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
2017 centres-etrangers

8 maths questions

QII Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Space is equipped with an orthonormal coordinate system $( O ; \vec { i } ; \vec { j } ; \vec { k } )$. We consider two lines $d _ { 1 }$ and $d _ { 2 }$ defined by the parametric representations:
$$d _ { 1 } : \left\{ \begin{array} { l } { x = 2 + t } \\ { y = 3 - t } \\ { z = t } \end{array} , t \in \mathbb { R } \text { and } \left\{ \begin{array} { l } x = - 5 + 2 t ^ { \prime } \\ y = - 1 + t ^ { \prime } \\ z = 5 \end{array} , t ^ { \prime } \in \mathbb { R } . \right. \right.$$
We admit that the lines $d _ { 1 }$ and $d _ { 2 }$ are non-coplanar. The purpose of this exercise is to determine, if it exists, a third line $\Delta$ that is simultaneously secant to both lines $d _ { 1 }$ and $d _ { 2 }$ and orthogonal to these two lines.
  1. Verify that the point $\mathrm { A } ( 2 ; 3 ; 0 )$ belongs to the line $d _ { 1 }$.
  2. Give a direction vector $\overrightarrow { u _ { 1 } }$ of the line $d _ { 1 }$ and a direction vector $\overrightarrow { u _ { 2 } }$ of the line $d _ { 2 }$. Are the lines $d _ { 1 }$ and $d _ { 2 }$ parallel?
  3. Verify that the vector $\vec { v } ( 1 ; - 2 ; - 3 )$ is orthogonal to the vectors $\overrightarrow { u _ { 1 } }$ and $\overrightarrow { u _ { 2 } }$.
  4. Let $P$ be the plane passing through point A, and directed by the vectors $\overrightarrow { u _ { 1 } }$ and $\vec { v }$. In this question we study the intersection of the line $d _ { 2 }$ and the plane $P$. a. Show that a Cartesian equation of the plane $P$ is: $5 x + 4 y - z - 22 = 0$. b. Show that the line $d _ { 2 }$ intersects the plane $P$ at the point $\mathrm { B } ( 3 ; 3 ; 5 )$.
  5. We now consider the line $\Delta$ directed by the vector $\vec { v} \left( \begin{array} { c } 1 \\ - 2 \\ - 3 \end{array} \right)$, and passing through the point $\mathrm { B } ( 3 ; 3 ; 5 )$. a. Give a parametric representation of this line $\Delta$. b. Are the lines $d _ { 1 }$ and $\Delta$ secant? Justify your answer. c. Explain why the line $\Delta$ answers the problem posed.
QIII 6 marks Exponential Functions Applied/Contextual Exponential Modeling View
Pharmacokinetics studies the evolution of a drug after its administration in the body, by measuring its plasma concentration, that is to say its concentration in the plasma. In this exercise we study the evolution of plasma concentration in a patient of the same dose of drug, considering different modes of administration.
Part A: administration by intravenous route
We denote $f ( t )$ the plasma concentration, expressed in microgram per litre ( $\mu \mathrm { g } . \mathrm { L } ^ { - 1 }$ ), of the drug, after $t$ hours following administration by intravenous route. The mathematical model is: $f ( t ) = 20 \mathrm { e } ^ { - 0,1 t }$, with $t \in [ 0 ; + \infty [$. The initial plasma concentration of the drug is therefore $f ( 0 ) = 20 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$.
  1. The half-life of the drug is the duration (in hours) after which the plasma concentration of the drug is equal to half the initial concentration. Determine this half-life, denoted $t _ { 0,5 }$.
  2. It is estimated that the drug is eliminated as soon as the plasma concentration is less than $0.2 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$. Determine the time from which the drug is eliminated. The result will be given rounded to the nearest tenth.
  3. In pharmacokinetics, we call AUC (or ``area under the curve''), in $\mu \mathrm { g } . \mathrm { L } ^ { - 1 }$, the number $\lim _ { x \rightarrow + \infty } \int _ { 0 } ^ { x } f ( t ) \mathrm { d } t$. Verify that for this model, the AUC is equal to $200 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$.

Part B: administration by oral route
We denote $g ( t )$ the plasma concentration of the drug, expressed in microgram per litre ( $\mu g.L^{-1}$ ), after $t$ hours following ingestion by oral route. The mathematical model is: $g ( t ) = 20 \left( \mathrm { e } ^ { - 0,1 t } - \mathrm { e } ^ { - t } \right)$, with $t \in [ 0 ; + \infty [$. In this case, the effect of the drug is delayed, since the initial plasma concentration is equal to: $g ( 0 ) = 0 \mu g . \mathrm { L } ^ { - 1 }$.
  1. Prove that, for all $t$ in the interval $[ 0 ; + \infty [$, we have: $g ^ { \prime } ( t ) = 20 \mathrm { e } ^ { - t } \left( 1 - 0,1 \mathrm { e } ^ { 0,9 t } \right)$.
  2. Study the variations of the function $g$ on the interval $[ 0 ; + \infty [$. (The limit at $+ \infty$ is not required.) Deduce the duration after which the plasma concentration of the drug is maximum. The result will be given to the nearest minute.

Part C: repeated administration by intravenous route
We decide to inject at regular time intervals the same dose of drug by intravenous route. The time interval (in hours) between two injections is chosen equal to the half-life of the drug, that is to say the number $t _ { 0,5 }$ which was calculated in A - 1. Each new injection causes an increase in plasma concentration of $20 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$. We denote $u _ { n }$ the plasma concentration of the drug immediately after the $n$-th injection. Thus, $u _ { 1 } = 20$ and, for all integer $n$ greater than or equal to 1, we have: $u _ { n + 1 } = 0,5 u _ { n } + 20$.
  1. Prove by induction that, for all integer $n \geqslant 1 : u _ { n } = 40 - 40 \times 0,5 ^ { n }$.
  2. Determine the limit of the sequence $( u _ { n } )$ as $n$ tends to $+ \infty$.
  3. We consider that equilibrium is reached as soon as the plasma concentration exceeds $38 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$. Determine the minimum number of injections necessary to reach this equilibrium.
QIV Radians, Arc Length and Sector Area View
The plane is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { u } , \vec { v }$ ). For all integer $n \geqslant 4$, we consider $P _ { n }$ a regular polygon with $n$ sides, with centre $O$ and whose area is equal to 1. We admit that such a polygon is made up of $n$ triangles superimposable to a given triangle $\mathrm { OA } _ { n } \mathrm {~B} _ { n }$, isosceles at O. We denote $r _ { n } = \mathrm { OA } _ { n }$ the distance between the centre O and the vertex $\mathrm { A } _ { n }$ of such a polygon.
Part A: study of the particular case $n = 6$
  1. Justify the fact that the triangle $\mathrm { OA } _ { 6 } \mathrm {~B} _ { 6 }$ is equilateral, and that its area is equal to $\frac { 1 } { 6 }$.
  2. Express as a function of $r _ { 6 }$ the height of the triangle $\mathrm { OA } _ { 6 } \mathrm {~B} _ { 6 }$ from the vertex $\mathrm { B } _ { 6 }$.
  3. Deduce that $r _ { 6 } = \sqrt { \frac { 2 } { 3 \sqrt { 3 } } }$.

Part B: general case with $n \geqslant 4$
In the method considered, we take as initial matrix the matrix $I = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$.
  1. Determine the two missing matrices $A$ and $B$, in the third row of the Stern-Brocot tree.
  2. We associate to a matrix $M = \left( \begin{array} { l l } a & c \\ b & d \end{array} \right)$ of the Stern-Brocot tree the fraction $\frac { a + c } { b + d }$. Show that, in this association, the path ``left-right-left'' starting from the initial matrix in the tree, leads to a matrix corresponding to the fraction $\frac { 3 } { 5 }$.
  3. Let $M = \left( \begin{array} { l l } a & c \\ b & d \end{array} \right)$ be a matrix of the tree. We recall that $a , b , c , d$ are integers. We denote $\Delta _ { M } = a d - b c$, the difference of the diagonal products of this matrix. a. Show that if $a d - b c = 1$, then $d ( a + c ) - c ( b + d ) = 1$. b. Deduce that if $M = \left( \begin{array} { l l } a & c \\ b & d \end{array} \right)$ is a matrix of the Stern-Brocot tree such that $\Delta _ { M } = a d - b c = 1$, then $\Delta _ { M \times G } = 1$, that is, the difference of the diagonal products of the matrix $M \times G$ is also equal to 1. We similarly admit that $\Delta _ { M \times D } = 1$, and that all other matrices $N$ of the Stern-Brocot tree satisfy the equality $\Delta _ { N } = 1$.
  4. Deduce from the previous question that every fraction associated with a matrix of the Stern-Brocot tree is in lowest terms.
  5. Let $m$ and $n$ be two non-zero natural integers that are coprime. Thus the fraction $\frac { m } { n }$ is in lowest terms. We consider the following algorithm: \begin{verbatim} VARIABLES : m and n are non-zero natural integers and coprime PROCESSING : While m = do If m
    Display$\ldots$$\ldots$$\ldots$$\ldots$
    $m$4$\ldots$$\ldots$$\ldots$$\ldots$
    $n$7$\ldots$$\ldots$$\ldots$$\ldots$

    b. Conjecture the role of this algorithm. Verify by a matrix calculation the result provided with the values $m = 4$ and $n = 7$.
Q1 1 marks Normal Distribution Multiple-Choice Conceptual Question on Normal Distribution Properties View
We study the production of a factory that manufactures sweets, packaged in bags. A bag is chosen at random from daily production. The mass of this bag, expressed in grams, is modelled by a random variable $X$ which follows a normal distribution with mean $\mu = 175$. Furthermore, statistical observation has shown that $2\%$ of bags have a mass less than or equal to 170 g, which is expressed in the model considered by: $P ( X \leqslant 170 ) = 0.02$.
What is the probability, rounded to the nearest hundredth, of the event ``the mass of the bag is between 170 and 180 grams''?
Answer a: 0.04 Answer b: 0.96 Answer c: 0.98 Answer d: We cannot answer because data is missing.
Q2 1 marks Binomial Distribution MCQ Selecting a Binomial Probability Expression or Value View
The different sweets in the bags are all coated with a layer of edible wax. This process, which deforms some sweets, is carried out by two machines A and B. When produced by machine A, the probability that a randomly selected sweet is deformed is equal to 0.05.
On a random sample of 50 sweets from machine A, what is the probability, rounded to the nearest hundredth, that at least 2 sweets are deformed?
Answer a: 0.72 Answer b: 0.28 Answer c: 0.54 Answer d: We cannot answer because data is missing
Q3 1 marks Conditional Probability Bayes' Theorem with Production/Source Identification View
Machine A produces one third of the factory's sweets. The rest of production is ensured by machine B. When produced by machine B, the probability that a randomly selected sweet is deformed is equal to 0.02. In a quality control test, a sweet is randomly selected from the entire production. It is deformed.
What is the probability, rounded to the nearest hundredth, that it was produced by machine B?
Answer a: 0.02 Answer b: 0.67 Answer c: 0.44 Answer d: 0.01
Q4 1 marks Exponential Distribution View
The operating lifetime, expressed in days, of a machine used for coating, is modelled by a random variable $Y$ which follows the exponential distribution with mean equal to 500 days.
What is the probability, rounded to the nearest hundredth, that the operating lifetime of the machine is less than or equal to 300 days?
Answer a: 0.45 Answer b: 1 Answer c: 0.55 Answer d: We cannot answer because data is missing
Q5 1 marks Confidence intervals Determine minimum sample size for a desired interval width View
The company wishes to estimate the proportion of people over 20 years old among its customers, at a confidence level of $95\%$, with an interval amplitude less than 0.05. It questions for this purpose a random sample of customers.
What is the minimum number of customers to question?
Answer a: 40 Answer b: 400 Answer c: 1600 Answer d: 20