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_j1

8 maths questions

QExercise 2 Applied differentiation Full function study (variation table, limits, asymptotes) View
We consider the function $f$ defined on $]0; +\infty[$ by $$f(x) = x^2 - 8\ln(x)$$ where ln denotes the natural logarithm function. We admit that $f$ is differentiable on $]0; +\infty[$, we denote by $f'$ its derivative function.
  1. Determine $\lim_{x \rightarrow 0} f(x)$.
  2. We admit that, for all $x > 0$, $f(x) = x^2\left(1 - 8\frac{\ln(x)}{x^2}\right)$.
    Deduce the limit: $\lim_{x \rightarrow +\infty} f(x)$.
  3. Show that, for all real $x$ in $]0; +\infty[$, $f'(x) = \frac{2(x^2 - 4)}{x}$.
  4. Study the variations of $f$ on $]0; +\infty[$ and draw up its complete variation table. We will specify the exact value of the minimum of $f$ on $]0; +\infty[$.
  5. Prove that, on the interval $]0; 2]$, the equation $f(x) = 0$ admits a unique solution $\alpha$ (we will not seek to determine the value of $\alpha$).
  6. We admit that, on the interval $[2; +\infty[$, the equation $f(x) = 0$ admits a unique solution $\beta$ (we will not seek to determine the value of $\beta$). Deduce the sign of $f$ on the interval $]0; +\infty[$.
  7. For any real number $k$, we consider the function $g_k$ defined on $]0; +\infty[$ by: $$g_k(x) = x^2 - 8\ln(x) + k$$ Using the variation table of $f$, determine the smallest value of $k$ for which the function $g_k$ is positive on the interval $]0; +\infty[$.
QExercise 3 Geometric Sequences and Series Applied Geometric Model with Contextual Interpretation View
A company has created a Frequently Asked Questions (``FAQ'') on its website. We study the number of questions asked there each month.
Part A: First modelling
In this part, we admit that, each month:
  • $90\%$ of questions already asked the previous month are kept on the FAQ;
  • 130 new questions are added to the FAQ.
During the first month, 300 questions were asked. To estimate the number of questions, in hundreds, present on the FAQ in the $n$-th month, we model the above situation using the sequence $(u_n)$ defined by: $$u_1 = 3 \text{ and, for all natural integer } n \geqslant 1, u_{n+1} = 0.9u_n + 1.3.$$
  1. Calculate $u_2$ and $u_3$ and propose an interpretation in the context of the exercise.
  2. Show by induction that for all natural integer $n \geqslant 1$: $$u_n = 13 - \frac{100}{9} \times 0.9^n.$$
  3. Deduce that the sequence $(u_n)$ is increasing.
  4. We consider the program opposite, written in Python language.
    Determine the value returned by entering \texttt{seuil(8.5)} and interpret it in the context of the exercise. \begin{verbatim} def seuil(p) : n=1 u=3 while u<=p : n=n+1 u=0.9*u+1.3 return n \end{verbatim}

Part B: Another modelling
In this part, we consider a second modelling using a new sequence $(v_n)$ defined for all natural integer $n \geqslant 1$ by: $$v_n = 9 - 6 \times \mathrm{e}^{-0.19 \times (n-1)}.$$ The term $v_n$ is an estimate of the number of questions, in hundreds, present in the $n$-th month on the FAQ.
  1. Specify the values rounded to the nearest hundredth of $v_1$ and $v_2$.
  2. Determine, by justifying the answer, the smallest value of $n$ such that $v_n > 8.5$.

Part C: Comparison of the two models
  1. The company considers that it must modify the presentation of its site when more than 850 questions are present on the FAQ. Of these two modellings, which leads to making this modification the soonest? Justify your answer.
  2. By justifying the answer, for which modelling is there the greatest number of questions on the FAQ in the long term?
QExercise 4 5 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
We consider the cube ABCDEFGH with edge 1. We call I the point of intersection of the plane (GBD) with the line (EC). The space is referred to the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.
  1. Give in this coordinate system the coordinates of points $\mathrm{E}, \mathrm{C}, \mathrm{G}$.
  2. Determine a parametric representation of the line (EC).
  3. Prove that the line (EC) is orthogonal to the plane (GBD).
    1. [a.] Justify that a Cartesian equation of the plane (GBD) is: $$x + y - z - 1 = 0.$$
    2. [b.] Show that point I has coordinates $\left(\frac{2}{3}; \frac{2}{3}; \frac{1}{3}\right)$.
    3. [c.] Deduce that the distance from point E to the plane (GBD) is equal to $\frac{2\sqrt{3}}{3}$.
    1. [a.] Prove that triangle BDG is equilateral.
    2. [b.] Calculate the area of triangle BDG. You may use point J, the midpoint of segment [BD].
  6. Justify that the volume of tetrahedron EGBD is equal to $\frac{1}{3}$.
    We recall that the volume of a tetrahedron is given by $V = \frac{1}{3}Bh$ where $B$ is the area of a base of the tetrahedron and $h$ is the height relative to this base.
Q1 Conditional Probability Direct Conditional Probability Computation from Definitions View
A technician controls the machines equipping a large company. All these machines are identical. We know that:
  • $20\%$ of machines are under warranty;
  • $0.2\%$ of machines are both defective and under warranty;
  • $8.2\%$ of machines are defective.
The technician tests a machine at random. We consider the following events:
  • G: ``the machine is under warranty'';
  • $D$: ``the machine is defective'';
  • $\bar{G}$ and $\bar{D}$ denote respectively the complementary events of $G$ and $D$.

The probability $p_{G}(D)$ of event $D$ given that $G$ is realized is equal to: a. 0.002 b. 0.01 c. 0.024 d. 0.2
Q2 Principle of Inclusion/Exclusion View
A technician controls the machines equipping a large company. All these machines are identical. We know that:
  • $20\%$ of machines are under warranty;
  • $0.2\%$ of machines are both defective and under warranty;
  • $8.2\%$ of machines are defective.
The technician tests a machine at random. We consider the following events:
  • G: ``the machine is under warranty'';
  • $D$: ``the machine is defective'';
  • $\bar{G}$ and $\bar{D}$ denote respectively the complementary events of $G$ and $D$.

The probability $p(\bar{G} \cap D)$ is equal to: a. 0.01 b. 0.08 c. 0.1 d. 0.21
Q3 Conditional Probability Bayes' Theorem with Production/Source Identification View
A technician controls the machines equipping a large company. All these machines are identical. We know that:
  • $20\%$ of machines are under warranty;
  • $0.2\%$ of machines are both defective and under warranty;
  • $8.2\%$ of machines are defective.
The technician tests a machine at random. We consider the following events:
  • G: ``the machine is under warranty'';
  • $D$: ``the machine is defective'';
  • $\bar{G}$ and $\bar{D}$ denote respectively the complementary events of $G$ and $D$.

The machine is defective. The probability that it is under warranty is approximately equal, to $10^{-3}$ near, to: a. 0.01 b. 0.024 c. 0.082 d. 0.1
Q4 Binomial Distribution MCQ Selecting a Binomial Probability Expression or Value View
We randomly choose, independently, $n$ machines from the company, where $n$ denotes a non-zero natural integer. We assimilate this choice to a sampling with replacement, and we denote by $X$ the random variable that associates to each batch of $n$ machines the number of defective machines in this batch. We admit that $X$ follows the binomial distribution with parameters $n$ and $p = 0.082$.
In this question, we take $n = 50$.
The value of the probability $p(X > 2)$, rounded to the nearest thousandth, is: a. 0.136 b. 0.789 c. 0.864 d. 0.924
Q5 Binomial Distribution MCQ Selecting a Binomial Probability Expression or Value View
We randomly choose, independently, $n$ machines from the company, where $n$ denotes a non-zero natural integer. We assimilate this choice to a sampling with replacement, and we denote by $X$ the random variable that associates to each batch of $n$ machines the number of defective machines in this batch. We admit that $X$ follows the binomial distribution with parameters $n$ and $p = 0.082$.
We consider an integer $n$ for which the probability that all machines in a batch of size $n$ function correctly is greater than 0.4.
The largest possible value for $n$ is equal to: a. 5 b. 6 c. 10 d. 11