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
2014 metropole-sept

5 maths questions

Q1 Differentiating Transcendental Functions Determine parameters from function or curve conditions View
On the graph below, we have drawn, in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, a curve $\mathscr{C}$ and the line $(\mathrm{AB})$ where A and B are the points with coordinates $(0;1)$ and $(-1;3)$ respectively.
We denote by $f$ the function differentiable on $\mathbb{R}$ whose representative curve is $\mathscr{C}$. We further assume that there exists a real number $a$ such that for all real $x$, $$f(x) = x + 1 + ax\mathrm{e}^{-x^{2}}$$
  1. a. Justify that the curve $\mathscr{C}$ passes through point A. b. Determine the slope of the line (AB). c. Prove that for all real $x$, $$f^{\prime}(x) = 1 - a\left(2x^{2} - 1\right)\mathrm{e}^{-x^{2}}$$ d. We assume that the line $(\mathrm{AB})$ is tangent to the curve $\mathscr{C}$ at point A. Determine the value of the real number $a$.
  2. According to the previous question, for all real $x$, $$f(x) = x + 1 - 3x\mathrm{e}^{-x^{2}} \text{ and } f^{\prime}(x) = 1 + 3\left(2x^{2} - 1\right)\mathrm{e}^{-x^{2}}.$$ a. Prove that for all real $x$ in the interval $]-1;0]$, $f(x) > 0$. b. Prove that for all real $x$ less than or equal to $-1$, $f^{\prime}(x) > 0$. c. Prove that there exists a unique real number $c$ in the interval $\left[-\frac{3}{2};-1\right]$ such that $f(c) = 0$. Justify that $c < -\frac{3}{2} + 2 \cdot 10^{-2}$.
  3. We denote by $\mathscr{A}$ the area, expressed in square units, of the region defined by: $$c \leqslant x \leqslant 0 \quad \text{and} \quad 0 \leqslant y \leqslant f(x)$$ a. Write $\mathscr{A}$ in the form of an integral. b. We admit that the integral $I = \int_{-\frac{3}{2}}^{0} f(x)\,\mathrm{d}x$ is an approximate value of $\mathscr{A}$ to within $10^{-3}$. Calculate the exact value of the integral $I$.
Q2 Exponential Distribution View
In this exercise, we are interested in the operating mode of two restaurants: without reservation or with prior reservation.
  1. The first restaurant operates without reservation but the waiting time to obtain a table is often a problem for customers. We model this waiting time in minutes by a random variable $X$ which follows an exponential distribution with parameter $\lambda$ where $\lambda$ is a strictly positive real number. We recall that the mathematical expectation of $X$ is equal to $\frac{1}{\lambda}$.
    A statistical study made it possible to observe that the average waiting time to obtain a table is 10 minutes. a. Determine the value of $\lambda$. b. What is the probability that a customer waits between 10 and 20 minutes to obtain a table? Round to $10^{-4}$. c. A customer has been waiting for 10 minutes. What is the probability that he must wait at least 5 more minutes to obtain a table? Round to $10^{-4}$.
  2. The second restaurant has a capacity of 70 seats and only serves people who have made a prior reservation. The probability that a person who has made a reservation appears at the restaurant is estimated at 0.8. We denote by $n$ the number of reservations taken by the restaurant and $Y$ the random variable corresponding to the number of people who have made a reservation and appear at the restaurant. We admit that the behaviours of people who have made a reservation are independent of each other. The random variable $Y$ then follows a binomial distribution. a. Specify, as a function of $n$, the parameters of the distribution of the random variable $Y$, its mathematical expectation $E(Y)$ and its standard deviation $\sigma(Y)$. b. In this question, we denote by $Z$ a random variable following the normal distribution $\mathscr{N}\left(\mu, \sigma^{2}\right)$ with mean $\mu = 64.8$ and standard deviation $\sigma = 3.6$. Calculate the probability $p_{1}$ of the event $\{Z \leqslant 71\}$ using a calculator. c. We admit that when $n = 81$, $p_{1}$ is an approximate value to within $10^{-2}$ of the probability $p(Y \leqslant 70)$ of the event $\{Z \leqslant 70\}$. The restaurant has received 81 reservations. What is the probability that it cannot accommodate some of the customers who have made a reservation and appear?
Q3 Geometric Sequences and Series Applied Geometric Model with Contextual Interpretation View
A patient is given a medication by intravenous injection. The amount of medication in the blood decreases as a function of time. The purpose of the exercise is to study, for different hypotheses, the evolution of this amount minute by minute.
  1. An injection of 10 mL of medication is performed at time 0. It is estimated that $20\%$ of the medication is eliminated per minute. For all natural integer $n$, we denote by $u_{n}$ the amount of medication, in mL, remaining in the blood after $n$ minutes. Thus $u_{0} = 10$. a. What is the nature of the sequence $\left(u_{n}\right)$? b. For all natural integer $n$, give the expression of $u_{n}$ as a function of $n$. c. After how much time does the amount of medication remaining in the blood become less than $1\%$ of the initial amount? Justify the answer.
  2. A machine performs an injection of 10 mL of medication at time 0. It is estimated that $20\%$ of the medication is eliminated per minute. When the amount of medication falls below 5 mL, the machine reinjects 4 mL of product. After 15 minutes, the machine is stopped. For all natural integer $n$, we denote by $v_{n}$ the amount of medication, in mL, remaining in the blood at minute $n$. The following algorithm gives the amount of medication remaining minute by minute.
    \begin{tabular}{l} Variables:
    Initialization:
    Processing:
    &
    $n$ is a natural integer.
    $v$ is a real number.
    Assign to $v$ the value 10.
    For $n$ going from 1 to 15
    Assign to $v$ the value $0.8 \times v$.
    If $v < 5$ then assign to $v$ the value $v + 4$
    Display $v$.
    End of loop.
    \hline \end{tabular}
    a. Calculate the missing elements of the table below giving, rounded to $10^{-2}$ and for $n$ greater than or equal to 1, the amount of medication remaining minute by minute obtained with the algorithm.
    $n$0123456789101112131415
    $v_{n}$1086.48.156.525.218.176.545.238.186.555.24

    b. After 15 minutes, what total amount of medication has been injected into the body? c. We wish to program the machine so that it injects 2 mL of product when the amount of medication in the blood is less than or equal to 6 mL and that it stops after 30 minutes. Rewrite the previous algorithm by modifying it so that it displays the amount of medication, in mL, remaining in the blood minute by minute with this new protocol.
  3. We program the machine so that:
    • at time 0, it injects 10 mL of medication,
    • every minute, it injects 1 mL of medication.
    It is estimated that $20\%$ of the medication present in the blood is eliminated per minute. For all natural integer $n$, we denote by $w_{n}$ the amount of medication, in mL, present in the blood of the patient after $n$ minutes. a. Justify that for all natural integer $n$, $w_{n+1} = 0.8w_{n} + 1$. b. For all natural integer $n$, we set $z_{n} = w_{n} - 5$. Prove that $(z_{n})$ is a geometric sequence whose ratio and first term we will specify. c. Deduce the expression of $w_{n}$ as a function of $n$. d. What is the limit of the sequence $\left(w_{n}\right)$? What interpretation can be given to this?
Q4a 5 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
(For candidates who have not followed the specialization course)
In space equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the tetrahedron ABCD whose vertices have coordinates: $$\mathrm{A}(1;-\sqrt{3};0);\quad \mathrm{B}(1;\sqrt{3};0);\quad \mathrm{C}(-2;0;0);\quad \mathrm{D}(0;0;2\sqrt{2}).$$
  1. Prove that the plane (ABD) has the Cartesian equation $4x + z\sqrt{2} = 4$.
  2. We denote by $\mathscr{D}$ the line whose parametric representation is $$\left\{\begin{array}{l} x = t \\ y = 0 \\ z = t\sqrt{2} \end{array}, t \in \mathbb{R}\right.$$ a. Prove that $\mathscr{D}$ is the line that is parallel to $(\mathrm{CD})$ and passes through O. b. Determine the coordinates of point G, the intersection of the line $\mathscr{D}$ and the plane (ABD).
  3. a. We denote by L the midpoint of segment $[\mathrm{AC}]$. Prove that the line (BL) passes through point O and is orthogonal to the line (AC). b. Prove that triangle ABC is equilateral and determine the centre of its circumscribed circle.
  4. Prove that the tetrahedron ABCD is regular, that is, a tetrahedron whose six edges all have the same length.
Q4b 5 marks Matrices Matrix Power Computation and Application View
(For candidates who have followed the specialization course)
As part of a study on social interactions between mice, researchers place laboratory mice in a cage with two compartments A and B. The door between these compartments is opened for ten minutes every day at noon. We study the distribution of mice in the two compartments. It is estimated that each day:
  • $20\%$ of the mice present in compartment A before the door opens are found in compartment B after the door closes,
  • $10\%$ of the mice that were in compartment B before the door opens are found in compartment A after the door closes.
We assume that initially, the two compartments A and B contain the same number of mice. We set $a_{0} = 0.5$ and $b_{0} = 0.5$. For all natural integer $n$ greater than or equal to 1, we denote by $a_{n}$ and $b_{n}$ the proportions of mice present respectively in compartments A and B after $n$ days, after the door closes. We denote by $U_{n}$ the matrix $\binom{a_{n}}{b_{n}}$.
  1. Let $n$ be a natural integer. a. Justify that $U_{1} = \binom{0.45}{0.55}$. b. Express $a_{n+1}$ and $b_{n+1}$ as functions of $a_{n}$ and $b_{n}$. c. Deduce that $U_{n+1} = MU_{n}$ where $M$ is a matrix that we will specify. We admit without proof that $U_{n} = M^{n}U_{0}$. d. Determine the distribution of mice in compartments A and B after 3 days.
  2. Let the matrix $P = \left(\begin{array}{cc} 1 & 1 \\ 2 & -1 \end{array}\right)$. a. Calculate $P^{2}$. Deduce that $P$ is invertible and $P^{-1} = \frac{1}{3}P$. b. Verify that $P^{-1}MP$ is a diagonal matrix $D$ that we will specify. c. Prove that for any natural integer $n$ greater than or equal to 1, $M^{n} = PD^{n}P^{-1}$. Using computer algebra software, we obtain $$M^{n} = \left(\begin{array}{cc} \frac{1 + 2 \times 0.7^{n}}{3} & \frac{1 - 0.7^{n}}{3} \\ \frac{2 - 2 \times 0.7^{n}}{3} & \frac{2 + 0.7^{n}}{3} \end{array}\right).$$
  3. Using the help of the previous questions, what can we say about the long-term distribution of mice in compartments A and B of the cage?