 \input{../preambules/preCorrection.tex}




\begin{document}

\chead{Correction Calculs A2}
\lfoot{}
\cfoot{Calcul}

\begin{center}
\fbox{ \Large Correction Calculs du 13/07 au 19/07}
\end{center}
\vspace{30pt}

\setcounter{theorem}{7}


\begin{exo}[Comparaison de fractions]
Comparer les fractions suivantes avec le signe >, < ou =.
\begin{qmulti}{3}
\item $ \dfrac{3}{5} = \dfrac{27}{45}  > \dfrac{5}{9} = \dfrac{25}{45}$
\item $ \dfrac{12}{11} > \dfrac{10}{12}$
\item $ \dfrac{125}{25} = 5 = \dfrac{105}{21}$
\end{qmulti}
\end{exo}

%%% Exercices d'approfondissement
\begin{exo}[Simplification de fractions]
Simplifier chacune des fractions suivantes.

\reponset{
A & = \dfrac{234}{288} \\
& = \dfrac{2 \times 117}{2 \times 144} \\
& = \boxed{\dfrac{117}{144}}}{
B &= \dfrac{(x^2 - x)(4-2x)}{x(2-x)} \\
& = \dfrac{x(x - 1)\times 2(2 - x)}{x(2-x)} \\
& = \boxed{2(x-1)}
}{C & =\dfrac{x^6 (1+x^3)}{x^3 + x^6} \\
& = \dfrac{x^3 \times x^3 (1+x^3)}{x^3(1+x^3)} \\
& = \boxed{x^3 }
}
\end{exo}

\begin{exo}[Produit de fraction]
Écrire sous la forme la plus simple possible.

\reponsed{
A & = \ds \frac{12}{42} \times \frac{7}{33} \times \frac{15}{21} \\
& = \dfrac{3 \times 2 \times 2}{2 \times 7 \times 3} \times \dfrac{7}{3 \times 11} \times \dfrac{5 \times 3}{3 \times 7} \\
& = \dfrac{2 \times 5}{11 \times 3 \times 7} \\
& =\boxed{ \dfrac{10}{231}}
}{B & = \displaystyle (x^2 - 2x) \frac{(x+ 3)}{2-x} \times \frac{x}{x^3 +3x^2} \\
& = \dfrac{x(x-2)(x+3)}{2-x} \times \dfrac{x}{x^2 (x+3)} \\
& = \boxed{-1}
}

\reponsed{
C & = \displaystyle - \frac{2x+4}{x} \times \frac{x}{-2x+4}\times \frac{2-x}{2+x} \\
& = - \dfrac{2(x+2)}{2(-x+2)} \times \dfrac{2-x}{2+x} \\
& = \boxed{-1}
}{
D & = \displaystyle \frac{18}{17} \times \frac{17}{16} \times \frac{16}{15} \times \frac{15}{14} \times \frac{14}{13} \times \frac{13}{12} \\
& = \dfrac{18}{12} \\
& = \boxed{\dfrac{3}{2}}
}

\end{exo}

\begin{exo}[Quotient de fractions]
Écrire sous forme de fractions irréductibles.
\def\columnseprulecolor{\color{white}}
\begin{multicols}{4}
\begin{enumerate}
\item $A = \displaystyle \frac{\frac{2}{5}}{2} = \boxed{\dfrac{1}{5}}$.
\item $B = \displaystyle \frac{\frac{2}{5}}{5} = \boxed{\dfrac{2}{25}}$.
\item $C = \displaystyle \frac{-1}{\frac{-1}{-2}} = \boxed{-2}$.
\item $D = \displaystyle \frac{\frac{x}{5}}{\frac{x}{2}} = \boxed{\dfrac{2}{5}} $.
\end{enumerate}
\end{multicols}
\end{exo}

\begin{exo}[Développement]
 Développer et réduire les expressions suivantes.

\reponset{
A & =  \displaystyle \frac{4}{5}\left( \frac{x}{2} - \frac{5}{4}\right)\\
& = \boxed{\dfrac{2x}{5} - 1}}{
B & = \displaystyle \left( \frac{x}{5} + \frac{4}{3}\right)\left(\frac{x}{5}	- \frac{2}{3}\right) \\		
& = \boxed{\dfrac{x^2}{25} + \dfrac{2x}{15} - \dfrac{8}{9}}
}{
C & = \displaystyle \left( \frac{x^2}{2} + \frac{x}{5} \right)^2 \\
& = \boxed{\dfrac{x^4}{4} + \dfrac{x^3}{5} + \dfrac{x^2}{25}}
} 
\end{exo}

\begin{exo}[Factorisation]
Factoriser les expressions suivantes.

\reponset{
A & = \displaystyle \frac{2x}{5} - \frac{6}{25} \\
& = \boxed{\dfrac{2}{5} \left( x - \dfrac{3}{5} \right)} 
}{
B & =  \displaystyle \frac{x^2}{25} - \frac{8x}{15} + \frac{16}{9} \\
& = \boxed{\left( \dfrac{x}{5} - \dfrac{4}{3} \right)^2}
}{
C & = \displaystyle \frac{x^2}{36} -  \frac{25}{49} \\
& = \boxed{\left( \dfrac{x}{6} + \dfrac{5}{7} \right) \left( \dfrac{x}{6} - \dfrac{5}{7} \right)}
}
\end{exo}

\begin{exo}[Plus difficile]
Écrire sous forme d'une fraction irréductible.
\begin{align*}
A & = \displaystyle \frac{n-1}{n} \times \frac{n-2}{n-1} \times \frac{n-3}{n-2} \times \ldots \times \frac{2}{3} \times \frac{1}{2} \\
& = \boxed{\dfrac{1}{n}}
\end{align*}

\begin{align*}
B & = \displaystyle \frac{n-2}{n} \times \frac{n-3}{n-1} \times \frac{n-4}{n-2} \times \ldots \times \frac{2}{4} \times \frac{1}{3} \\
& = \dfrac{1}{n} \times \dfrac{1}{n-1} \times 2 \times 1 \\
& = \boxed{\dfrac{2}{n(n-1)}}
\end{align*}

\begin{align*}
C & = \displaystyle  \frac{\dfrac{6(n+1)}{n(n-1)(2n-2)}}{\dfrac{2n+2}{n^2(n-1)^2}} \\
& = \dfrac{6(n+1)}{n(n-1)(2n-2)} \times \dfrac{n^2 (n-1)^2}{2n+2} \\
& = \dfrac{6(n+1)}{2(n-1)} \times \dfrac{n(n-1)}{2(n+1)} \\
& = \boxed{\dfrac{3}{2} n}
\end{align*}
\end{exo}


\end{document}
