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
2014 metropole

5 maths questions

Q1 5 marks Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Part A
In the plane with an orthonormal coordinate system, we denote by $\mathscr { C } _ { 1 }$ the curve representing the function $f _ { 1 }$ defined on $\mathbb { R }$ by: $$f _ { 1 } ( x ) = x + \mathrm { e } ^ { - x } .$$
  1. Justify that $\mathscr { C } _ { 1 }$ passes through point A with coordinates $( 0 ; 1 )$.
  2. Determine the variation table of the function $f _ { 1 }$. Specify the limits of $f _ { 1 }$ at $+ \infty$ and at $- \infty$.

Part B
The purpose of this part is to study the sequence $\left( I _ { n } \right)$ defined on $\mathbb { N }$ by: $$I _ { n } = \int _ { 0 } ^ { 1 } \left( x + \mathrm { e } ^ { - n x } \right) \mathrm { d } x .$$
  1. In the plane with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ), for every natural integer $n$, we denote by $\mathscr { C } _ { n }$ the curve representing the function $f _ { n }$ defined on $\mathbb { R }$ by $$f _ { n } ( x ) = x + \mathrm { e } ^ { - n x } .$$ a. Give a geometric interpretation of the integral $I _ { n }$. b. Using this interpretation, formulate a conjecture about the direction of variation of the sequence ( $I _ { n }$ ) and its possible limit. Specify the elements on which you base your conjecture.
  2. Prove that for every natural integer $n$ greater than or equal to 1, $$I _ { n + 1 } - I _ { n } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { - ( n + 1 ) x } \left( 1 - \mathrm { e } ^ { x } \right) \mathrm { d } x$$ Deduce the sign of $I _ { n + 1 } - I _ { n }$ and then prove that the sequence ( $I _ { n }$ ) is convergent.
  3. Determine the expression of $I _ { n }$ as a function of $n$ and determine the limit of the sequence $\left( I _ { n } \right)$.
Q2 5 marks Conditional Probability Bayes' Theorem with Diagnostic/Screening Test View
Parts A and B can be treated independently.
Part A
A pharmaceutical laboratory offers screening tests for various diseases. Its communications department highlights the following characteristics:
  • the probability that a sick person tests positive is 0.99;
  • the probability that a healthy person tests positive is 0.001.

  1. For a disease that has just appeared, the laboratory develops a new test. A statistical study makes it possible to estimate that the percentage of sick people among the population of a metropolis is equal to $0.1 \%$. A person is chosen at random from this population and undergoes the test. We denote by $M$ the event ``the chosen person is sick'' and $T$ the event ``the test is positive''. a. Translate the statement in the form of a weighted tree. b. Prove that the probability $p ( T )$ of event $T$ is equal to $$1.989 \times 10 ^ { - 3 } .$$ c. Is the following statement true or false? Justify your answer. Statement: ``If the test is positive, there is less than one chance in two that the person is sick''.
  2. The laboratory decides to market a test as soon as the probability that a person who tests positive is sick is greater than or equal to 0.95. We denote by $x$ the proportion of people affected by a certain disease in the population. From what value of $x$ does the laboratory market the corresponding test?

Part B
The laboratory's production line manufactures, in very large quantities, tablets of a medicine.
  1. A tablet is compliant if its mass is between 890 and 920 mg. We assume that the mass in milligrams of a tablet taken at random from production can be modeled by a random variable $X$ that follows the normal distribution $\mathscr { N } \left( \mu , \sigma ^ { 2 } \right)$, with mean $\mu = 900$ and standard deviation $\sigma = 7$. a. Calculate the probability that a tablet drawn at random is compliant. Round to $10 ^ { - 2 }$. b. Determine the positive integer $h$ such that $P ( 900 - h \leqslant X \leqslant 900 + h ) \approx 0.99$ to within $10 ^ { - 3 }$.
  2. The production line has been adjusted to obtain at least $97 \%$ compliant tablets. To evaluate the effectiveness of the adjustments, a check is performed by taking a sample of 1000 tablets from production. The size of the production is assumed to be large enough that this sample can be treated as 1000 successive draws with replacement. The check made it possible to count 53 non-compliant tablets in the sample taken. Does this check call into question the adjustments made by the laboratory? An asymptotic fluctuation interval at the $95 \%$ threshold can be used.
Q3 Complex numbers 2 Solving Polynomial Equations in C View
We denote by (E) the equation $$z ^ { 4 } + 4 z ^ { 2 } + 16 = 0$$ of unknown complex number $z$.
  1. Solve in $\mathbb { C }$ the equation $Z ^ { 2 } + 4 Z + 16 = 0$. Write the solutions of this equation in exponential form.
  2. We denote by $a$ the complex number whose modulus is equal to 2 and one of whose arguments is equal to $\frac { \pi } { 3 }$. Calculate $a ^ { 2 }$ in algebraic form. Deduce the solutions in $\mathbb { C }$ of the equation $z ^ { 2 } = - 2 + 2 \mathrm { i } \sqrt { 3 }$. Write the solutions in algebraic form.
  3. Organized presentation of knowledge We assume it is known that for every complex number $z = x + \mathrm { i } y$ where $x \in \mathbb { R }$ and $y \in \mathbb { R }$, the conjugate of $z$ is the complex number $\bar{z}$ defined by $\bar{z} = x - \mathrm { i } y$. Prove that:
    • For all complex numbers $z _ { 1 }$ and $z _ { 2 } , \overline { z _ { 1 } z _ { 2 } } = \overline { z _ { 1 } } \cdot \overline { z _ { 2 } }$.
    • For every complex number $z$ and every non-zero natural integer $n , \overline { z ^ { n } } = ( \bar { z } ) ^ { n }$.
  4. Prove that if $z$ is a solution of equation (E) then its conjugate $\bar { z }$ is also a solution of (E). Deduce the solutions in $\mathbb { C }$ of equation (E). We will assume that (E) has at most four solutions.
Q4A 5 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
Exercise 4 — Candidates who have not followed the specialization course
In space, we consider a tetrahedron ABCD whose faces ABC, ACD and ABD are right-angled and isosceles triangles at A. We denote by E, F and G the midpoints of sides $[\mathrm{AB}]$, $[\mathrm{BC}]$ and $[\mathrm{CA}]$ respectively. We choose AB as the unit of length and we place ourselves in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AC}}, \overrightarrow{\mathrm{AD}})$ of space.
  1. We denote by $\mathscr { P }$ the plane that passes through A and is perpendicular to the line $(\mathrm{DF})$. We denote by H the point of intersection of plane $\mathscr { P }$ and line (DF). a. Give the coordinates of points D and F. b. Give a parametric representation of line (DF). c. Determine a Cartesian equation of plane $\mathscr { P }$. d. Calculate the coordinates of point H. e. Prove that the angle $\widehat{\mathrm{EHG}}$ is a right angle.
  2. We denote by $M$ a point on line (DF) and by $t$ the real number such that $\overrightarrow{\mathrm{DM}} = t \overrightarrow{\mathrm{DF}}$. We denote by $\alpha$ the measure in radians of the geometric angle $\widehat{\mathrm{EMG}}$. The purpose of this question is to determine the position of point $M$ so that $\alpha$ is maximum. a. Prove that $ME^{2} = \frac{3}{2} t^{2} - \frac{5}{2} t + \frac{5}{4}$. b. Prove that triangle $M\mathrm{EG}$ is isosceles at $M$. Deduce that $ME \sin\left(\frac{\alpha}{2}\right) = \frac{1}{2\sqrt{2}}$. c. Justify that $\alpha$ is maximum if and only if $\sin\left(\frac{\alpha}{2}\right)$ is maximum. Deduce that $\alpha$ is maximum if and only if $ME^{2}$ is minimum. d. Conclude.
Q4B 5 marks Matrices Matrix Power Computation and Application View
Exercise 4 — Candidates who have followed the specialization course
A fish farmer has two basins A and B for raising his fish. Every year at the same time:
  • he empties basin B and sells all the fish it contained and transfers all the fish from basin A to basin B;
  • the sale of each fish allows the purchase of two small fish intended for basin A.
Furthermore, the fish farmer buys an additional 200 fish for basin A and 100 fish for basin B. For every natural integer greater than or equal to 1, we denote respectively by $a _ { n }$ and $b _ { n }$ the numbers of fish in basins A and B after $n$ years. At the beginning of the first year, the number of fish in basin A is $a _ { 0 } = 200$ and that in basin B is $b _ { 0 } = 100$.
  1. Justify that $a _ { 1 } = 400$ and $b _ { 1 } = 300$ then calculate $a _ { 2 }$ and $b _ { 2 }$.
  2. We denote by $A$ and $B$ the matrices such that $A = \left( \begin{array} { l l } 0 & 2 \\ 1 & 0 \end{array} \right)$ and $B = \binom { 200 } { 100 }$ and for every natural integer $n$, we set $X _ { n } = \binom { a _ { n } } { b _ { n } }$. a. Explain why for every natural integer $n , X _ { n + 1 } = A X _ { n } + B$. b. Determine the real numbers $x$ and $y$ such that $\binom { x } { y } = A \binom { x } { y } + B$. c. For every natural integer $n$, we set $Y _ { n } = \binom { a _ { n } + 400 } { b _ { n } + 300 }$. Prove that for every natural integer $n , Y _ { n + 1 } = A Y _ { n }$.
  3. For every natural integer $n$, we set $Z _ { n } = Y _ { 2 n }$. a. Prove that for every natural integer $n , Z _ { n + 1 } = A ^ { 2 } Z _ { n }$. Deduce that for every natural integer $n , Z _ { n + 1 } = 2 Z _ { n }$. b. We admit that this recurrence relation allows us to conclude that for every natural integer $n$, $$Y _ { 2 n } = 2 ^ { n } Y _ { 0 }$$ Deduce that $Y _ { 2 n + 1 } = 2 ^ { n } Y _ { 1 }$ then prove that for every natural integer $n$, $$a _ { 2 n } = 600 \times 2 ^ { n } - 400 \text { and } a _ { 2 n + 1 } = 800 \times 2 ^ { n } - 400 .$$
  4. Basin A has a capacity limited to 10000 fish. a. An algorithm is given that, for a given value of $p$, computes the number of fish $a$ in basin A after $p$ years using the formulas $a_{2n} = 600 \times 2^n - 400$ and $a_{2n+1} = 800 \times 2^n - 400$. Use this algorithm to determine from which year the capacity of basin A is exceeded.