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
2019 integrale-annuelle

4 maths questions

Q1 5 marks Normal Distribution Direct Probability Calculation from Given Normal Distribution View
In this exercise and unless otherwise stated, results should be rounded to $10^{-3}$.
A factory manufactures tubes.
Part A
Questions 1. and 2. are independent. We are interested in two types of tubes, called type 1 tubes and type 2 tubes.
  1. A type 1 tube is accepted at inspection if its thickness is between 1.35 millimetres and 1.65 millimetres. a. Let $X$ denote the random variable which, for each type 1 tube randomly selected from the day's production, gives its thickness expressed in millimetres. We assume that the random variable $X$ follows a normal distribution with mean 1.5 and standard deviation 0.07.
    A type 1 tube is randomly selected from the day's production. Calculate the probability that the tube is accepted at inspection. b. The company wishes to improve the quality of type 1 tube production. To do this, the settings of the machines producing these tubes are modified. Let $X_1$ denote the random variable which, for each type 1 tube selected from the production of the modified machine, gives its thickness. We assume that the random variable $X_1$ follows a normal distribution with mean 1.5 and standard deviation $\sigma_1$.
    A type 1 tube is randomly selected from the production of the modified machine. Determine an approximate value to $10^{-3}$ of $\sigma_1$ so that the probability that this tube is accepted at inspection equals 0.98. (You may use the random variable $Z$ defined by $Z = \frac{X_1 - 1.5}{\sigma_1}$ which follows the standard normal distribution.)
  2. A machine produces type 2 tubes. A type 2 tube is said to be ``compliant for length'' when its length, in millimetres, belongs to the interval [298; 302]. The specifications establish that, in the production of type 2 tubes, a proportion of $2\%$ of tubes that are not ``compliant for length'' is acceptable.
    It is desired to decide whether the production machine should be serviced. To do this, a random sample of 250 tubes is taken from the production of type 2 tubes, in which 10 tubes are found to be not ``compliant for length''. a. Give an asymptotic confidence interval at $95\%$ for the frequency of tubes not ``compliant for length'' in a sample of 250 tubes. b. Should the machine be serviced? Justify your answer.

Part B
Adjustment errors in the production line can affect the thickness or length of type 2 tubes.
A study conducted on the production revealed that: --- $96\%$ of type 2 tubes have compliant thickness; --- among type 2 tubes that have compliant thickness, $95\%$ have compliant length; --- $3.6\%$ of type 2 tubes have non-compliant thickness and compliant length.
A type 2 tube is randomly selected from the production and we consider the events: --- $E$: ``the tube's thickness is compliant''; --- $L$: ``the tube's length is compliant''.
We model the random experiment with a probability tree.
  1. Copy and complete this tree entirely.
  2. Show that the probability of event $L$ equals 0.948.
Q2 4 marks Complex Numbers Arithmetic True/False or Property Verification Statements View
The complex plane is equipped with a direct orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). In what follows, $z$ denotes a complex number.
For each of the statements below, indicate on your answer sheet whether it is true or false. Justify. Any unjustified answer receives no points.
Statement 1: The equation $z - \mathrm{i} = \mathrm{i}(z + 1)$ has solution $\sqrt{2}\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$.
Statement 2: For all real $x \in ]-\frac{\pi}{2}; \frac{\pi}{2}[$, the complex number $1 + \mathrm{e}^{2\mathrm{i}x}$ has exponential form $2\cos x\, \mathrm{e}^{-\mathrm{i}x}$.
Statement 3: A point M with affix $z$ such that $|z - \mathrm{i}| = |z + 1|$ belongs to the line with equation $y = -x$.
Statement 4: The equation $z^5 + z - \mathrm{i} + 1 = 0$ has a real solution.
Q3 6 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
Part A: establishing an inequality
On the interval $[0; +\infty[$, we define the function $f$ by $f(x) = x - \ln(x+1)$.
  1. Study the monotonicity of the function $f$ on the interval $[0; +\infty[$.
  2. Deduce that for all $x \in [0; +\infty[,\; \ln(x+1) \leqslant x$.

Part B: application to the study of a sequence
We set $u_0 = 1$ and for all natural number $n$, $u_{n+1} = u_n - \ln(1 + u_n)$. We admit that the sequence with general term $u_n$ is well defined.
  1. Calculate an approximate value to $10^{-3}$ of $u_2$.
  2. a. Prove by induction that for all natural number $n$, $u_n \geqslant 0$. b. Prove that the sequence $(u_n)$ is decreasing, and deduce that for all natural number $n$, $u_n \leqslant 1$. c. Show that the sequence $(u_n)$ is convergent.
  3. Let $\ell$ denote the limit of the sequence $(u_n)$ and we admit that $\ell = f(\ell)$, where $f$ is the function defined in part A. Deduce the value of $\ell$.
  4. a. Write an algorithm which, for a given natural number $p$, allows us to determine the smallest rank $N$ from which all terms of the sequence $(u_n)$ are less than $10^{-p}$. b. Determine the smallest natural number $n$ from which all terms of the sequence $(u_n)$ are less than $10^{-15}$.
Q4 5 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
(For candidates who have not followed the specialization course)
We connect the centres of each face of a cube ABCDEFGH to form a solid IJKLMN. More precisely, the points I, J, K, L, M and N are the centres respectively of the square faces ABCD, BCGF, CDHG, ADHE, ABFE and EFGH (thus the midpoints of the diagonals of these squares).
  1. Without using a coordinate system (and thus coordinates) in the reasoning, justify that the lines (IN) and (ML) are orthogonal.

In what follows, we consider the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}; \overrightarrow{\mathrm{AD}}; \overrightarrow{\mathrm{AE}})$ in which, for example, the point N has coordinates $\left(\frac{1}{2}; \frac{1}{2}; 1\right)$.
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  1. a. Give the coordinates of the vectors $\overrightarrow{\mathrm{NC}}$ and $\overrightarrow{\mathrm{ML}}$. b. Deduce that the lines (NC) and (ML) are orthogonal. c. From the previous questions, deduce a Cartesian equation of the plane (NCI).
  2. a. Show that a Cartesian equation of the plane (NJM) is: $x - y + z = 1$. b. Is the line (DF) perpendicular to the plane (NJM)? Justify. c. Show that the intersection of the planes (NJM) and (NCI) is a line for which you will give a point and a direction vector. Name the line thus obtained using two points from the figure.