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
2024 bac-spe-maths__amerique-sud_j1

4 maths questions

Q1 Differential equations First-Order Linear DE: General Solution View
Exercise 1
PART A Consider the differential equation $$( E ) : \quad y ^ { \prime } + \frac { 1 } { 4 } y = 20 \mathrm { e } ^ { - \frac { 1 } { 4 } x } ,$$ with unknown $y$, a function defined and differentiable on the interval $[ 0 ; + \infty [$.
  1. Determine the value of the real number $a$ such that the function $g$ defined on the interval $[ 0 ; + \infty [$ by $g ( x ) = a x \mathrm { e } ^ { - \frac { 1 } { 4 } x }$ is a particular solution of the differential equation $( E )$.
  2. Consider the differential equation $$\left( E ^ { \prime } \right) : \quad y ^ { \prime } + \frac { 1 } { 4 } y = 0 ,$$ with unknown $y$, a function defined and differentiable on the interval $[ 0 ; + \infty [$. Determine the solutions of the differential equation ( $E ^ { \prime }$ ).
  3. Deduce the solutions of the differential equation ( $E$ ).
  4. Determine the solution $f$ of the differential equation ( $E$ ) such that $f ( 0 ) = 8$.

PART B Consider the function $f$ defined on the interval $[ 0 ; + \infty [$ by $$f ( x ) = ( 20 x + 8 ) \mathrm { e } ^ { - \frac { 1 } { 4 } x }$$ It is admitted that the function $f$ is differentiable on the interval $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $f ^ { \prime }$ its derivative function on the interval $\left[ 0 ; + \infty \left[ \right. \right.$. Moreover, it is admitted that $\lim _ { x \rightarrow + \infty } f ( x ) = 0$.
  1. a. Justify that, for every positive real number $x$, $$f ^ { \prime } ( x ) = ( 18 - 5 x ) \mathrm { e } ^ { - \frac { 1 } { 4 } x }$$ b. Deduce the table of variations of the function $f$. The exact value of the maximum of the function $f$ on the interval $[ 0 ; + \infty [$ will be specified.
  2. In this question we are interested in the equation $f ( x ) = 8$. a. Justify that the equation $f ( x ) = 8$ admits a unique solution, denoted $\alpha$, in the interval [14; 15]. b. Copy and complete the table below by running step by step the solution\_equation function opposite, written in Python language
    $a$14
    $b$15
    $b - a$1
    $m$14,5
    \begin{tabular}{ l } Condition
    $f ( m ) > 8$
    & FALSE & & & & \hline \end{tabular}
    \begin{verbatim} from math import exp def f(x) : return (20* x +8)*exp(-1/4* x) def solution_equation() : a,b = 14,15 while b-a>0.1: m = (a+b)/2 if f (m) > 8 : |a=m else : | b=m return a,b \end{verbatim}
    c. What is the objective of the solution\_equation function in the context of the question?
Q2 Conditional Probability Sequential/Multi-Stage Conditional Probability View
Exercise 2
We have two opaque urns $\mathrm { U } _ { 1 }$ and $\mathrm { U } _ { 2 }$. Urn $\mathrm { U } _ { 1 }$ contains 4 black balls and 6 white balls. Urn $\mathrm { U } _ { 2 }$ contains 1 black ball and 3 white balls. Consider the following random experiment: We randomly draw a ball from $U _ { 1 }$ which we place in $U _ { 2 }$, then we randomly draw a ball from $\mathrm { U } _ { 2 }$. We denote:
  • $N _ { 1 }$ the event ``Drawing a black ball from urn $\mathrm { U } _ { 1 }$''.
  • $N _ { 2 }$ the event ``Drawing a black ball from urn $\mathrm { U } _ { 2 }$''.
For any event $A$, we denote $\bar { A }$ its complementary event.
PART A
  1. Consider the probability tree opposite. a. Justify that the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ given that a white ball was drawn from urn $\mathrm { U } _ { 1 }$ is 0.2. b. Copy and complete the probability tree opposite, showing on each branch the probabilities of the events concerned, in decimal form.
  2. Calculate the probability of drawing a black ball from urn $U _ { 1 }$ and a black ball from urn $\mathrm { U } _ { 2 }$.
  3. Justify that the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ is equal to 0.28.
  4. A black ball was drawn from urn $\mathrm { U } _ { 2 }$. Calculate the probability of having drawn a white ball from urn $\mathrm { U } _ { 1 }$. The result will be given in decimal form rounded to $10 ^ { - 2 }$.

PART B $n$ denotes a non-zero natural number. The previous random experiment is repeated $n$ times in an identical and independent manner, that is, urns $\mathrm { U } _ { 1 }$ and $\mathrm { U } _ { 2 }$ are returned to their initial configuration, with respectively 4 black balls and 6 white balls in urn $U _ { 1 }$ and 1 black ball and 3 white balls in urn $\mathrm { U } _ { 2 }$, between each experiment. We denote $X$ the random variable that counts the number of times a black ball is drawn from urn $\mathrm { U } _ { 2 }$. We recall that the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ is equal to 0.28 and that of drawing a white ball from urn $\mathrm { U } _ { 2 }$ is equal to 0.72.
  1. Determine the probability distribution followed by $X$. Justify your answer.
  2. Determine by calculation the smallest natural number $n$ such that: $$1 - 0{,}72 ^ { n } \geqslant 0{,}9$$
  3. Interpret the previous result in the context of the experiment.

PART C In this part urns $\mathrm { U } _ { 1 }$ and $\mathrm { U } _ { 2 }$ are returned to their initial configuration, with respectively 4 black balls and 6 white balls in urn $U _ { 1 }$ and 1 black ball and 3 white balls in urn $\mathrm { U } _ { 2 }$.
Consider the following new random experiment: We simultaneously draw two balls from urn $\mathrm { U } _ { 1 }$ which we place in urn $\mathrm { U } _ { 2 }$, then we randomly draw a ball from urn $\mathrm { U } _ { 2 }$.
  1. How many possible draws of two balls simultaneously from urn $\mathrm { U } _ { 1 }$ are there?
  2. How many possible draws of two balls simultaneously from urn $\mathrm { U } _ { 1 }$ containing exactly one white ball and one black ball are there?
  3. Is the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ with this new experiment greater than the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ with the experiment in part A? Justify your answer. You may use a weighted tree diagram modeling this experiment.
Q3 4 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
Exercise 3
Answer TRUE or FALSE to each of the following statements and justify your answer. Any answer without justification will not be taken into account in the grading. All questions in this exercise are independent.
  1. Consider the sequence $( u _ { n } )$ defined for every non-zero natural number $n$ by $$u _ { n } = \frac { 25 + ( - 1 ) ^ { n } } { n }$$ Statement 1: The sequence $\left( u _ { n } \right)$ is divergent.
  2. Consider the sequence $\left( w _ { n } \right)$ defined for every natural number $n$ by $\left\{ \begin{aligned} w _ { 0 } & = 1 \\ w _ { n + 1 } & = \frac { w _ { n } } { 1 + w _ { n } } \end{aligned} \right.$
    It is admitted that for every natural number $n , w _ { n } > 0$. Consider the sequence $( t _ { n } )$ defined for every natural number $n$ by $t _ { n } = \frac { k } { w _ { n } }$ where $k$ is a strictly positive real number. Statement 2: The sequence $\left( t _ { n } \right)$ is a strictly increasing arithmetic sequence.
  3. Consider the sequence $\left( v _ { n } \right)$ defined for every natural number $n$ by $\left\{ \begin{array} { l l l } v _ { 0 } & = 1 \\ v _ { n + 1 } & = & \ln \left( 1 + v _ { n } \right) \end{array} \right.$ It is admitted that for every natural number $n , v _ { n } > 0$. Statement 3: The sequence $( v _ { n } )$ is decreasing.
  4. Consider the sequence $\left( I _ { n } \right)$ defined for every natural number $n$ by $I _ { n } = \int _ { 1 } ^ { \mathrm { e } } [ \ln ( x ) ] ^ { n } \mathrm {~d} x$.
    Statement 4: For every natural number $n , I _ { n + 1 } = \mathrm { e } - ( n + 1 ) I _ { n }$.
Q4 5 marks Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
Exercise 4
The objective of this exercise is to determine the distance between two non-coplanar lines. By definition, the distance between two non-coplanar lines in space, $( d _ { 1 } )$ and $( d _ { 2 } )$ is the length of the segment $[\mathrm { EF }]$, where E and F are points belonging respectively to $\left( d _ { 1 } \right)$ and to $( d _ { 2 } )$ such that the line (EF) is orthogonal to $( d _ { 1 } )$ and $( d _ { 2 } )$. The space is equipped with an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$. Let $\left( d _ { 1 } \right)$ be the line passing through $\mathrm { A } ( 1 ; 2 ; - 1 )$ with direction vector $\overrightarrow { u _ { 1 } } \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)$ and $\left( d _ { 2 } \right)$ the line with parametric representation: $\left\{ \begin{array} { l } x = 0 \\ y = 1 + t \\ z = 2 + t \end{array} , t \in \mathbb { R } \right.$.
  1. Give a parametric representation of the line $\left( d _ { 1 } \right)$.
  2. Prove that the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$ are non-coplanar.
  3. Let $\mathscr { P }$ be the plane passing through A and directed by the non-collinear vectors $\overrightarrow { u _ { 1 } }$ and $\vec { w } \left( \begin{array} { c } 2 \\ - 1 \\ 1 \end{array} \right)$. Justify that a Cartesian equation of the plane $\mathscr { P }$ is: $- 2 x + y + 5 z + 5 = 0$.
  4. a. Without seeking to calculate the coordinates of the intersection point, justify that the line $( d _ { 2 } )$ and the plane $\mathscr { P }$ are secant. b. We denote F the intersection point of the line $( d _ { 2 } )$ and the plane $\mathscr { P }$. Verify that the point F has coordinates $\left( 0 ; - \frac { 5 } { 3 } ; - \frac { 2 } { 3 } \right)$. Let $( \delta )$ be the line passing through F with direction vector $\vec { w }$. It is admitted that the lines $( \delta )$ and $( d _ { 1 } )$ are secant at a point E with coordinates $\left( - \frac { 2 } { 3 } ; - \frac { 4 } { 3 } ; - 1 \right)$.
  5. a. Justify that the distance EF is the distance between the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$. b. Calculate the distance between the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$.