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-nord_j1

5 maths questions

Q1A Conditional Probability Bayes' Theorem with Production/Source Identification View
A video game rewards players who have won a challenge with a randomly drawn object. The drawn object can be ``common'' or ``rare''. Two types of objects, common or rare, are available: swords and shields.
The video game designers have planned that:
  • the probability of drawing a rare object is $7\%$;
  • if a rare object is drawn, the probability that it is a sword is $80\%$;
  • if a common object is drawn, the probability that it is a sword is $40\%$.

Part A
A player has just won a challenge and draws an object at random. We denote:
  • R the event ``the player draws a rare object'';
  • $E$ the event ``the player draws a sword'';
  • $\bar{R}$ and $\bar{E}$ the complementary events of events $R$ and $E$.

  1. Draw a probability tree modelling the situation, then calculate $P(R \cap E)$.
  2. Calculate the probability of drawing a sword.
  3. The player has drawn a sword. Determine the probability that it is a rare object. Round the result to the nearest thousandth.
Q1B Binomial Distribution Justify Binomial Model and State Parameters View
A video game rewards players who have won a challenge with a randomly drawn object. The drawn object can be ``common'' or ``rare''. Two types of objects, common or rare, are available: swords and shields.
The video game designers have planned that:
  • the probability of drawing a rare object is $7\%$;
  • if a rare object is drawn, the probability that it is a sword is $80\%$;
  • if a common object is drawn, the probability that it is a sword is $40\%$.

Part B
A player wins 30 challenges. We denote $X$ the random variable corresponding to the number of rare objects the player obtains after winning 30 challenges. The successive draws are considered independent.
  1. Determine, by justifying, the probability distribution followed by the random variable $X$. Specify its parameters, as well as its expected value.
  2. Determine $P(X < 6)$. Round the result to the nearest thousandth.
  3. Determine the largest value of $k$ such that $P(X \geqslant k) \geqslant 0.5$. Interpret the result in the context of the exercise.
  4. The video game developers want to offer players the option to buy a ``gold ticket'' which allows them to draw $N$ objects. The probability of drawing a rare object remains $7\%$. The developers would like that by buying a gold ticket, the probability that a player obtains at least one rare object in these $N$ draws is greater than or equal to $0.95$. Determine the minimum number of objects to draw to achieve this objective. Care should be taken to detail the approach used.
Q2 4 marks Vectors 3D & Lines Parametric Representation of a Line View
This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, multiple answers, or the absence of an answer to a question earns neither points nor deducts points. The four questions are independent.
Space is referred to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$.
  1. Consider the points $A(1; 0; 3)$ and $B(4; 1; 0)$.
    A parametric representation of the line (AB) is: a. $\left\{ \begin{aligned} x & = 3 + t \\ y & = 1 \\ z & = -3 + 3t \end{aligned} \right.$ with $t \in \mathbb{R}$ b. $\left\{ \begin{array}{l} x = 1 + 4t \\ y = 3 \\ z = 3 \end{array} \right.$ with $t \in \mathbb{R}$ c. $\left\{ \begin{aligned} x & = 1 + 3t \\ y & = t \\ z & = 3 - 3t \end{aligned} \right.$ with $t \in \mathbb{R}$ d. $\left\{ \begin{aligned} x & = 4 + t \\ y & = 1 \\ z & = 3 - 3t \end{aligned} \right.$ with $t \in \mathbb{R}$
  2. Consider the line (d) with parametric representation $\left\{ \begin{aligned} x & = 3 + 4t \\ y & = 6t \\ z & = 4 - 2t \end{aligned} \right.$ with $t \in \mathbb{R}$
    Among the following points, which one belongs to the line (d)? a. $M(7; 6; 6)$ b. $N(3; 6; 4)$ c. $P(4; 6; -2)$ d. $R(-3; -9; 7)$
  3. Consider the line $(d')$ with parametric representation $\left\{ \begin{aligned} x & = -2 + 3k \\ y & = -1 - 2k \\ z & = 1 + k \end{aligned} \right.$ with $k \in \mathbb{R}$
    The lines $(d)$ and $(d')$ are: a. secant b. non-coplanar c. parallel d. coincident
  4. Consider the plane $(P)$ passing through the point $I(2; 1; 0)$ and perpendicular to the line (d).
    An equation of the plane $(P)$ is: a. $2x + 3y - z - 7 = 0$ b. $-x + y - 4z + 1 = 0$ c. $4x + 6y - 2z + 9 = 0$ d. $2x + y + 1 = 0$
Q3 Applied differentiation Convexity and inflection point analysis View
The purpose of this exercise is to study the function $f$ defined on the interval $]0; +\infty[$ by:
$$f(x) = x \ln(x^2) - \frac{1}{x}$$
Part A: graphical readings
Below is the representative curve $(\mathscr{C}_f)$ of the function $f$, as well as the line $(T)$, tangent to the curve $(\mathscr{C}_f)$ at point A with coordinates $(1; -1)$. This tangent also passes through the point $B(0; -4)$.
  1. Read graphically $f'(1)$ and give the reduced equation of the tangent $(T)$.
  2. Give the intervals on which the function $f$ appears to be convex or concave. What does point A appear to represent for the curve $(\mathscr{C}_f)$?

Part B: analytical study
  1. Determine, by justifying, the limit of $f$ at $+\infty$, then its limit at 0.
  2. It is admitted that the function $f$ is twice differentiable on the interval $]0; +\infty[$. a. Determine $f'(x)$ for $x$ belonging to the interval $]0; +\infty[$. b. Show that for all $x$ belonging to the interval $]0; +\infty[$, $$f''(x) = \frac{2(x+1)(x-1)}{x^3}.$$
  3. a. Study the convexity of the function $f$ on the interval $]0; +\infty[$. b. Study the variations of the function $f'$, then the sign of $f'(x)$ for $x$ belonging to the interval $]0; +\infty[$. Deduce the direction of variation of the function $f$ on the interval $]0; +\infty[$.
  4. a. Show that the equation $f(x) = 0$ has a unique solution $\alpha$ on the interval $]0; +\infty[$. b. Give the value of $\alpha$ rounded to the nearest hundredth and show that $\alpha$ satisfies: $$\alpha^2 = \exp\left(\frac{1}{\alpha^2}\right)$$
Q4 Reduction Formulae Compute a Base Case or Specific Value of a Parametric Integral View
For every natural number $n$, we consider the following integrals:
$$I_n = \int_0^{\pi} \mathrm{e}^{-nx} \sin(x) \mathrm{d}x, \quad J_n = \int_0^{\pi} \mathrm{e}^{-nx} \cos(x) \mathrm{d}x$$
  1. Calculate $I_0$.
  2. a. Justify that, for every natural number $n$, we have $I_n \geqslant 0$. b. Show that, for every natural number $n$, we have $I_{n+1} - I_n \leqslant 0$. c. Deduce from the two previous questions that the sequence $(I_n)$ converges.
  3. a. Show that, for every natural number $n$, we have: $$I_n \leqslant \int_0^{\pi} \mathrm{e}^{-nx} \mathrm{d}x$$ b. Show that, for every natural number $n \geqslant 1$, we have: $$\int_0^{\pi} \mathrm{e}^{-nx} \mathrm{d}x = \frac{1 - \mathrm{e}^{-n\pi}}{n}.$$ c. Deduce from the two previous questions the limit of the sequence $(I_n)$.
  4. a. By integrating by parts the integral $I_n$ in two different ways, establish the two following relations, for every natural number $n \geqslant 1$: $$I_n = 1 + \mathrm{e}^{-n\pi} - nJ_n \quad \text{and} \quad I_n = \frac{1}{n}J_n$$ b. Deduce that, for every natural number $n \geqslant 1$, we have $$I_n = \frac{1 + \mathrm{e}^{-n\pi}}{n^2 + 1}$$
  5. It is desired to obtain the rank $n$ from which the sequence $(I_n)$ becomes less than $0.1$. Copy and complete the fifth line of the Python script below with the appropriate command. \begin{verbatim} from math import * def seuil() : n = 0 I = 2 ... n=n+1 I =(1+exp(-n*pi))/(n*n+1) return n \end{verbatim}