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\begin{document}

\devpers{Selectividad Blanca}{}{Matemáticas II,\ abril 2014}{}


\section*{Modelo A}

\begin{exercice}
Resuelva, dependiendo del valor $\lambda$ el siguiente sistema

\[
  S_{\lambda}\equiv \left\{ 
\begin{array}{lcc}
             -6y-9z-9x\lambda = 0\\
             2x - 2z -2y\lambda = 0\\
             2 y -2x+ 2z = -2\lambda
             \end{array}
   \right.
\]
\end{exercice} 


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


\begin{exercice}
 Calcule $\ds\int\dfrac{2x^2+x+8}{x^3+4x} \,dx$.
\end{exercice}

\begin{exercice}
Dado el plano $\pi$ de ecuaci\'on general $-x + 2y - 3z + 2 = 0$, determinar la
ecuaci\'on general de cada uno de los planos que distan 1 unidades del plano
$\pi$.
\end{exercice}

\hrule
\vspace{1cm}

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


\begin{exercice}
Estudie el rango de la matriz $A$ para todos los  n\'umeros reales $\alpha$ y los  n\'umeros enteros $n$.  

$A=\left( \begin{matrix} \alpha   & n &  \alpha \\ n & \alpha  & n+1 \\ \alpha & n+1& \alpha \end{matrix} \right)$
\end{exercice}



\begin{exercice}
Represente la gr\'afica de la funci\'on $f(x)=\dfrac{x^3-3x^2-1x+3}{x^2-5x+4}$
\end{exercice}


\begin{exercice}
Calcule $\ds\int x^2\sen (2x) \,dx$.
\end{exercice}


\begin{exercice}
Determine la ecuaci\'on general de la recta $s$ que est\'a contenida en los planos $\pi_1$ y $\pi_2$. $\pi_1$ es el plano que contiene al origen y es perpendicular a la recta  $r \equiv  \sysd{x-2y+z+2=0}{x+3y-z-3=0}$ y $\pi_2$,
el plano que pasa por el origen y es paralelo al plano $\pi \equiv -2x+3y+z+2=0$. 
\end{exercice}


\end{document}