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
2022 bac-spe-maths__polynesie-sept

4 maths questions

Q1 7 marks Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
Exercise 1 (7 points) -- Probabilities
Among sore throats, one quarter requires taking antibiotics, the others do not. In order to avoid unnecessarily prescribing antibiotics, doctors have a diagnostic test with the following characteristics:
  • when the sore throat requires taking antibiotics, the test is positive in $90\%$ of cases;
  • when the sore throat does not require taking antibiotics, the test is negative in $95\%$ of cases.

The probabilities requested in the rest of the exercise will be rounded to $10^{-4}$ if necessary.
Part 1
A patient with a sore throat who has undergone the test is chosen at random. Consider the following events:
  • $A$: ``the patient has a sore throat requiring taking antibiotics'';
  • $T$: ``the test is positive'';
  • $\bar{A}$ and $\bar{T}$ are respectively the complementary events of $A$ and $T$.

  1. Calculate $P(A \cap T)$. You may use a probability tree.
  2. Prove that $P(T) = 0.2625$.
  3. A patient with a positive test is chosen. Calculate the probability that they have a sore throat requiring taking antibiotics.
  4. a. Among the following events, determine which correspond to an incorrect test result: $A \cap T,\ \bar{A} \cap T,\ A \cap \bar{T},\ \bar{A} \cap \bar{T}$. b. Define the event $E$: ``the test gives an incorrect result''. Prove that $P(E) = 0.0625$.

Part 2
A sample of $n$ patients who have been tested is selected at random. We assume that this sample selection can be treated as sampling with replacement. Let $X$ be the random variable giving the number of patients in this sample with an incorrect test result.
  1. Suppose that $n = 50$. a. Justify that the random variable $X$ follows a binomial distribution $\mathscr{B}(n, p)$ with parameters $n = 50$ and $p = 0.0625$. b. Calculate $P(X = 7)$. c. Calculate the probability that there is at least one patient in the sample whose test is incorrect.
  2. What is the minimum sample size needed so that $P(X \geqslant 10)$ is greater than $0.95$?
Q2 7 marks Sequences and series, recurrence and convergence Conjecture from numerical data or computation View
Exercise 2 (7 points) -- Sequences, functions
Let $k$ be a real number. Consider the sequence $\left(u_n\right)$ defined by its first term $u_0$ and for every natural number $n$, $$u_{n+1} = k u_n \left(1 - u_n\right)$$
The two parts of this exercise are independent. We study two cases depending on the values of $k$.
Part 1
In this part, $k = 1.9$ and $u_0 = 0.1$. Therefore, for every natural number $n$, $u_{n+1} = 1.9 u_n \left(1 - u_n\right)$.
  1. Consider the function $f$ defined on $[0; 1]$ by $f(x) = 1.9x(1 - x)$. a. Study the variations of $f$ on the interval $[0; 1]$. b. Deduce that if $x \in [0; 1]$ then $f(x) \in [0; 1]$.
  2. Below are represented the first terms of the sequence $\left(u_n\right)$ constructed from the curve $\mathscr{C}_f$ of the function $f$ and the line $D$ with equation $y = x$. Conjecture the direction of variation of the sequence $(u_n)$ and its possible limit.
  3. a. Using the results from question 1, prove by induction that for every natural number $n$: $$0 \leqslant u_n \leqslant u_{n+1} \leqslant \frac{1}{2}$$ b. Deduce that the sequence $(u_n)$ converges. c. Determine its limit.

Part 2
In this part, $k = \frac{1}{2}$ and $u_0 = \frac{1}{4}$. Therefore, for every natural number $n$, $u_{n+1} = \frac{1}{2} u_n \left(1 - u_n\right)$ and $u_0 = \frac{1}{4}$. We admit that for every natural number $n$: $0 \leqslant u_n \leqslant \left(\frac{1}{2}\right)^n$.
  1. Prove that the sequence $(u_n)$ converges and determine its limit.
  2. Consider the Python function \texttt{algo(p)} where \texttt{p} denotes a non-zero natural number: \begin{verbatim} def algo(p) : u = 1/4 n = 0 while u > 10**(-p): u = 1/2*u*(1 - u) n = n+1 return(n) \end{verbatim} Explain why, for every non-zero natural number $p$, the while loop does not run indefinitely, which allows the command \texttt{algo(p)} to return a value.
Q3 7 marks Differentiating Transcendental Functions Full function study with transcendental functions View
Exercise 3 (7 points)
Part 1
Let $g$ be the function defined for every real number $x$ in the interval $]0; +\infty[$ by: $$g(x) = \frac{2\ln x}{x}$$
  1. Let $g'$ denote the derivative of $g$. Prove that for every strictly positive real $x$: $$g'(x) = \frac{2 - 2\ln x}{x^2}$$
  2. We have the following variation table for the function $g$ on the interval $]0; +\infty[$:
    $x$01e$+\infty$
    \begin{tabular}{ c } Variations
    of $g$
    & & & ${}^{\frac{2}{\mathrm{e}}}$ & & & & & & & & & \hline \end{tabular}
    Justify the following information read from this table: a. the value $\frac{2}{\mathrm{e}}$; b. the variations of the function $g$ on its domain; c. the limits of the function $g$ at the boundaries of its domain.
  3. Deduce the sign table of the function $g$ on the interval $]0; +\infty[$.

Part 2
Let $f$ be the function defined on the interval $]0; +\infty[$ by $$f(x) = [\ln(x)]^2.$$ In this part, each study is carried out on the interval $]0; +\infty[$.
  1. Prove that on the interval $]0; +\infty[$, the function $f$ is a primitive of the function $g$.
  2. Using Part 1, study: a. the convexity of the function $f$; b. the variations of the function $f$.
  3. a. Give an equation of the tangent line to the curve representing the function $f$ at the point with abscissa $e$. b. Deduce that, for every real $x$ in $]0; e]$: $$[\ln(x)]^2 \geqslant \frac{2}{\mathrm{e}} x - 1$$
Q4 7 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 4 (7 points) -- Geometry in the plane and in space
Consider the cube ABCDEFGH. Let I be the midpoint of segment [EH] and consider the triangle CFI. The space is equipped with the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$ and we admit that point I has coordinates $\left(0; \frac{1}{2}; 1\right)$ in this coordinate system.
  1. a. Give without justification the coordinates of points C, F and G. b. Prove that the vector $\vec{n}\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ is normal to the plane (CFI). c. Verify that a Cartesian equation of the plane (CFI) is: $x + 2y + 2z - 3 = 0$.
  2. Let $d$ be the line passing through G and perpendicular to the plane (CFI). a. Determine a parametric representation of the line $d$. b. Prove that the point $\mathrm{K}\left(\frac{7}{9}; \frac{5}{9}; \frac{5}{9}\right)$ is the orthogonal projection of point G onto the plane (CFI). c. Deduce from the previous questions that the distance from point G to the plane (CFI) is equal to $\frac{2}{3}$.
  3. Consider the pyramid GCFI. Recall that the volume $V$ of a pyramid is given by the formula $$V = \frac{1}{3} \times b \times h$$ where $b$ is the area of a base and $h$ is the height associated with this base. a. Prove that the volume of the pyramid GCFI is equal to $\frac{1}{6}$, expressed in cubic units. b. Deduce the area of triangle CFI, in square units.