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
2019 liban

9 maths questions

Q1 Differentiating Transcendental Functions Monotonicity or convexity of transcendental functions View
Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
Consider the function $f$ defined on the interval $]0; 1]$ by $$f(x) = x(1 - \ln x)^2.$$
a. Determine an expression for the derivative of $f$ and verify that for all $x \in ]0; 1]$, $f'(x) = (\ln x + 1)(\ln x - 1)$.
b. Study the variations of the function $f$ and draw its variation table on the interval $]0; 1]$ (it will be admitted that the limit of the function $f$ at 0 is zero).
Q2 Addition & Double Angle Formulae Function Analysis via Identity Transformation View
Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
We denote by $\Gamma$ the representative curve of the function $g$ defined on the interval $]0; 1]$ by $g(x) = \ln x$. Let $a$ be a real number in the interval $]0; 1]$. We denote by $M_a$ the point on the curve $\Gamma$ with abscissa $a$ and $d_a$ the tangent line to the curve $\Gamma$ at the point $M_a$. This line $d_a$ intersects the $x$-axis at point $N_a$ and the $y$-axis at point $P_a$. We are interested in the area of triangle $\mathrm{O}N_aP_a$ as the real number $a$ varies in the interval $]0; 1]$.
In this question, we study the particular case where $a = 0.2$.
a. Determine graphically an estimate of the area of triangle $\mathrm{O}N_{0.2}P_{0.2}$ in square units.
b. Determine an equation of the tangent line $d_{0.2}$.
c. Calculate the exact value of the area of triangle $\mathrm{O}N_{0.2}P_{0.2}$.
Q3 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
It is admitted that, for all real $a$ in the interval $]0; 1]$, the area of triangle $\mathrm{O}N_aP_a$ in square units is given by $\mathscr{A}(a) = \frac{1}{2}a(1 - \ln a)^2$.
Using the previous questions, determine for which value of $a$ the area $\mathscr{A}(a)$ is maximum. Determine this maximum area.
Q4 Complex numbers 2 Complex Mappings and Transformations View
Exercise 2
The complex plane is equipped with a direct orthonormal coordinate system $(\mathrm{O}; \vec{u}; \vec{v})$ with unit 2 cm. We call $f$ the function that, to any point $M$, distinct from point O and with affixe a complex number $z$, associates the point $M'$ with affixe $z'$ such that $$z' = -\frac{1}{z}$$
1. Consider the points A and B with affixes respectively $z_{\mathrm{A}} = -1 + \mathrm{i}$ and $z_{\mathrm{B}} = \frac{1}{2}\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$.
a. Determine the algebraic form of the affixe of point $\mathrm{A}'$ image of point A by the function $f$.
b. Determine the exponential form of the affixe of point $\mathrm{B}'$ image of point B by the function $f$.
c. On your paper, place the points $\mathrm{A}, \mathrm{B}, \mathrm{A}'$ and $\mathrm{B}'$ in the direct orthonormal coordinate system $(\mathrm{O}; \vec{u}; \vec{v})$. For points B and $\mathrm{B}'$, construction lines should be left visible.
2. Let $r$ be a strictly positive real number and $\theta$ a real number. Consider the complex number $z$ defined by $z = r\mathrm{e}^{\mathrm{i}\theta}$.
a. Show that $z' = \frac{1}{r}\mathrm{e}^{\mathrm{i}(\pi - \theta)}$.
b. Is it true that if a point $M$, distinct from O, belongs to the disk with center O and radius 1 without belonging to the circle with center O and radius 1, then its image $M'$ by the function $f$ is outside this disk? Justify.
3. Let the circle $\Gamma$ with center K with affixe $z_{\mathrm{K}} = -\frac{1}{2}$ and radius $\frac{1}{2}$.
a. Show that a Cartesian equation of the circle $\Gamma$ is $x^2 + x + y^2 = 0$.
b. Let $z = x + \mathrm{i}y$ with $x$ and $y$ not both zero. Determine the algebraic form of $z'$ as a function of $x$ and $y$.
c. Let $M$ be a point, distinct from O, on the circle $\Gamma$. Show that the image $M'$ of point $M$ by the function $f$ belongs to the line with equation $x = 1$.
Q5 Vectors: Lines & Planes Prove Perpendicularity/Orthogonality of Line and Plane View
Exercise 3 — Part A
In a plane P, consider a triangle ABC right-angled at A. Let $d$ be the line orthogonal to plane P and passing through point B. Consider a point D on this line distinct from point B.
1. Show that the line (AC) is orthogonal to the plane (BAD).
A bicoin is called a tetrahedron whose four faces are right triangles.
2. Show that the tetrahedron ABCD is a bicoin.
3. a. Justify that the edge $[CD]$ is the longest edge of the bicoin ABCD.
b. Let I be the midpoint of edge $[CD]$. Show that point I is equidistant from the 4 vertices of the bicoin ABCD.
Q6 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 3 — Part B
In an orthonormal coordinate system of space, consider the point $\mathrm{A}(3; 1; -5)$ and the line $d$ with parametric representation $\left\{\begin{array}{rl} x &= 2t + 1 \\ y &= -2t + 9 \\ z &= t - 3 \end{array}\right.$ where $t \in \mathbb{R}$.
1. Determine a Cartesian equation of the plane $P$ orthogonal to the line $d$ and passing through point A.
2. Show that the intersection point of plane $P$ and line $d$ is point $\mathrm{B}(5; 5; -1)$.
3. Justify that point $\mathrm{C}(7; 3; -9)$ belongs to plane $P$ then show that triangle ABC is a right isosceles triangle at A.
4. Let $t$ be a real number different from 2 and $M$ the point with parameter $t$ belonging to line $d$.
a. Justify that triangle $\mathrm{AB}M$ is right-angled.
b. Show that triangle $\mathrm{AB}M$ is isosceles at B if and only if the real number $t$ satisfies the equation $t^2 - 4t = 0$.
c. Deduce the coordinates of points $M_1$ and $M_2$ on line $d$ such that the right triangles $\mathrm{AB}M_1$ and $\mathrm{AB}M_2$ are isosceles at B.
Q7 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 3 — Part C
We are given the point $\mathrm{D}(9; 1; 1)$ which is one of the two solution points from question 4.c. of Part B. The four vertices of tetrahedron ABCD are located on a sphere.
Using the results from the questions in Parts A and B above, determine the coordinates of the center of this sphere and calculate its radius.
Q8 5 marks Conditional Probability Markov Chain / Day-to-Day Transition Probabilities View
Exercise 4 (5 points) — Candidates who have not followed the specialization course
Each week, a farmer offers for direct sale to each of his customers a basket of fresh products that contains a single bottle of fruit juice. A statistical study carried out gives the following results:
  • at the end of the first week, the probability that a customer returns the bottle from his basket is 0.9;
  • if the customer returned the bottle from his basket one week, then the probability that he brings back the bottle from the basket the following week is 0.95;
  • if the customer did not return the bottle from his basket one week, then the probability that he brings back the bottle from the basket the following week is 0.2.
A customer is chosen at random from the farmer's clientele. For any non-zero natural number $n$, we denote by $R_n$ the event ``the customer returns the bottle from his basket in the $n$-th week''.
1. a. Model the situation studied for the first two weeks using a weighted tree that will involve the events $R_1$ and $R_2$.
b. Determine the probability that the customer returns the bottles from the baskets of the first and second weeks.
c. Show that the probability that the customer returns the bottle from the basket of the second week is equal to 0.875.
d. Given that the customer returned the bottle from his basket in the second week, what is the probability that he did not return the bottle from his basket in the first week? Round the result to $10^{-3}$.
2. For any non-zero natural number $n$, we denote by $r_n$ the probability that the customer returns the bottle from the basket in the $n$-th week. We then have $r_n = p(R_n)$.
a. Copy and complete the weighted tree (no justification is required).
b. Justify that for any non-zero natural number $n$, $r_{n+1} = 0.75r_n + 0.2$.
c. Prove that for any non-zero natural number $n$, $r_n = 0.1 \times 0.75^{n-1} + 0.8$.
d. Calculate the limit of the sequence $(r_n)$. Interpret the result in the context of the exercise.
Q9 5 marks Sequences and series, recurrence and convergence Applied/contextual sequence problem View
Exercise 4 (5 points) — Candidates who have followed the specialization course
In a public garden, an artist must install an aquatic artwork. This artwork will consist of two basins A and B as well as a filtering reserve R. Initially, the two basins each contain 100 liters of water. A system of pipes allows the following water transfers to be carried out, every hour and in this order:
  • first, half of basin A empties into reserve R;
  • then, three quarters of basin B empty into basin A;
  • finally, 200 liters of water are added to basin A and 300 liters of water are added to basin B.
The quantities of water in the two basins A and B are modeled using two sequences $(a_n)$ and $(b_n)$: for any natural number $n$, we denote by $a_n$ and $b_n$ the quantities of water in hundreds of liters that will be respectively contained in basins A and B after $n$ hours. For any natural number $n$, we denote by $U_n$ the column matrix $U_n = \binom{a_n}{b_n}$. Thus $U_0 = \binom{1}{1}$.
1. Justify that, for any natural number $n$, $U_{n+1} = MU_n + C$ where $M = \left(\begin{array}{cc} 0.5 & 0.75 \\ 0 & 0.25 \end{array}\right)$ and $C = \binom{2}{3}$.
2. Consider the matrix $P = \left(\begin{array}{cc} 1 & 3 \\ 0 & -1 \end{array}\right)$.
a. Calculate $P^2$. Deduce that the matrix $P$ is invertible and specify its inverse matrix.
b. Show that $PMP$ is a diagonal matrix $D$ that you will specify.
c. Calculate $PDP$.
d. Prove by induction that, for any natural number $n$, $M^n = PD^nP$.
It is admitted henceforth that for any natural number $n$, $M^n = \left(\begin{array}{cc} 0.5^n & 3 \times 0.5^n - 3 \times 0.25^n \\ 0 & 0.25^n \end{array}\right)$.
3. Show that the matrix $X = \binom{10}{4}$ satisfies $X = MX + C$.
4. For any natural number $n$, we define the matrix $V_n$ by $V_n = U_n - X$.
a. Show that for any natural number $n$, $V_{n+1} = MV_n$.
b. It is admitted that, for any non-zero natural number $n$, $V_n = M^n V_0$. Show that for any non-zero natural number $n$, $$U_n = \binom{-18 \times 0.5^n + 9 \times 0.25^n + 10}{-3 \times 0.25^n + 4}.$$
5. a. Show that the sequence $(b_n)$ is increasing and bounded above. Determine its limit.
b. Determine the limit of the sequence $(a_n)$.
c. It is admitted that the sequence $(a_n)$ is increasing. Deduce the capacity of the two basins A and B that must be planned for the feasibility of the project, that is, to avoid any overflow.