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__amerique-sud_j1

4 maths questions

Q1 7 marks Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
Exercise 1 Probability
The alarm system of a company operates in such a way that, if a danger presents itself, the alarm activates with a probability of 0.97. The probability that a danger presents itself is 0.01 and the probability that the alarm activates is 0.01465. We denote $A$ the event ``the alarm activates'' and $D$ the event ``a danger presents itself''. We denote $\bar{M}$ the opposite event of an event $M$ and $P(M)$ the probability of the event $M$.
PART A
  1. Represent the situation with a weighted tree diagram that will be completed as the exercise progresses.
  2. a. Calculate the probability that a danger presents itself and the alarm activates. b. Deduce from this the probability that a danger presents itself given that the alarm activates. Round the result to $10^{-3}$.
  3. Show that the probability that the alarm activates given that no danger has presented itself is 0.005.
  4. An alarm is considered not to function normally when a danger presents itself and it does not activate, or when no danger presents itself and it activates. Show that the probability that the alarm does not function normally is less than 0.01.

PART B
A factory manufactures alarm systems in large quantities. We successively and randomly select 5 alarm systems from the factory's production. This selection is treated as sampling with replacement. We denote $S$ the event ``the alarm does not function normally'' and we admit that $P(S) = 0.00525$. We consider $X$ the random variable that gives the number of alarm systems not functioning normally among the 5 alarm systems selected. Results should be rounded to $10^{-4}$.
  1. Give the probability distribution followed by the random variable $X$ and specify its parameters.
  2. Calculate the probability that, in the selected batch, only one alarm system does not function normally.
  3. Calculate the probability that, in the selected batch, at least one alarm system does not function normally.

PART C
Let $n$ be a non-zero natural integer. We successively and randomly select $n$ alarm systems. This selection is treated as sampling with replacement. Determine the smallest integer $n$ such that the probability of having, in the selected batch, at least one alarm system that does not function normally is greater than 0.07.
Q2 7 marks Geometric Sequences and Series Prove a Transformed Sequence is Geometric View
Exercise 2 Sequences
Let $\left(u_{n}\right)$ be the sequence defined by $u_{0} = 4$ and, for every natural integer $n$, $u_{n+1} = \frac{1}{5} u_{n}^{2}$.
  1. a. Calculate $u_{1}$ and $u_{2}$. b. Copy and complete the function below written in Python language. This function is named suite\_u and takes as parameter the natural integer $p$. It returns the value of the term of rank $p$ of the sequence $(u_{n})$. \begin{verbatim} def suite_u(p) : u= ... for i in range(1,...) : u =... return u \end{verbatim}
  2. a. Prove by induction that for every natural integer $n$, $0 < u_{n} \leqslant 4$. b. Prove that the sequence $(u_{n})$ is decreasing. c. Deduce from this that the sequence $(u_{n})$ is convergent.
  3. a. Justify that the limit $\ell$ of the sequence $(u_{n})$ satisfies the equality $\ell = \frac{1}{5} \ell^{2}$. b. Deduce from this the value of $\ell$.
  4. For every natural integer $n$, we set $v_{n} = \ln\left(u_{n}\right)$ and $w_{n} = v_{n} - \ln(5)$. a. Show that, for every natural integer $n$, $v_{n+1} = 2v_{n} - \ln(5)$. b. Show that the sequence $(w_{n})$ is geometric with common ratio 2. c. For every natural integer $n$, give the expression of $w_{n}$ as a function of $n$ and show that $v_{n} = \ln\left(\frac{4}{5}\right) \times 2^{n} + \ln(5)$.
  5. Calculate $\lim_{n \rightarrow +\infty} v_{n}$ and find again $\lim_{n \rightarrow +\infty} u_{n}$.
Q3 7 marks Differentiating Transcendental Functions Full function study with transcendental functions View
Exercise 3 Functions, logarithm function
Let $g$ be the function defined on the interval $]0; +\infty[$ by $$g(x) = 1 + x^{2}[1 - 2\ln(x)]$$ The function $g$ is differentiable on the interval $]0; +\infty[$ and we denote $g'$ its derivative function. We call $\mathscr{C}$ the representative curve of the function $g$ in an orthonormal coordinate system of the plane.
PART A
  1. Justify that $g(\mathrm{e})$ is strictly negative.
  2. Justify that $\lim_{x \rightarrow +\infty} g(x) = -\infty$.
  3. a. Show that, for all $x$ belonging to the interval $]0; +\infty[$, $g'(x) = -4x\ln(x)$. b. Study the direction of variation of the function $g$ on the interval $]0; +\infty[$. c. Show that the equation $g(x) = 0$ admits a unique solution, denoted $\alpha$, on the interval $[1; +\infty[$. d. Give an interval for $\alpha$ with amplitude $10^{-2}$.
  4. Deduce from the above the sign of the function $g$ on the interval $[1; +\infty[$.

PART B
  1. We admit that, for all $x$ belonging to the interval $[1; \alpha]$, $g''(x) = -4[\ln(x) + 1]$. Justify that the function $g$ is concave on the interval $[1; \alpha]$.
  2. In the figure opposite, A and B are points on the curve $\mathscr{C}$ with abscissae respectively 1 and $\alpha$. a. Determine the reduced equation of the line (AB). b. Deduce from this that for all real $x$ belonging to the interval $[1; \alpha]$, $$g(x) \geqslant \frac{-2}{\alpha - 1} x + \frac{2\alpha}{\alpha - 1}.$$
Q4 7 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 4 Geometry in space
In the figure below, ABCDEFGH is a rectangular parallelepiped such that $\mathrm{AB} = 5$, $\mathrm{AD} = 3$ and $\mathrm{AE} = 2$. The space is equipped with an orthonormal coordinate system with origin A in which the points B, D and E have coordinates respectively $(5; 0; 0)$, $(0; 3; 0)$ and $(0; 0; 2)$.
  1. a. Give, in the coordinate system considered, the coordinates of points H and G. b. Give a parametric representation of the line (GH).
  2. Let M be a point of the segment $[\mathrm{GH}]$ such that $\overrightarrow{\mathrm{HM}} = k\overrightarrow{\mathrm{HG}}$ with $k$ a real number in the interval $[0; 1]$. a. Justify that the coordinates of M are $(5k; 3; 2)$. b. Deduce from this that $\overrightarrow{\mathrm{AM}} \cdot \overrightarrow{\mathrm{CM}} = 25k^{2} - 25k + 4$. c. Determine the values of $k$ for which AMC is a triangle right-angled at M.

For the rest of the exercise, we consider that point M has coordinates $(1; 3; 2)$. We admit that triangle AMC is right-angled at M. We recall that the volume of a tetrahedron is given by the formula $\frac{1}{3} \times$ Area of the base $\times h$ where $h$ is the height relative to the base.
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  1. We consider the point K with coordinates $(1; 3; 0)$. a. Determine a Cartesian equation of the plane (ACD). b. Justify that point K is the orthogonal projection of point M onto the plane (ACD). c. Deduce from this the volume of the tetrahedron MACD.
  2. We denote P the orthogonal projection of point D onto the plane (AMC). Calculate the distance DP; give a value rounded to $10^{-1}$.