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
2025 bac-spe-maths__amerique-sud_j2

7 maths questions

Q1A Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
In tennis, the player who is serving can, in case of failure on the first serve, serve a second ball. In match play, Abel succeeds with his first serve in $70\%$ of cases. When the first serve is successful, he wins the point in $80\%$ of cases. On the other hand, after a failure on his first serve, Abel wins the point in $45\%$ of cases. Abel is serving. Consider the following events:
  • S: ``Abel succeeds with his first serve''
  • G: ``Abel wins the point''.

  1. Describe the event $S$ then translate the situation with a probability tree.
  2. Calculate $P(S \cap G)$.
  3. Justify that the probability of event $G$ is equal to 0.695.
  4. Abel has won the point. What is the probability that he succeeded with his first serve?
  5. Are events $S$ and $G$ independent? Justify.
Q1B Binomial Distribution Justify Binomial Model and State Parameters View
At the output of a tennis ball manufacturing factory, a ball is judged to be compliant in $85\%$ of cases.
  1. We test successively 20 balls. We consider that the number of balls is large enough to assimilate these tests to sampling with replacement. We denote $X$ the random variable that counts the number of compliant balls among the 20 tested. a. What is the distribution followed by $X$ and what are its parameters? Justify. b. Calculate $P(X \leqslant 18)$. c. What is the probability that at least two balls are not compliant among the 20 balls tested? d. Determine the expectation of $X$.
  2. We now test $n$ balls successively. We consider the $n$ tests as a sample of $n$ independent random variables $X$ following the Bernoulli distribution with parameter 0.85. We consider the random variable $$M_n = \sum_{i=1}^{n} \frac{X_i}{n} = \frac{X_1}{n} + \frac{X_2}{n} + \frac{X_3}{n} + \ldots + \frac{X_n}{n}$$ a. Determine the expectation and variance of $M_n$. b. After recalling the Bienaymé-Chebyshev inequality, show that, for every natural integer $n$, $P\left(0.75 < M_n < 0.95\right) \geqslant 1 - \frac{12.75}{n}$. c. Deduce an integer $n$ such that the average number of compliant balls for a sample of size $n$ belongs to the interval $]0.75 ; 0.95[$ with a probability greater than 0.9.
Q2 4 marks Vectors: Lines & Planes MCQ: Identify Correct Equation or Representation View
This exercise is a multiple choice questionnaire. For each question, only one of the three propositions is correct.
In all the following questions, space is referred to an orthonormal coordinate system.
  1. Consider the line $\Delta_1$ with parametric representation $\left\{ \begin{aligned} x &= 1 - 3t \\ y &= 4 + 2t \\ z &= t \end{aligned} \right.$, where $t \in \mathbb{R}$ as well as the line $\Delta_2$ with parametric representation $\left\{ \begin{aligned} x &= -4 + s \\ y &= 2 + 2s \\ z &= -1 + s \end{aligned} \right.$, where $s \in \mathbb{R}$. a. The lines $\Delta_1$ and $\Delta_2$ are parallel. b. The lines $\Delta_1$ and $\Delta_2$ are orthogonal. c. The lines $\Delta_1$ and $\Delta_2$ are secant.
  2. Consider the line $d$ with parametric representation $\left\{ \begin{aligned} x &= 1 + t \\ y &= 3 - t \\ z &= 1 + 2t \end{aligned} \right.$, where $t \in \mathbb{R}$, and the plane $P$ with Cartesian equation: $4x + 2y - z + 3 = 0$. a. The line $d$ is contained in the plane $P$. b. The line $d$ is strictly parallel to the plane $P$. c. The line $d$ is secant to the plane $P$.
  3. Consider the points $\mathrm{A}(3;2;1)$, $\mathrm{B}(7;3;1)$, $\mathrm{C}(-1;4;5)$ and $\mathrm{D}(-3;3;5)$. a. The points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and D are not coplanar. b. The points $\mathrm{A}$, $\mathrm{B}$ and C are collinear. c. $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{CD}}$ are collinear.
  4. Consider the planes $Q$ and $Q'$ with respective Cartesian equations $3x - 2y + z + 1 = 0$ and $4x + y - z + 3 = 0$. a. The point $\mathrm{R}(1;1;-2)$ belongs to both planes. b. The two planes are orthogonal. c. The two planes are secant with intersection the line with parametric representation $$\left\{ \begin{aligned} x &= t \\ y &= 7t + 4, \text{ where } t \in \mathbb{R}. \\ z &= 11t + 7 \end{aligned} \right.$$
Q3 4 marks Sequences and series, recurrence and convergence Conjecture from numerical data or computation View
Consider the sequences $\left(v_n\right)$ and $\left(w_n\right)$ defined for every natural integer $n$ by:
$$\left\{ \begin{array}{ll} v_0 &= \ln(4) \\ v_{n+1} &= \ln\left(-1 + 2\mathrm{e}^{v_n}\right) \end{array} \quad \text{and} \quad w_n = -1 + \mathrm{e}^{v_n} \right.$$
We admit that the sequence $\left(v_n\right)$ is well defined and strictly positive.
  1. Give the exact values of $v_1$ and $w_0$.
  2. a. Among the three formulas below, choose the formula which, entered in cell B3 then copied downward, will allow you to obtain the values of the sequence $(v_n)$ in column B.
    Formula 1$\mathrm{LN}\left(-1 + 2^*\operatorname{EXP}(\mathrm{B}2)\right)$
    Formula 2$=\mathrm{LN}\left(-1 + 2^*\operatorname{EXP}(\mathrm{B}2)\right)$
    Formula 3$=\mathrm{LN}\left(-1 + 2^*\operatorname{EXP}(\mathrm{A}2)\right)$

    b. Conjecture the direction of variation of the sequence $\left(v_n\right)$. c. Using a proof by induction, validate your conjecture concerning the direction of variation of the sequence $(v_n)$.
  3. a. Prove that the sequence $(w_n)$ is geometric. b. Deduce that for every natural integer $n$, $v_n = \ln\left(1 + 3 \times 2^n\right)$. c. Determine the limit of the sequence $\left(v_n\right)$.
  4. Justify that the following algorithm written in Python language returns a result regardless of the choice of the value of the number S. \begin{verbatim} from math import* def seuil(S): V=ln(4) n=0 while V < S : n=n+1 V=ln(2*exp(V)-1) return(n) \end{verbatim}
Q4A Discriminant and conditions for roots Probability involving discriminant conditions View
Consider the set of non-zero relative integers between $-30$ and $30$; this set can be written as follows: $\{-30; -29; -28; \ldots -1; 1; \ldots; 28; 29; 30\}$. It contains 60 elements. We choose from this set successively and without replacement a relative integer $a$ then a relative integer $c$.
  1. How many different pairs $(a; c)$ can we obtain?

Consider the event $M$: ``the equation $ax^2 + 2x + c = 0$ has two distinct real solutions'', where $a$ and $c$ are the relative integers previously chosen.
    \setcounter{enumi}{1}
  1. Show that event $M$ occurs if and only if $ac < 1$.
  2. Explain why the opposite event $\bar{M}$ contains 1740 outcomes.
  3. What is the probability of event $M$? Round the result to $10^{-2}$.
Q4B Second order differential equations Solving non-homogeneous second-order linear ODE View
Consider the differential equation
$$(E): \quad y' + 10y = \left(30x^2 + 22x - 8\right)\mathrm{e}^{-5x+1} \quad \text{with} \quad x \in \mathbb{R}$$
where $y$ is a function defined and differentiable on $\mathbb{R}$.
  1. Solve on $\mathbb{R}$ the differential equation: $y' + 10y = 0$.
  2. Let the function $f$ defined on $\mathbb{R}$ by $$f(x) = \left(6x^2 + 2x - 2\right)\mathrm{e}^{-5x+1}$$ We admit that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ the derivative function of the function $f$. Justify that $f$ is a particular solution of $(E)$.
  3. Give the expression of all solutions of $(E)$.
Q4C Applied differentiation Existence and number of solutions via calculus View
We propose to study in this part the function $f$ encountered in Part B question 2. We recall that, for every real $x$, $f(x) = \left(6x^2 + 2x - 2\right)\mathrm{e}^{-5x+1}$. We denote $f'$ the derivative function of the function $f$. We call $\mathscr{C}_f$ the representative curve of $f$ in a coordinate system of the plane.
  1. We admit that $\lim_{x \rightarrow +\infty} f(x) = 0$. Determine the limit of the function $f$ at $-\infty$.
  2. Using Part A (Exercise 4), show that $\mathscr{C}_f$ intersects the $x$-axis at two points (the coordinates of these points are not expected).
  3. Using Parts A and B (Exercise 4), show that $\mathscr{C}_f$ has two horizontal tangent lines.
  4. Draw the complete variation table of the function $f$.
  5. Determine by justifying the number of solution(s) of the equation $f(x) = 1$.
  6. For every real $m$ strictly greater than 0.2, we define $I_m$ by $I_m = \int_{0.2}^{m} f(x)\,\mathrm{d}x$. a. Verify that the function $F$ defined on $\mathbb{R}$ by $$F(x) = \left(-\frac{6}{5}x^2 - \frac{22}{25}x + \frac{28}{125}\right)\mathrm{e}^{-5x+1}$$ is a primitive of the function $f$ on $\mathbb{R}$. b. Does there exist a value of $m$ for which $I_m = 0$? Interpret this result graphically.