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__metropole-juin_j2

5 maths questions

QA Differentiating Transcendental Functions Full function study with transcendental functions View
Main topics covered: Logarithm function; differentiation
Part 1
Let $h$ denote the function defined on the interval $]0; +\infty[$ by: $$h(x) = 1 + \frac{\ln(x)}{x^2}.$$ It is admitted that the function $h$ is differentiable on $]0; +\infty[$ and we denote $h'$ its derivative function.
  1. Determine the limits of $h$ at 0 and at $+\infty$.
  2. Show that, for every real number $x$ in $]0; +\infty[$, $h'(x) = \frac{1 - 2\ln(x)}{x^3}$.
  3. Deduce the variations of the function $h$ on the interval $]0; +\infty[$.
  4. Show that the equation $h(x) = 0$ admits a unique solution $\alpha$ belonging to $]0; +\infty[$ and verify that: $\frac{1}{2} < \alpha < 1$.
  5. Determine the sign of $h(x)$ for $x$ belonging to $]0; +\infty[$.

Part 2
Let $f_1$ and $f_2$ denote the functions defined on $]0; +\infty[$ by: $$f_1(x) = x - 1 - \frac{\ln(x)}{x^2} \quad \text{and} \quad f_2(x) = x - 2 - \frac{2\ln(x)}{x^2}.$$ We denote $\mathscr{C}_1$ and $\mathscr{C}_2$ the respective graphs of $f_1$ and $f_2$ in a reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.
  1. Show that, for every real number $x$ belonging to $]0; +\infty[$, we have: $$f_1(x) - f_2(x) = h(x).$$ \setcounter{enumi}{1}
  2. Deduce from the results of Part 1 the relative position of the curves $\mathscr{C}_1$ and $\mathscr{C}_2$.
You will justify that their unique point of intersection has coordinates $(\alpha; \alpha)$. Recall that $\alpha$ is the unique solution of the equation $h(x) = 0$.
QB Differentiating Transcendental Functions Full function study with transcendental functions View
Main topics covered: Exponential function; differentiation; convexity
Part 1
Below is given, in the plane referred to an orthonormal reference frame, the curve representing the derivative function $f'$ of a function $f$ differentiable on $\mathbb{R}$. Using this curve, conjecture, by justifying the answers:
  1. The direction of variation of the function $f$ on $\mathbb{R}$.
  2. The convexity of the function $f$ on $\mathbb{R}$.

Part 2
It is admitted that the function $f$ mentioned in Part 1 is defined on $\mathbb{R}$ by: $$f(x) = (x+2)\mathrm{e}^{-x}.$$ We denote $\mathscr{C}$ the representative curve of $f$ in an orthonormal reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. It is admitted that the function $f$ is twice differentiable on $\mathbb{R}$, and we denote $f'$ and $f''$ the first and second derivative functions of $f$ respectively.
  1. Show that, for every real number $x$, $$f(x) = \frac{x}{\mathrm{e}^x} + 2\mathrm{e}^{-x}.$$ Deduce the limit of $f$ at $+\infty$. Justify that the curve $\mathscr{C}$ admits an asymptote which you will specify. It is admitted that $\lim_{x \rightarrow -\infty} f(x) = -\infty$.
  2. a. Show that, for every real number $x$, $f'(x) = (-x-1)\mathrm{e}^{-x}$. b. Study the variations on $\mathbb{R}$ of the function $f$ and draw up its variation table. c. Show that the equation $f(x) = 2$ admits a unique solution $\alpha$ on the interval $[-2;-1]$ and give an approximate value to the nearest $10^{-1}$.
  3. Determine, for every real number $x$, the expression of $f''(x)$ and study the convexity of the function $f$.

What does point A with abscissa 0 represent for the curve $\mathscr{C}$?
Q1 4 marks Vectors 3D & Lines MCQ: Point Membership on a Line View
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct.
A correct answer earns one point. An incorrect answer, a multiple answer or the absence of an answer to a question earns or loses no points.
Space is referred to an orthonormal reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.
We consider:
  • The line $\mathscr{D}$ passing through the points $\mathrm{A}(1;1;-2)$ and $\mathrm{B}(-1;3;2)$.
  • The line $\mathscr{D}'$ with parametric representation: $\left\{ \begin{aligned} x &= -4 + 3t \\ y &= 6 - 3t \\ z &= 8 - 6t \end{aligned} \right.$ with $t \in \mathbb{R}$.
  • The plane $\mathscr{P}$ with Cartesian equation $x + my - 2z + 8 = 0$ where $m$ is a real number.

Question 1: Among the following points, which one belongs to the line $\mathscr{D}'$? a. $\mathrm{M}_1(-1;3;-2)$ b. $\mathrm{M}_2(11;-9;-22)$ c. $\mathrm{M}_3(-7;9;2)$ d. $\mathrm{M}_4(-2;3;4)$
Question 2: A direction vector of the line $\mathscr{D}'$ is: a. $\overrightarrow{u_1}\left(\begin{array}{c}-4\\6\\8\end{array}\right)$ b. $\overrightarrow{u_2}\left(\begin{array}{l}3\\3\\6\end{array}\right)$ c. $\overrightarrow{u_3}\left(\begin{array}{c}3\\-3\\-6\end{array}\right)$ d. $\overrightarrow{u_4}\left(\begin{array}{c}-1\\3\\2\end{array}\right)$
Question 3: The lines $\mathscr{D}$ and $\mathscr{D}'$ are: a. intersecting b. strictly parallel c. non-coplanar d. coincident
Question 4: The value of the real number $m$ for which the line $\mathscr{D}$ is parallel to the plane $\mathscr{P}$ is: a. $m = -1$ b. $m = 1$ c. $m = 5$ d. $m = -2$
Q2 6 marks Conditional Probability Bayes' Theorem with Diagnostic/Screening Test View
In this exercise, the results of the probabilities requested will be, if necessary, rounded to the nearest thousandth.
Feline leukaemia is a disease affecting cats; it is caused by a virus. In a large veterinary centre, the proportion of cats carrying the disease is estimated at $40\%$. A screening test for the disease is carried out among the cats present in this veterinary centre. This test has the following characteristics.
  • When the cat carries the disease, its test is positive in $90\%$ of cases.
  • When the cat does not carry the disease, its test is negative in $85\%$ of cases.

A cat is chosen at random from the veterinary centre and the following events are considered:
  • $M$: ``The cat carries the disease'';
  • $T$: ``The cat's test is positive'';
  • $\bar{M}$ and $\bar{T}$ denote the complementary events of events $M$ and $T$ respectively.

  1. a. Represent the situation with a probability tree. b. Calculate the probability that the cat carries the disease and that its test is positive. c. Show that the probability that the cat's test is positive is equal to 0.45. d. A cat is chosen from among those whose test is positive. Calculate the probability that it carries the disease.
  2. A sample of 20 cats is chosen at random from the veterinary centre. It is assumed that this choice can be treated as sampling with replacement.

Let $X$ be the random variable giving the number of cats presenting a positive test in the chosen sample. a. Determine, by justifying, the distribution followed by the random variable $X$. b. Calculate the probability that there are exactly 5 cats with a positive test in the sample. c. Calculate the probability that there are at most 8 cats with a positive test in the sample. d. Determine the expected value of the random variable $X$ and interpret the result in the context of the exercise.
3. In this question, a sample of $n$ cats is chosen from the centre, which is again treated as sampling with replacement. Let $p_n$ be the probability that there is at least one cat presenting a positive test in this sample. a. Show that $p_n = 1 - 0{,}55^n$. b. Describe the role of the program below written in Python language, in which the variable $n$ is a natural integer and the variable $P$ is a real number. \begin{verbatim} def seuil() : n = 0 P = 0 while P < 0,99 : n = n + 1 P = 1 - 0,55**n return n \end{verbatim} c. Determine, by specifying the method used, the value returned by this program.
Q3 Sequences and series, recurrence and convergence Conjecture from numerical data or computation View
Consider the sequence $(u_n)$ defined by: $u_0 = 1$ and, for every natural integer $n$, $$u_{n+1} = \frac{4u_n}{u_n + 4}.$$
1. The screenshot below presents the values, calculated using a spreadsheet, of the terms of the sequence $(u_n)$ for $n$ varying from 0 to 12, as well as those of the quotient $\frac{4}{u_n}$, (with, for the values of $u_n$, display of two digits for the decimal parts). Using these values, conjecture the expression of $\frac{4}{u_n}$ as a function of $n$. The purpose of this exercise is to prove this conjecture (question 5.) and to deduce the limit of the sequence $(u_n)$ (question 6.).
$n$$u_n$$\frac{4}{u_n}$
01,004
10,805
20,676
30,577
40,508
50,449
60,4010
70,3611
80,3312
90,3113
100,2914
110,2715
120,2516

    \setcounter{enumi}{1}
  1. Prove by induction that, for every natural integer $n$, we have: $u_n > 0$.
  2. Prove that the sequence $(u_n)$ is decreasing.
  3. What can be concluded from questions 2. and 3. concerning the sequence $(u_n)$?
  4. Consider the sequence $(v_n)$ defined for every natural integer $n$ by: $v_n = \frac{4}{u_n}$.

Prove that $(v_n)$ is an arithmetic sequence. Specify its common difference and its first term. Deduce, for every natural integer $n$, the expression of $v_n$ as a function of $n$.
6. Determine, for every natural integer $n$, the expression of $u_n$ as a function of $n$.
Deduce the limit of the sequence $(u_n)$.