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	\Large {Brevet de Technicien\\ Supérieur\\
	\textbf{Groupement D1}}
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\textsc{\textbf{Mathématiques} \\ \rule{4cm}{0.5pt} \vspace{10mm} \\ Session 2024}\\
\vspace{10mm}
Durée: 2 heures\\
\vspace{20mm}
\textsc{
\begin{tabular}{|c|c|}
\hline 
Spécialités & Coeff \\ 
\hline
Analyses de biologie médicale  & 1 \\ \hline 
Bio analyses et contrôles   & 2 \\ \hline 
Biotechnologies   & 1 \\ \hline 
europlastics et composites & 1,5 \\ \hline 
Bioqualité & 2 \\ \hline 
\end{tabular} 
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Ce sujet comporte 6 pages numérotées de 1/6 à 6/6.

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\newpage
\begin{Large}
\textbf{Exercice n$^\circ$1 (10 points)}
\end{Large}

\textbf{Les questions 1) à 5) de cet exercice peuvent être traitées indépendamment les unes des autres. Les probabilités seront arrondies à $10^{-3}$.}

De par la diversité des coccinelles, on peut utiliser deux critères pour reconnaître certaines espèces de coccinelles~:

\begin{itemize}
    \item la taille de la coccinelle~;
    \item le type de pronotum (le pronotum est la face dorsale d'une partie du thorax de la coccinelle~; voir dessin ci-dessous).
\end{itemize}

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;
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;
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;
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;
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;
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;
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;
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;
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;
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;
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;
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;
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;
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;
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 -- (76.998pt, -213.3087pt) .. controls (83.1765pt, -213.3087pt) and (77.31375pt, -219.224pt) .. (77.31375pt, -219.224pt) -- cycle
;
\draw[->,ultra thick,>=latex] (-0.5,-2.2) to (2.5,-2.2);
\node at (-2.5,-2.2) {\footnotesize{pronotum}}; 
\end{tikzpicture}
\end{center}


Un type de pronotum répandu est le pronotum dit de type P.

\begin{enumerate}
\item On s'intéresse à toutes les coccinelles présentes dans une forêt d'Europe. Des études statistiques montrent que~:

\begin{itemize}
    \item 25~\% des coccinelles présentes dans cette forêt ont une taille inférieure ou égale à 5~mm~; parmi les coccinelles qui ont une taille inférieure ou égale à 5~mm, 40~\% ont un pronotum de type P~;
    \item parmi les coccinelles présentes dans cette forêt de taille strictement supérieure à 5~mm, 68~\% ont un pronotum de type P. 
\end{itemize}

Dans cette forêt, on prélève au hasard une coccinelle. On appelle~:

\begin{itemize}
    \item $T$ l'\evt{}~: \og{}~la coccinelle a une taille inférieure ou égale à 5~mm~\fg{}
~;
\item $P$ l'\evt{}~: \og{}~la coccinelle a un pronotum de type P~\fg{}.
\end{itemize}

On note $\overline{T}$ l'\evt{} contraire de l'\evt{} $T$ et $\overline{P}$ l'\evt{} contraire de l'\evt{} $P$.

\begin{enumerate}
\item Sur \textbf{l'annexe qui sera à rendre avec la copie}, compléter l'arbre modélisant la situation décrite ci-dessus.

Pour les questions b) à d), on prélève au hasard une coccinelle dans la forêt.

\item Déterminer la \pr{} de l'\evt{} A~: \og{}~la coccinelle prélevée a une taille inférieure ou égale à 5~mm et n'a pas un pronotum de type P~\fg{}.
\item Déterminer la \pr{} de l'\evt{} B~: \og{}~la coccinelle prélevée dans la forêt n'a pas un pronotum de type P~\fg{}.
    \item Déterminer la \pr{} que la coccinelle prélevée soit de taille inférieure ou égale à 5~mm sachant qu'elle n'a pas un pronotum de type P.
\end{enumerate}
\newpage
\item Les critères \og{}~avoir une taille moyenne inférieure ou égale à 5~mm~\fg{} et \og{}~ne pas avoir un pronotum de type P~\fg{} caractérisent une espèce particulière~: la coccinelle à deux points.

On appelle $X$ la \va{} qui, à tout \ech{} de 10 coccinelles prélevées au hasard dans cette forêt, associe le nombre de coccinelles à deux points dans l'\ech{}. La population de coccinelles de la forêt est suffisamment importante pour que ce \prel{}
 soit assimilable à un tirage avec remise. On admet que la \pr{} de prélever au hasard dans la forêt étudiée une coccinelle à deux points est égale à 0,15.

\begin{enumerate}
\item Donner la loi suivie par la \va{} $X$ et préciser ses paramètres. On ne justifiera pas la réponse.
    \item On prélève au hasard 10 coccinelles dans la forêt étudiée. Calculer la \pr{}
 d'avoir au moins une coccinelle à deux points dans ce prélèvement.
\end{enumerate}
\item On considère un élevage de coccinelles. On note $L$ la \va{} qui, à chaque coccinelle de l'élevage associe sa taille. On estime que cette taille exprimée en mm suit la loi uniforme sur l'\iv{} $\left[3 ; 6 \right]$. 
\begin{enumerate}
    \item On prélève au hasard une coccinelle dans cet élevage. Quelle est la \pr{} que sa taille soit comprise entre 4~mm et 5~mm~?
    \item Quelle est la taille moyenne des coccinelles dans cet élevage~?
\end{enumerate}
\item On s'intéresse ici à la ponte d'une espèce particulière de coccinelles~: les coccinelles à deux points. On modélise le nombre d'œufs par ponte à l'aide d'une \va{} $U$ qui suit la \N{} d'\moy{} égale à 35 et d'\sd{} égal à 6.

\begin{enumerate}
    \item Donner le nombre moyen d'œufs par ponte d'une coccinelle à deux points.
    \item Calculer la \pr{} $P\left( 31 \leqslant U \leqslant 39\right)$.
    \item Déterminer un nombre entier $h$ tel que~: $P\left( 35-h \leqslant U \leqslant 35+h \right) \approx 0,95$ à $10^{-2}$ près.

    Interpréter sous forme d'une phrase la \pr{} correspondante dans le contexte.
\end{enumerate}
    \item Une autre espèce spécifique est utilisée dans la lutte contre l'infestation des pucerons.

    Michel a créé un mur végétalisé. Avec effroi, il constate une infestation de pucerons. Sachant que les larves de cette espèce spécifique de coccinelles dévorent un grand nombre de pucerons, il en commande un lot de 80 larves pour traiter le mur. Le vendeur annonce que la durée de vie de ces larves, en jours, suit la \N{} d'\moy{} $m=20$ et d'\sd{} $\sigma = 3$.

    Michel constate que la durée de vie moyenne des larves de son lot n'est que de 18 jours. Peut-il remettre en question l'annonce du vendeur au seuil de confiance de 95~\%~?

    \emph{On s'aidera d'un test bilatéral relatif à une moyenne pour répondre à cette question. On rappelle la formule de l'\iv{} de confiance de la moyenne~: $\left[ m-1,96\frac{\sigma}{\sqrt{n}} ; m+1,96\frac{\sigma}{\sqrt{n}}\right]$.}
\end{enumerate}


\newpage
\begin{Large}
\textbf{Exercice n$^\circ$2 (10 points)}
\end{Large}

Cet exercice est composé de trois parties indépendantes entre elles.

Dans une usine agroalimentaire, il y a une seule ligne de production, qui fonctionne en continu. Elle peut fabriquer deux variétés M1 et M2 d'un même produit, qui diffèrent par leur teneur en matière grasse~:

\begin{itemize}
    \item la variété M1 correspond à une teneur idéale en matière grasse de 8,6~\%, soit 0,086~;
    \item la variété M2 correspond à une teneur idéale en matière grasse de 15,2~\%, soit 0,152.
\end{itemize}

La ligne était réglée jusqu'alors pour produire la variété M1. On injecte continûment de la matière grasse dans le produit en fabrication, sans arrêter la ligne de production, jusqu'à obtenir la variété M2. Durant cette phase de transition, la production est recyclée.

La variété M2 est considérée commercialisable par le service qualité dès que sa teneur en matière grasse dépasse 14,9~\% soit 0,149.

\textbf{Partie A}

La phase de transition pour passer de la variété M1 à la variété M2 est lancée. Un \prel{} est réalisé toutes les minutes pour mesurer la teneur en matière grasse du produit (le temps $t=0$ correspond au démarrage de la période de transition). Les résultats sont rassemblés dans le tableau ci-dessous~:


\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\begin{tabular}{l}
Temps $t_i$ \\
(en minutes)
\end{tabular}
  & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
\begin{tabular}{l}
Teneur en  \\
matière grasse $y_i$
\end{tabular}
 & 
0,086 &
0,108 &
0,121 &
0,131 &
0,14 &
0,144 &
0,147 &
0,149 &
0,149 &
0,151 & 
0,151
\\
\hline 
\end{tabular}    
\end{center}

\begin{enumerate}

\item Le nuage de points $(t_i,y_i)$ est représenté sur le graphique ci-dessous.

\begin{center}
\begin{tikzpicture}
\begin{axis}[
  width=0.4\textwidth,
  grid=both,
  grid style={line width=.1pt, draw=gray!50},major grid style={line width=.2pt,draw=gray},
  xmin=-1,
  xmax=11,
  ymin=0.06,
  ymax=0.16,
  minor tick num=1,
  xtick={-1,...,11},
  ytick={0.06,0.07,...,0.16},
  yticklabels={0.06,0.07,0.08,0.09,0.1,0.11,0.12,0.13,0.14,0.15,0.16}
]
\addplot[only marks,mark=+] coordinates {
(0,0.086)
(1,0.108)
(2,0.121)
(3,0.131)
(4,0.14)
(5,0.144)
(6,0.147)
(7,0.149)
(8,0.149)
(9,0.151)
(10,0.151)
};

\end{axis}
\end{tikzpicture}
\end{center}
Un ajustement affine paraît-il approprié~? Pourquoi~?


\item On pose $z_i = ln(0,152-y_i)$. Le tableau ci-dessous donne certaines valeurs de $z_i$ arrondies à $10^{-3}$~:

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$t_i$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
$z_i$ & -2,718 & $z_1$ & $z_2$ & -3,863 & -4,423 & -4,828 & -5,298 & -5,809 & -5,809 & -6,908 & -6,908 \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}
    \item Indiquer sur la copie les deux valeurs manquantes $z_1$ et $z_2$ du tableau, arrondies à $10^{-3}$.
    \item À l'aide de la calculatrice, déterminer par la \mmc{} une équation de la droite d'ajustement du nuage de points $(t_i,z_i)$ sous la forme $z = a t +b$, les réels $a$ et $b$ étant arrondis à $10^{-3}$.
    \item Pour cette question, on prendra $a = -0,44$ et $b = -2,7$. Déterminer une expression de la fonction $y$ ajustant le nuage de points $(t_i,y_i)$ dans le cadre de cette modélisation.
\end{enumerate}

\end{enumerate}

\newpage

\textbf{Partie B}

\begin{enumerate}

\item L'étude expérimentale menée conduit à considérer la fonction $C$ définie sur $\left[ 0 ; 10 \right]$ par~:

$$C(t) = 0,152 - 0,067\text{e}^{-0,44 t}.$$

On admet que la fonction $C$ modélise la teneur en matière grasse du produit (nombre entre 0 et 1) en fonction du temps $t$ en minutes, décompté depuis le démarrage de la période de transition.

\begin{enumerate}
    \item Calculer l'arrondi de $C(2)$ à $10^{-3}$. Interpréter par rapport à la situation étudiée.
    \item Calculer la dérivée de la fonction $C$ et en déduire le sens de variation de la fonction $C$ sur $\left[ 0 ; 10 \right]$.

    Ce résultat est-il cohérent avec la situation étudiée~?
\end{enumerate}

\item On considère l'algorithme ci-dessous rédigé en langage naturel.

\begin{center}
\begin{tabular}{|l|}
\hline
M $\leftarrow$ 0
\\
D $\leftarrow$ 0,085
\\
Tant que D $\leqslant$ 0,149
\\
$\qquad$ M $\leftarrow$ M+1
\\
$\qquad$ D $\leftarrow$ $C$(M)
\\
Fin du tant que
\\
Renvoyer M
\\
\hline
\end{tabular}
\end{center}

Quelle est la valeur numérique renvoyée par cet algorithme lorsqu'il a été exécuté~?

\item 

\begin{enumerate}
    \item On admet que l'équation $C(t) = 0,149$ admet une unique solution, notée $T$, sur l'\iv{} $\left[ 0 ; 10 \right]$. Déterminer une valeur approchée de $T$ à $10^{-2}$ près.
    \item Déduire de la question précédente le temps, à la seconde près, qu'il faut attendre après le démarrage de la phase de transition afin d'obtenir une variété M2 commercialisable.
\end{enumerate}

\end{enumerate}

\textbf{Partie C}

On considère la fonction $f$ définie sur l'\iv{} $\left[ 0 ; 10 \right]$ par $f(t) = 0,152 - 0,062\text{e}^{-0,5 t}$.

Le génie des procédés permet de considérer que la fonction $f$ constitue une autre modélisation acceptable de la teneur en matière grasse du produit en fonction du temps $t$ en minutes qui s'est écoulé à partir du démarrage de la période de transition.

On admet que le plus grand nombre réel $t$ appartenant à l'\iv{} $\left[ 0 ; 10 \right]$ tel que $f(t) \leqslant 0,149$ vaut approximativement 6 au dixième près.

\begin{enumerate}

\item On considère la fonction $F$ définie sur l'\iv{} $\left[ 0 ; 10 \right]$ par $F(t) = 0,152 t + 0,124 \text{e}^{-0,5 t}$.

Vérifier que la fonction $F$ est une primitive de la fonction $f$ sur l'\iv{} $\left[ 0 ; 10 \right]$ .

\item
\begin{enumerate}
    \item Déduire de la question précédente que l'arrondi à $10^{-4}$ de $\int_0^6 f(t) \text{d}t$ est 0,7942.
    \item Hachurer sur le graphique fourni \textbf{en annexe à rendre avec la copie} le domaine dont l'aire est égale à la valeur de cette intégrale.
\end{enumerate}

\item On note $m = \frac{1}{6} \int_0^6 f(t) \text{d}t$.

Calculer l'arrondi du réel $m$ à $10^{-3}$ et interpréter la valeur obtenue dans le contexte de l'exercice.
\end{enumerate}


\newpage

\textbf{\sc{Annexe à rendre avec la copie}}

\textbf{Exercice 1~-~Question 1) a)}

\begin{center}
\begin{tikzpicture}
\node (lh) at (0,2) {$T$};
\node (lb) at (0,-2) {$\overline{T}$};
\node (f) at (4,3) {$P$};
\node (a) at (4,1) {$\overline{P}$};
\node (b) at (4,-1) {$P$};
\node (g) at (4,-3) {$\overline{P}$};
\draw (lh) to node[sloped,above]{$\ldots$} (f) ;
\draw (lh) to node[sloped,below]{$\ldots$} (a) ;
\draw (lb) to node[sloped,above]{0,68} (b) ;
\draw (lb) to node[sloped,below]{$\ldots$} (g) ;
\draw (-4,0) to node[sloped,above]{0,25} (lh) ;
\draw (-4,0) to node[sloped,below]{$\ldots$} (lb) ;
\end{tikzpicture}

\end{center}


\textbf{Exercice 2 - Partie C~-~Question 2) b)}

\begin{center}
\begin{tikzpicture}%[domain=0:10]
\begin{axis}[
  width=0.9\textwidth,
  grid=both,
  grid style={line width=.1pt, draw=gray!50},major grid style={line width=.2pt,draw=gray},
  xmin=0,
  xmax=10,
  ymin=0,
  ymax=0.19,
  minor tick num=4,
  xtick={0,...,10},
  ytick={0,0.01,...,0.19},
  yticklabels={0,0.01,0.02,0.03,0.04,0.05,0.06,0.07,0.08,0.1,0.11,0.12,0.13,0.14,0.15,0.16,0.17,0.18,0.19}
]
\addplot[mark=none, draw=blue!50!black, line width=1pt] expression[domain=0:10] {0.152-0.062*exp(-0.5*x)};
\end{axis}
\end{tikzpicture}
\end{center}


 


\end{document}
