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
2022 bac-spe-maths__centres-etrangers_j2

4 maths questions

Q1 7 marks Exponential Functions MCQ on Function Properties View
Exercise 1 — Multiple Choice (Exponential function)
For each of the following questions, only one of the four proposed answers is correct.
  1. Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = \frac{x}{\mathrm{e}^{x}}$$ We assume that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function. a. $f'(x) = \mathrm{e}^{-x}$ b. $f'(x) = x\mathrm{e}^{-x}$ c. $f'(x) = (1-x)\mathrm{e}^{-x}$ d. $f'(x) = (1+x)\mathrm{e}^{-x}$
  2. Let $f$ be a function twice differentiable on the interval $[-3;1]$. The graphical representation of its second derivative function $f''$ is given. We can then affirm that: a. The function $f$ is convex on the interval $[-2;0]$ b. The function $f$ is concave on the interval $[-1;1]$ c. The function $f'$ is decreasing on the interval $[-2;0]$ d. The function $f'$ admits a maximum at $x = -1$
  3. We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = x^3 \mathrm{e}^{-x^2}$$ If $F$ is an antiderivative of $f$ on $\mathbb{R}$, a. $F(x) = -\frac{1}{6}\left(x^3+1\right)\mathrm{e}^{-x^2}$ b. $F(x) = -\frac{1}{4}x^4 \mathrm{e}^{-x^2}$ c. $F(x) = -\frac{1}{2}\left(x^2+1\right)\mathrm{e}^{-x^2}$ d. $F(x) = x^2\left(3-2x^2\right)\mathrm{e}^{-x^2}$
  4. What is the value of: $$\lim_{x \rightarrow +\infty} \frac{\mathrm{e}^x + 1}{\mathrm{e}^x - 1}$$ a. $-1$ b. $1$ c. $+\infty$ d. does not exist
  5. We consider the function $f$ defined on $\mathbb{R}$ by $f(x) = \mathrm{e}^{2x+1}$. The only antiderivative $F$ on $\mathbb{R}$ of the function $f$ such that $F(0) = 1$ is the function: a. $x \longmapsto 2\mathrm{e}^{2x+1} - 2\mathrm{e} + 1$ b. $x \longmapsto 2\mathrm{e}^{2x+1} - \mathrm{e}$ c. $x \longmapsto \frac{1}{2}\mathrm{e}^{2x+1} - \frac{1}{2}\mathrm{e} + 1$ d. $x \longmapsto \mathrm{e}^{x^2+x}$
  6. In a coordinate system, the representative curve of a function $f$ defined and twice differentiable on $[-2;4]$ is drawn. Among the following curves (a, b, c, d), which one represents the function $f''$, the second derivative of $f$?
Q2 7 marks Differentiating Transcendental Functions Full function study with transcendental functions View
Exercise 2 — 7 points Themes: Logarithm function and sequence Let $f$ be the function defined on the interval $]0;+\infty[$ by $$f(x) = x\ln(x) + 1$$ We denote $\mathscr{C}_f$ its representative curve in a coordinate system of the plane.
  1. Determine the limit of the function $f$ at $0$ as well as its limit at $+\infty$.
    1. [a.] We admit that $f$ is differentiable on $]0;+\infty[$ and we denote $f'$ its derivative function. Show that for every strictly positive real number $x$: $$f'(x) = 1 + \ln(x).$$
    2. [b.] Deduce the variation table of the function $f$ on $]0;+\infty[$. The exact value of the extremum of $f$ on $]0;+\infty[$ and the limits must be shown.
    3. [c.] Justify that for all $x \in ]0;1[$, $f(x) \in ]0;1[$.

    1. [a.] Determine an equation of the tangent line $(T)$ to the curve $\mathscr{C}_f$ at the point with abscissa $1$.
    2. [b.] Study the convexity of the function $f$ on $]0;+\infty[$.
    3. [c.] Deduce that for every strictly positive real number $x$: $$f(x) \geqslant x$$

  4. The sequence $(u_n)$ is defined by its first term $u_0$ element of the interval $]0;1[$ and for every natural number $n$: $$u_{n+1} = f(u_n)$$
    1. [a.] Prove by induction that for every natural number $n$, we have: $0 < u_n < 1$.
    2. [b.] Deduce from question 3.c. the increasing nature of the sequence $(u_n)$.
    3. [c.] Deduce that the sequence $(u_n)$ is convergent.
Q3 7 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 3 — 7 points Theme: Geometry in space Space is equipped with an orthonormal coordinate system $(\mathrm{O};\vec{\imath},\vec{\jmath},\vec{k})$. We consider the points $$\mathrm{A}(3;-2;2), \quad \mathrm{B}(6;1;5), \quad \mathrm{C}(6;-2;-1) \quad \text{and} \quad \mathrm{D}(0;4;-1).$$ We recall that the volume of a tetrahedron is given by the formula: $$V = \frac{1}{3}\mathscr{A} \times h$$ where $\mathscr{A}$ is the area of the base and $h$ is the corresponding height.
  1. Prove that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$ are not coplanar.
    1. [a.] Show that the triangle ABC is right-angled.
    2. [b.] Show that the line (AD) is perpendicular to the plane (ABC).
    3. [c.] Deduce the volume of the tetrahedron ABCD.

  3. We consider the point $\mathrm{H}(5;0;1)$.
    1. [a.] Show that there exist real numbers $\alpha$ and $\beta$ such that $\overrightarrow{\mathrm{BH}} = \alpha\overrightarrow{\mathrm{BC}} + \beta\overrightarrow{\mathrm{BD}}$.
    2. [b.] Prove that H is the orthogonal projection of point A onto the plane (BCD).
    3. [c.] Deduce the distance from point A to the plane (BCD).

  4. Deduce from the previous questions the area of triangle BCD.
Q4 7 marks Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
Exercise 4 — 7 points Theme: Probability An urn contains white and black tokens all indistinguishable to the touch.
A game consists of drawing at random successively and with replacement two tokens from this urn. The following game rule is established:
  • a player loses 9 euros if the two tokens drawn are white;
  • a player loses 1 euro if the two tokens drawn are black;
  • a player wins 5 euros if the two tokens drawn are of different colors.

  1. We consider that the urn contains 2 black tokens and 3 white tokens.
    1. [a.] Model the situation using a probability tree.
    2. [b.] Calculate the probability of losing $9\,\text{\euro}$ in one game.

  2. We now consider that the urn contains 3 white tokens and at least two black tokens but we do not know the exact number of black tokens. We will call $N$ the number of black tokens.
    1. [a.] Let $X$ be the random variable giving the gain of the game for one game. Determine the probability distribution of this random variable.
    2. [b.] Solve the inequality for real $x$: $$-x^2 + 30x - 81 > 0$$
    3. [c.] Using the result of the previous question, determine the number of black tokens the urn must contain so that this game is favorable to the player.
    4. [d.] How many black tokens should the player request in order to obtain a maximum average gain?

  3. We observe 10 players who try their luck by playing one game of this game, independently of each other. We assume that 7 black tokens have been placed in the urn (with 3 white tokens). What is the probability of having at least 1 player winning 5 euros?