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
2021 bac-spe-maths__asie_j2

5 maths questions

QA Differential equations Applied Modeling with Differential Equations View
Exercise A (Main topics: Sequences, Differential equations)
In this exercise, we are interested in the growth of Moso bamboo with maximum height 20 meters. Ludwig von Bertalanffy's growth model assumes that the growth rate for such bamboo is proportional to the difference between its height and the maximum height.
Part I: discrete model
In this part, we observe a bamboo with initial height 1 meter. For every natural integer $n$, we denote $u_n$ the height, in meters, of the bamboo $n$ days after the start of observation. Thus $u_0 = 1$. Von Bertalanffy's model for bamboo growth between two consecutive days is expressed by the equality: $$u_{n+1} = u_n + 0.05\left(20 - u_n\right) \text{ for every natural integer } n.$$
  1. Verify that $u_1 = 1.95$.
  2. a. Show that for every natural integer $n$, $u_{n+1} = 0.95 u_n + 1$. b. We set for every natural integer $n$, $v_n = 20 - u_n$. Prove that the sequence $(v_n)$ is a geometric sequence and specify its initial term $v_0$ and its common ratio. c. Deduce that, for every natural integer $n$, $u_n = 20 - 19 \times 0.95^n$.
  3. Determine the limit of the sequence $(u_n)$.

Part II: continuous model
In this part, we wish to model the height of the same Moso bamboo by a function giving its height, in meters, as a function of time $t$ expressed in days. According to von Bertalanffy's model, this function is a solution of the differential equation $$(E) \quad y^{\prime} = 0.05(20 - y)$$ where $y$ denotes a function of the variable $t$, defined and differentiable on $[0; +\infty[$ and $y^{\prime}$ denotes its derivative function. Let the function $L$ defined on the interval $[0; +\infty[$ by $$L(t) = 20 - 19\mathrm{e}^{-0.05t}$$
  1. Verify that the function $L$ is a solution of $(E)$ and that we also have $L(0) = 1$.
  2. We take this function $L$ as our model and we admit that, if we denote $L^{\prime}$ its derivative function, $L^{\prime}(t)$ represents the growth rate of the bamboo at time $t$. a. Compare $L^{\prime}(0)$ and $L^{\prime}(5)$. b. Calculate the limit of the derivative function $L^{\prime}$ at $+\infty$. Is this result consistent with the description of the growth model presented at the beginning of the exercise?
QB Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
Exercise B (Main topics: Sequences, function study, Logarithm function)
Let the function $f$ defined on the interval $]1; +\infty[$ by $$f(x) = x - \ln(x-1).$$ Consider the sequence $(u_n)$ with initial term $u_0 = 10$ and such that $u_{n+1} = f(u_n)$ for every natural integer $n$.
Part I:
The spreadsheet below was used to obtain approximate values of the first terms of the sequence $(u_n)$.
AB
1$n$$u_n$
2010
317.80277542
425.88544474
534.29918442
643.10550913
752.36095182
862.0527675
972.00134509
1082.0000009

  1. What formula was entered in cell B3 to allow the calculation of approximate values of $(u_n)$ by copying downward?
  2. Using these values, conjecture the direction of variation and the limit of the sequence $(u_n)$.

Part II:
We recall that the function $f$ is defined on the interval $]1; +\infty[$ by $$f(x) = x - \ln(x-1).$$
  1. Calculate $\lim_{x \rightarrow 1} f(x)$. We will admit that $\lim_{x \rightarrow +\infty} f(x) = +\infty$.
  2. a. Let $f^{\prime}$ be the derivative function of $f$. Show that for all $x \in ]1; +\infty[$, $f^{\prime}(x) = \frac{x-2}{x-1}$. b. Deduce the table of variations of $f$ on the interval $]1; +\infty[$, completed by the limits. c. Justify that for all $x \geqslant 2$, $f(x) \geqslant 2$.

Part III:
  1. Using the results of Part II, prove by induction that $u_n \geqslant 2$ for every natural integer $n$.
  2. Show that the sequence $(u_n)$ is decreasing.
  3. Deduce that the sequence $(u_n)$ is convergent. We denote its limit by $\ell$.
  4. We admit that $\ell$ satisfies $f(\ell) = \ell$. Give the value of $\ell$.
Q1 Curve Sketching Multi-Statement Verification (Remarks/Options) View
This exercise is a multiple choice questionnaire (MCQ). For each question, three statements are proposed, only one of these statements is correct.
  1. Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \left(x^2 - 2x - 1\right)\mathrm{e}^x$$ A. The derivative function of $f$ is the function defined by $f^{\prime}(x) = (2x-2)\mathrm{e}^x$.
    B. The function $f$ is decreasing on the interval $]-\infty; 2]$.
    C. $\lim_{x \rightarrow -\infty} f(x) = 0$.
  2. Consider the function $f$ defined on $\mathbb{R}$ by $f(x) = \frac{3}{5 + \mathrm{e}^x}$.
    Its representative curve in a coordinate system has:
    A. only one horizontal asymptote;
    B. one horizontal asymptote and one vertical asymptote;
    C. two horizontal asymptotes.
  3. Below is the curve $\mathcal{C}_{f^{\prime\prime}}$ representing the second derivative function $f^{\prime\prime}$ of a function $f$ defined and twice differentiable on the interval $[-3.5; 6]$.
    A. The function $f$ is convex on the interval $[-3; 3]$.
    B. The function $f$ has three inflection points.
    C. The derivative function $f^{\prime}$ of $f$ is decreasing on the interval $[0; 2]$.
  4. Consider the sequence $(u_n)$ defined for every natural integer $n$ by $u_n = n^2 - 17n + 20$.
    A. The sequence $(u_n)$ is bounded below.
    B. The sequence $(u_n)$ is decreasing.
    C. One of the terms of the sequence $(u_n)$ equals 2021.
  5. Consider the sequence $(u_n)$ defined by $u_0 = 2$ and, for every natural integer $n$, $u_{n+1} = 0.75 u_n + 5$. Consider the following ``threshold'' function written in Python: \begin{verbatim} def seuil() : u = 2 n = 0 while u < 45 : u = 0,75*u + 5 n = n+1 return n \end{verbatim} This function returns:
    A. the smallest value of $n$ such that $u_n \geqslant 45$;
    B. the smallest value of $n$ such that $u_n < 45$;
    C. the largest value of $n$ such that $u_n \geqslant 45$.
Q2 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Consider a rectangular parallelepiped ABCDEFGH such that $\mathrm{AB} = \mathrm{AD} = 1$ and $\mathrm{AE} = 2$. Point I is the midpoint of segment [AE]. Point K is the midpoint of segment [DC]. Point L is defined by: $\overrightarrow{\mathrm{DL}} = \frac{3}{2}\overrightarrow{\mathrm{AI}}$. N is the orthogonal projection of point D onto the plane (AKL).
We use the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AI}})$. We admit that point L has coordinates $\left(0; 1; \frac{3}{2}\right)$.
  1. Determine the coordinates of vectors $\overrightarrow{\mathrm{AK}}$ and $\overrightarrow{\mathrm{AL}}$.
  2. a. Prove that the vector $\vec{n}$ with coordinates $(6; -3; 2)$ is a normal vector to the plane (AKL). b. Deduce a Cartesian equation of the plane (AKL). c. Determine a system of parametric equations of the line $\Delta$ passing through D and perpendicular to the plane (AKL). d. Deduce that the point N with coordinates $\left(\frac{18}{49}; \frac{40}{49}; \frac{6}{49}\right)$ is the orthogonal projection of point D onto the plane (AKL).

We recall that the volume $\mathcal{V}$ of a tetrahedron is given by the formula: $$\mathcal{V} = \frac{1}{3} \times (\text{area of the base}) \times \text{height.}$$
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  1. a. Calculate the volume of tetrahedron ADKL using triangle ADK as the base. b. Calculate the distance from point D to the plane (AKL). c. Deduce from the previous questions the area of triangle AKL.
Q3 5 marks Combinations & Selection Combinatorial Probability View
An online gaming company offers a new smartphone application called ``Heart Tickets!''. Each participant generates on their smartphone a ticket containing a $3 \times 3$ grid on which three hearts are placed randomly. The ticket is winning if the three hearts are positioned side by side on the same line, on the same column or on the same diagonal.
  1. Justify that there are exactly 84 different ways to position the three hearts on a grid.
  2. Show that the probability that a ticket is winning equals $\frac{2}{21}$.
  3. When a player generates a ticket, the company deducts \euro{}1 from their bank account. If the ticket is winning, the company then gives the player \euro{}5. Is the game favorable to the player?
  4. A player decides to generate 20 tickets on this application. We assume that the generations of tickets are independent of each other. a. Give the probability distribution of the random variable $X$ which counts the number of winning tickets among the 20 tickets generated. b. Calculate the probability, rounded to $10^{-3}$, of the event $(X = 5)$. c. Calculate the probability, rounded to $10^{-3}$, of the event $(X \geqslant 1)$ and interpret the result in the context of the exercise.