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 amerique-sud

6 maths questions

Q1 Differentiating Transcendental Functions Determine parameters from function or curve conditions View
The curves $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$ given in appendix 1 are the graphical representations, in an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ), of two functions $f$ and $g$ defined on $[ 0 ; + \infty [$. We consider the points $\mathrm { A } ( 0,5 ; 1 )$ and $\mathrm { B } ( 0 ; - 1 )$ in the coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ). We know that O belongs to $\mathscr { C } _ { f }$ and that the line (OA) is tangent to $\mathscr { C } _ { f }$ at point O.
  1. We assume that the function $f$ is written in the form $f ( x ) = ( a x + b ) \mathrm { e } ^ { - x ^ { 2 } }$ where $a$ and $b$ are real numbers. Determine the exact values of the real numbers $a$ and $b$, detailing the approach. From now on, we consider that $\boldsymbol { f } ( \boldsymbol { x } ) = \mathbf { 2 } \boldsymbol { x } \mathrm { e } ^ { - \boldsymbol { x } ^ { \mathbf { 2 } } }$ for all $\boldsymbol { x }$ belonging to $[ \mathbf { 0 } ; + \infty [$
  2. a. We will admit that, for all real $x$ strictly positive, $f ( x ) = \frac { 2 } { x } \times \frac { x ^ { 2 } } { \mathrm { e } ^ { x ^ { 2 } } }$.
    Calculate $\lim _ { x \rightarrow + \infty } f ( x )$. b. Draw up, justifying it, the table of variations of the function $f$ on $[ 0 ; + \infty [$.
  3. The function $g$ whose representative curve $\mathscr { C } _ { g }$ passes through the point $\mathrm { B } ( 0 ; - 1 )$ is a primitive of the function $f$ on $[ 0 ; + \infty [$. a. Determine the expression of $g ( x )$. b. Let $m$ be a strictly positive real number.
    Calculate $I _ { m } = \int _ { 0 } ^ { m } f ( t ) \mathrm { d } t$ as a function of $m$. c. Determine $\lim _ { m \rightarrow + \infty } I _ { m }$.
  4. a. Justify that $f$ is a probability density function on $[ 0 ; + \infty [$. b. Let $X$ be a continuous random variable that admits the function $f$ as its probability density function. Justify that, for all real $x$ in $[ 0 ; + \infty [$, $P ( X \leqslant x ) = g ( x ) + 1$. c. Deduce the exact value of the real number $\alpha$ such that $P ( X \leqslant \alpha ) = 0,5$. d. Without using an approximate value of $\alpha$, construct in the coordinate system of appendix 1 the point with coordinates ( $\alpha ; 0$ ) leaving the construction lines visible. Then shade the region of the plane corresponding to $P ( X \leqslant \alpha )$.
Q2 Complex Numbers Argand & Loci True/False or Multiple-Statement Verification View
We equip the complex plane with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ).
For each of the three following propositions, indicate whether it is true or false and justify the chosen answer. One point is awarded for each correct answer properly justified. An unjustified answer is not taken into account.
Proposition 1 The set of points in the plane with affixe $z$ such that $| z - 4 | = | z + 2 \mathrm { i } |$ is a line that passes through the point A with affixe 3i.
Proposition 2 Let ( $E$ ) be the equation $( z - 1 ) \left( z ^ { 2 } - 8 z + 25 \right) = 0$ where $z$ belongs to the set $\mathbb { C }$ of complex numbers. The points in the plane whose affixes are the solutions in $\mathbb { C }$ of the equation ( $E$ ) are the vertices of a right triangle.
Proposition 3 $\frac { \pi } { 3 }$ is an argument of the complex number $( - \sqrt { 3 } + \mathrm { i } ) ^ { 8 }$.
Q3 3 marks Sequences and series, recurrence and convergence Closed-form expression derivation View
The sequence ( $u _ { n }$ ) is defined by:
$$u _ { 0 } = 0 \quad \text { and, for all natural integer } n , u _ { n + 1 } = \frac { 1 } { 2 - u _ { n } } .$$
  1. a. Using the calculation of the first terms of the sequence ( $u _ { n }$ ), conjecture the explicit form of $u _ { n }$ as a function of $n$. Prove this conjecture. b. Deduce the value of the limit $\ell$ of the sequence $\left( u _ { n } \right)$.
  2. Complete, in appendix 2, the algorithm to determine the value of the smallest integer $n$ such that $\left| u _ { n + 1 } - u _ { n } \right| \leqslant 10 ^ { - 3 }$.
Q4 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Part A: a volume calculation without a coordinate system We consider an equilateral pyramid SABCD (pyramid with a square base whose lateral faces are all equilateral triangles). The diagonals of the square ABCD measure 24 cm. We denote O the center of the square ABCD. We will admit that $\mathrm { OS } = \mathrm { OA }$.
  1. Without using a coordinate system, prove that the line (SO) is orthogonal to the plane (ABC).
  2. Deduce the volume, in $\mathrm { cm } ^ { 3 }$, of the pyramid SABCD.

Part B: in a coordinate system We consider the orthonormal coordinate system ( $\mathrm { O } ; \overrightarrow { \mathrm { OA } } , \overrightarrow { \mathrm { OB } } , \overrightarrow { \mathrm { OS } }$ ).
  1. We denote P and Q the midpoints of the segments [AS] and [BS] respectively. a. Justify that $\vec { n } ( 1 ; 1 ; - 3 )$ is a normal vector to the plane (PQC). b. Deduce a Cartesian equation of the plane (PQC).
  2. Let H be the point of the plane (PQC) such that the line (SH) is orthogonal to the plane (PQC). a. Give a parametric representation of the line (SH). b. Calculate the coordinates of the point H. c. Show then that the length SH, in unit of length, is $\frac { 2 \sqrt { 11 } } { 11 }$.
  3. We will admit that the area of the quadrilateral PQCD, in unit of area, is equal to $\frac { 3 \sqrt { 11 } } { 8 }$. Calculate the volume of the pyramid SPQCD, in unit of volume.

Part C: fair sharing For the birthday of her twin daughters Anne and Fanny, Mrs. Nova has made a beautiful cake in the shape of an equilateral pyramid whose diagonals of the square base measure 24 cm. She is about to share it equally by placing her knife on the apex. That is when Anne stops her and proposes a more original cut: ``Place the blade on the midpoint of an edge, parallel to a side of the base, then cut towards the opposite side''. Is this the case? Justify the answer.
Q5a 5 marks Normal Distribution Finding Unknown Standard Deviation from a Given Probability Condition View
Exercise 5 — Candidates who have not followed the specialization course In this exercise, all requested probabilities will be rounded to $10 ^ { - 4 }$. We study a model of automobile air conditioner composed of a mechanical module and an electronic module. If a module fails, it is replaced.
Part A: Study of mechanical module failures An automobile maintenance company has found, through a statistical study, that the operating time (in months) of the mechanical module can be modeled by a random variable $D$ that follows a normal distribution with mean $\mu = 50$ and standard deviation $\sigma$:
  1. Determine the rounding to $10 ^ { - 4 }$ of $\sigma$ knowing that the statistical service indicates that $P ( D \geqslant 48 ) = 0,7977$.

For the rest of this exercise, we will take $\sigma = 2,4$.
    \setcounter{enumi}{1}
  1. Determine the probability that the operating time of the mechanical module is between 45 and 52 months.
  2. Determine the probability that the mechanical module of an air conditioner that has been operating for 48 months will continue to function for at least 6 more months.

Part B: Study of electronic module failures On the same air conditioner model, the automobile maintenance company has found that the operating time (in months) of the electronic module can be modeled by a random variable $T$ that follows an exponential distribution with parameter $\lambda$.
  1. Determine the exact value of $\lambda$, knowing that the statistical service indicates that $P ( 0 \leqslant T \leqslant 24 ) = 0,03$.

For the rest of this exercise, we will take $\boldsymbol { \lambda } = 0,00127$.
    \setcounter{enumi}{1}
  1. Determine the probability that the operating time of the electronic module is between 24 and 48 months.
  2. a. Prove that, for all positive real numbers $t$ and $h$, we have: $P _ { T \geqslant t } ( T \geqslant t + h ) = P ( T \geqslant h )$, that is, the random variable $T$ is memoryless. b. The electronic module of the air conditioner has been operating for 36 months. Determine the probability that it will continue to function for the next 12 months.

Part C: Mechanical and electronic failures We admit that the events ( $D \geqslant 48$ ) and ( $T \geqslant 48$ ) are independent. Determine the probability that the air conditioner does not fail before 48 months.
Part D: Special case of a company garage
A garage of the company has studied the maintenance records of 300 air conditioners over 4 years old. It finds that 246 of them have their mechanical module in working order for 4 years. Should this report call into question the result given by the company's statistical service, namely that $P ( D \geqslant 48 ) = 0,7977$? Justify the answer.
Q5b Number Theory Congruence Reasoning and Parity Arguments View
Exercise 5 — Candidates who have followed the specialization course The natural integers $1, 11, 111, 1111, \ldots$ are rep-units. These are natural integers written only with 1s. For all non-zero natural integer $p$, we denote $N _ { p }$ the rep-unit written with $p$ times the digit 1:
$$N _ { p } = \underbrace { 11 \ldots 1 } _ { \substack { p \text { repetitions } \\ \text { of the digit 1 } } } = \sum _ { k = 0 } ^ { k = p - 1 } 10 ^ { k }$$
Throughout the exercise, $p$ denotes a non-zero natural integer. The purpose of this exercise is to study some properties of rep-units.
Part A: divisibility of rep-units in some particular cases
  1. Show that $N _ { p }$ is divisible neither by 2 nor by 5.
  2. In this question, we study the divisibility of $N _ { p }$ by 3. a. Prove that, for all natural integer $j$, $10 ^ { j } \equiv 1 \bmod 3$. b. Deduce that $N _ { p } \equiv p \bmod 3$. c. Determine a necessary and sufficient condition for the rep-unit $N _ { p }$ to be divisible by 3.
  3. In this question, we study the divisibility of $N _ { p }$ by 7. a. Copy and complete the congruence table below, where $a$ is the unique relative integer belonging to $\{ - 3 ; - 2 ; - 1 ; 0 ; 1 ; 2 ; 3 \}$ such that $10 ^ { m } \equiv a \bmod 7$. No justification is required.
    $m$0123456
    $a$

    b. Let $p$ be a non-zero natural integer. Show that $10 ^ { p } \equiv 1 \bmod 7$ if and only if $p$ is a multiple of 6. You may use the Euclidean division of $p$ by 6. c. Justify that, for all natural integer $p$ non-zero, $N _ { p } = \frac { 10 ^ { p } - 1 }{ 9 }$.