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__metropole-sept_j1

7 maths questions

QExercise 2 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. Consider the points $$\mathrm{A}(1;0;-1), \quad \mathrm{B}(3;-1;2), \quad \mathrm{C}(2;-2;-1) \quad \text{and} \quad \mathrm{D}(4;-1;-2).$$ We denote by $\Delta$ the line with parametric representation $$\left\{\begin{aligned} x &= 0 \\ y &= 2+t, \text{ with } t \in \mathbb{R}. \\ z &= -1+t \end{aligned}\right.$$
  1. a. Show that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ define a plane which we will denote $\mathscr{P}$. b. Show that the line (CD) is orthogonal to the plane $\mathscr{P}$. On the plane $\mathscr{P}$, what does point C represent with respect to D? c. Show that a Cartesian equation of the plane $\mathscr{P}$ is: $2x + y - z - 3 = 0$.
  2. a. Calculate the distance CD. b. Does there exist a point M on the plane $\mathscr{P}$ different from C satisfying $\mathrm{MD} = \sqrt{6}$? Justify your answer.
  3. a. Show that the line $\Delta$ is contained in the plane $\mathscr{P}$. Let H be the orthogonal projection of point D onto the line $\Delta$. b. Show that H is the point of $\Delta$ associated with the value $t = -2$ in the parametric representation of $\Delta$ given above. c. Deduce the distance from point D to the line $\Delta$.
QExercise 3 4 marks Conditional Probability Conditional Probability as a Function of a Parameter View
4 points Parts $\boldsymbol{A}$ and $\boldsymbol{B}$ are independent. The requested probabilities will be given to $10^{-3}$ near. To help detect certain allergies, a blood test can be performed whose result is either positive or negative. In a population, this test gives the following results:
  • If an individual is allergic, the test is positive in $97\%$ of cases;
  • If an individual is not allergic, the test is negative in $95{,}7\%$ of cases.
Furthermore, $20\%$ of individuals in the concerned population have a positive test. We randomly choose an individual from the population, and we denote:
  • $A$ the event ``the individual is allergic'';
  • $T$ the event ``the individual has a positive test''.
We denote by $\bar{A}$ and $\bar{T}$ the complementary events of $A$ and $T$. We also call $x$ the probability of event $A$: $x = p(A)$.
Part A
  1. Reproduce and complete the tree describing the situation, indicating on each branch the corresponding probability.
  2. a. Prove the equality: $p(T) = 0{,}927x + 0{,}043$. b. Deduce the probability that the chosen individual is allergic.
  3. Justify by a calculation the following statement: ``If the test of an individual chosen at random is positive, there is more than $80\%$ chance that this individual is allergic''.

Part B
A survey on allergies is conducted in a city by interviewing 150 randomly chosen residents, and we assume that this choice amounts to successive independent draws with replacement. We know that the probability that a randomly chosen resident in this city is allergic is equal to $0{,}08$. We denote by $X$ the random variable that associates to a sample of 150 randomly chosen residents the number of allergic people in this sample.
  1. What is the probability distribution followed by the random variable $X$? Specify its parameters.
  2. Determine the probability that exactly 20 people among the 150 interviewed are allergic.
  3. Determine the probability that at least $10\%$ of the people among the 150 interviewed are allergic.
QExercise 4 Applied differentiation Existence and number of solutions via calculus View
PART A We define on the interval $]0;+\infty[$ the function $g$ by: $$g(x) = \frac{2}{x} - \frac{1}{x^2} + \ln x \text{ where ln denotes the natural logarithm function.}$$ We admit that the function $g$ is differentiable on $]0;+\infty[ = I$ and we denote by $g'$ its derivative function.
  1. Show that for $x > 0$, the sign of $g'(x)$ is that of the quadratic trinomial $(x^2 - 2x + 2)$.
  2. Deduce that the function $g$ is strictly increasing on $]0;+\infty[$.
  3. Show that the equation $g(x) = 0$ admits a unique solution on the interval $[0{,}5; 1]$, which we will denote $\alpha$.
  4. We are given the sign table of $g$ on the interval $]0;+\infty[ = I$:
    $x$0$\alpha$$+\infty$
    $g(x)$$-$0$+$

    Justify this sign table using the results obtained in the previous questions.

PART B We consider the function $f$ defined on the interval $]0;+\infty[ = I$ by: $$f(x) = \mathrm{e}^x \ln x.$$ We denote by $\mathscr{C}_f$ the representative curve of $f$ in an orthonormal coordinate system.
  1. We admit that the function $f$ is twice differentiable on $]0;+\infty[$, we denote by $f'$ its derivative function, $f''$ its second derivative function and we admit that: for every real number $x > 0$, $f'(x) = \mathrm{e}^x\left(\frac{1}{x} + \ln x\right)$. Prove that, for every real number $x > 0$, we have: $f''(x) = \mathrm{e}^x\left(\frac{2}{x} - \frac{1}{x^2} + \ln x\right)$.
  2. We may note that for every real $x > 0$, $f''(x) = \mathrm{e}^x \times g(x)$, where $g$ denotes the function studied in part A.
  3. a. Draw the sign table of the function $f''$ on $]0;+\infty[$. Justify. b. Justify that the curve $\mathscr{C}_f$ admits a unique inflection point A. c. Study the convexity of the function $f$ on the interval $]0;+\infty[$. Justify.
  4. a. Calculate the limits of $f$ at the boundaries of its domain of definition. b. Show that $f'(\alpha) = \frac{\mathrm{e}^\alpha}{\alpha^2}(1-\alpha)$. We recall that $\alpha$ is the unique solution of the equation $g(x) = 0$. c. Prove that $f'(\alpha) > 0$ and deduce the sign of $f'(x)$ for $x$ belonging to $]0;+\infty[$. d. Deduce the complete variation table of the function $f$ on $]0;+\infty[$.
Q1 1 marks Standard Integrals and Reverse Chain Rule Reverse Chain Rule Antiderivative (MCQ) View
Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{x^2-3}$$ One of the antiderivatives $F$ of the function $f$ on $\mathbb{R}$ is defined by: a. $F(x) = 2x\mathrm{e}^{x^2-3}$ b. $F(x) = \left(2x^2+1\right)\mathrm{e}^{x^2-3}$ c. $F(x) = \frac{1}{2}x\mathrm{e}^{x^2-3}$ d. $F(x) = \frac{1}{2}\mathrm{e}^{x^2-3}$
Q2 1 marks Geometric Sequences and Series True/False or Multiple-Statement Verification View
Consider the sequence $(u_n)$ defined for every natural number $n$ by: $$u_n = \mathrm{e}^{2n+1}$$ The sequence $(u_n)$ is: a. arithmetic with common difference 2; b. geometric with common ratio e; c. geometric with common ratio $\mathrm{e}^2$; d. convergent to e.
Q3 1 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
For questions 3. and 4., consider the sequence $(u_n)$ defined on $\mathbb{N}$ by: $$u_0 = 15 \text{ and for every natural number } n : u_{n+1} = 1{,}2\, u_n + 12.$$
The following Python function, whose line 4 is incomplete, must return the smallest value of the integer $n$ such that $u_n > 10000$. \begin{verbatim} def seuil() : n=0 u=15 while ......: n=n+1 u=1,2*u+12 return(n) \end{verbatim} On line 4, we complete with: a. $\mathrm{u} \leqslant 10000$; b. $\mathrm{u} = 10000$ c. $\mathrm{u} > 10000$; d. $n \leqslant 10000$.
Q4 1 marks Geometric Sequences and Series True/False or Multiple-Statement Verification View
For questions 3. and 4., consider the sequence $(u_n)$ defined on $\mathbb{N}$ by: $$u_0 = 15 \text{ and for every natural number } n : u_{n+1} = 1{,}2\, u_n + 12.$$
Consider the sequence $(v_n)$ defined on $\mathbb{N}$ by: $v_n = u_n + 60$.
The sequence $(v_n)$ is: a. a decreasing sequence; b. a geometric sequence with common ratio 1,2; c. an arithmetic sequence with common difference 60; d. a sequence that is neither geometric nor arithmetic.