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
2024 bac-spe-maths__amerique-sud_j2

4 maths questions

Q1 Conditional Probability Direct Conditional Probability Computation from Definitions View
Here is the distribution of the main blood groups of the inhabitants of France. $A+, O+, B+, A-, O-, AB+, B-$ and $AB-$ are the different blood groups combined with the Rh factor.
A random experiment consists of choosing a person at random from the French population and determining their blood group and Rh factor.
We adopt notations of the type: $A+$ is the event ``the person has blood group A and Rh factor $+$'' $A-$ is the event ``the person has blood group A and Rh factor $-$'' $A$ is the event ``the person has blood group A''
Parts 1 and 2 are independent.
Part 1
We denote $Rh+$ the event ``The person has positive Rh factor''.
  1. Justify that the probability that the chosen person has positive Rh factor is equal to 0.849.
  2. Demonstrate using the data from the problem statement that $P_{Rh_+}(A) = 0.450$ to 0.001 near.
  3. A person remembers that their blood group is AB but has forgotten their Rh factor. What is the probability that their Rh factor is negative? Round the result to 0.001 near.

Part 2
In this part, results will be rounded to 0.001 near.
A universal blood donor is a person with blood group O and negative Rh factor. Recall that $6.5\%$ of the French population has blood group $O-$.
  1. We consider 50 people chosen at random from the French population and we denote $X$ the random variable that counts the number of universal donors. a. Determine the probability that 8 people are universal donors. Justify your answer. b. Consider the function below named \texttt{proba} with argument \texttt{k} written in Python language. \begin{verbatim} def proba(k) : p = 0 for i in range(k+1) : p = p + binomiale(i,50,0.065) return p \end{verbatim} This function uses the binomial function with arguments $i, n$ and $p$, created for this purpose, which returns the value of the probability $P(X=i)$ in the case where $X$ follows a binomial distribution with parameters $n$ and $p$. Determine the numerical value returned by the \texttt{proba} function when you enter \texttt{proba(8)} in the Python console. Interpret this result in the context of the exercise.
  2. What is the minimum number of people to choose at random from the French population so that the probability that at least one of the chosen people is a universal donor is greater than 0.999.
Q2 5 marks Differential equations Verification that a Function Satisfies a DE View
This exercise contains 5 statements. For each statement, answer TRUE or FALSE by justifying the answer. Any lack of justification or incorrect justification will not be taken into account in the grading.
Part 1
We consider the sequence $(u_n)$ defined by: $$u_0 = 10 \text{ and for all natural integer } n,\ u_{n+1} = \frac{1}{3}u_n + 2.$$
  1. Statement 1: The sequence $(u_n)$ is decreasing and bounded below by 0.
  2. Statement 2: $\lim_{n \rightarrow +\infty} u_n = 0$.
  3. Statement 3: The sequence $(v_n)$ defined for all natural integer $n$ by $v_n = u_n - 3$ is geometric.

Part 2
We consider the differential equation $(E): y' = \frac{3}{2}y + 2$ with unknown $y$, a function defined and differentiable on $\mathbb{R}$.
  1. Statement 4: There exists a constant function that is a solution to the differential equation $(E)$.
  2. In an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$ we denote $\mathscr{C}_f$ the representative curve of the function $f$ solution of $(E)$ such that $f(0) = 0$. Statement 5: The tangent line at the point with abscissa 1 of $\mathscr{C}_f$ has slope $2\mathrm{e}^{\frac{3}{2}}$.
Q3 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View
Part 1
We consider the function $f$ defined on the set of real numbers $\mathbb{R}$ by: $$f(x) = \left(x^2 - 4\right)\mathrm{e}^{-x}$$ We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function.
  1. Determine the limits of the function $f$ at $-\infty$ and at $+\infty$.
  2. Justify that for all real $x$, $f'(x) = \left(-x^2 + 2x + 4\right)\mathrm{e}^{-x}$.
  3. Deduce the variations of the function $f$ on $\mathbb{R}$.

Part 2
We consider the sequence $(I_n)$ defined for all natural integer $n$ by $I_n = \int_{-2}^{0} x^n \mathrm{e}^{-x}\,\mathrm{d}x$.
  1. Justify that $I_0 = \mathrm{e}^2 - 1$.
  2. Using integration by parts, demonstrate the equality: $$I_{n+1} = (-2)^{n+1}\mathrm{e}^2 + (n+1)I_n$$
  3. Deduce the exact values of $I_1$ and $I_2$.

Part 3
  1. Determine the sign on $\mathbb{R}$ of the function $f$ defined in Part 1.
  2. The curve $\mathscr{C}_f$ of the function $f$ is represented in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. The domain $D$ of the shaded region is bounded by the curve $\mathscr{C}_f$, the $x$-axis and the $y$-axis. Calculate the exact value, in square units, of the area $S$ of the domain $D$.
Q4 5 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the three points $\mathrm{A}(3;0;0)$, $\mathrm{B}(0;2;0)$ and $\mathrm{C}(0;0;2)$.
The objective of this exercise is to demonstrate the following property: ``The square of the area of triangle ABC is equal to the sum of the squares of the areas of the three other faces of the tetrahedron OABC''.
Part 1: Distance from point O to the plane (ABC)
  1. Demonstrate that the vector $\vec{n}(2;3;3)$ is normal to the plane (ABC).
  2. Demonstrate that a Cartesian equation of the plane (ABC) is: $2x + 3y + 3z - 6 = 0$.
  3. Give a parametric representation of the line $d$ passing through O and with direction vector $\vec{n}$.
  4. We denote H the point of intersection of the line $d$ and the plane (ABC). Determine the coordinates of point H.
  5. Deduce that the distance from point O to the plane (ABC) is equal to $\dfrac{3\sqrt{22}}{11}$.

Part 2: Demonstration of the property
  1. Demonstrate that the volume of the tetrahedron OABC is equal to 2.
  2. Deduce that the area of triangle ABC is equal to $\sqrt{22}$.
  3. Demonstrate that for the tetrahedron OABC, ``the square of the area of triangle ABC is equal to the sum of the squares of the areas of the three other faces of the tetrahedron''. Recall that the volume of a tetrahedron is given by $V = \dfrac{1}{3}B \times h$ where $B$ is the area of a base of the tetrahedron and $h$ is the height relative to this base.