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
2023 bac-spe-maths__metropole-sept_j2

4 maths questions

Q1 Conditional Probability Bayes' Theorem with Diagnostic/Screening Test View
Paratuberculosis is an infectious digestive disease that affects cows. It is caused by the presence of a bacterium in the cow's intestine.
A study is conducted in a region where $0.4\%$ of the cow population is infected.
There is a test that reveals the immune response of an organism infected by the bacterium. The result of this test can be either ``positive'' or ``negative''.
A cow is chosen at random from the region. Given the characteristics of the test, we know that:
  • If the cow is affected by the infection, the probability that its test is positive is 0.992;
  • If the cow is not affected by the infection, the probability that its test is negative is 0.984.

We denote by $I$ the event ``the cow is affected by the infection'' and $T$ the event ``the cow presents a positive test''. We denote by $\bar{I}$ the complementary event of $I$ and $\bar{T}$ the complementary event of $T$.
Part A
  1. Reproduce and complete the weighted tree below modelling the situation.
  2. a. Calculate the probability that the cow is not affected by the infection and that its test is negative. Give the result to $10^{-3}$ near. b. Show that the probability, to $10^{-3}$ near, that the cow presents a positive test is approximately equal to 0.020. c. The ``positive predictive value of the test'' is the probability that the cow is affected by the infection given that its test is positive. Calculate the positive predictive value of this test. Give the result to $10^{-3}$ near. d. The test gives incorrect information about the cow's state of health when the cow is not infected and presents a positive test result or when the cow is infected and presents a negative test result. Calculate the probability that this test gives incorrect information about the cow's state of health. Give a result to $10^{-3}$ near.

Part B
    \setcounter{enumi}{2}
  1. When a sample of 100 cows is chosen at random from the region, this choice is treated as a draw with replacement. Recall that, for a cow chosen at random from the region, the probability that the test is positive is equal to 0.02. We denote by $X$ the random variable that associates to a sample of 100 cows from the region chosen at random the number of cows presenting a positive test in this sample. a. What is the probability distribution followed by the random variable $X$? Justify the answer and specify the parameters of this distribution. b. Calculate the probability that in a sample of 100 cows, there are exactly 3 cows presenting a positive test. Give a result to $10^{-3}$ near. c. Calculate the probability that in a sample of 100 cows, there are at most 3 cows presenting a positive test. Give a result to $10^{-3}$ near.
  2. We now choose a sample of $n$ cows from this region, $n$ being a non-zero natural integer. We admit that this choice can be treated as a draw with replacement. Determine the minimum value of $n$ so that the probability that there is, in the sample, at least one cow tested positive, is greater than or equal to 0.99.
Q2 Applied differentiation Full function study (variation table, limits, asymptotes) View
We consider the function $f$ defined on the interval $]0; +\infty[$ by $$f(x) = (2 - \ln x) \times \ln x,$$ where ln denotes the natural logarithm function.
We admit that the function $f$ is twice differentiable on $]0; +\infty[$.
We denote by $C$ the representative curve of the function $f$ in an orthogonal coordinate system and $C'$ the representative curve of the function $f'$, the derivative function of the function $f$.
The curve $\boldsymbol{C}'$ is given (with its unique horizontal tangent (T)).
  1. By graphical reading, with the precision that the above diagram allows, give: a. the slope of the tangent to $C$ at the point with abscissa 1. b. the largest interval on which the function $f$ is convex.
  2. a. Calculate the limit of the function $f$ at $+\infty$. b. Calculate $\lim_{x \rightarrow 0} f(x)$. Interpret this result graphically.
  3. Show that the curve $C$ intersects the x-axis at exactly two points, whose coordinates you will specify.
  4. a. Show that for all real $x$ belonging to $]0; +\infty[$, $f'(x) = \dfrac{2(1 - \ln x)}{x}$. b. Deduce, by justifying, the table of variations of the function $f$ on $]0; +\infty[$.
  5. We denote by $f''$ the second derivative of $f$ and we admit that for all real $x$ belonging to $]0; +\infty[$, $f''(x) = \dfrac{2(\ln x - 2)}{x^2}$. Determine by calculation the largest interval on which the function $f$ is convex and specify the coordinates of the inflection point of the curve $C$.
Q3 Proof by induction Prove a closed-form expression for a sequence by induction View
We consider the sequence $(u_n)$ defined by: $$\left\{\begin{aligned} u_1 &= \frac{1}{\mathrm{e}} \\ u_{n+1} &= \frac{1}{\mathrm{e}}\left(1 + \frac{1}{n}\right)u_n \text{ for all integer } n \geqslant 1. \end{aligned}\right.$$
  1. Calculate the exact values of $u_2$ and $u_3$. Details of calculations should be shown.
  2. We consider a function written in Python language which, for a given natural integer $n$, displays the term $u_n$. Complete lines $L_2$ and $L_4$ of this program. \begin{verbatim} L1 def u(n): L2 ................. L3 for i in range(1, n): L4 u=................. L5 return u \end{verbatim}
  3. We admit that all terms of the sequence $(u_n)$ are strictly positive. a. Show that for all non-zero natural integer $n$, we have: $1 + \dfrac{1}{n} \leqslant \mathrm{e}$. b. Deduce that the sequence $(u_n)$ is decreasing. c. Is the sequence $(u_n)$ convergent? Justify your answer.
  4. a. Show by induction that for all non-zero natural integer, we have: $u_n = \dfrac{n}{\mathrm{e}^n}$. b. Deduce, if it exists, the limit of the sequence $(u_n)$.
Q4 5 marks 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, a multiple answer, or the absence of an answer to a question earns neither points nor deducts points.
Space is referred to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$.
We consider:
  • the points $A(-1; -2; 3)$, $B(1; -2; 7)$ and $C(1; 0; 2)$;
  • the line $\Delta$ with parametric representation: $\left\{\begin{array}{l} x = 1 - t \\ y = 2 \\ z = -4 + 3t \end{array}\right.$, where $t \in \mathbb{R}$;
  • the plane $\mathscr{P}$ with Cartesian equation: $3x + 2y + z - 4 = 0$;
  • the plane $\mathscr{Q}$ with Cartesian equation: $-6x - 4y - 2z + 7 = 0$.

  1. Which of the following points belongs to the plane $\mathscr{P}$? a. $R(1; -3; 1)$; b. $S(1; 2; -1)$; c. $T(1; 0; 1)$; d. $U(2; -1; 1)$.
  2. Triangle ABC is: a. equilateral; b. right isosceles; c. isosceles non-right; d. right non-isosceles.
  3. The line $\Delta$ is: a. orthogonal to the plane $\mathscr{P}$; b. secant to the plane $\mathscr{P}$; c. included in the plane $\mathscr{P}$; d. strictly parallel to the plane $\mathscr{P}$.
  4. We are given the dot product $\overrightarrow{BA} \cdot \overrightarrow{BC} = 20$.
    A measure to the nearest degree of the angle $\widehat{ABC}$ is: a. $34°$; b. $120°$; c. $90°$; d. $0°$.
  5. The intersection of planes $\mathscr{P}$ and $\mathscr{Q}$ is: a. a plane; b. the empty set; c. a line; d. reduced to a point.