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
2025 bac-spe-maths__asie_j2

4 maths questions

Q1 5 marks Conditional Probability Conditional Probability as a Function of a Parameter View
All probabilities, unless otherwise indicated, will be rounded to $10^{-3}$ in this exercise.
A test was developed for the detection of the chikungunya virus. The laboratory manufacturing this test provides the following characteristics:
  • the probability that an individual affected by the virus has a positive test is 0.999;
  • the probability that an individual not affected by the virus has a positive test is 0.005.
An individual is chosen at random from this population. We call:
  • $M$ the event: ``the chosen individual is affected by chikungunya''.
  • $T$ the event: ``the test of the chosen individual is positive''.
The test is considered reliable when the probability that an individual with a positive test is affected by the virus is greater than 0.95.
Part A: Study of an example
  1. Give the probabilities $P_{M}(T)$ and $P_{\bar{M}}(T)$. ``In March 2005, the epidemic spread rapidly on the island of Réunion, with a major outbreak between late April and early June followed by persistence of viral transmission during the austral winter. In total, 270,000 people were infected out of a total population of 750,000 individuals''. At the end of 2005, the laboratory conducted a mass screening test of the population of the island of Réunion. In this part, the target population is therefore the population of the island of Réunion.
  2. Give the exact value of $P(M)$.
  3. Copy and complete the weighted tree.
  4. Calculate the probability that an individual is affected by the virus and has a positive test.
  5. Calculate the probability that an individual has a positive test.
  6. Calculate the probability that an individual with a positive test is affected by the virus.
  7. Can we estimate that this test is reliable? Argue.

Part B: Screening on a target population
In this part, we denote by $p$ the proportion of people affected by chikungunya virus in a target population. We seek here to test the reliability of this laboratory's test as a function of $p$.
  1. Copy, adapting it, the weighted tree from question A3 taking into account the new data.
  2. Express the probability $P(T)$ as a function of $p$.
  3. Show that $P_{T}(M) = \frac{999p}{994p + 5}$.
  4. For which values of $p$ can we consider that this test is reliable?

Part C: Study on a sample
During the epidemic, we assume that the probability of being affected by chikungunya on the island of Réunion is 0.36. We consider a sample of $n$ individuals chosen at random, assimilating this choice to a random draw with replacement. We denote by $X$ the random variable counting the number of infected individuals in this sample among the $n$ drawn. We assume that $X$ follows a binomial distribution with parameters $n$ and $p = 0.36$. Determine from how many individuals $n$ the probability of the event ``at least one of the $n$ inhabitants of this sample is affected by the virus'' is greater than 0.99. Explain the approach.
Q2 Sequences and series, recurrence and convergence Auxiliary sequence transformation View
Part A
Let $\left(u_{n}\right)$ be the sequence defined by $u_{0} = 30$ and, for every natural integer $n$, $u_{n+1} = \frac{1}{2} u_{n} + 10$. Let $(v_{n})$ be the sequence defined for every natural integer $n$ by $v_{n} = u_{n} - 20$.
  1. Calculate the exact values of $u_{1}$ and $u_{2}$.
  2. Prove that the sequence $(v_{n})$ is geometric with common ratio $\frac{1}{2}$.
  3. Express $v_{n}$ as a function of $n$ for every natural integer $n$.
  4. Deduce that, for every natural integer $n$, $u_{n} = 20 + 10\left(\frac{1}{2}\right)^{n}$.
  5. Determine the limit of the sequence $\left(u_{n}\right)$. Justify the answer.

Part B
Let $(w_{n})$ be the sequence defined for every natural integer $n$ by: $$\left\{\begin{array}{l} w_{0} = 45 \\ w_{n+1} = \frac{1}{2} w_{n} + \frac{1}{2} u_{n} + 7 \end{array}\right.$$
  1. Show that $w_{1} = 44.5$.
We wish to write a function \texttt{suite}, in Python language, which returns the value of the term $w_{n}$ for a given value of $n$. We give below a proposal for this function \texttt{suite}. \begin{verbatim} def suite(n) : U=30 W=45 for i in range (1,n+1) : U=U/2+10 W=W/2+U/2+7 return W \end{verbatim}
    \setcounter{enumi}{1}
  1. The execution of \texttt{suite(1)} does not return the term $w_{1}$. How should the function \texttt{suite} be modified so that the execution of \texttt{suite(n)} returns the value of the term $w_{n}$?
  2. a. Show, by induction on $n$, that for every natural integer $n$ we have: $$w_{n} = 10n\left(\frac{1}{2}\right)^{n} + 11\left(\frac{1}{2}\right)^{n} + 34$$ b. We admit that for every natural integer $n \geqslant 4$, we have: $0 \leqslant 10n\left(\frac{1}{2}\right)^{n} \leqslant \frac{10}{n}$. What can we deduce about the convergence of the sequence $\left(w_{n}\right)$?
Q3 Vectors: Lines & Planes True/False or Verify a Given Statement View
Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider:
  • the points $\mathrm{C}(3; 0; 0)$, $\mathrm{D}(0; 2; 0)$, $\mathrm{H}(-6; 2; 2)$ and $\mathrm{J}\left(\frac{-54}{13}; \frac{62}{13}; 0\right)$;
  • the plane $P$ with Cartesian equation $2x + 3y + 6z - 6 = 0$;
  • the plane $P'$ with Cartesian equation $x - 2y + 3z - 3 = 0$;
  • the line $(d)$ with a parametric representation: $\left\{\begin{array}{l} x = -8 + \frac{1}{3}t \\ y = -1 + \frac{1}{2}t \\ z = -4 + t \end{array}, t \in \mathbb{R}\right.$
For each of the following statements, specify whether it is true or false, then justify the answer given. An answer without justification will not be taken into account.
Statement 1: The line $(d)$ is orthogonal to the plane $P$ and intersects this plane at $H$.
Statement 2: The measure in degrees of the angle $\widehat{\mathrm{DCH}}$, rounded to $10^{-1}$, is $17.3^{\circ}$.
Statement 3: The planes $P$ and $P'$ are secant and their intersection is the line $\Delta$ with a parametric representation: $\left\{\begin{array}{l} x = 3 - 3t \\ y = 0 \\ z = t \end{array}, t \in \mathbb{R}\right.$.
Statement 4: Point J is the orthogonal projection of point H onto the line (CD).
Q4 Second order differential equations Solving non-homogeneous second-order linear ODE View
In a laboratory, a chemical reaction is studied in a closed reactor under certain conditions. The numerical processing of experimental data made it possible to model the evolution of the temperature of this chemical reaction as a function of time. Temperature is expressed in degrees Celsius and time is expressed in minutes. Throughout the exercise, we place ourselves on the time interval $[0;10]$.
Part A
In an orthogonal coordinate system of the plane, we give below the representative curve of the temperature function as a function of time on the interval $[0; 10]$.
  1. Determine, by graphical reading, after how much time the temperature returns to its initial value at time $t = 0$.
We call $f$ the temperature function represented by the curve above. We specify that the function $f$ is defined and differentiable on the interval $[0; 10]$. We admit that the function $f$ can be written in the form $f(t) = (at + b)\mathrm{e}^{-0.5t}$ where $a$ and $b$ are two real constants.
2. We admit that the exact value of $f(0)$ is 40. Deduce the value of $b$.
3. We admit that $f$ satisfies the differential equation (E): $y' + 0.5y = 60\mathrm{e}^{-0.5t}$. Determine the value of $a$.
Part B: Study of the function $f$
We admit that the function $f$ is defined for every real $t$ in the interval $[0; 10]$ by $$f(t) = (60t + 40)\mathrm{e}^{-0.5t}$$
  1. Show that for every real $t$ in the interval $[0; 10]$, we have: $f'(t) = (40 - 30t)\mathrm{e}^{-0.5t}$.
  2. a. Study the direction of variation of the function $f$ on the interval $[0; 10]$. Draw the variation table of the function $f$ showing the images of the values present in the table. b. Show that the equation $f(t) = 40$ has a unique solution $\alpha$ strictly positive on the interval $]0; 10]$. c. Give an approximate value of $\alpha$ to the nearest tenth and give an interpretation in the context of the exercise.
  3. We define the average temperature, expressed in degrees Celsius, of this chemical reaction between two times $t_{1}$ and $t_{2}$, expressed in minutes, by $$\frac{1}{t_{2} - t_{1}} \int_{t_{1}}^{t_{2}} f(t)\,\mathrm{dt}$$ a. Using integration by parts, show that $$\int_{0}^{4} f(t)\,\mathrm{dt} = 320 - \frac{800}{\mathrm{e}^{2}}$$ b. Deduce an approximate value, to the nearest degree Celsius, of the average temperature of this chemical reaction during the first 4 minutes.