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
2017 asie

8 maths questions

Q1A Exponential Functions Applied/Contextual Exponential Modeling View
A treatment protocol for a disease in children involves long-term infusion of an appropriate medication. The concentration of the medication in the blood over time is modeled by the function $C$ defined on the interval $[0; +\infty[$ by:
$$C ( t ) = \frac { d } { a } \left( 1 - \mathrm { e } ^ { - \frac { a } { 80 } t } \right)$$
The clearance $a$ of a certain patient is 7, and we choose an infusion rate $d$ equal to 84. In this part, the function $C$ is therefore defined on $[0; +\infty[$ by:
$$C ( t ) = 12 \left( 1 - \mathrm { e } ^ { - \frac { 7 } { 80 } t } \right)$$
  1. Study the monotonicity of the function $C$ on $[0; +\infty[$.
  2. For the treatment to be effective, the plateau must equal 15. Is the treatment of this patient effective?
Q1B Stationary points and optimisation Construct or complete a full variation table View
A treatment protocol for a disease in children involves long-term infusion of an appropriate medication. The concentration of the medication in the blood over time is modeled by the function $C$ defined on the interval $[0; +\infty[$ by:
$$C ( t ) = \frac { d } { a } \left( 1 - \mathrm { e } ^ { - \frac { a } { 80 } t } \right)$$
Part B: study of functions
  1. Let $f$ be the function defined on $]0; +\infty[$ by:
    $$f ( x ) = \frac { 105 } { x } \left( 1 - \mathrm { e } ^ { - \frac { 3 } { 40 } x } \right)$$
    Prove that, for every real $x$ in $]0; +\infty[$, $f ^ { \prime } ( x ) = \frac { 105 g ( x ) } { x ^ { 2 } }$, where $g$ is the function defined on $[0; +\infty[$ by:
    $$g ( x ) = \frac { 3 x } { 40 } \mathrm { e } ^ { - \frac { 3 } { 40 } x } + \mathrm { e } ^ { - \frac { 3 } { 40 } x } - 1$$
  2. The variation table of the function $g$ is given:
    $x$0$+\infty$
    0
    $g ( x )$- 1

    Deduce the monotonicity of the function $f$. The limits of the function $f$ are not required.
  3. Show that the equation $f ( x ) = 5.9$ has a unique solution on the interval $[1; 80]$. Deduce that this equation has a unique solution on the interval $]0; +\infty[$. Give an approximate value of this solution to the nearest tenth.
Q1C Exponential Equations & Modelling Threshold or Tipping-Point Calculation in Applied Exponential Models View
A treatment protocol for a disease in children involves long-term infusion of an appropriate medication. The concentration of the medication in the blood over time is modeled by the function $C$ defined on the interval $[0; +\infty[$ by:
$$C ( t ) = \frac { d } { a } \left( 1 - \mathrm { e } ^ { - \frac { a } { 80 } t } \right)$$
where $C$ denotes the concentration of the medication in the blood (in micromoles per liter), $t$ the time elapsed since the start of the infusion (in hours), $d$ the infusion rate (in micromoles per hour), $a$ a strictly positive real parameter called clearance (in liters per hour).
Part C: determination of appropriate treatment
The purpose of this part is to determine, for a given patient, the value of the infusion rate that allows the treatment to be effective, that is, the plateau to equal 15. The infusion rate $d$ is provisionally set to 105.
  1. We seek to determine the clearance $a$ of a patient. The infusion rate is provisionally set to 105. a. Express as a function of $a$ the concentration of the medication 6 hours after the start of the infusion. b. After 6 hours, analyses allow us to know the concentration of the medication in the blood; it is equal to 5.9 micromoles per liter. Determine an approximate value, to the nearest tenth of a liter per hour, of the clearance of this patient.
  2. Determine the value of the infusion rate $d$ guaranteeing the effectiveness of the treatment.
Q2 3 marks Geometric Sequences and Series Prove a Transformed Sequence is Geometric View
Consider the sequence $(u_n)$ defined by:
$$\left\{ \begin{aligned} u _ { 0 } & = 1 \text{ and, for every natural number } n, \\ u _ { n + 1 } & = \left( \frac { n + 1 } { 2 n + 4 } \right) u _ { n } . \end{aligned} \right.$$
Define the sequence $(v_n)$ by: for every natural number $n$, $v_n = (n + 1) u_n$.
  1. The spreadsheet below presents the values of the first terms of the sequences $(u_n)$ and $(v_n)$, rounded to the hundred-thousandth. What formula, then extended downward, can be written in cell B3 of the spreadsheet to obtain the successive terms of $(u_n)$?
    ABC
    1$n$$u_n$$v_n$
    201.000001.00000
    310.250000.50000
    420.083330.25000
    530.031250.12500
    640.012500.06250
    750.005210.03125
    860.002230.01563
    970.000980.00781
    1080.000430.00391
    1190.000200.00195

  2. a. Conjecture the expression of $v_n$ as a function of $n$. b. Prove this conjecture.
  3. Determine the limit of the sequence $(u_n)$.
Q3 4 marks Vectors: Lines & Planes True/False or Verify a Given Statement View
For each of the four statements below, indicate whether it is true or false, by justifying the answer. One point is awarded for each correct answer with proper justification. An answer without justification is not taken into account. An absence of answer is not penalized.
  1. We have two dice, identical in appearance, one of which is biased so that 6 appears with probability $\frac{1}{2}$. We take one of the two dice at random, roll it, and obtain 6. Statement 1: the probability that the die rolled is the biased die is equal to $\frac{2}{3}$.
  2. In the complex plane, consider the points M and N with affixes respectively $z_{\mathrm{M}} = 2 \mathrm{e}^{-\mathrm{i} \frac{\pi}{3}}$ and $z_{\mathrm{N}} = \frac{3 - \mathrm{i}}{2 + \mathrm{i}}$. Statement 2: the line $(MN)$ is parallel to the imaginary axis.
  3. In an orthonormal frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$ of space, consider the line $d$ with parametric representation: $\left\{ \begin{array}{l} x = 1 + t \\ y = 2 \\ z = 3 + 2t \end{array} \quad t \in \mathbf{R} \right.$. Consider the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ with $\mathrm{A}(-2; 2; 3)$, $\mathrm{B}(0; 1; 2)$ and $\mathrm{C}(4; 2; 0)$. We admit that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear. Statement 3: the line $d$ is orthogonal to the plane (ABC).
  4. In an orthonormal frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$ of space, consider the line $d$ with parametric representation: $\left\{ \begin{array}{l} x = 1 + t \\ y = 2 \\ z = 3 + 2t \end{array} \quad t \in \mathbf{R} \right.$. Consider the line $\Delta$ passing through the point $\mathrm{D}(1; 4; 1)$ and with direction vector $\vec{v}(2; 1; 3)$. Statement 4: the line $d$ and the line $\Delta$ are not coplanar.
Q4 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The purpose of the problem is the study of the integrals $I$ and $J$ defined by:
$$I = \int_{0}^{1} \frac{1}{1 + x} \mathrm{~d}x \quad \text{and} \quad J = \int_{0}^{1} \frac{1}{1 + x^{2}} \mathrm{~d}x$$
Part A: exact value of the integral $I$
  1. Give a geometric interpretation of the integral $I$.
  2. Calculate the exact value of $I$.

Part B: estimation of the value of $J$
Let $g$ be the function defined on the interval $[0; 1]$ by $g(x) = \frac{1}{1 + x^{2}}$. We denote $\mathscr{C}_{g}$ its representative curve in an orthonormal frame of the plane. We therefore have: $J = \int_{0}^{1} g(x) \mathrm{d}x$. The purpose of this part is to evaluate the integral $J$ using the probabilistic method described below. We choose at random a point $\mathrm{M}(x; y)$ by drawing independently its coordinates $x$ and $y$ at random according to the uniform distribution on $[0; 1]$. We admit that the probability $p$ that a point drawn in this manner is located below the curve $\mathscr{C}_{g}$ is equal to the integral $J$. In practice, we initialize a counter $c$ to 0, we fix a natural number $n$ and we repeat $n$ times the following process:
  • we choose at random and independently two numbers $x$ and $y$, according to the uniform distribution on $[0; 1]$;
  • if $\mathrm{M}(x; y)$ is below the curve $\mathscr{C}_{g}$ we increment the counter $c$ by 1.
We admit that $f = \frac{c}{n}$ is an approximate value of $J$. This is the principle of the so-called Monte-Carlo method.
Q5A Matrices Matrix Power Computation and Application View
Several transmission lines are assembled end to end, and we assume they introduce errors independently of one another. Each line transmits a bit correctly with probability $p$, and incorrectly with probability $1 - p$. We study the transmission of a single bit, which has value 1 at the beginning of transmission. After passing through $n$ transmission lines, we denote:
  • $p _ { n }$ the probability that the received bit has value 1;
  • $q _ { n }$ the probability that the received bit has value 0.
We therefore have $p _ { 0 } = 1$ and $q _ { 0 } = 0$. We define the following matrices:
$$A = \left( \begin{array} { c c } p & 1 - p \\ 1 - p & p \end{array} \right) \quad X _ { n } = \binom { p _ { n } } { q _ { n } } \quad P = \left( \begin{array} { c c } 1 & 1 \\ 1 & - 1 \end{array} \right) .$$
We admit that, for every integer $n$, we have: $X _ { n + 1 } = A X _ { n }$ and therefore, $X _ { n } = A ^ { n } X _ { 0 }$.
  1. a. Show that $P$ is invertible and determine $P ^ { - 1 }$. b. We set: $D = \left( \begin{array} { c c } 1 & 0 \\ 0 & 2 p - 1 \end{array} \right)$. Verify that: $A = P D P ^ { - 1 }$. c. Show that, for every integer $n \geqslant 1$, $$A ^ { n } = P D ^ { n } P ^ { - 1 } .$$ d. Using the screenshot of a computer algebra system given below, determine the expression of $q _ { n }$ as a function of $n$.
    1$X 0 : = [ [ 1 ] , [ 0 ] ]$
    $\left[ \begin{array} { l } 1 \\ 0 \end{array} \right]$M
    2$\mathrm { P } : = [ [ 1,1 ] , [ 1 , - 1 ] ]$
    $\left[ \begin{array} { c c } 1 & 1 \\ 1 & - 1 \end{array} \right]$M
    3$\mathrm { D } : = [ [ 1,0 ] , [ 0,2 * p - 1 ] ]$
    $\left[ \begin{array} { c c } 1 & 0 \\ 0 & 2 * p - 1 \end{array} \right]$M
    4$P * \left( D ^ { \wedge } n \right) * P ^ { \wedge } ( - 1 ) * X 0$
    $\left[ \frac { ( 2 * p - 1 ) ^ { n } + 1 } { 2 } \right]$
    $\frac { - ( 2 * p - 1 ) ^ { n } + 1 } { 2 }$M

  2. We assume in this question that $p$ equals 0.98. We recall that the bit before transmission has value 1. We wish the probability that the received bit has value 0 be less than or equal to 0.25. What is the maximum number of such transmission lines that can be aligned?
Q5B Number Theory Congruence Reasoning and Parity Arguments View
We consider a ``word'' formed of 4 bits which we denote $b _ { 1 } , b _ { 2 } , b _ { 3 }$ and $b _ { 4 }$. We add to this list a check key $c _ { 1 } c _ { 2 } c _ { 3 }$ formed of three bits:
  • $c _ { 1 }$ is the remainder of the Euclidean division of $b _ { 2 } + b _ { 3 } + b _ { 4 }$ by 2;
  • $c _ { 2 }$ is the remainder of the Euclidean division of $b _ { 1 } + b _ { 3 } + b _ { 4 }$ by 2;
  • $c _ { 3 }$ is the remainder of the Euclidean division of $b _ { 1 } + b _ { 2 } + b _ { 4 }$ by 2.
We then call ``message'' the sequence of 7 bits formed of the 4 bits of the word and the 3 control bits.
  1. Preliminaries a. Justify that $c _ { 1 } , c _ { 2 }$ and $c _ { 3 }$ can only take the values 0 or 1. b. Calculate the check key associated with the word 1001.
  2. Let $b _ { 1 } b _ { 2 } b _ { 3 } b _ { 4 }$ be a word of 4 bits and $c _ { 1 } c _ { 2 } c _ { 3 }$ the associated key. Prove that if we change the value of $b _ { 1 }$ and recalculate the key, then:
    • the value of $c _ { 1 }$ is unchanged;
    • the value of $c _ { 2 }$ is modified;
    • the value of $c _ { 3 }$ is modified.

  3. We assume that, during the transmission of the message, at most one of the 7 bits was transmitted incorrectly. From the first four bits of the received message, we recalculate the 3 control bits, and compare them with the received control bits. Without justification, copy and complete the table below. The letter $F$ means that the received control bit does not match the calculated control bit, and $J$ means that these two bits are equal.
    \backslashbox{Calculated control bit}{Erroneous bit}$b _ { 1 }$$b _ { 2 }$$b _ { 3 }$$b _ { 4 }$$c _ { 1 }$$c _ { 2 }$$c _ { 3 }$None
    $c _ { 1 }$$J$
    $c _ { 2 }$$F$
    $c _ { 3 }$$F$

  4. Justify briefly, using the table, that if a single received bit is erroneous, we can in all cases determine which one it is, and correct the error.
  5. Here are two messages of 7 bits: $$A = 0100010 \quad \text { and } \quad B = 1101001 .$$ We admit that each of them contains at most one transmission error. Say whether they contain an error, and correct it if necessary.