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
2016 liban

6 maths questions

Q1 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Consider a solid ADECBF consisting of two identical pyramids with the square ABCD as common base with centre I. A perspective representation of this solid is given in the appendix (to be returned with the answer sheet). All edges have length 1. The space is referred to the orthonormal coordinate system ( $\mathrm { A } ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AK } }$ ).
  1. a) Show that $\mathrm { IE } = \frac { \sqrt { 2 } } { 2 }$. Deduce the coordinates of points I, E and F. b) Show that the vector $\vec { n } \left( \begin{array} { c } 0 \\ - 2 \\ \sqrt { 2 } \end{array} \right)$ is normal to the plane (ABE). c) Determine a Cartesian equation of the plane (ABE).
  2. Let M be the midpoint of segment [DF] and N the midpoint of segment [AB]. a) Prove that the planes $( \mathrm { FDC } )$ and $( \mathrm { ABE } )$ are parallel. b) Determine the intersection of planes (EMN) and (FDC). c) Construct on the appendix (to be returned with the answer sheet) the cross-section of solid ADECBF by plane (EMN).
Q2 Binomial Distribution Compute Cumulative or Complement Binomial Probability View
On a tennis court, a ball launcher allows a player to train alone. This device sends balls one by one at a regular rate. The player then hits the ball and the next ball arrives. According to the manufacturer's manual, the ball launcher sends the ball randomly to the right or to the left with equal probability.
Throughout the exercise, results will be rounded to $10 ^ { - 3 }$ near.
Part A
The player is about to receive a series of 20 balls.
  1. What is the probability that the ball launcher sends 10 balls to the right?
  2. What is the probability that the ball launcher sends between 5 and 10 balls to the right?

Part B
The ball launcher is equipped with a reservoir that can hold 100 balls. Over a sequence of 100 launches, 42 balls were launched to the right. The player then doubts the proper functioning of the device. Are his doubts justified?
Part C
To increase the difficulty, the player configures the ball launcher to give spin to the balls launched. They can be either ``topspin'' or ``slice''. The probability that the ball launcher sends a ball to the right is still equal to the probability that the ball launcher sends a ball to the left. The device settings allow us to state that:
  • the probability that the ball launcher sends a topspin ball to the right is 0.24;
  • the probability that the ball launcher sends a slice ball to the left is 0.235.

If the ball launcher sends a slice ball, what is the probability that it is sent to the right?
Q3 4 marks Standard Integrals and Reverse Chain Rule Verify or Prove an Antiderivative/Integral Identity View
Consider the function $f$ defined on the interval $[ 0 ; 1 ]$ by:
$$f ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { 1 - x } }$$
Part A
  1. Study the direction of variation of the function $f$ on the interval $[ 0 ; 1 ]$.
  2. Prove that for all real $x$ in the interval $[ 0 ; 1 ] , f ( x ) = \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + \mathrm { e } }$ (recall that $\mathrm { e } = \mathrm { e } ^ { 1 }$ ).
  3. Show then that $\int _ { 0 } ^ { 1 } f ( x ) \mathrm { d } x = \ln ( 2 ) + 1 - \ln ( 1 + \mathrm { e } )$.

Part B
Let $n$ be a natural number. Consider the functions $f _ { n }$ defined on $[ 0 ; 1 ]$ by:
$$f _ { n } ( x ) = \frac { 1 } { 1 + n \mathrm { e } ^ { 1 - x } }$$
We denote $\mathscr { C } _ { n }$ the representative curve of the function $f _ { n }$ in the plane with an orthonormal coordinate system. Consider the sequence with general term
$$u _ { n } = \int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x$$
  1. The representative curves of the functions $f _ { n }$ for $n$ varying from 1 to 5 are drawn in the appendix. Complete the graph by drawing the curve $\mathscr { C } _ { 0 }$ representative of the function $f _ { 0 }$.
  2. Let $n$ be a natural number, interpret graphically $u _ { n }$ and specify the value of $u _ { 0 }$.
  3. What conjecture can be made regarding the direction of variation of the sequence $\left( u _ { n } \right)$ ?

Prove this conjecture.
4. Does the sequence ( $u _ { n }$ ) have a limit?
Q4a Normal Distribution Multiple-Choice Conceptual Question on Normal Distribution Properties View
Exercise 4 — Candidates who have NOT followed the specialization course
For each of the following statements, say whether it is true or false by justifying the answer. One point is awarded for each correct justified answer. An unjustified answer will not be taken into account and the absence of an answer is not penalized.
  • In the diagram below, the density curve of a random variable $X$ following a normal distribution with mean $\mu = 20$ is represented. The probability that the random variable $X$ is between 20 and 21.6 is equal to 0.34.

Statement 1: The probability that the random variable $X$ belongs to the interval $[23.2; + \infty [$ is approximately 0.046.
  • Let $z$ be a complex number different from 2. We set:

$$Z = \frac { \mathrm { i } z } { z - 2 }$$
Statement 2: The set of points in the complex plane with affixe $z$ such that $| Z | = 1$ is a line passing through point $\mathrm { A } ( 1 ; 0 )$. Statement 3: $Z$ is a pure imaginary number if and only if $z$ is real.
  • Let $f$ be the function defined on $\mathbb { R }$ by:

$$f ( x ) = \frac { 3 } { 4 + 6 \mathrm { e } ^ { - 2 x } }$$
Statement 4: The equation $f ( x ) = 0.5$ has a unique solution on $\mathbb { R }$. Statement 5: The following algorithm displays as output the value 0.54.
\begin{tabular}{l} Variables: Initialization:
Processing:
Output:
&
$X$ and $Y$ are real numbers
$X$ takes the value 0
$Y$ takes the value $\frac { 3 } { 10 }$
While $Y < 0.5$
$X$ takes the value $X + 0.01$
$Y$ takes the value $\frac { 3 } { 4 + 6 \mathrm { e } ^ { - 2 X } }$
End While
Display $X$
\hline \end{tabular}
Q4b Discrete Probability Distributions Markov Chain and Transition Matrix Analysis View
Exercise 4 — Candidates who have followed the specialization course
For each of the following statements, say whether it is true or false by justifying the answer. One point is awarded for each correct justified answer. An unjustified answer will not be taken into account and the absence of an answer is not penalized.
  • Consider the system $\left\{ \begin{array} { l l l l } n & \equiv & 1 & { [ 5 ] } \\ n & \equiv & 3 & { [ 4 ] } \end{array} \right.$ with unknown $n$ a relative integer.

Statement 1: If $n$ is a solution of this system then $n - 11$ is divisible by 4 and by 5. Statement 2: For all relative integer $k$, the integer $11 + 20 k$ is a solution of the system. Statement 3: If a relative integer $n$ is a solution of the system then there exists a relative integer $k$ such that $n = 11 + 20 k$.
  • An automaton can be in one of two states A or B. At each second it can either remain in the state it is in or change it, with probabilities given by the probabilistic graph below. For all natural number $n$, we denote $a _ { n }$ the probability that the automaton is in state A after $n$ seconds and $b _ { n }$ the probability that the automaton is in state B after $n$ seconds. Initially, the automaton is in state B.

Consider the following algorithm:
\begin{tabular}{l} Variables: Initialization:
Processing:
Output:
&
$a$ and $b$ are real numbers
$a$ takes the value 0
$b$ takes the value 1
For $k$ going from 1 to 10
$a$ takes the value $0.8 a + 0.3 b$
$b$ takes the value $1 - a$
End For
Display $a$
Display $b$
\hline \end{tabular}
Statement 4: On output, this algorithm displays the values of $a _ { 10 }$ and $b _ { 10 }$. Statement 5: After 4 seconds, the automaton has an equal chance of being in state $A$ or being in state $B$.
Q5 3 marks Complex numbers 2 Complex Recurrence Sequences View
Consider the sequence ( $z _ { n }$ ) of complex numbers defined for all natural number $n$ by:
$$\left\{ \begin{array} { l } z _ { 0 } = 0 \\ z _ { n + 1 } = \frac { 1 } { 2 } \mathrm { i } \times z _ { n } + 5 \end{array} \right.$$
In the plane with an orthonormal coordinate system, we denote $M _ { n }$ the point with affixe $z _ { n }$. Consider the complex number $z _ { \mathrm { A } } = 4 + 2 \mathrm { i }$ and A the point in the plane with affixe $z _ { \mathrm { A } }$.
  1. Let ( $u _ { n }$ ) be the sequence defined for all natural number $n$ by $u _ { n } = z _ { n } - z _ { \mathrm { A } }$. a) Show that, for all natural number $n , u _ { n + 1 } = \frac { 1 } { 2 } \mathrm { i } \times u _ { n }$. b) Prove that, for all natural number $n$:
    $$u _ { n } = \left( \frac { 1 } { 2 } \mathrm { i } \right) ^ { n } ( - 4 - 2 \mathrm { i } )$$
  2. Prove that, for all natural number $n$, the points $\mathrm { A } , M _ { n }$ and $M _ { n + 4 }$ are collinear.