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
2023 bac-spe-maths__caledonie_j1

4 maths questions

Q1 5 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
A boat rental company for tourism offers its clients two types of boats: sailboat and motorboat.
Furthermore, a client can take the PILOT option. In this case, the boat, whether sailboat or motorboat, is rented with a pilot.
We know that:
  • $60\%$ of clients choose a sailboat; among them, $20\%$ take the PILOT option.
  • $42\%$ of clients take the PILOT option.

A client is chosen at random and we consider the events:
  • $V$: ``the client chooses a sailboat'';
  • $L$: ``the client takes the PILOT option''.

Part A
  1. Represent the situation with a probability tree that you will complete as you go.
  2. Calculate the probability that the client chooses a sailboat and does not take the PILOT option.
  3. Prove that the probability that the client chooses a motorboat and takes the PILOT option is equal to 0.30.
  4. Deduce $P_{\bar{V}}(L)$, the probability of $L$ given that $V$ is not realized.
  5. A client has taken the PILOT option. What is the probability that he chose a sailboat? Round to 0.01.

Part B
When a client does not take the PILOT option, the probability that his boat suffers a breakdown is equal to 0.12. This probability is only 0.005 if the client takes the PILOT option. We consider a client. We denote by $A$ the event: ``his boat suffers a breakdown''.
  1. Determine $P(L \cap A)$ and $P(\bar{L} \cap A)$.
  2. The company rents 1000 boats. How many breakdowns can it expect?

Part C
We recall that the probability that a given client takes the PILOT option is equal to 0.42. We consider a random sample of 40 clients. We denote by $X$ the random variable counting the number of clients in the sample taking the PILOT option.
  1. We admit that the random variable $X$ follows a binomial distribution. Give its parameters without justification.
  2. Calculate the probability, rounded to $10^{-3}$, that at least 15 clients take the PILOT option.
Q2 5 marks Proof by induction Prove a sequence bound or inequality by induction View
We consider the sequence $(u_n)$ defined by $u_0 = 3$ and, for every natural integer $n$, by:
$$u_{n+1} = 5u_n - 4n - 3$$
  1. a. Prove that $u_1 = 12$. b. Determine $u_2$ by detailing the calculation. c. Using a calculator, conjecture the direction of variation and the limit of the sequence $(u_n)$.
  2. a. Prove by induction that, for every natural integer $n$, we have: $$u_n \geqslant n + 1.$$ b. Deduce the limit of the sequence $(u_n)$.
  3. We consider the sequence $(v_n)$ defined for every natural integer $n$ by: $$v_n = u_n - n - 1$$ a. Prove that the sequence $(v_n)$ is geometric. Give its common ratio and its first term $v_0$. b. Deduce, for every natural integer $n$, the expression of $v_n$ as a function of $n$. c. Deduce that for every natural integer $n$: $$u_n = 2 \times 5^n + n + 1$$ d. Deduce the direction of variation of the sequence $(u_n)$.
  4. We consider the function below, written incompletely in Python language and intended to return the smallest natural integer $n$ such that $u_n \geqslant 10^7$. a. Copy the program and complete the two missing instructions. b. What is the value returned by this function? \begin{verbatim} def suite() : u = 3 n = 0 while...: u = ... n = n + 1 return n \end{verbatim}
Q3 5 marks Vectors: Lines & Planes MCQ: Identify Correct Equation or Representation View
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
A wrong answer, no answer, or multiple answers, neither earn nor lose points.
  1. We consider the function $f$ defined on $\mathbb{R}$ by: $f(x) = (x + 1)e^x$.
    An antiderivative $F$ of $f$ on $\mathbb{R}$ is defined by: a. $F(x) = 1 + xe^x$ b. $F(x) = (1 + x)e^x$ c. $F(x) = (2 + x)e^x$ d. $F(x) = \left(\frac{x^2}{2} + x\right)e^x$.
  2. We consider the lines $(d_1)$ and $(d_2)$ whose parametric representations are respectively: $$\left(d_1\right) \left\{\begin{array}{l} x = 2 + r \\ y = 1 + r \\ z = -r \end{array} \quad (r \in \mathbb{R}) \quad \text{and} \quad (d_2) \left\{\begin{array}{rl} x & = 1 - s \\ y & = -1 + s \\ z & = 2 - s \end{array} \quad (s \in \mathbb{R})\right.\right.$$ The lines $(d_1)$ and $(d_2)$ are: a. secant. b. strictly parallel. c. coincident. d. non-coplanar.
  3. We consider the plane $(P)$ whose Cartesian equation is: $$2x - y + z - 1 = 0$$ We consider the line $(\Delta)$ whose parametric representation is: $$\left\{\begin{array}{l} x = 2 + u \\ y = 4 + u \quad (u \in \mathbb{R}) \\ z = 1 - u \end{array}\right.$$ The line $(\Delta)$ is: a. secant and non-orthogonal to the plane $(P)$. b. included in the plane $(P)$. c. strictly parallel to the plane $(P)$. d. orthogonal to the plane $(P)$.
  4. We consider the plane $(P_1)$ whose Cartesian equation is $x - 2y + z + 1 = 0$, as well as the plane $(P_2)$ whose Cartesian equation is $2x + y + z - 6 = 0$. The planes $(P_1)$ and $(P_2)$ are: a. secant and perpendicular. b. coincident. c. secant and non-perpendicular. d. strictly parallel.
  5. We consider the points $E(1; 2; 1)$, $F(2; 4; 3)$ and $G(-2; 2; 5)$.
    We can affirm that the measure $\alpha$ of the angle $\widehat{FEG}$ satisfies: a. $\alpha = 90°$ b. $\alpha > 90°$ c. $\alpha = 0°$ d. $\alpha \approx 71°$
Q4 5 marks Differentiating Transcendental Functions Full function study with transcendental functions View
We consider the function $f$ defined for every real $x$ in the interval $]0; +\infty[$ by:
$$f(x) = 5x^2 + 2x - 2x^2\ln(x).$$
We denote by $\mathscr{C}_f$ the representative curve of $f$ in an orthogonal reference frame of the plane. We admit that $f$ is twice differentiable on the interval $]0; +\infty[$. We denote by $f'$ its derivative and $f''$ its second derivative.
  1. a. Prove that the limit of the function $f$ at 0 is equal to 0. b. Determine the limit of the function $f$ at $+\infty$.
  2. Determine $f'(x)$ for every real $x$ in the interval $]0; +\infty[$.
  3. a. Prove that for every real $x$ in the interval $]0; +\infty[$ $$f''(x) = 4(1 - \ln(x)).$$ b. Deduce the largest interval on which the curve $\mathscr{C}_f$ is above its tangent lines. c. Draw the variation table of the function $f'$ on the interval $]0; +\infty[$. (We will admit that $\lim_{\substack{x \to 0 \\ x > 0}} f'(x) = 2$ and that $\lim_{x \to +\infty} f'(x) = -\infty$.)
  4. a. Show that the equation $f'(x) = 0$ admits in the interval $]0; +\infty[$ a unique solution $\alpha$ for which we will give an enclosure of amplitude $10^{-2}$. b. Deduce the sign of $f'(x)$ on the interval $]0; +\infty[$ as well as the variation table of the function $f$ on the interval $]0; +\infty[$.
  5. a. Using the equality $f'(\alpha) = 0$, prove that: $$\ln(\alpha) = \frac{4\alpha + 1}{2\alpha}.$$ Deduce that $f(\alpha) = \alpha^2 + \alpha$. b. Deduce an enclosure of amplitude $10^{-1}$ of the maximum of the function $f$.