the evolution of the two populations.

Part B - A model more in line with reality
We construct another model using the following relations: $$\left\{\begin{array}{l} u_{n+1} = 1{,}1\, u_n - 0{,}001\, u_n \times v_n \\ v_{n+1} = 2 \times 10^{-7}\, u_n \times v_n + 0{,}6\, v_n \end{array}\right. \quad \text{for all integers } n \geqslant 0, \text{ with } u_0 = 2000000 \text{ and } v_0 = 120.$$
  1. What formulas must be written in cells B4 and C4 and copied downwards to fill columns B and C?
  2. With the second model, from what year do we observe the phenomenon described (decrease in foxes and increase in voles)?

Part C
In this part we use the model from Part B. Is it possible to give $u_0$ and $v_0$ values such that the two populations remain stable from one year to the next, that is, such that for all natural integers $n$ we have $u_{n+1} = u_n$ and $v_{n+1} = v_n$? (We then speak of a stable state.)