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__metropole-sept_j2

5 maths questions

QA Sequences and series, recurrence and convergence Auxiliary sequence transformation View
Main topics covered: Numerical sequences; proof by induction.
We consider the sequences $(u_n)$ and $(v_n)$ defined by:
$$u_0 = 16 \quad; \quad v_0 = 5$$
and for any natural number $n$:
$$\left\{\begin{aligned} u_{n+1} & = \frac{3u_n + 2v_n}{5} \\ v_{n+1} & = \frac{u_n + v_n}{2} \end{aligned}\right.$$
  1. Calculate $u_1$ and $v_1$.
  2. We consider the sequence $(w_n)$ defined for any natural number $n$ by: $w_n = u_n - v_n$. a. Prove that the sequence $(w_n)$ is geometric with ratio 0.1.
    Deduce from this, for any natural number $n$, the expression of $w_n$ as a function of $n$. b. Specify the sign of the sequence $(w_n)$ and the limit of this sequence.
  3. a. Prove that, for any natural number $n$, we have: $u_{n+1} - u_n = -0.4w_n$. b. Deduce that the sequence $(u_n)$ is decreasing.
    It can be shown in the same way that the sequence $(v_n)$ is increasing. We admit this result, and we note that we then have: for any natural number $n$, $v_n \geq v_0 = 5$. c. Prove by induction that, for any natural number $n$, we have: $u_n \geq 5$.
    Deduce that the sequence $(u_n)$ is convergent. We call $\ell$ the limit of $(u_n)$. It can be shown in the same way that the sequence $(v_n)$ is convergent. We admit this result, and we call $\ell'$ the limit of $(v_n)$.
  4. a. Prove that $\ell = \ell'$. b. We consider the sequence $(c_n)$ defined for any natural number $n$ by: $c_n = 5u_n + 4v_n$. Prove that the sequence $(c_n)$ is constant, that is, for any natural number $n$, we have: $c_{n+1} = c_n$. Deduce that, for any natural number $n$, $c_n = 100$. c. Determine the common value of the limits $\ell$ and $\ell'$.
QB Differentiating Transcendental Functions Full function study with transcendental functions View
Main topics covered: Logarithm function, limits, differentiation.
Part 1
The graph below gives the graphical representation in an orthonormal coordinate system of the function $f$ defined on the interval $]0; +\infty[$ by:
$$f(x) = \frac{2\ln(x) - 1}{x}$$
  1. Determine by calculation the unique solution $\alpha$ of the equation $f(x) = 0$.
    Give the exact value of $\alpha$ as well as the value rounded to the nearest hundredth.
  2. Specify, by graphical reading, the sign of $f(x)$ when $x$ varies in the interval $]0; +\infty[$.

Part II
We consider the function $g$ defined on the interval $]0; +\infty[$ by:
$$g(x) = [\ln(x)]^2 - \ln(x)$$
  1. a. Determine the limit of the function $g$ at 0. b. Determine the limit of the function $g$ at $+\infty$.
  2. We denote by $g'$ the derivative function of the function $g$ on the interval $]0; +\infty[$.
    Prove that, for any real number $x$ in $]0; +\infty[$, we have: $g'(x) = f(x)$, where $f$ denotes the function defined in Part I.
  3. Draw up the variation table of the function $g$ on the interval $]0; +\infty[$.
    This table should include the limits of the function $g$ at 0 and at $+\infty$, as well as the value of the minimum of $g$ on $]0; +\infty[$.
  4. Prove that, for any real number $m > -0.25$, the equation $g(x) = m$ has exactly two solutions.
  5. Determine by calculation the two solutions of the equation $g(x) = 0$.
Q1 4 marks Conditional Probability Bayes' Theorem with Diagnostic/Screening Test View
A company receives numerous emails daily. Among these emails, $8\%$ are ``spam'', that is, emails with advertising or malicious intent, which it is desirable not to open. An email received by the company is chosen at random. The properties of the email software used in the company allow us to state that:
  • The probability that the chosen email is classified as ``undesirable'' given that it is spam is equal to 0.9.
  • The probability that the chosen email is classified as ``undesirable'' given that it is not spam is equal to 0.01.

We denote:
  • S the event ``the chosen email is spam'';
  • I the event ``the chosen email is classified as undesirable by the email software''.
  • $\bar{S}$ and $\bar{I}$ the complementary events of $S$ and $I$ respectively.

  1. Model the situation studied using a probability tree, on which the probabilities associated with each branch should appear.
  2. a. Prove that the probability that the chosen email is a spam message and is classified as undesirable is equal to 0.072. b. Calculate the probability that the chosen message is classified as undesirable. c. The chosen message is classified as undesirable. What is the probability that it is actually a spam message? Give the answer rounded to the nearest hundredth.
  3. A random sample of 50 emails is chosen from those received by the company. It is assumed that this choice amounts to a random draw with replacement of 50 emails from the set of all emails received by the company. Let $Z$ be the random variable counting the number of spam emails among the 50 chosen. a. What is the probability distribution followed by the random variable $Z$, and what are its parameters? b. What is the probability that, among the 50 chosen emails, at least two are spam? Give the answer rounded to the nearest hundredth.
Q2 Vectors: Lines & Planes MCQ: Identify Correct Equation or Representation View
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A correct answer earns one point. An incorrect answer, multiple answers, or no answer to a question earns or loses no points.
In space with respect to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points $A(1; 0; 2)$, $B(2; 1; 0)$, $C(0; 1; 2)$ and the line $\Delta$ whose parametric representation is: $$\left\{\begin{array}{rl}x & = 1 + 2t \\ y & = -2 + t \\ z & = 4 - t\end{array}, t \in \mathbb{R}\right.$$
  1. Which of the following points belongs to the line $\Delta$?
    Answer A: $M(2; 1; -1)$; Answer B: $N(-3; -4; 6)$; Answer C: $P(-3; -4; 2)$; Answer D: $Q(-5; -5; 1)$.
  2. The vector $\overrightarrow{AB}$ has coordinates:
    $$\begin{array}{ll} \text{Answer A}: \left(\begin{array}{c} 1.5 \\ 0.5 \\ 1 \end{array}\right); & \text{Answer B}: \left(\begin{array}{c} -1 \\ -1 \\ 2 \end{array}\right); \\ \text{Answer C}: \left(\begin{array}{c} 1 \\ 1 \\ -2 \end{array}\right) & \text{Answer D}: \left(\begin{array}{l} 3 \\ 1 \\ 2 \end{array}\right). \end{array}$$
  3. A parametric representation of the line (AB) is:
    $$\begin{array}{ll} \text{Answer A}: \left\{\begin{array}{l} x = 1 + 2t \\ y = t \\ z = 2 \end{array}, t \in \mathbb{R}\right. & \text{Answer B}: \left\{\begin{array}{l} x = 2 - t \\ y = 1 - t \\ z = 2t \end{array}, t \in \mathbb{R}\right. \\ \text{Answer C}: \left\{\begin{array}{l} x = 2 + t \\ y = 1 + t \\ z = 2t \end{array}, t \in \mathbb{R}\right. & \text{Answer D}: \left\{\begin{array}{l} x = 1 + t \\ y = 1 + t \\ z = 2 - 2t \end{array}, t \in \mathbb{R}\right. \end{array}$$
  4. A Cartesian equation of the plane passing through point C and orthogonal to the line $\Delta$ is: Answer A: $x - 2y + 4z - 6 = 0$; Answer B: $2x + y - z + 1 = 0$; Answer C: $2x + y - z - 1 = 0$; Answer D: $y + 2z - 5 = 0$.
  5. We consider the point D defined by the vector relation $\overrightarrow{OD} = 3\overrightarrow{OA} - \overrightarrow{OB} - \overrightarrow{OC}$.
    Answer A: $\overrightarrow{AD}$, $\overrightarrow{AB}$, $\overrightarrow{AC}$ are coplanar; Answer B: $\overrightarrow{AD} = \overrightarrow{BC}$; Answer C: D has coordinates $(3; -1; -1)$; Answer D: the points A, B, C and D are collinear.
Q3 6 marks Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
Part I
We consider the function $f$ defined on $\mathbb{R}$ by
$$f(x) = x - \mathrm{e}^{-2x}$$
We call $\Gamma$ the representative curve of the function $f$ in an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath})$.
  1. Determine the limits of the function $f$ at $-\infty$ and at $+\infty$.
  2. Study the monotonicity of the function $f$ on $\mathbb{R}$ and draw up its variation table.
  3. Show that the equation $f(x) = 0$ has a unique solution $\alpha$ on $\mathbb{R}$, and give an approximate value to $10^{-2}$ precision.
  4. Deduce from the previous questions the sign of $f(x)$ according to the values of $x$.

Part II
In the orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath})$, we call $\mathscr{C}$ the representative curve of the function $g$ defined on $\mathbb{R}$ by:
$$g(x) = \mathrm{e}^{-x}$$
The curves $\mathscr{C}$ and the curve $\Gamma$ (which represents the function $f$ from Part I) are drawn on the graph provided in the appendix which is to be completed and returned with your paper. The purpose of this part is to determine the point on the curve $\mathscr{C}$ closest to the origin O of the coordinate system and to study the tangent to $\mathscr{C}$ at this point.
  1. For any real number $t$, we denote by $M$ the point with coordinates $(t; \mathrm{e}^{-t})$ on the curve $\mathscr{C}$.
    We consider the function $h$ which, to the real number $t$, associates the distance $OM$. We therefore have: $h(t) = OM$, that is:
    $$h(t) = \sqrt{t^2 + \mathrm{e}^{-2t}}$$
    a. Show that, for any real number $t$,
    $$h'(t) = \frac{f(t)}{\sqrt{t^2 + \mathrm{e}^{-2t}}}$$
    where $f$ denotes the function studied in Part I. b. Prove that the point A with coordinates $(\alpha; \mathrm{e}^{-\alpha})$ is the point on the curve $\mathscr{C}$ for which the length $OM$ is minimal. Place this point on the graph provided in the appendix, to be returned with your paper.
  2. We call $T$ the tangent to the curve $\mathscr{C}$ at A. a. Express in terms of $\alpha$ the slope of the tangent $T$.
    We recall that the slope of the line (OA) is equal to $\frac{\mathrm{e}^{-\alpha}}{\alpha}$. We also recall the following result which may be used without proof: In an orthonormal coordinate system of the plane, two lines $D$ and $D'$ with slopes $m$ and $m'$ respectively are perpendicular if and only if the product $mm'$ is equal to $-1$. b. Prove that the line (OA) and the tangent $T$ are perpendicular.
    Draw these lines on the graph provided in the appendix, to be returned with your paper.