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
2021 bac-spe-maths__centres-etrangers_j1

9 maths questions

QExercise 2 Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
According to a study, regular users of public transport represent $17\%$ of the French population. Among these regular users, $32\%$ are young people aged 18 to 24 years old.
A person is randomly interviewed and we note:
  • $R$ the event: ``The person interviewed regularly uses public transport''.
  • $J$ the event: ``The person interviewed is aged 18 to 24 years old''.

Part A:
  1. Represent the situation using a probability tree, reporting the data from the problem statement.
  2. Calculate the probability $P(R \cap J)$.
  3. According to this same study, young people aged 18 to 24 represent $11\%$ of the French population. Show that the probability that the person interviewed is a young person aged 18 to 24 who does not regularly use public transport is 0.056 to $10^{-3}$ precision.
  4. Deduce the proportion of young people aged 18 to 24 among non-regular users of public transport.

Part B: During a census of the French population, a census taker randomly interviews 50 people in one day about their use of public transport. The French population is large enough to assimilate this census to sampling with replacement. Let $X$ be the random variable counting the number of people regularly using public transport among the 50 people interviewed.
  1. Determine, by justifying, the distribution of $X$ and specify its parameters.
  2. Calculate $P(X = 5)$ and interpret the result.
  3. The census taker indicates that there is more than a $95\%$ chance that, among the 50 people interviewed, fewer than 13 of them regularly use public transport. Is this statement true? Justify your answer.
  4. What is the average number of people regularly using public transport among the 50 people interviewed?
QExercise 3 5 marks Geometric Sequences and Series Prove a Transformed Sequence is Geometric View
In May 2020, a company chose to develop telework. It proposed to its 5000 employees in France to choose between telework and working at the company's premises. In May 2020, only 200 of them chose telework. Each month, since the implementation of this measure, $85\%$ of those who had chosen telework the previous month choose to continue, and each month, 450 additional employees choose telework. The number of company employees working from home is modeled by the sequence $(a_n)$. The term $a_n$ designates an estimate of the number of employees working from home in the $n$-th month after May 2020. Thus $a_0 = 200$.
Part A:
  1. Calculate $a_1$.
  2. Justify that for every natural number $n$, $a_{n+1} = 0.85a_n + 450$.
  3. Consider the sequence $(v_n)$ defined for every natural number $n$ by: $v_n = a_n - 3000$. a. Prove that the sequence $(v_n)$ is a geometric sequence with common ratio 0.85. b. Express $v_n$ as a function of $n$ for every natural number $n$. c. Deduce that, for every natural number $n$, $a_n = -2800 \times 0.85^n + 3000$.
  4. Determine the number of months after which the number of teleworkers will be strictly greater than 2500, after the implementation of this measure in the company.

Part B: The company's managers modeled the number of employees satisfied with this system using the sequence $(u_n)$ defined by $u_0 = 1$ and, for every natural number $n$, $$u_{n+1} = \frac{5u_n + 4}{u_n + 2}$$ where $u_n$ denotes the number of thousands of employees satisfied with this new measure after $n$ months following May 2020.
  1. Prove that the function $f$ defined for all $x \in [0;+\infty[$ by $f(x) = \dfrac{5x+4}{x+2}$ is strictly increasing on $[0;+\infty[$.
  2. a. Prove by induction that for every natural number $n$: $$0 \leqslant u_n \leqslant u_{n+1} \leqslant 4.$$ b. Justify that the sequence $(u_n)$ is convergent.
  3. We admit that for every natural number $n$, $$0 \leqslant 4 - u_n \leqslant 3 \times \left(\frac{1}{2}\right)^n.$$ Deduce the limit of the sequence $(u_n)$ and interpret it in the context of the modeling.
QExercise A Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
In an orthonormal coordinate system of space, we consider the following points: $$\mathrm{A}(2;-1;0),\quad \mathrm{B}(3;-1;2),\quad \mathrm{C}(0;4;1)\quad \text{and}\quad \mathrm{S}(0;1;4).$$
  1. Show that triangle ABC is right-angled at A.
  2. a. Show that the vector $\vec{n}\begin{pmatrix}2\\1\\-1\end{pmatrix}$ is orthogonal to the plane (ABC). b. Deduce a Cartesian equation of the plane (ABC). c. Show that the points A, B, C and S are not coplanar.
  3. Let (d) be the line perpendicular to the plane (ABC) passing through S. It intersects the plane (ABC) at H. a. Determine a parametric representation of the line (d). b. Show that the coordinates of point H are $\mathrm{H}(2;2;3)$.
  4. We recall that the volume $V$ of a tetrahedron is $V = \dfrac{\text{area of base} \times \text{height}}{3}$. Calculate the volume of tetrahedron SABC.
  5. a. Calculate the length SA. b. We are told that $\mathrm{SB} = \sqrt{17}$. Deduce an approximate measure of the angle $\widehat{\mathrm{ASB}}$ to the nearest tenth of a degree.
QExercise B Differential equations First-Order Linear DE: General Solution View
Part A: Let $g$ be the function defined on $\mathbb{R}$ by: $$g(x) = 2\mathrm{e}^{\frac{-1}{3}x} + \frac{2}{3}x - 2$$
  1. We admit that the function $g$ is differentiable on $\mathbb{R}$ and we denote $g^{\prime}$ its derivative function. Show that, for every real number $x$: $$g^{\prime}(x) = \frac{-2}{3}e^{-\frac{1}{3}x} + \frac{2}{3}.$$
  2. Deduce the direction of variation of the function $g$ on $\mathbb{R}$.
  3. Determine the sign of $g(x)$, for every real $x$.

Part B:
  1. Consider the differential equation $$(E):\quad 3y^{\prime} + y = 0.$$ Solve the differential equation (E).
  2. Determine the particular solution whose representative curve, in a coordinate system of the plane, passes through the point $\mathrm{M}(0;2)$.
  3. Let $f$ be the function defined on $\mathbb{R}$ by: $$f(x) = 2\mathrm{e}^{-\frac{1}{3}x}$$ and $\mathscr{C}_f$ its representative curve. a. Show that the tangent line $(\Delta_0)$ to the curve $\mathscr{C}_f$ at the point $\mathrm{M}(0;2)$ has an equation of the form: $$y = -\frac{2}{3}x + 2$$ b. Study, on $\mathbb{R}$, the position of this curve $\mathscr{C}_f$ relative to the tangent line $(\Delta_0)$.

Part C:
  1. Let A be the point on the curve $\mathscr{C}_f$ with abscissa $a$, where $a$ is any real number. Show that the tangent line $(\Delta_a)$ to the curve $\mathscr{C}_f$ at point A intersects the $x$-axis at a point P with abscissa $a+3$.
  2. Explain the construction of the tangent line $(\Delta_{-2})$ to the curve $\mathscr{C}_f$ at point B with abscissa $-2$.
Q1 1 marks Differentiating Transcendental Functions Higher-order or nth derivative computation View
Consider the function defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{-2x}$$ Let $f^{\prime\prime}$ denote the second derivative of the function $f$. For any real number $x$, $f^{\prime\prime}(x)$ is equal to: a. $(1-2x)\mathrm{e}^{-2x}$ b. $4(x-1)\mathrm{e}^{-2x}$ c. $4\mathrm{e}^{-2x}$ d. $(x+2)\mathrm{e}^{-2x}$
Q2 1 marks Combinations & Selection Basic Combination Computation View
A first-year general education student chooses three specializations from the twelve offered. The number of possible combinations is: a. 1728 b. 1320 c. 220 d. 33
Q3 1 marks Stationary points and optimisation Analyze function behavior from graph or table of derivative View
Below is the graphical representation of $f^{\prime}$, the derivative function of a function $f$ defined on [0;7].
The variation table of $f$ on the interval [0; 7] is:
a.
$x$03,257
$f(x)$

b.
$x$0257
$f(x)$

c.
$x$0257
$f(x)$$\nearrow$

d.
$x$027
$f(x)$
Q4 1 marks Principle of Inclusion/Exclusion View
A company manufactures microchips. Each chip can have two defects denoted A and B.
A statistical study shows that $2.8\%$ of chips have defect A, $2.2\%$ of chips have defect B, and fortunately, $95.4\%$ of chips have neither of the two defects.
The probability that a randomly selected chip has both defects is: a. 0.05 b. 0.004 c. 0.046 d. We cannot know
Q5 1 marks Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
We are given a function $f$, assumed to be differentiable on $\mathbb{R}$, and we denote $f^{\prime}$ its derivative function.
Below is the variation table of $f$:
$x$$-\infty$$-1$$+\infty$
$f(x)$
$-\infty$0

According to this variation table: a. $f^{\prime}$ is positive on $\mathbb{R}$. b. $f^{\prime}$ is positive on $\left.]-\infty;-1\right]$ c. $f^{\prime}$ is negative on $\mathbb{R}$ d. $f^{\prime}$ is positive on $[-1;+\infty[$