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
2013 caledonie

5 maths questions

Q1 5 marks Stationary points and optimisation Construct or complete a full variation table View
Let $f$ be the differentiable function defined on the interval $] 0 ; + \infty [$ by
$$f ( x ) = \mathrm { e } ^ { x } + \frac { 1 } { x }$$
1. Study of an auxiliary function
a. Let the function $g$ be differentiable, defined on $[ 0 ; + \infty [$ by
$$g ( x ) = x ^ { 2 } \mathrm { e } ^ { x } - 1 .$$
Study the direction of variation of the function $g$.
b. Prove that there exists a unique real number $a$ belonging to $[ 0 ; + \infty [$ such that $g ( a ) = 0$.
Prove that $a$ belongs to the interval $[ 0{,}703 ; 0{,}704 [$.
c. Determine the sign of $g ( x )$ on $[ 0 ; + \infty [$.
2. Study of the function $f$
a. Determine the limits of the function $f$ at 0 and at $+ \infty$.
b. Let $f ^ { \prime }$ denote the derivative function of $f$ on the interval $] 0 ; + \infty [$.
Prove that for every strictly positive real number $x , f ^ { \prime } ( x ) = \frac { g ( x ) } { x ^ { 2 } }$.
c. Deduce the direction of variation of the function $f$ and draw its variation table on the interval $] 0 ; + \infty [$.
d. Prove that the function $f$ has as its minimum the real number
$$m = \frac { 1 } { a ^ { 2 } } + \frac { 1 } { a } .$$
e. Justify that $3{,}43 < m < 3{,}45$.
Q2 5 marks Sequences and series, recurrence and convergence Algorithm and programming for sequences View
Let two sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ be defined by $u _ { 0 } = 2$ and $v _ { 0 } = 10$ and for every natural number $n$,
$$u _ { n + 1 } = \frac { 2 u _ { n } + v _ { n } } { 3 } \quad \text { and } \quad v _ { n + 1 } = \frac { u _ { n } + 3 v _ { n } } { 4 } .$$
PART A
Consider the following algorithm:
\begin{verbatim} Variables: N is an integer U,V,W are real numbers K is an integer Start: Assign 0 to K Assign 2 to U Assign 10 to V Input N While KExecute this algorithm by inputting $N = 2$. Copy and complete the table given below showing the state of the variables during the execution of the algorithm.
$K$$W$$U$$V$
0
1
2

PART B
1. a. Show that for every natural number $n , v _ { n + 1 } - u _ { n + 1 } = \frac { 5 } { 12 } \left( v _ { n } - u _ { n } \right)$.
b. For every natural number $n$ let $w _ { n } = v _ { n } - u _ { n }$.
Show that for every natural number $n , w _ { n } = 8 \left( \frac { 5 } { 12 } \right) ^ { n }$.
2. a. Prove that the sequence $( u _ { n } )$ is increasing and that the sequence $( v _ { n } )$ is decreasing.
b. Deduce from the results of questions 1. b. and 2. a. that for every natural number $n$ we have $u _ { n } \leqslant 10$ and $v _ { n } \geqslant 2$.
c. Deduce that the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ are convergent.
3. Show that the sequences $( u _ { n } )$ and $( v _ { n } )$ have the same limit.
4. Show that the sequence $( t _ { n } )$ defined by $t _ { n } = 3 u _ { n } + 4 v _ { n }$ is constant.
Deduce that the common limit of the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ is $\frac { 46 } { 7 }$.
Q3 5 marks Normal Distribution Normal Distribution Combined with Total Probability or Bayes' Theorem View
A factory manufactures spherical balls whose diameter is expressed in millimetres. A ball is said to be out of specification when its diameter is less than 9 mm or greater than 11 mm.
Part A
1. Let $X$ be the random variable that associates to each ball chosen at random from production its diameter expressed in mm.
It is assumed that the random variable $X$ follows the normal distribution with mean 10 and standard deviation 0.4.
Show that an approximate value to 0.0001 of the probability that a ball is out of specification is 0.0124. You may use the table of values given in the appendix.
2. A production control is put in place such that 98\% of out-of-specification balls are rejected and 99\% of correct balls are kept.
A ball is chosen at random from production. Let $N$ denote the event: ``the chosen ball is within specification'', and $A$ the event: ``the chosen ball is accepted after the control''.
a. Construct a weighted tree diagram that incorporates the data from the problem statement.
b. Calculate the probability of event $A$.
c. What is the probability that an accepted ball is out of specification?
Part B
This production control proving too costly for the company, it is abandoned: henceforth, all balls produced are kept, and they are packaged in bags of 100 balls.
It is considered that the probability that a ball is out of specification is 0.0124.
It will be assumed that taking a bag of 100 balls at random is equivalent to performing a sampling with replacement of 100 balls from the set of manufactured balls.
Let $Y$ be the random variable that associates to every bag of 100 balls the number of out-of-specification balls in that bag.
1. What is the distribution followed by the random variable $Y$?
2. What are the mean and standard deviation of the random variable $Y$?
3. What is the probability that a bag of 100 balls contains exactly two out-of-specification balls?
4. What is the probability that a bag of 100 balls contains at most one out-of-specification ball?
Q4a 5 marks Complex Numbers Argand & Loci True/False or Multiple-Statement Verification View
Exercise 4 — For candidates who have NOT followed the specialization course
The plane is referred to an orthonormal direct coordinate system $( \mathrm { O } , \vec { u } , \vec { v } )$.
Let $\mathbb { C }$ denote the set of complex numbers.
For each of the following propositions, state whether it is true or false by justifying the answer.
1. Proposition: For every natural number $n$ : $( 1 + \mathrm { i } ) ^ { 4 n } = ( - 4 ) ^ { n }$.
2. Let (E) be the equation $( z - 4 ) \left( z ^ { 2 } - 4 z + 8 \right) = 0$ where $z$ denotes a complex number.
Proposition: The points whose affixes are the solutions, in $\mathbb { C }$, of (E) are the vertices of a triangle with area 8.
3. Proposition: For every real number $\alpha , 1 + \mathrm { e } ^ { 2 i \alpha } = 2 \mathrm { e } ^ { \mathrm { i } \alpha } \cos ( \alpha )$.
4. Let A be the point with affix $z _ { \mathrm { A } } = \frac { 1 } { 2 } ( 1 + \mathrm { i } )$ and $M _ { n }$ the point with affix $\left( z _ { \mathrm { A } } \right) ^ { n }$ where $n$ denotes a natural number greater than or equal to 2.
Proposition: if $n - 1$ is divisible by 4, then the points O, A and $M _ { n }$ are collinear.
5. Let j be the complex number with modulus 1 and argument $\frac { 2 \pi } { 3 }$.
Proposition: $1 + \mathrm { j } + \mathrm { j } ^ { 2 } = 0$.
Q4b 5 marks Number Theory Modular Arithmetic Computation View
Exercise 4 — For candidates who have followed the specialization course
Let $E$ denote the set of twenty-seven integers between 0 and 26.
Let $A$ denote the set whose elements are the twenty-six letters of the alphabet and a separator between two words, denoted ``$\star$'' and considered as a character.
To encode the elements of $A$, we proceed as follows:
  • First: We associate to each of the letters of the alphabet, arranged in alphabetical order, a natural integer between 0 and 25, arranged in increasing order. We thus have $a \rightarrow 0 , b \rightarrow 1 , \ldots z \rightarrow 25$. We associate to the separator ``$\star$'' the number 26.
  • Second: to each element $x$ of $E$, the function $g$ associates the remainder of the Euclidean division of $4 x + 3$ by 27. Note that for every $x$ in $E$, $g ( x )$ belongs to $E$.
  • Third: The initial character is then replaced by the character of rank $g ( x )$.

Example: $s \rightarrow 18 , \quad g ( 18 ) = 21$ and $21 \rightarrow v$. So the letter $s$ is replaced during encoding by the letter $v$.
1. Find all integers $x$ of $E$ such that $g ( x ) = x$, that is, invariant under $g$.
Deduce the invariant characters in this encoding.
2. Prove that, for every natural number $x$ belonging to $E$ and every natural number $y$ belonging to $E$, if $y \equiv 4 x + 3$ modulo 27 then $x \equiv 7 y + 6$ modulo 27.
Deduce that two distinct characters are encoded by two distinct characters.
3. Propose a decoding method.
4. Decode the word ``vfv''.