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_j2

5 maths questions

QA Differentiating Transcendental Functions Full function study with transcendental functions View
Main topics covered: Exponential function; differentiation.
The graph below represents, in an orthogonal coordinate system, the curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ of the functions $f$ and $g$ defined on $\mathbb{R}$ by:
$$f(x) = x^{2}\mathrm{e}^{-x} \text{ and } g(x) = \mathrm{e}^{-x}.$$
Question 3 is independent of questions 1 and 2.
  1. a. Determine the coordinates of the intersection points of $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$. b. Study the relative position of the curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$.
  2. For every real number $x$ in the interval $[-1; 1]$, we consider the points $M$ with coordinates $(x; f(x))$ and $N$ with coordinates $(x; g(x))$, and we denote by $d(x)$ the distance $MN$. We assume that: $d(x) = \mathrm{e}^{-x} - x^{2}\mathrm{e}^{-x}$. We assume that the function $d$ is differentiable on the interval $[-1; 1]$ and we denote by $d^{\prime}$ its derivative function. a. Show that $d^{\prime}(x) = \mathrm{e}^{-x}\left(x^{2} - 2x - 1\right)$. b. Deduce the variations of the function $d$ on the interval $[-1; 1]$. c. Determine the common abscissa $x_{0}$ of the points $M_{0}$ and $N_{0}$ allowing to obtain a maximum distance $d(x_{0})$, and give an approximate value to 0.1 of the distance $M_{0}N_{0}$.
  3. Let $\Delta$ be the line with equation $y = x + 2$. We consider the function $h$ differentiable on $\mathbb{R}$ and defined by: $h(x) = \mathrm{e}^{-x} - x - 2$. By studying the number of solutions of the equation $h(x) = 0$, determine the number of intersection points of the line $\Delta$ and the curve $\mathscr{C}_{g}$.
QB Differentiating Transcendental Functions Full function study with transcendental functions View
Main topics covered: Logarithm function; differentiation.
Part I: Study of an auxiliary function
Let $g$ be the function defined on $]0; +\infty[$ by:
$$g(x) = \ln(x) + 2x - 2.$$
  1. Determine the limits of $g$ at $+\infty$ and 0.
  2. Determine the direction of variation of the function $g$ on $]0; +\infty[$.
  3. Prove that the equation $g(x) = 0$ admits a unique solution $\alpha$ on $]0; +\infty[$.
  4. Calculate $g(1)$ then determine the sign of $g$ on $]0; +\infty[$.

Part II: Study of a function $f$
We consider the function $f$, defined on $]0; +\infty[$ by:
$$f(x) = \left(2 - \frac{1}{x}\right)(\ln(x) - 1)$$
  1. a. We assume that the function $f$ is differentiable on $]0; +\infty[$ and we denote by $f^{\prime}$ its derivative. Prove that, for every $x$ in $]0; +\infty[$, we have: $$f^{\prime}(x) = \frac{g(x)}{x^{2}}$$ b. Draw the variation table of the function $f$ on $]0; +\infty[$. The calculation of limits is not required.
  2. Solve the equation $f(x) = 0$ on $]0; +\infty[$ then draw the sign table of $f$ on the interval $]0; +\infty[$.

Part III: Study of a function $F$ whose derivative is the function $f$
We assume that there exists a function $F$ differentiable on $]0; +\infty[$ whose derivative $F^{\prime}$ is the function $f$. Thus, we have: $F^{\prime} = f$. We denote by $\mathscr{C}_{F}$ the representative curve of the function $F$ in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$. We will not seek to determine an expression for $F(x)$.
  1. Study the variations of $F$ on $]0; +\infty[$.
  2. Does the curve $\mathscr{C}_{F}$ representative of $F$ admit tangent lines parallel to the x-axis? Justify the answer.
Q1 5 marks Binomial Distribution Compute Exact Binomial Probability 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, a multiple answer, or the absence of an answer to a question earns or deducts no points.
PART I
In a mail processing centre, a machine is equipped with an automatic optical reader for recognizing postal addresses. This reading system correctly recognizes $97\%$ of addresses; the remaining mail, which will be described as unreadable for the machine, is directed to a centre employee responsible for reading the addresses. This machine has just read nine addresses. We denote by $X$ the random variable that gives the number of unreadable addresses among these nine addresses. We assume that $X$ follows the binomial distribution with parameters $n = 9$ and $p = 0.03$.
  1. The probability that none of the nine addresses is unreadable is equal, to the nearest hundredth, to: a. 0 b. 1 c. 0.24 d. 0.76
  2. The probability that exactly two of the nine addresses are unreadable for the machine is: a. $\binom{9}{2} \times 0.97^{2} \times 0.03^{7}$ b. $\binom{7}{2} \times 0.97^{2} \times 0.03^{7}$ c. $\binom{9}{2} \times 0.97^{7} \times 0.03^{2}$ d. $\binom{7}{2} \times 0.97^{7} \times 0.03^{2}$
  3. The probability that at least one of the nine addresses is unreadable for the machine is: a. $P(X < 1)$ b. $P(X \leqslant 1)$ c. $P(X \geqslant 2)$ d. $1 - P(X = 0)$

PART II
An urn contains 5 green balls and 3 white balls, indistinguishable to the touch. We draw at random successively and without replacement two balls from the urn. We consider the following events:
  • $V_{1}$: "the first ball drawn is green";
  • $B_{1}$: "the first ball drawn is white";
  • $V_{2}$: "the second ball drawn is green";
  • $B_{2}$: "the second ball drawn is white".

    \setcounter{enumi}{3}
  1. The probability of $V_{2}$ given that $V_{1}$ is realized, denoted $P_{V_{1}}\left(V_{2}\right)$, is equal to: a. $\frac{5}{8}$ b. $\frac{4}{7}$ c. $\frac{5}{14}$ d. $\frac{20}{56}$
  2. The probability of event $V_{2}$ is equal to: a. $\frac{5}{8}$ b. $\frac{5}{7}$ c. $\frac{3}{28}$ d. $\frac{9}{7}$
Q2 Sequences and series, recurrence and convergence Convergence proof and limit determination View
We consider the sequences $(u_{n})$ and $(v_{n})$ defined for every natural integer $n$ by:
$$\left\{ \begin{array}{l} u_{0} = v_{0} = 1 \\ u_{n+1} = u_{n} + v_{n} \\ v_{n+1} = 2u_{n} + v_{n} \end{array} \right.$$
Throughout the rest of the exercise, we assume that the sequences $(u_{n})$ and $(v_{n})$ are strictly positive.
  1. a. Calculate $u_{1}$ and $v_{1}$. b. Prove that the sequence $(v_{n})$ is strictly increasing, then deduce that for every natural integer $n$, $v_{n} \geqslant 1$. c. Prove by induction that for every natural integer $n$, we have: $u_{n} \geqslant n + 1$. d. Deduce the limit of the sequence $(u_{n})$.
  2. We set, for every natural integer $n$: $$r_{n} = \frac{v_{n}}{u_{n}}.$$ We assume that: $$r_{n}^{2} = 2 + \frac{(-1)^{n+1}}{u_{n}^{2}}$$ a. Prove that for every natural integer $n$: $$-\frac{1}{u_{n}^{2}} \leqslant \frac{(-1)^{n+1}}{u_{n}^{2}} \leqslant \frac{1}{u_{n}^{2}}.$$ b. Deduce: $$\lim_{n \rightarrow +\infty} \frac{(-1)^{n+1}}{u_{n}^{2}}$$ c. Determine the limit of the sequence $\left(r_{n}^{2}\right)$ and deduce that $\left(r_{n}\right)$ converges to $\sqrt{2}$. d. Prove that for every natural integer $n$, $$r_{n+1} = \frac{2 + r_{n}}{1 + r_{n}}$$ e. Consider the following program written in Python language: \begin{verbatim} def seuil() : n = 0 r = l while abs(r-sqrt(2)) > 10**(-4) : r = (2+r)/(1+r) n = n+1 return n \end{verbatim} (abs denotes absolute value, sqrt the square root and $10^{**}(-4)$ represents $10^{-4}$). The value of $n$ returned by this program is 5. What does it correspond to?
Q3 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
In space with respect to an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points: A with coordinates $(2; 0; 0)$, B with coordinates $(0; 3; 0)$ and C with coordinates $(0; 0; 1)$.
The objective of this exercise is to calculate the area of triangle ABC.
  1. a. Show that the vector $\vec{n}\left(\begin{array}{l}3\\2\\6\end{array}\right)$ is normal to the plane (ABC). b. Deduce that a Cartesian equation of the plane (ABC) is: $3x + 2y + 6z - 6 = 0$.
  2. We denote by $d$ the line passing through O and perpendicular to the plane (ABC). a. Determine a parametric representation of the line $d$. b. Show that the line $d$ intersects the plane (ABC) at the point H with coordinates $\left(\frac{18}{49}; \frac{12}{49}; \frac{36}{49}\right)$. c. Calculate the distance OH.
  3. We recall that the volume of a pyramid is given by: $V = \frac{1}{3}\mathscr{B}h$, where $\mathscr{B}$ is the area of a base and $h$ is the height of the pyramid corresponding to this base. By calculating in two different ways the volume of the pyramid OABC, determine the area of triangle ABC.