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\begin{document}

\devpers{Selectividad Blanca}{}{Matemáticas II,\ abril 2013}{}


\section*{Modelo A}

\begin{exercice}
Estudie y represente la gráfica de la función 
\[
 f(x)=\dfrac{3x^3-9x^2}{x^2-2x-3}
\]

\end{exercice}

\begin{exercice}%{\bf Matematicas II - Juno 2010}\par
Discuta y resuelva, seg\'un los valores del par\'ametro $a$ el siguiente sistema de ecuaciones.
\[
\syst{2x+y-z=0}{(a+1)x+y-az=a}{x+(a+1)y=2a}
\]

\end{exercice}
\begin{exercice}
 Calcule $\ds\int \dfrac{x^2-2x-1}{x^3-x} \,dx$.
\end{exercice}
\begin{exercice}
Se considera la recta $r  \equiv \sysd{x+y-z-2=0}{ x-y-z+1=0}$
y el plano $\pi \equiv    x+y+mz=-5$.
Determine $m$ para que $r$ sea paralela a $\pi$.
Calcule para dicho valor de $m$, la distancia entre $r$ y $\pi$.
\end{exercice}

\hrule
\vspace{1cm}

\setcounter{exercice}{0}
\section*{Modelo B}

\begin{exercice}
Estudie la gr\'afica de la funci\'on $f(x)=\ln(x^2+x+1)$. Observaci\'on : $\ln x$ es el logaritmo nepariano de $x$.

\end{exercice}

\begin{exercice}
 Calcule $\ds\int x^2\sin(2x) \,dx$.
\end{exercice}
\begin{exercice}
Estudie y determine en caso de existir, el plano que contiene a los puntos $A = (1,3,2)$ y $B =
(0,3,0)$ y que es perpendicular a la recta $r \equiv \sysd{x+2y+2z-5=0}{2x-y-3z+3=0}$.
\end{exercice}



\begin{exercice}
Dada la matriz $A=
\left( \begin{array}{ccc} 0 & 3 & 4\\ 
 1 &-4 & -5 \\ -1 & 3& 4 \end{array} \right).$
 
 Demuestra  que 
$A^{3}+I=O$ siendo $I$ la matriz identidad $3\times 3$ y $O$ la matriz nula. Deducir la matriz $A^{10}$.
\end{exercice}


\end{document}