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\begin{document}

\devpers{Prueba 10 : Repaso general}{}{TS,\ 9 de mayo 2013}{}

\section*{Opción A}

\begin{exercice}
Estudie la existencia de rectas as\'intotas correspondiente a la gr\'afica de la funci\'on.
\[
 f(x)=\dfrac{\ln x^2}{x+\sqrt[3]{x}}
\]
\end{exercice}

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

\begin{exercice}
Sean la recta $r \equiv \syst{x=-1-\lambda}{y=-\lambda}{z=2\lambda}$. y el plano $\pi \equiv 2x-y+z-1=0$. Halle las ecuaciones param\'etricas de la recta $s$, proyecci\'on ortogonal de $r$ sobre $\pi$.
\end{exercice}

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


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


\begin{exercice}
Discutir y resolver en los casos de compatibilidad el siguiente sistema
\[
\syst{x+\phantom{3y}+z=1}{\phantom{x+}y+(a-1)z=0}{x+(a-1)y+az=a}
\]
\end{exercice}



\begin{exercice}
Hallar el volumen del tetraedro cuyos vértices son el punto $V(1, 1, 1)$
y los puntos de corte del plano $\pi \equiv 2x+3y+z-12=0$, con los ejes de
coordenadas. 
\end{exercice}

\end{document}