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
2022 bac-spe-maths__caledonie_j1

4 maths questions

Q1 7 marks Differentiating Transcendental Functions Full function study with transcendental functions View
Consider the function $f$ defined on the interval $] 0 ; + \infty [$ by
$$f ( x ) = x ^ { 2 } - 6 x + 4 \ln ( x )$$
It is admitted that the function $f$ is twice differentiable on the interval $] 0 ; + \infty [$. We denote $f ^ { \prime }$ its derivative and $f ^ { \prime \prime }$ its second derivative. We denote $\mathscr { C } _ { f }$ the representative curve of the function $f$ in an orthogonal coordinate system.
  1. a. Determine $\lim _ { x \rightarrow 0 } f ( x )$.
    Interpret this result graphically. b. Determine $\lim _ { x \rightarrow + \infty } f ( x )$.
  2. a. Determine $f ^ { \prime } ( x )$ for all real $x$ belonging to $] 0 ; + \infty [$. b. Study the sign of $f ^ { \prime } ( x )$ on the interval $] 0 ; + \infty [$.
    Deduce the variation table of $f$.
  3. Show that the equation $f ( x ) = 0$ has a unique solution in the interval $[4;5]$.
  4. It is admitted that, for all $x$ in $] 0 ; + \infty [$, we have:
    $$f ^ { \prime \prime } ( x ) = \frac { 2 x ^ { 2 } - 4 } { x ^ { 2 } }$$
    a. Study the convexity of the function $f$ on $] 0 ; + \infty [$.
    The exact coordinates of any inflection points of $\mathscr { C } _ { f }$ will be specified. b. We denote A the point with coordinates $( \sqrt { 2 } ; f ( \sqrt { 2 } ) )$.
    Let $t$ be a strictly positive real number such that $t \neq \sqrt { 2 }$. Let $M$ be the point with coordinates $( t ; f ( t ) )$. Using question 4. a, indicate, according to the value of $t$, the relative positions of the segment [AM] and the curve $\mathscr { C } _ { f }$.
Q2 7 marks Differentiating Transcendental Functions Full function study with transcendental functions View
Consider the function $f$ defined on $\mathbb { R }$ by
$$f ( x ) = x ^ { 3 } \mathrm { e } ^ { x }$$
It is admitted that the function $f$ is differentiable on $\mathbb { R }$ and we denote $f ^ { \prime }$ its derivative function.
  1. The sequence $(u _ { n })$ is defined by $u _ { 0 } = - 1$ and, for all natural integer $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$. a. Calculate $u _ { 1 }$ then $u _ { 2 }$.
    Exact values will be given, then approximate values to $10 ^ { - 3 }$. b. Consider the function fonc, written in Python language below.
    Recall that in Python language, ``i in range (n)'' means that i varies from 0 to n -1.
    \begin{verbatim} def fonc (n): u =- 1 for i in range(n) : u=u**3*exp(u) return u \end{verbatim}
    Determine, without justification, the value returned by fonc (2) rounded to $10 ^ { - 3 }$.
  2. a. Prove that, for all real $x$, we have $f ^ { \prime } ( x ) = x ^ { 2 } \mathrm { e } ^ { x } ( x + 3 )$. b. Justify that the variation table of $f$ on $\mathbb { R }$ is the one represented below:
    $x$$- \infty$- 3$+ \infty$
    0$+ \infty$
    $f$$+ 27 \mathrm { e } ^ { - 3 }$

    c. Prove, by induction, that for all natural integer $n$, we have:
    $$- 1 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 0$$
    d. Deduce that the sequence $(u _ { n })$ is convergent. e. We denote $\ell$ the limit of the sequence $(u _ { n })$.
    Recall that $\ell$ is a solution of the equation $f ( x ) = x$. Determine $\ell$. (For this, it will be admitted that the equation $x ^ { 2 } \mathrm { e } ^ { x } - 1 = 0$ has only one solution in $\mathbb { R }$ and that this solution is strictly greater than $\frac { 1 } { 2 }$).
Q3 7 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
A house consists of a rectangular parallelepiped ABCDEFGH topped with a prism EFIHGJ whose base is the triangle EIF isosceles at I.
We have $\mathrm { AB } = 3 , \quad \mathrm { AD } = 2 , \quad \mathrm { AE } = 1$. We define the vectors $\vec { \imath } = \frac { 1 } { 3 } \overrightarrow { \mathrm { AB } } , \vec { \jmath } = \frac { 1 } { 2 } \overrightarrow { \mathrm { AD } } , \vec { k } = \overrightarrow { \mathrm { AE } }$. We thus equip space with the orthonormal coordinate system $( \mathrm { A } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$.
  1. Give the coordinates of point G.
  2. The vector $\vec { n }$ with coordinates $( 2 ; 0 ; - 3 )$ is a normal vector to the plane (EHI).
    Determine a Cartesian equation of the plane (EHI).
  3. Determine the coordinates of point I.
  4. Determine a measure to the nearest degree of the angle $\widehat { \mathrm { EIF } }$.
  5. In order to connect the house to the electrical network, it is desired to dig a straight trench from an electrical relay located below the house.
    The relay is represented by the point R with coordinates $( 6 ; - 3 ; - 1 )$. The trench is assimilated to a segment of a line $\Delta$ passing through R and directed by the vector $\vec { u }$ with coordinates $( - 3 ; 4 ; 1 )$. It is desired to verify that the trench will reach the house at the level of the edge [BC]. a. Give a parametric representation of the line $\Delta$. b. It is admitted that an equation of the plane (BFG) is $x = 3$.
    Let K be the point of intersection of the line $\Delta$ with the plane (BFG). Determine the coordinates of point K. c. Does the point K indeed belong to the edge $[ \mathrm { BC } ]$?
Q4 7 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
Consider a binary communication system transmitting 0s and 1s. Each 0 or 1 is called a bit. Due to interference, there may be transmission errors: a 0 can be received as a 1 and, likewise, a 1 can be received as a 0. For a bit chosen at random in the message, we note the events:
  • $E _ { 0 }$: ``the bit sent is a 0'';
  • $E _ { 1 }$: ``the bit sent is a 1'';
  • $R _ { 0 }$: ``the bit received is a 0''
  • $R _ { 1 }$: ``the bit received is a 1''.

We know that: $p \left( E _ { 0 } \right) = 0{,}4 ; \quad p _ { E _ { 0 } } \left( R _ { 1 } \right) = 0{,}01 ; \quad p _ { E _ { 1 } } \left( R _ { 0 } \right) = 0{,}02$. Recall that the conditional probability of $A$ given $B$ is denoted $p _ { B } ( A )$.
  1. The probability that the bit sent is a 0 and the bit received is a 0 is equal to: a. 0,99 b. 0,396 c. 0,01 d. 0,4
  2. The probability $p \left( R _ { 0 } \right)$ is equal to: a. 0,99 b. 0,02 c. 0,408 d. 0,931
  3. A value, approximated to the nearest thousandth, of the probability $p _ { R _ { 1 } } \left( E _ { 0 } \right)$ is equal to: a. 0,004 b. 0,001 c. 0,007 d. 0,010
  4. The probability of the event ``there is a transmission error'' is equal to: a. 0,03 b. 0,016 c. 0,16 d. 0,015

A message of length eight bits is called a byte. It is admitted that the probability that a byte is transmitted without error is equal to 0,88.
    \setcounter{enumi}{4}
  1. 10 bytes are transmitted successively in an independent manner.
    The probability, to $10 ^ { - 3 }$ near, that exactly 7 bytes are transmitted without error is equal to: a. 0,915 b. 0,109 c. 0,976 d. 0,085
  2. 10 bytes are transmitted successively in an independent manner.
    The probability that at least 1 byte is transmitted without error is equal to: a. $1 - 0{,}12 ^ { 10 }$ b. $0{,}12 ^ { 10 }$ c. $0{,}88 ^ { 10 }$ d. $1 - 0{,}88 ^ { 10 }$
  3. Let $N$ be a natural integer. $N$ bytes are transmitted successively in an independent manner. Let $N _ { 0 }$ be the largest value of $N$ for which the probability that all $N$ bytes are transmitted without error is greater than or equal to 0,1. We can affirm that: a. $N _ { 0 } = 17$ b. $N _ { 0 } = 18$ c. $N _ { 0 } = 19$ d. $N _ { 0 } = 20$