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 antilles-guyane

6 maths questions

Q1 3 marks Complex numbers 2 Solving Polynomial Equations in C View
The complex plane is equipped with a direct orthonormal coordinate system. Consider the equation
$$( E ) : \quad z ^ { 4 } + 2 z ^ { 3 } - z - 2 = 0$$
with unknown complex number $z$.
  1. Give an integer solution of ( $E$ ).
  2. Prove that, for every complex number $z$, $$z ^ { 4 } + 2 z ^ { 3 } - z - 2 = \left( z ^ { 2 } + z - 2 \right) \left( z ^ { 2 } + z + 1 \right) .$$
  3. Solve equation ( $E$ ) in the set of complex numbers.
  4. The solutions of equation ( $E$ ) are the affixes of four points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ in the complex plane such that ABCD is a non-crossed quadrilateral. Is quadrilateral ABCD a rhombus? Justify.
Q2 4 marks Normal Distribution Conditional Probability Involving Normal Distribution View
In an automobile factory, certain metal parts are covered with a thin layer of nickel that protects them against corrosion and wear. The process used is electroplating with nickel.
It is assumed that the random variable $X$, which associates to each treated part the thickness of nickel deposited, follows a normal distribution with mean $\mu _ { 1 } = 25$ micrometers ( $\mu \mathrm { m }$ ) and standard deviation $\sigma _ { 1 }$.
A part is compliant if the thickness of nickel deposited is between $22.8 \mu \mathrm {~m}$ and $27.2 \mu \mathrm {~m}$.
The probability density function of $X$ is represented below. It was determined that $P ( X > 27.2 ) = 0.023$.
  1. a. Determine the probability that a part is compliant. b. Justify that 1.1 is an approximate value of $\sigma _ { 1 }$ to within $10 ^ { - 1 }$. c. Given that a part is compliant, calculate the probability that the thickness of nickel deposited on it is less than $24 \mu \mathrm {~m}$. Round to $10 ^ { - 3 }$.
  2. A team of engineers proposes another nickel plating process, obtained by chemical reaction without any current source. The team claims that this new process theoretically allows obtaining $98 \%$ of compliant parts. The random variable $Y$ which, for each part treated with this new process, associates the thickness of nickel deposited follows a normal distribution with mean $\mu _ { 2 } = 25 \mu \mathrm {~m}$ and standard deviation $\sigma _ { 2 }$. a. Assuming the above claim, compare $\sigma _ { 1 }$ and $\sigma _ { 2 }$. b. A quality control evaluates the new process; it reveals that out of 500 parts tested, 15 are not compliant. At the $95 \%$ confidence level, can we reject the team's claim?
Q3 3 marks Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View
Let $f$ and $g$ be the functions defined on the set $\mathbb { R }$ of real numbers by
$$f ( x ) = \mathrm { e } ^ { x } \quad \text { and } \quad g ( x ) = \mathrm { e } ^ { - x } .$$
We denote by $\mathscr { C } _ { f }$ the representative curve of function $f$ and $\mathscr { C } _ { g }$ that of function $g$ in an orthonormal coordinate system of the plane.
For every real number $a$, we denote by $M$ the point of $\mathscr { C } _ { f }$ with abscissa $a$ and $N$ the point of $\mathscr { C } _ { g }$ with abscissa $a$.
The tangent line to $\mathscr { C } _ { f }$ at $M$ intersects the $x$-axis at $P$, the tangent line to $\mathscr { C } _ { g }$ at $N$ intersects the $x$-axis at $Q$.
Questions 1 and 2 can be treated independently.
  1. Prove that the tangent line to $\mathscr { C } _ { f }$ at $M$ is perpendicular to the tangent line to $\mathscr { C } _ { g }$ at $N$.
  2. a. What can be conjectured about the length $PQ$? b. Prove this conjecture.
Q4 5 marks Curve Sketching Variation Table and Monotonicity from Sign of Derivative View
Throughout the exercise, $n$ denotes a strictly positive natural number. The purpose of the exercise is to study the equation
$$\left( E _ { n } \right) : \quad \frac { \ln ( x ) } { x } = \frac { 1 } { n }$$
with unknown strictly positive real number $x$.
Part A
Let $f$ be the function defined on the interval $] 0$; $+ \infty [$ by
$$f ( x ) = \frac { \ln ( x ) } { x }$$
It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$.
  1. Study the variations of function $f$.
  2. Determine its maximum.

Part B
  1. Show that, for $n \geqslant 3$, the equation $f ( x ) = \frac { 1 } { n }$ has a unique solution on $[ 1 ; e]$ denoted $\alpha _ { n }$.
  2. From the above, for every integer $n \geqslant 3$, the real number $\alpha _ { n }$ is a solution of equation $\left( E _ { n } \right)$. a. On the graph are drawn the lines $D _ { 3 } , D _ { 4 }$ and $D _ { 5 }$ with equations respectively $y = \frac { 1 } { 3 } , y = \frac { 1 } { 4 }$, $y = \frac { 1 } { 5 }$. Conjecture the direction of variation of the sequence ( $\alpha _ { n }$ ). b. Compare, for every integer $n \geqslant 3 , f \left( \alpha _ { n } \right)$ and $f \left( \alpha _ { n + 1 } \right)$. Determine the direction of variation of the sequence $\left( \alpha _ { n } \right)$. c. Deduce that the sequence ( $\alpha _ { n }$ ) converges. It is not asked to calculate its limit.
  3. It is admitted that, for every integer $n \geqslant 3$, equation $\left( E _ { n } \right)$ has another solution $\beta _ { n }$ such that $$1 \leqslant \alpha _ { n } \leqslant \mathrm { e } \leqslant \beta _ { n }$$ a. It is admitted that the sequence ( $\beta _ { n }$ ) is increasing. Establish that, for every natural number $n$ greater than or equal to 3, $$\beta _ { n } \geqslant n \frac { \beta _ { 3 } } { 3 } .$$ b. Deduce the limit of the sequence ( $\beta _ { n }$ ).
Q5a Number Theory GCD, LCM, and Coprimality View
(Candidates who followed the specialization course)
Consider the sequence defined by its first term $u _ { 0 } = 3$ and, for every natural number $n$, by
$$u _ { n + 1 } = 2 u _ { n } + 6$$
  1. Prove that, for every natural number $n$, $$u _ { n } = 9 \times 2 ^ { n } - 6$$
  2. Prove that, for every integer $n \geqslant 1 , u _ { n }$ is divisible by 6. Define the sequence of integers ( $\nu _ { n }$ ) by, for every natural number $n \geqslant 1 , \nu _ { n } = \frac { u _ { n } } { 6 }$.
  3. Consider the statement: ``for every non-zero natural number $n$, $v _ { n }$ is a prime number''. Indicate whether this statement is true or false by justifying the answer.
  4. a. Prove that, for every integer $n \geqslant 1 , v _ { n + 1 } - 2 v _ { n } = 1$. b. Deduce that, for every integer $n \geqslant 1 , v _ { n }$ and $v _ { n + 1 }$ are coprime. c. Deduce, for every integer $n \geqslant 1$, the GCD of $u _ { n }$ and $u _ { n + 1 }$.
  5. a. Verify that $2 ^ { 4 } \equiv 1 [ 5 ]$. b. Deduce that if $n$ is of the form $4 k + 2$ with $k$ a natural number, then $u _ { n }$ is divisible by 5. c. Is the number $u _ { n }$ divisible by 5 for the other values of the natural number $n$? Justify.
Q5b Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
(Candidates who did not follow the specialization course)
We denote by $\mathbb { R }$ the set of real numbers. The space is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). Consider the points $\mathrm { A } ( - 1 ; 2 ; 0 ) , \mathrm { B } ( 1 ; 2 ; 4 )$ and $\mathrm { C } ( - 1 ; 1 ; 1 )$.
  1. a. Prove that points $\mathrm { A } , \mathrm { B }$ and C are not collinear. b. Calculate the dot product $\overrightarrow { \mathrm { AB } } \cdot \overrightarrow { \mathrm { AC } }$. c. Deduce the measure of angle $\widehat { \mathrm { BAC } }$, rounded to the nearest degree.
  2. Let $\vec { n }$ be the vector with coordinates $\left( \begin{array} { c } 2 \\ - 1 \\ - 1 \end{array} \right)$. a. Prove that $\vec { n }$ is a normal vector to plane (ABC). b. Determine a Cartesian equation of plane ( ABC ).
  3. Let $\mathscr { P } _ { 1 }$ be the plane with equation $3 x + y - 2 z + 3 = 0$ and $\mathscr { P } _ { 2 }$ the plane passing through O and parallel to the plane with equation $x - 2 z + 6 = 0$. a. Prove that plane $\mathscr { P } _ { 2 }$ has equation $x = 2z$. b. Prove that planes $\mathscr{P}_1$, $\mathscr{P}_2$ and (ABC) have a common point, and determine its coordinates.