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
2016 pondichery

6 maths questions

Q1A Normal Distribution Finding Unknown Mean from a Given Probability Condition View
Statistical studies have made it possible to model the weekly time, in hours, of internet connection for young people in France aged 16 to 24 years by a random variable $T$ following a normal distribution with mean $\mu = 13.9$ and standard deviation $\sigma$.
  1. We know that $p ( T \geqslant 22 ) = 0.023$.
    By exploiting this information: a. shade on the graph provided in the appendix, two distinct regions whose area is equal to 0.023; b. determine $P ( 5.8 \leqslant T \leqslant 22 )$. Justify the result. Show that an approximate value of $\sigma$ to one decimal place is 4.1.
  2. A young person in France is chosen at random.
    Determine the probability that they are connected to the internet for more than 18 hours per week. Round to the nearest hundredth.
Q1B Confidence intervals Compute confidence interval for a proportion (estimation) View
We denote by $p$ the unknown proportion of young people aged 16 to 24 years who practice illegal downloading on the internet at least once a week.
A young person participating in protocol $( \mathscr { P } )$ is randomly selected. The protocol $( \mathscr { P } )$ is as follows: each young person rolls a fair 6-sided die; if the result is even, the young person answers sincerely; if the result is ``1'', the young person must answer ``Yes''; if the result is ``3 or 5'', the young person must answer ``No''.
We denote: $R$ the event ``the result of the roll is even'', $O$ the event ``the young person answered Yes''.
1. Probability calculations
Reproduce and complete the weighted tree diagram.
Deduce that the probability $q$ of the event ``the young person answered Yes'' is: $$q = \frac { 1 } { 2 } p + \frac { 1 } { 6 }$$
2. Confidence interval
a. At the request of Hadopi, a polling institute conducts a survey according to protocol $( \mathscr { P } )$. On a sample of size 1500, it counts 625 ``Yes'' responses. Give a confidence interval, at the 95\% confidence level, for the proportion $q$ of young people who answer ``Yes'' to such a survey, among the population of young French people aged 16 to 24 years. b. What can be concluded about the proportion $p$ of young people who practice illegal downloading on the internet at least once a week?
Q2 3 marks Complex numbers 2 Roots of Unity and Cyclotomic Properties View
The objective of this exercise is to find a method to construct a regular pentagon with straightedge and compass. In the complex plane equipped with a direct orthonormal coordinate system ( $\mathrm { O } , \vec { u } , \vec { v }$ ), we consider the regular pentagon $A _ { 0 } A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$, with center $O$ such that $\overrightarrow { O A _ { 0 } } = \vec { u }$. We recall that in the regular pentagon $A _ { 0 } A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$:
  • the five sides have the same length;
  • the points $A _ { 0 } , A _ { 1 } , A _ { 2 } , A _ { 3 }$ and $A _ { 4 }$ belong to the unit circle;
  • for any integer $k$ belonging to $\{ 0 ; 1 ; 2 ; 3 \}$ we have $\left( \overrightarrow { O A _ { k } } ; \overrightarrow { O A _ { k + 1 } } \right) = \frac { 2 \pi } { 5 }$.

  1. We consider the points $B$ with affix $-1$ and $J$ with affix $\frac { \mathrm { i } } { 2 }$.
    The circle $\mathscr { C }$ with center $J$ and radius $\frac { 1 } { 2 }$ intersects the segment $[ B J ]$ at a point $K$. Calculate $B J$, then deduce $B K$.
  2. a. Give in exponential form the affix of point $A _ { 2 }$. Justify briefly. b. Prove that $B A _ { 2 } { } ^ { 2 } = 2 + 2 \cos \left( \frac { 4 \pi } { 5 } \right)$. c. A computer algebra system displays the results below, which may be used without justification:
    \multicolumn{2}{|l|}{Formal calculation}
    1\begin{tabular}{ l } $\cos \left( 4 ^ { * } \mathrm { pi } / 5 \right)$
    $\rightarrow \frac { 1 } { 4 } ( - \sqrt { 5 } - 1 )$
    \hline 2 & $\operatorname { sqrt } ( ( 3 - \operatorname { sqrt } ( 5 ) ) / 2 )$ \hline & $\rightarrow \frac { 1 } { 2 } ( \sqrt { 5 } - 1 )$ \hline \end{tabular}
    ``sqrt'' means ``square root'' Deduce, using these results, that $B A _ { 2 } = B K$.
  3. In the coordinate system ( $\mathrm { O} , \vec { u } , \vec { v }$ ) provided in the appendix, construct a regular pentagon with straightedge and compass. Do not use a protractor or the ruler's graduations and leave the construction lines visible.
Q3 (non-specialization) 5 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
ABCDEFGH designates a cube with side length 1. Point I is the midpoint of segment [BF]. Point J is the midpoint of segment [BC]. Point K is the midpoint of segment [CD].
Part A
In this part, no justification is required. We admit that the lines (IJ) and (CG) intersect at a point L. Construct, on the figure provided in the appendix and leaving the construction lines visible:
  • the point L;
  • the intersection $\mathscr { D }$ of the planes (IJK) and (CDH);
  • the cross-section of the cube by the plane (IJK).

Part B
Space is referred to the coordinate system ( $\mathrm { A } ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } }$ ).
  1. Give the coordinates of $\mathrm { A } , \mathrm { G } , \mathrm { I } , \mathrm { J }$ and K in this coordinate system.
  2. a. Show that the vector $\overrightarrow { \mathrm { AG } }$ is normal to the plane (IJK). b. Deduce a Cartesian equation of the plane (IJK).
  3. We denote by $M$ a point of the segment [AG] and $t$ the real number in the interval $[ 0 ; 1 ]$ such that $\overrightarrow { \mathrm { AM } } = t \overrightarrow { \mathrm { AG } }$. a. Prove that $M \mathrm { I } ^ { 2 } = 3 t ^ { 2 } - 3 t + \frac { 5 } { 4 }$. b. Prove that the distance $M I$ is minimal for the point $\mathrm { N } \left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; \frac { 1 } { 2 } \right)$.
  4. Prove that for this point $\mathrm { N } \left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; \frac { 1 } { 2 } \right)$: a. N belongs to the plane (IJK). b. The line (IN) is perpendicular to the lines (AG) and (BF).
Q3 (specialization) 5 marks Matrices Determinant and Rank Computation View
Part A
We consider matrices $M$ of the form $M = \left( \begin{array} { l l } a & b \\ 5 & 3 \end{array} \right)$ where $a$ and $b$ are integers. The number $3 a - 5 b$ is called the determinant of $M$. We denote it $\operatorname { det } ( M )$. Thus $\operatorname { det } ( M ) = 3 a - 5 b$.
  1. In this question we assume that $\operatorname { det } ( M ) \neq 0$ and we set $N = \frac { 1 } { \operatorname { det } ( M ) } \left( \begin{array} { c c } 3 & - b \\ - 5 & a \end{array} \right)$. Justify that $N$ is the inverse of $M$.
  2. We consider the equation $( E ) : \quad \operatorname { det } ( M ) = 3$.
    We wish to determine all pairs of integers ( $a ; b$ ) that are solutions of equation ( $E$ ). a. Verify that the pair (6; 3) is a solution of $( E )$. b. Show that the pair of integers ( $a$; $b$ ) is a solution of ( $E$ ) if and only if $3 ( a - 6 ) = 5 ( b - 3 )$. Deduce the set of solutions of equation ( $E$ ).

Part B
  1. We set $Q = \left( \begin{array} { l l } 6 & 3 \\ 5 & 3 \end{array} \right)$.
    Using Part A, determine the inverse matrix of $Q$.
  2. Encoding with matrix $Q$
    To encode a two-letter word using the matrix $Q = \left( \begin{array} { l l } 6 & 3 \\ 5 & 3 \end{array} \right)$ we use the following procedure: Step 1: We associate with the word the matrix $X = \binom { x _ { 1 } } { x _ { 2 } }$ where $x _ { 1 }$ is the integer corresponding to the first letter of the word and $x _ { 2 }$ the integer corresponding to the second letter of the word according to the correspondence table below:
    ABCDEFGHIJKLM
    0123456789101112
    NOPQRSTUVWXYZ
    13141516171819202122232425

    Step 2: The matrix $X$ is transformed into the matrix $Y = \binom { y _ { 1 } } { y _ { 2 } }$ such that $Y = Q X$. Step 3: The matrix $Y$ is transformed into the matrix $R = \binom { r _ { 1 } } { r _ { 2 } }$ such that $r _ { 1 }$ is the remainder of the Euclidean division of $y _ { 1 }$ by 26 and $r _ { 2 }$ is the remainder of the Euclidean division of $y _ { 2 }$ by 26.
Q5 Areas by integration View
We denote $\mathscr { A } ( \theta )$ the region bounded by the lines with equations $t = 10 , t = \theta$, $y = 85$ and the curve $\mathscr { C } _ { f }$ representing $f$. We consider that sterilization is complete after a time $\theta$, if the area, expressed in square units of the region $\mathscr { A } ( \theta )$ is greater than 80.
  1. [a.] Justify, using the graph given in the appendix, that we have $\mathscr { A } ( 25 ) > 80$.
  2. [b.] Justify that, for $\theta \geqslant 10$, we have $\mathscr { A } ( \theta ) = 15 ( \theta - 10 ) - 75 \int _ { 10 } ^ { \theta } \mathrm { e } ^ { - \frac { \ln 5 } { 10 } t } \mathrm {~d} t$.
  3. [c.] Is sterilization complete after 20 minutes?