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
2022 bac-spe-maths__amerique-nord_j1

4 maths questions

Q1 7 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
Every day he works, Paul must go to the station to reach his workplace by train. To do this, he takes his bicycle two times out of three and, if he does not take his bicycle, he takes his car.
  1. When he takes his bicycle to reach the station, Paul misses the train only once in 50, whereas when he takes his car to reach the station, Paul misses his train once in 10. We consider a random day on which Paul will be at the station to catch the train that will take him to work. We denote:
    • V the event ``Paul takes his bicycle to reach the station'';
    • R the event ``Paul misses his train''. a. Draw a weighted tree summarizing the situation. b. Show that the probability that Paul misses his train is equal to $\frac { 7 } { 150 }$. c. Paul has missed his train. Determine the exact value of the probability that he took his bicycle to reach the station.

  2. A random month is chosen during which Paul went to the station 20 days to reach his workplace according to the procedures described in the preamble. We assume that, for each of these 20 days, the choice between bicycle and car is independent of the choices on other days. We denote $X$ the random variable giving the number of days Paul takes his bicycle over these 20 days. a. Determine the distribution followed by the random variable $X$. Specify its parameters. b. What is the probability that Paul takes his bicycle exactly 10 days out of these 20 days to reach the station? Round the probability sought to $10 ^ { - 3 }$. c. What is the probability that Paul takes his bicycle at least 10 days out of these 20 days to reach the station? Round the probability sought to $10 ^ { - 3 }$. d. On average, how many days over a randomly chosen period of 20 days to reach the station does Paul take his bicycle? Round the answer to the nearest integer.
  3. In the case where Paul goes to the station by car, we denote $T$ the random variable giving the travel time needed to reach the station. The duration of the journey is given in minutes, rounded to the minute. The probability distribution of $T$ is given by the table below:

$k$ (in minutes)101112131415161718
$P ( T = k )$0,140,130,130,120,120,110,100,080,07

Determine the expected value of the random variable $T$ and interpret this value in the context of the exercise.
Q2 7 marks Geometric Sequences and Series Prove a Transformed Sequence is Geometric View
In this exercise, we consider the sequence ( $T _ { n }$ ) defined by:
$$T _ { 0 } = 180 \mathrm { and } , \text { for all natural integer } n , T _ { n + 1 } = 0,955 T _ { n } + 0,9$$
  1. a. Prove by induction that, for all natural integer $n , T _ { n } \geqslant 20$. b. Verify that for all natural integer $n , T _ { n + 1 } - T _ { n } = - 0,045 \left( T _ { n } - 20 \right)$. Deduce the direction of variation of the sequence ( $T _ { n }$ ). c. Conclude from the above that the sequence ( $T _ { n }$ ) is convergent. Justify.
  2. For all natural integer $n$, we set: $u _ { n } = T _ { n } - 20$. a. Show that the sequence ( $u _ { n }$ ) is a geometric sequence and specify its common ratio. b. Deduce that for all natural integer $n , T _ { n } = 20 + 160 \times 0,955 ^ { n }$. c. Calculate the limit of the sequence ( $T _ { n }$ ). d. Solve the inequality $T _ { n } \leqslant 120$ with unknown $n$ a natural integer.
  3. In this part, we are interested in the evolution of temperature at the center of a cake after it comes out of the oven. We consider that when the cake comes out of the oven, the temperature at the center of the cake is $180 ^ { \circ } \mathrm { C }$ and that of the ambient air is $20 ^ { \circ } \mathrm { C }$. Newton's law of cooling allows us to model the temperature at the center of the cake by the previous sequence ( $T _ { n }$ ). More precisely, $T _ { n }$ represents the temperature at the center of the cake, expressed in degrees Celsius, $n$ minutes after it comes out of the oven. a. Explain why the limit of the sequence ( $T _ { n }$ ) determined in question 2. c. was foreseeable in the context of the exercise. b. We consider the following Python function:

\begin{verbatim} def temp(x) : T = 180 n = 0 while T > x : T=0.955*T+0.9 n=n+1 return n \end{verbatim}
Give the result obtained by executing the command temp(120). Interpret the result in the context of the exercise.
Q3 7 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
In space equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ) with unit 1 cm, we consider the following points:
$$\mathrm { J } ( 2 ; 0 ; 1 ) , \quad \mathrm { K } ( 1 ; 2 ; 1 ) \text { and } \quad \mathrm { L } ( - 2 ; - 2 ; - 2 )$$
  1. a. Show that triangle JKL is right-angled at J. b. Calculate the exact value of the area of triangle JKL in $\mathrm { cm } ^ { 2 }$. c. Determine an approximate value to the nearest tenth of the geometric angle $\widehat { \mathrm { JKL } }$.
  2. a. Prove that the vector $\vec { n }$ with coordinates $\left( \begin{array} { c } 6 \\ 3 \\ - 10 \end{array} \right)$ is a normal vector to the plane (JKL). b. Deduce a Cartesian equation of the plane (JKL).

In the following, T denotes the point with coordinates ( $10 ; 9 ; - 6$ ).
3. a. Determine a parametric representation of the line $\Delta$ perpendicular to the plane (JKL) and passing through T. b. Determine the coordinates of point H, the orthogonal projection of point T onto the plane (JKL). c. We recall that the volume $V$ of a tetrahedron is given by the formula:
$$V = \frac { 1 } { 3 } \mathscr { B } \times h \text { where } \mathscr { B } \text { denotes the area of a base and } h \text { the corresponding height }$$
Calculate the exact value of the volume of tetrahedron JKLT in $\mathrm { cm } ^ { 3 }$.
Q4 7 marks Exponential Functions True/False or Multiple-Statement Verification View
For each of the following statements, indicate whether it is true or false. Justify each answer.
  1. Statement 1: For all real $x : 1 - \frac { 1 - \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } } = \frac { 2 } { 1 + \mathrm { e } ^ { - x } }$.
  2. We consider the function $g$ defined on $\mathbb { R }$ by $g ( x ) = \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 }$. Statement 2: The equation $g ( x ) = \frac { 1 } { 2 }$ admits a unique solution in $\mathbb { R }$.
  3. We consider the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x ^ { 2 } \mathrm { e } ^ { - x }$ and we denote $\mathscr { C }$ its curve in an orthonormal coordinate system. Statement 3: The $x$-axis is tangent to the curve $\mathscr { C }$ at only one point.
  4. We consider the function $h$ defined on $\mathbb { R }$ by $h ( x ) = \mathrm { e } ^ { x } \left( 1 - x ^ { 2 } \right)$. Statement 4: In the plane equipped with an orthonormal coordinate system, the curve representing the function $h$ does not admit an inflection point.
  5. Statement 5: $\lim _ { x \rightarrow + \infty } \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + x } = 0$.
  6. Statement 6: For all real $x , 1 + \mathrm { e } ^ { 2 x } \geqslant 2 \mathrm { e } ^ { x }$.