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__caledonie_j2

4 maths questions

Q1 7 marks Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
Exercise 1 (7 points) — Main topics covered: Probability
In basketball, there are two types of shots:
  • two-point shots: taken near the basket and score two points if successful.
  • three-point shots: taken far from the basket and score three points if successful.
Stéphanie is practising shooting. We have the following data:
  • One quarter of her shots are two-point shots. Among these, $60\%$ are successful.
  • Three quarters of her shots are three-point shots. Among these, $35\%$ are successful.

  1. Stéphanie takes a shot. Consider the following events: $D$: ``It is a two-point shot''. $R$: ``the shot is successful''. a. Represent the situation using a probability tree. b. Calculate the probability $p(\bar{D} \cap R)$. c. Prove that the probability that Stéphanie successfully makes a shot is equal to 0.4125. d. Stéphanie successfully makes a shot. Calculate the probability that it is a three-point shot. Round the result to the nearest hundredth.
  2. Stéphanie now takes a series of 10 three-point shots. Let $X$ be the random variable that counts the number of successful shots. Consider that the shots are independent. Recall that the probability that Stéphanie successfully makes a three-point shot is equal to 0.35. a. Justify that $X$ follows a binomial distribution. Specify its parameters. b. Calculate the expected value of $X$. Interpret the result in the context of the exercise. c. Determine the probability that Stéphanie misses 4 or more shots. Round the result to the nearest hundredth. d. Determine the probability that Stéphanie misses at most 4 shots. Round the result to the nearest hundredth.
  3. Let $n$ be a non-zero natural number. Stéphanie wishes to take a series of $n$ three-point shots. Consider that the shots are independent. Recall that the probability that she successfully makes a three-point shot is equal to 0.35. Determine the minimum value of $n$ so that the probability that Stéphanie successfully makes at least one shot among the $n$ shots is greater than or equal to 0.99.
Q2 Differentiating Transcendental Functions Full function study with transcendental functions View
Exercise 2 — Main topics covered: functions, logarithm function.
Let $f$ be the function defined on the interval $]0; +\infty[$ by: $$f(x) = x\ln(x) - x - 2.$$ We admit that the function $f$ is twice differentiable on $]0; +\infty[$. We denote $f'$ its derivative, $f''$ its second derivative and $\mathcal{C}_f$ its representative curve in a coordinate system.
  1. a. Prove that, for all $x$ belonging to $]0; +\infty[$, we have $f'(x) = \ln(x)$. b. Determine an equation of the tangent line $T$ to the curve $\mathcal{C}_f$ at the point with abscissa $x = \mathrm{e}$. c. Justify that the function $f$ is convex on the interval $]0; +\infty[$. d. Deduce the relative position of the curve $\mathcal{C}_f$ and the tangent line $T$.
  2. a. Calculate the limit of the function $f$ at 0. b. Prove that the limit of the function $f$ at $+\infty$ is equal to $+\infty$.
  3. Draw up the table of variations of the function $f$ on the interval $]0; +\infty[$.
  4. a. Prove that the equation $f(x) = 0$ has a unique solution in the interval $]0; +\infty[$. We denote this solution by $\alpha$. b. Justify that the real number $\alpha$ belongs to the interval $]4{,}3; 4{,}4[$. c. Deduce the sign of the function $f$ on the interval $]0; +\infty[$.
  5. Consider the following threshold function written in Python: Recall that the \texttt{log} function of the \texttt{math} module (which we assume is imported) denotes the natural logarithm function $\ln$. \begin{verbatim} def seuil(pas) : x=4.3 while x*log(x) - x - 2 < 0: x=x+pas return x \end{verbatim} What is the value returned when calling the function \texttt{seuil(0.01)}? Interpret this result in the context of the exercise.
Q3 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 3 — Main topics covered: geometry in space.
A house is modelled by a rectangular parallelepiped ABCDEFGH topped with a pyramid EFGHS. We have $\mathrm{DC} = 6$, $\mathrm{DA} = \mathrm{DH} = 4$. Let the points I, J and K be such that $$\overrightarrow{\mathrm{DI}} = \frac{1}{6}\overrightarrow{\mathrm{DC}}, \quad \overrightarrow{\mathrm{DJ}} = \frac{1}{4}\overrightarrow{\mathrm{DA}}, \quad \overrightarrow{\mathrm{DK}} = \frac{1}{4}\overrightarrow{\mathrm{DH}}.$$ We denote $\vec{\imath} = \overrightarrow{\mathrm{DI}}$, $\vec{\jmath} = \overrightarrow{\mathrm{DJ}}$, $\vec{k} = \overrightarrow{\mathrm{DK}}$. We use the orthonormal coordinate system $(\mathrm{D}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We admit that point S has coordinates $(3; 2; 6)$.
  1. Give, without justification, the coordinates of points $\mathrm{B}$, $\mathrm{E}$, $\mathrm{F}$ and G.
  2. Prove that the volume of the pyramid EFGHS represents one seventh of the total volume of the house. Recall that the volume $V$ of a tetrahedron is given by the formula: $$V = \frac{1}{3} \times (\text{area of the base}) \times \text{height}.$$
  3. a. Prove that the vector $\vec{n}$ with coordinates $\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}$ is normal to the plane (EFS). b. Deduce that a Cartesian equation of the plane (EFS) is $y + z - 8 = 0$.
  4. An antenna is installed on the roof, represented by the segment $[\mathrm{PQ}]$. We have the following data:
    • point P belongs to the plane (EFS);
    • point Q has coordinates $(2; 3; 5{,}5)$;
    • the line (PQ) is directed by the vector $\vec{k}$.
    a. Justify that a parametric representation of the line (PQ) is: $$\left\{\begin{aligned} x &= 2 \\ y &= 3 \\ z &= 5{,}5 + t \end{aligned} \quad (t \in \mathbb{R})\right.$$ b. Deduce the coordinates of point $P$. c. Deduce the length PQ of the antenna.
  5. A bird flies following a trajectory modelled by the line $\Delta$ whose parametric representation is: $$\left\{\begin{aligned} x &= -4 + 6s \\ y &= 7 - 4s \\ z &= 2 + 4s \end{aligned} \quad (s \in \mathbb{R})\right.$$ Determine the relative position of the lines (PQ) and $\Delta$. Will the bird collide with the antenna represented by the segment $[\mathrm{PQ}]$?
Q4 7 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
Exercise 4 (7 points) — Main topics covered: sequences, functions, antiderivatives.
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers or the absence of an answer to a question neither awards nor deducts points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
  1. Consider the sequence $(u_n)$ defined for all natural number $n$ by $$u_n = \frac{(-1)^n}{n+1}.$$ We can affirm that: a. the sequence $(u_n)$ diverges to $+\infty$. b. the sequence $(u_n)$ diverges to $-\infty$. c. the sequence $(u_n)$ has no limit. d. the sequence $(u_n)$ converges.
    In questions 2 and 3, we consider two sequences $(v_n)$ and $(w_n)$ satisfying the relation: $$w_n = \mathrm{e}^{-2v_n} + 2.$$
  2. Let $a$ be a strictly positive real number. We have $v_0 = \ln(a)$. a. $w_0 = \dfrac{1}{a^2} + 2$ b. $w_0 = \dfrac{1}{a^2 + 2}$ c. $w_0 = -2a + 2$ d. $w_0 = \dfrac{1}{-2a} + 2$
  3. We know that the sequence $(v_n)$ is increasing. We can affirm that the sequence $(w_n)$ is: a. decreasing and bounded above by 3. b. decreasing and bounded below by 2. c. increasing and bounded above by 3. d. increasing and bounded below by 2.
  4. Consider the sequence $(a_n)$ defined as follows: $$a_0 = 2 \text{ and, for all natural number } n, \quad a_{n+1} = \frac{1}{3}a_n + \frac{8}{3}.$$ For all natural number $n$, we have: a. $a_n = 4 \times \left(\dfrac{1}{3}\right)^n - 2$ b. $a_n = -\dfrac{2}{3^n} + 4$ c. $a_n = 4 - \left(\dfrac{1}{3}\right)^n$ d. $a_n = 2 \times \left(\dfrac{1}{3}\right)^n + \dfrac{8n}{3}$
  5. Consider a sequence $(b_n)$ such that, for all natural number $n$, we have: $$b_{n+1} = b_n + \ln\left(\frac{2}{(b_n)^2 + 3}\right)$$ We can affirm that: a. the sequence $(b_n)$ is increasing. b. the sequence $(b_n)$ is decreasing. c. the sequence $(b_n)$ is not monotone. d. the direction of variation of the sequence $(b_n)$ depends on $b_0$.
  6. Consider the function $g$ defined on the interval $]0; +\infty[$ by: $$g(x) = \frac{\mathrm{e}^x}{x}$$ We denote $\mathcal{C}_g$ the representative curve of the function $g$ in an orthogonal coordinate system. The curve $\mathcal{C}_g$ has: a. a vertical asymptote and a horizontal asymptote. b. a vertical asymptote and no horizontal asymptote. c. no vertical asymptote and a horizontal asymptote. d. no vertical asymptote and no horizontal asymptote.
  7. Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{x^2+1}$$ Let $F$ be an antiderivative on $\mathbb{R}$ of the function $f$. For all real $x$, we have: a. $F(x) = \dfrac{1}{2}x^2\mathrm{e}^{x^2+1}$ b. $F(x) = \left(1 + 2x^2\right)\mathrm{e}^{x^2+1}$ c. $F(x) = \mathrm{e}^{x^2+1}$ d. $F(x) = \dfrac{1}{2}\mathrm{e}^{x^2+1}$