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
2024 bac-spe-maths__metropole_j2

4 maths questions

Q1 Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
The director of a school wishes to conduct a study among students who took the final examination to analyze how they think they performed on this exam. For this study, students are asked at the end of the exam to answer individually the question: ``Do you think you passed the exam?''.
Only the answers ``yes'' or ``no'' are possible, and it is observed that $91.7\%$ of the students surveyed answered ``yes''. Following the publication of exam results, it is discovered that:
  • $65\%$ of students who failed answered ``no'';
  • $98\%$ of students who passed answered ``yes''.
A student who took the exam is randomly selected. We denote by $R$ the event ``the student passed the exam'' and $Q$ the event ``the student answered ``yes'' to the question''. For any event $A$, we denote by $P(A)$ its probability and $\bar{A}$ its complementary event.
Throughout the exercise, probabilities are, if necessary, rounded to $10^{-3}$ near.
  1. Specify the values of the probabilities $P(Q)$ and $P_{\bar{R}}(\bar{Q})$.
  2. Let $x$ be the probability that the randomly selected student passed the exam. a. Copy and complete the weighted tree below. b. Show that $x = 0.9$.
  3. The student selected answered ``yes'' to the question. What is the probability that he passed the exam?
  4. The grade obtained by a randomly selected student is an integer between 0 and 20. It is assumed to be modeled by a random variable $N$ that follows the binomial distribution with parameters $(20; 0.615)$.
    The director wishes to award a prize to students with the best results.
    Starting from which grade should she award prizes so that $65\%$ of students are rewarded?
  5. Ten students are randomly selected.
    The random variables $N_1, N_2, \ldots, N_{10}$ model the grade out of 20 obtained on the exam by each of them. It is admitted that these variables are independent and follow the same binomial distribution with parameters $(20; 0.615)$. Let $S$ be the variable defined by $S = N_1 + N_2 + \cdots + N_{10}$. Calculate the expectation $E(S)$ and the variance $V(S)$ of the random variable $S$.
  6. Consider the random variable $M = \frac{S}{10}$. a. What does this random variable $M$ model in the context of the exercise? b. Justify that $E(M) = 12.3$ and $V(M) = 0.47355$. c. Using the Bienaymé-Chebyshev inequality, justify the statement below. ``The probability that the average grade of ten randomly selected students is strictly between 10.3 and 14.3 is at least $80\%$''.
Q2 5 marks Sequences and series, recurrence and convergence Applied/contextual sequence problem View
Alain owns a swimming pool that contains $50\mathrm{~m}^3$ of water. Recall that $1\mathrm{~m}^3 = 1000\mathrm{~L}$. To disinfect the water, he must add chlorine. The chlorine level in the water, expressed in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, is defined as the mass of chlorine per unit volume of water. Pool specialists recommend a chlorine level between 1 and $3\mathrm{~mg}\cdot\mathrm{L}^{-1}$. Under the action of the ambient environment, particularly ultraviolet rays, chlorine decomposes and gradually disappears. On Wednesday, June 19, he measures a chlorine level of $0.70\mathrm{~mg}\cdot\mathrm{L}^{-1}$.
Part A: study of a discrete model.
To maintain the chlorine level in his pool, Alain decides, starting from Thursday, June 20, to add 15 g of chlorine each day. It is admitted that this chlorine mixes uniformly in the pool water.
  1. Justify that this addition of chlorine increases the level by $0.3\mathrm{~mg}\cdot\mathrm{L}^{-1}$.
  2. For any natural number $n$, we denote by $v_n$ the chlorine level, in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, obtained with this new protocol $n$ days after Wednesday, June 19. Thus $v_0 = 0.7$. It is admitted that for any natural number $n$, $$v_{n+1} = 0.92 v_n + 0.3.$$ a. Show by induction that for any natural number $n$, $\quad v_n \leqslant v_{n+1} \leqslant 4$. b. Show that the sequence $(v_n)$ is convergent and calculate its limit.
  3. In the long term, will the chlorine level comply with the pool specialists' recommendation? Justify your answer.
  4. Reproduce and complete the algorithm below written in Python language so that the function \texttt{alerte\_chlore} returns, when it exists, the smallest integer $n$ such that $v_n > s$. \begin{verbatim} def alerte_chlore(s) : n = 0 u=0.7 while...: n = ... u=... return n \end{verbatim}
  5. What value is obtained by entering the instruction \texttt{alerte\_chlore(3)}? Interpret this result in the context of the exercise.

Part B: study of a continuous model.
Alain decides to call on a specialized engineering firm. This firm uses a continuous model to describe the chlorine level in the pool. In this model, for a duration $x$ (in days elapsed from Wednesday, June 19), $f(x)$ represents the chlorine level, in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, in the pool. It is admitted that the function $f$ is a solution of the differential equation $(E): y' = -0.08y + \frac{q}{50}$, where $q$ is the quantity of chlorine, in grams, added to the pool each day.
  1. Justify that the function $f$ is of the form $f(x) = C\mathrm{e}^{-0.08x} + \frac{q}{4}$ where $C$ is a real constant.
  2. a. Express as a function of $q$ the limit of $f$ at $+\infty$. b. Recall that the chlorine level observed on Wednesday, June 19 is equal to $0.7\mathrm{~mg}\cdot\mathrm{L}^{-1}$. It is desired that the chlorine level stabilizes in the long term around $2\mathrm{~mg}\cdot\mathrm{L}^{-1}$. Determine the values of $C$ and $q$ so that these two conditions are satisfied.
Q3 Applied differentiation Full function study (variation table, limits, asymptotes) View
Consider a function $f$ defined and twice differentiable on $]-2; +\infty[$. We denote by $\mathscr{C}_f$ its representative curve in an orthogonal coordinate system of the plane, $f'$ its derivative and $f''$ its second derivative. The curve $\mathscr{C}_f$ and its tangent $T$ at point B with abscissa $-1$ are drawn. It is specified that the line $T$ passes through the point $\mathrm{A}(0; -1)$.
Part A: exploitation of the graph.
Using the graph, answer the questions below.
  1. Specify $f(-1)$ and $f'(-1)$.
  2. Is the function $f$ convex on its domain of definition? Justify.
  3. Conjecture the number of solutions of the equation $f(x) = 0$ and give a value rounded to $10^{-1}$ near a solution.

Part B: study of the function $f$
Consider that the function $f$ is defined on $]-2; +\infty[$ by: $$f(x) = x^2 + 2x - 1 + \ln(x+2),$$ where ln denotes the natural logarithm function.
  1. Determine by calculation the limit of the function $f$ at $-2$. Interpret this result graphically.

It is admitted that $\lim_{x \rightarrow +\infty} f(x) = +\infty$.
    \setcounter{enumi}{1}
  1. Show that for all $x > -2$, $\quad f'(x) = \frac{2x^2 + 6x + 5}{x+2}$.
  2. Study the variations of the function $f$ on $]-2; +\infty[$ then draw up its complete variation table.
  3. Show that the equation $f(x) = 0$ admits a unique solution $\alpha$ on $]-2; +\infty[$ and give a value of $\alpha$ rounded to $10^{-2}$ near.
  4. Deduce the sign of $f(x)$ on $]-2; +\infty[$.
  5. Show that $\mathscr{C}_f$ admits a unique inflection point and determine its abscissa.

Part C: a minimum distance.
Let $g$ be the function defined on $]-2; +\infty[$ by $\quad g(x) = \ln(x+2)$. We denote by $\mathscr{C}_g$ its representative curve in an orthonormal coordinate system $(O; I, J)$. Let $M$ be a point of $\mathscr{C}_g$ with abscissa $x$. The purpose of this part is to determine for which value of $x$ the distance $JM$ is minimal. Consider the function $h$ defined on $]-2; +\infty[$ by $\quad h(x) = JM^2$.
  1. Justify that for all $x > -2$, we have: $\quad h(x) = x^2 + [\ln(x+2) - 1]^2$.
  2. It is admitted that the function $h$ is differentiable on $]-2; +\infty[$ and we denote by $h'$ its derivative function. It is also admitted that for all real $x > -2$, $$h'(x) = \frac{2f(x)}{x+2}$$ where $f$ is the function studied in part B. a. Draw up the variation table of $h$ on $]-2; +\infty[$. The limits are not required. b. Deduce that the value of $x$ for which the distance $JM$ is minimal is $\alpha$ where $\alpha$ is the real number defined in question 4 of part B.
  3. We will denote by $M_\alpha$ the point of $\mathscr{C}_g$ with abscissa $\alpha$. a. Show that $\ln(\alpha + 2) = 1 - 2\alpha - \alpha^2$. b. Deduce that the tangent to $\mathscr{C}_g$ at point $M_\alpha$ and the line $(JM_\alpha)$ are perpendicular. One may use the fact that, in an orthonormal coordinate system, two lines are perpendicular when the product of their slopes is equal to $-1$.
Q4 4 marks Vectors: Lines & Planes True/False or Verify a Given Statement View
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
In space with an orthonormal coordinate system, consider the following points: $$\mathrm{A}(2;0;0), \quad \mathrm{B}(0;4;3), \quad \mathrm{C}(4;4;1), \quad \mathrm{D}(0;0;4) \text{ and } \mathrm{H}(-1;1;2)$$
Statement 1: the points A, C and D define a plane $\mathscr{P}$ with equation $8x - 5y + 4z - 16 = 0$. Statement 2: the points A, B, C and D are coplanar. Statement 3: the lines $(\mathrm{AC})$ and $(\mathrm{BH})$ are secant. It is admitted that the plane (ABC) has the Cartesian equation $x - y + 2z - 2 = 0$. Statement 4: the point H is the orthogonal projection of point D onto the plane (ABC).