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 amerique-nord

5 maths questions

Q1 5 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
We work in space with an orthonormal coordinate system. We consider the points $\mathrm { A } ( 0 ; 4 ; 1 ) , \mathrm { B } ( 1 ; 3 ; 0 ) , \mathrm { C } ( 2 ; - 1 ; - 2 )$ and $\mathrm { D } ( 7 ; - 1 ; 4 )$.
  1. Prove that the points $\mathrm { A } , \mathrm { B }$ and C are not collinear.
  2. Let $\Delta$ be the line passing through point D with direction vector $\vec { u } ( 2 ; - 1 ; 3 )$. a. Prove that the line $\Delta$ is orthogonal to the plane ( ABC ). b. Deduce a Cartesian equation of the plane (ABC). c. Determine a parametric representation of the line $\Delta$. d. Determine the coordinates of point H, the intersection of the line $\Delta$ and the plane (ABC).
  3. Let $\mathscr { P } _ { 1 }$ be the plane with equation $x + y + z = 0$ and $\mathscr { P } _ { 2 }$ be the plane with equation $x + 4 y + 2 = 0$. a. Prove that the planes $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$ are secant. b. Verify that the line $d$, the intersection of the planes $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$, has the parametric representation $$\left\{ \begin{array} { l } x = - 4 t - 2 \\ y = t \\ z = 3 t + 2 \end{array} , t \in \mathbb { R } . \right.$$ c. Are the line $d$ and the plane ( ABC ) secant or parallel?
Q2a 5 marks Sequences and series, recurrence and convergence Algorithm and programming for sequences View
Exercise 2 — Candidates WHO HAVE NOT FOLLOWED the mathematics specialization course
We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = 1$ and, for every natural integer $n$, $$u _ { n + 1 } = \sqrt { 2 u _ { n } } .$$
  1. We consider the following algorithm:

Variables:$n$ is a natural integer
$u$ is a positive real number
Initialization:Request the value of $n$
Assign to $u$ the value 1
Processing:For $i$ varying from 1 to $n :$
$\mid$ Assign to $u$ the value $\sqrt { 2 u }$
End of For
Output :Display $u$

a. Give an approximate value to $10 ^ { - 4 }$ of the result displayed by this algorithm when $n = 3$ is chosen. b. What does this algorithm allow us to calculate? c. The table below gives approximate values obtained using this algorithm for certain values of $n$.
$n$15101520
Displayed value1,41421,95711,99861,99991,9999

What conjectures can be made concerning the sequence $\left( u _ { n } \right)$ ?
2. a. Prove that, for every natural integer $n , 0 < u _ { n } \leqslant 2$. b. Determine the direction of variation of the sequence $( u _ { n } )$. c. Prove that the sequence $( u _ { n } )$ is convergent. We do not ask for the value of its limit.
3. We consider the sequence $( v _ { n } )$ defined, for every natural integer $n$, by $v _ { n } = \ln u _ { n } - \ln 2$. a. Prove that the sequence $\left( v _ { n } \right)$ is a geometric sequence with ratio $\frac { 1 } { 2 }$ and first term $v _ { 0 } = - \ln 2$. b. Determine, for every natural integer $n$, the expression of $v _ { n }$ as a function of $n$, then of $u _ { n }$ as a function of $n$. c. Determine the limit of the sequence $\left( u _ { n } \right)$. d. Copy the algorithm below and complete it with the instructions for processing and output, so as to display as output the smallest value of $n$ such that $u _ { n } > 1,999$.
Variables:$n$ is a natural integer
$u$ is a real number
Initialization :Assign to $n$ the value 0
Assign to $u$ the value 1
Processing:
Output :
Q2b 5 marks Number Theory Modular Arithmetic Computation View
Exercise 2 — Candidates WHO HAVE FOLLOWED the mathematics specialization course
Part A
We consider the following algorithm:
Variables :$a$ is a natural integer $b$ is a natural integer $c$ is a natural integer
Initialization :Assign to $c$ the value 0 Request the value of $a$ Request the value of $b$
Processing :While $a > b$ Assign to $c$ the value $c + 1$ Assign to $a$ the value $a - b$ End of while
Output :Display $c$ Display $a$

  1. Run this algorithm with $a = 13$ and $b = 4$ by indicating the values of the variables at each step.
  2. What does this algorithm allow us to calculate?

Part B
To each letter of the alphabet, we associate, using the table below, an integer between 0 and 25.
ABCDEFGHIJKLM
0123456789101112
NOPQRSTUVWXYZ
13141516171819202122232425

We define an encoding process as follows: Step 1: To the letter we want to encode, we associate the corresponding number $m$ in the table. Step 2: We calculate the remainder of the Euclidean division of $9 m + 5$ by 26 and denote it $p$. Step 3 : To the number $p$, we associate the corresponding letter in the table.
  1. Encode the letter U.
  2. Modify the algorithm from Part A so that for a value of $m$ entered by the user, it displays the value of $p$, calculated using the above encoding process.

Part C
  1. Find an integer $x$ such that $9 x \equiv 1$ [26].
  2. Then prove the equivalence: $$9 m + 5 \equiv p \quad [ 26 ] \Longleftrightarrow m \equiv 3 p - 15$$
  3. Then decode the letter B.
Q3 Normal Distribution Direct Probability Calculation from Given Normal Distribution View
An industrial bakery uses a machine to manufacture loaves of country bread weighing on average 400 grams. To be sold to customers, these loaves must weigh at least 385 grams. A loaf whose mass is strictly less than 385 grams is non-marketable, a loaf whose mass is greater than or equal to 385 grams is marketable. The mass of a loaf manufactured by the machine can be modeled by a random variable $X$ following the normal distribution with mean $\mu = 400$ and standard deviation $\sigma = 11$.
Probabilities will be rounded to the nearest thousandth.
Part A
You may use the following table in which values are rounded to the nearest thousandth.
$x$380385390395400405410415420
$P ( X \leqslant x )$0,0350,0860,1820,3250,50,6750,8180,9140,965

  1. Calculate $P ( 390 \leqslant X \leqslant 410 )$.
  2. Calculate the probability $p$ that a loaf chosen at random from production is marketable.
  3. The manufacturer finds this probability $p$ too low. He decides to modify his production methods in order to vary the value of $\sigma$ without changing that of $\mu$. For what value of $\sigma$ is the probability that a loaf is marketable equal to $96\%$ ? Round the result to the nearest tenth. You may use the following result: when $Z$ is a random variable that follows the normal distribution with mean 0 and standard deviation 1, we have $P ( Z \leqslant - 1,751 ) \approx 0,040$.

Part B
The production methods have been modified with the aim of obtaining $96\%$ marketable loaves. To evaluate the effectiveness of these modifications, a quality control is performed on a sample of 300 loaves manufactured.
  1. Determine the asymptotic confidence interval at the $95\%$ confidence level for the proportion of marketable loaves in a sample of size 300.
  2. Among the 300 loaves in the sample, 283 are marketable.

In light of the confidence interval obtained in question 1, can we decide that the objective has been achieved?
Part C
The baker uses an electronic scale. The operating time without malfunction, in days, of this electronic scale is a random variable $T$ that follows an exponential distribution with parameter $\lambda$.
  1. We know that the probability that the electronic scale does not malfunction before 30 days is 0,913. Deduce the value of $\lambda$ rounded to the nearest thousandth.

Throughout the rest, we will take $\lambda = 0,003$.
2. What is the probability that the electronic scale continues to function without malfunction after 90 days, given that it has functioned without malfunction for 60 days?
3. The seller of this electronic scale assured the baker that there was a one in two chance that the scale would not malfunction before a year. Is he right? If not, for how many days is this true?
Q4 5 marks Differentiating Transcendental Functions Full function study with transcendental functions View
Let $f$ be the function defined on the interval $] 0 ; + \infty [$ by
$$f ( x ) = \frac { 1 + \ln ( x ) } { x ^ { 2 } }$$
and let $\mathscr { C }$ be the representative curve of the function $f$ in a coordinate system of the plane.
  1. a. Study the limit of $f$ at 0. b. What is $\lim _ { x \rightarrow + \infty } \frac { \ln ( x ) } { x }$ ? Deduce the limit of the function $f$ at $+ \infty$. c. Deduce the possible asymptotes to the curve $\mathscr { C }$.
  2. a. Let $f ^ { \prime }$ denote the derivative function of the function $f$ on the interval $] 0 ; + \infty [$. Prove that, for every real $x$ belonging to the interval $] 0 ; + \infty [$, $$f ^ { \prime } ( x ) = \frac { - 1 - 2 \ln ( x ) } { x ^ { 3 } }$$ b. Solve on the interval $] 0 ; + \infty [$ the inequality $- 1 - 2 \ln ( x ) > 0$. Deduce the sign of $f ^ { \prime } ( x )$ on the interval $] 0 ; + \infty [$. c. Draw up the table of variations of the function $f$.
  3. a. Prove that the curve $\mathscr { C }$ has a unique point of intersection with the $x$-axis, whose coordinates you will specify. b. Deduce the sign of $f ( x )$ on the interval $] 0 ; + \infty [$.
  4. For every integer $n \geqslant 1$, we denote by $I _ { n }$ the area, expressed in square units, of the region bounded by the $x$-axis, the curve $\mathscr { C }$ and the lines $x = 1$ and $x = n$.