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\begin{document}

\devpers{Selectividad Blanca}{}{Matemáticas II,\ abril 2015}{}


\section*{Modelo A}
\begin{exercice}
Resuelva, dependiendo del valor $\lambda$ el siguiente sistema
\[
 \left\{
\begin{array}{ll}
   2y + 2z -2x\lambda &=-2\lambda \\
  2y +x\lambda+z\lambda &=1 \\
 2x\lambda -2z-2y\lambda  &= 4 \\
\end{array} \right.
\]
\end{exercice} 

\begin{exercice}%{\bf Matematicas II - Juno 2010}\par
Se consideran las matrices $A=\left(
  \begin{matrix}
  1&1\\
1& 2 
  \end{matrix}
\right)
$ y $C=\left(
  \begin{matrix}
  1&1\\
3& 1 
  \end{matrix}
\right)
$. H\'allese une matriz $X$ que sea soluci\'on de la ecuaci\'on matricial $AXA^{-1}=C$.
\end{exercice}

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

\begin{exercice}
 Calcule $\ds\int 
\dfrac{2x^2-x+1}{x^3-x^2+x-1} \,dx$.
\end{exercice}



\hrule
\vspace{1cm}

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

\begin{exercice}%{\bf Matematicas II - Juno 2010}\par
Sea $\lambda \in \R$ y la matriz $A=\left(
  \begin{matrix}
  1&0&-1\\
0&\lambda&3\\
4&1&-\lambda
  \end{matrix}
\right)
$. Para que valores de $\lambda$ la matriz $A$ no tiene inversa ?
\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}
Sea la func\'ion  
\[
 f(x)=\begin{cases}
       -x-1 &\text{si $x<-3$} \\
      p-2x^2 &\text{si $-3\leq x \leq 3$} \\
     \dfrac{q}{x} &\text{si $3 < x$} 
      \end{cases}
\]
determine los valores de $p$ y $q$ para que $f$ sea continua en toda la recta real.\par
Una vez determinados los valores de $p$ y $q$ en el apartado anterior, estudie la derivabilidad de $f$.
\end{exercice}

\begin{exercice}
 Calcule $\ds\int_{1}^\E (\ln x^2+\ln^2 x) \,dx$.
\end{exercice}
 Observación: $\ln x$ representa el logaritmo neperiano de $x$.

\end{document}