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
2022 bac-spe-maths__centres-etrangers_j1

4 maths questions

Q1 7 marks Laws of Logarithms Analyze a Logarithmic Function (Limits, Monotonicity, Zeros, Extrema) View
Exercise 1 — Multiple Choice Questionnaire (Logarithmic function)
For each of the following questions, only one of the four proposed answers is correct. The six questions are independent.
  1. Consider the function $f$ defined for all real $x$ by $f(x) = \ln\left(1 + x^2\right)$.
    On $\mathbb{R}$, the equation $f(x) = 2022$ a. has no solution. b. has exactly one solution. c. has exactly two solutions. d. has infinitely many solutions.
  2. Let the function $g$ defined for all strictly positive real $x$ by: $$g(x) = x\ln(x) - x^2$$ We denote $\mathscr{C}_g$ its representative curve in a coordinate system of the plane. a. The function $g$ is convex on $]0; +\infty[$. b. The function $g$ is concave on $]0; +\infty[$. c. The curve $\mathscr{C}_g$ has exactly one inflection point on $]0; +\infty[$. d. The curve $\mathscr{C}_g$ has exactly two inflection points on $]0; +\infty[$.
  3. Consider the function $f$ defined on $]-1; 1[$ by $$f(x) = \frac{x}{1 - x^2}$$ An antiderivative of the function $f$ is the function $g$ defined on the interval $]-1; 1[$ by: a. $g(x) = -\frac{1}{2}\ln\left(1 - x^2\right)$ b. $g(x) = \frac{1 + x^2}{\left(1 - x^2\right)^2}$ c. $g(x) = \frac{x^2}{2\left(x - \frac{x^3}{3}\right)}$ d. $g(x) = \frac{x^2}{2}\ln\left(1 - x^2\right)$
  4. The function $x \longmapsto \ln\left(-x^2 - x + 6\right)$ is defined on a. $]-3; 2[$ b. $]-\infty; 6]$ c. $]0; +\infty[$ d. $]2; +\infty[$
  5. Consider the function $f$ defined on $]0.5; +\infty[$ by $$f(x) = x^2 - 4x + 3\ln(2x - 1)$$ An equation of the tangent line to the representative curve of $f$ at the point with abscissa 1 is: a. $y = 4x - 7$ b. $y = 2x - 4$ c. $y = -3(x - 1) + 4$ d. $y = 2x - 1$
  6. The set $S$ of solutions in $\mathbb{R}$ of the inequality $\ln(x + 3) < 2\ln(x + 1)$ is: a. $S = ]-\infty; -2[ \cup ]1; +\infty[$ b. $S = ]1; +\infty[$ c. $S = \varnothing$ d. $S = ]-1; 1[$
Q2 7 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 2 — 7 points
Theme: Geometry in space
In space, referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points: $$\mathrm{A}(2; 0; 3),\ \mathrm{B}(0; 2; 1),\ \mathrm{C}(-1; -1; 2)\ \text{and}\ \mathrm{D}(3; -3; -1).$$
1. Calculation of an angle
a. Calculate the coordinates of the vectors $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{AC}}$ and deduce that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear. b. Calculate the lengths AB and AC. c. Using the dot product $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AC}}$, determine the value of the cosine of the angle $\widehat{\mathrm{BAC}}$ then give an approximate value of the measure of the angle $\widehat{\mathrm{BAC}}$ to the nearest tenth of a degree.
2. Calculation of an area
a. Determine an equation of the plane $\mathscr{P}$ passing through point C and perpendicular to the line (AB). b. Give a parametric representation of the line (AB). c. Deduce the coordinates of the orthogonal projection E of point C onto the line $(\mathrm{AB})$, that is to say the point of intersection of the line (AB) and the plane $\mathscr{P}$. d. Calculate the area of triangle ABC.
3. Calculation of a volume
a. Let the point $\mathrm{F}(1; -1; 3)$. Show that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{F}$ are coplanar. b. Verify that the line (FD) is orthogonal to the plane (ABC). c. Knowing that the volume of a tetrahedron is equal to one third of the area of its base multiplied by its height, calculate the volume of the tetrahedron ABCD.
Q3 7 marks Exponential Functions Variation and Monotonicity Analysis View
Exercise 3 — 7 points
Themes: Exponential function and sequence
Part A:
Let $h$ be the function defined on $\mathbb{R}$ by $$h(x) = \mathrm{e}^x - x$$
  1. Determine the limits of $h$ at $-\infty$ and $+\infty$.
  2. Study the variations of $h$ and draw up its variation table.
  3. Deduce that: if $a$ and $b$ are two real numbers such that $0 < a < b$ then $h(a) - h(b) < 0$.

Part B:
Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = \mathrm{e}^x$$ We denote $\mathscr{C}_f$ its representative curve in a coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.
  1. Determine an equation of the tangent line $T$ to $\mathscr{C}_f$ at the point with abscissa 0.

In the rest of the exercise we are interested in the gap between $T$ and $\mathscr{C}_f$ in the neighbourhood of 0. This gap is defined as the difference of the ordinates of the points of $T$ and $\mathscr{C}_f$ with the same abscissa. We are interested in points with abscissa $\frac{1}{n}$, with $n$ a non-zero natural number. We then consider the sequence $(u_n)$ defined for all non-zero natural numbers $n$ by: $$u_n = \exp\left(\frac{1}{n}\right) - \frac{1}{n} - 1$$
    \setcounter{enumi}{1}
  1. Determine the limit of the sequence $(u_n)$.
  2. a. Prove that, for all non-zero natural numbers $n$, $$u_{n+1} - u_n = h\left(\frac{1}{n+1}\right) - h\left(\frac{1}{n}\right)$$ where $h$ is the function defined in Part A. b. Deduce the direction of variation of the sequence $(u_n)$.
  3. The table below gives approximate values to $10^{-9}$ of the first terms of the sequence $(u_n)$.
    $n$$u_n$
    10.718281828
    20.148721271
    30.062279092
    40.034025417
    50.021402758
    60.014693746
    70.010707852
    80.008148453
    90.006407958
    100.005170918

    Give the smallest value of the natural number $n$ for which the gap between $T$ and $\mathscr{C}_f$ appears to be less than $10^{-2}$.
Q4 7 marks Principle of Inclusion/Exclusion View
Exercise 4 — 7 points
Theme: Probability
During the manufacture of a pair of glasses, the pair of lenses must undergo two treatments denoted T1 and T2.
Part A
A pair of lenses is randomly selected from production. We denote by $A$ the event: ``the pair of lenses has a defect for treatment T1''. We denote by $B$ the event: ``the pair of lenses has a defect for treatment T2''. We denote by $\bar{A}$ and $\bar{B}$ respectively the complementary events of $A$ and $B$.
A study has shown that:
  • the probability that a pair of lenses has a defect for treatment T1, denoted $P(A)$, is equal to 0.1.
  • the probability that a pair of lenses has a defect for treatment T2, denoted $P(B)$, is equal to 0.2.
  • the probability that a pair of lenses has neither of the two defects is 0.75.

  1. Copy and complete the following table with the corresponding probabilities.
    $A$$\bar{A}$Total
    $B$
    $\bar{B}$
    Total1

  2. a. Determine, by justifying the answer, the probability that a pair of lenses, randomly selected from production, has a defect for at least one of the two treatments T1 or T2. b. Give the probability that a pair of lenses, randomly selected from production, has two defects, one for each treatment T1 and T2. c. Are the events $A$ and $B$ independent? Justify the answer.
  3. Calculate the probability that a pair of lenses, randomly selected from production, has a defect for only one of the two treatments.
  4. Calculate the probability that a pair of lenses, randomly selected from production, has a defect for treatment T2, given that this pair of lenses has a defect for treatment T1.

Part B
A sample of 50 pairs of lenses is randomly selected from production. We assume that the production is large enough to assimilate this selection to a draw with replacement. We denote by $X$ the random variable which, to each sample of this type, associates the number of pairs of lenses that have the defect for treatment T1.
  1. Justify that the random variable $X$ follows a binomial distribution and specify the parameters of this distribution.
  2. Give the expression allowing the calculation of the probability of having, in such a sample, exactly 10 pairs of lenses that have this defect. Perform this calculation and round the result to $10^{-3}$.
  3. On average, how many pairs of lenses with this defect can be found in a sample of 50 pairs?