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 amerique-nord

5 maths questions

Q1 5 marks Normal Distribution Direct Probability Calculation from Given Normal Distribution View
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 that, 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 that, 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 thickness of the tube is compliant''; --- $L$: ``the length of the tube 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 answer without justification earns 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 every real $x \in ] - \frac { \pi } { 2 } ; \frac { \pi } { 2 } \left[ \right.$, 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 every natural number $n , u _ { n + 1 } = u _ { n } - \ln \left( 1 + u _ { n } \right)$. 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 every natural number $n , \quad u _ { n } \geqslant 0$. b. Prove that the sequence $( u _ { n } )$ is decreasing, and deduce that for every natural number $n , \quad u _ { n } \leqslant 1$. 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 $10 ^ { - 15 }$.
Q4a 5 marks Vectors 3D & Lines Perpendicularity Proof in 3D Geometry View
Exercise 4 (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 $\mathrm { I } , \mathrm { J } , \mathrm { K } , \mathrm { L } , \mathrm { M }$ 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 $\left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; 1 \right)$.
    \setcounter{enumi}{1}
  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.
Q4b 5 marks Matrices Structured Matrix Characterization View
Exercise 4 (Candidates who have followed the specialization course)
Two column matrices $\binom { x } { y }$ and $\binom { x ^ { \prime } } { y ^ { \prime } }$ with integer coefficients are said to be congruent modulo 5 if and only if $\left\{ \begin{array} { l } x \equiv x ^ { \prime } [ 5 ] \\ y \equiv y ^ { \prime } [ 5 ] \end{array} \right.$. Two square matrices of order $2 \left( \begin{array} { l l } a & c \\ b & d \end{array} \right)$ and $\left( \begin{array} { l l } a ^ { \prime } & c ^ { \prime } \\ b ^ { \prime } & d ^ { \prime } \end{array} \right)$ with integer coefficients are said to be congruent modulo 5 if and only if $\left\{ \begin{array} { l } a \equiv a ^ { \prime } [ 5 ] \\ b \equiv b ^ { \prime } [ 5 ] \\ c \equiv c ^ { \prime } [ 5 ] \\ d \equiv d ^ { \prime } [ 5 ] \end{array} \right.$.
Alice and Bob want to exchange messages using the procedure described below.
  • They choose a square matrix M of order 2, with integer coefficients.
  • Their initial message is written in capital letters without accents.
  • Each letter of this message is replaced by a column matrix $\binom { x } { y }$ deduced from the table below: $x$ is the digit located at the top of the column and $y$ is the digit located to the left of the row; for example, the letter T in an initial message corresponds to the column matrix $\binom { 4 } { 3 }$.
  • We calculate a new matrix $\binom { x ^ { \prime } } { y ^ { \prime } }$ by multiplying $\binom { x } { y }$ on the left by the matrix $M$:
$$\binom { x ^ { \prime } } { y ^ { \prime } } = \mathrm { M } \binom { x } { y } .$$
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0ABCDE
1FGHIJ
2KLMNO
3PQRST
4UVXYZ