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\begin{document}
\raggedright


\renewcommand{\thesection}{}
\section{Grupa A, 05.09.2013, ispit pisati hemiskom olovkom}
\setcounter{zad}{0}


\begin{zad}
Odrediti sve vrijednosti parametra $m$ tako da vektori 
$\vec{a}=\left(
\begin{matrix} 
m-2\\ 1\\ 2\\
\end{matrix}
\right)$,
$\vec{b}=\left(
\begin{matrix} 
m-2\\
m-2\\
3\\
\end{matrix}
\right)$,
$\vec{c}=\left(
\begin{matrix} 
m-2&
1&
m-2\\
\end{matrix}
\right)^\top$, nisu baza (ne čine bazu) vektorskog prostora $\R^3$. Za najveću dobijenu vrijednost parametra $m$ izraziti vektor $\vec{c}$ kao linearnu kombinaciju vektora $\vec{a}$ i $\vec{b}$.
\end{zad}


\begin{zad}
Ispitati i grafički predstaviti funkciju 
$\displaystyle{
y=\frac{\ln^2x+1}{x^2}
}$.
\end{zad}


\begin{zad}
Odrediti površinu figure ograničene hiperbolom $xy=4$ i pravom $y=-x+5$.
\end{zad}


\begin{zad}
Riješiti diferencijalnu jednačinu 
$\displaystyle{
y-xy'-\frac{1}{2}y'^2=0
}$.
\end{zad}


















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\section{Grupa B, 05.09.2013, ispit pisati hemiskom olovkom}
\setcounter{zad}{0}


\begin{zad}
Ako je $\{\vec{a}_1,\vec{a}_2,\vec{a}_3\}$ jedna baza vektorskog prostora $\R^3$, dokazati da i vektori $\vec{b}_1=\vec{a}_2+3\vec{a}_3$, $\vec{b}_2=\vec{a}_1+\vec{a}_2+2\vec{a}_3$ i $\vec{b}_3=2\vec{a}_1+2\vec{a}_2+6\vec{a}_3$ također čine bazu prostora $\R^3$ i izraziti vektor $\vec{c}=-\vec{a}_1+\vec{a}_2+2\vec{a}_3$ preko vektora vaze $\{\vec{b}_1,\vec{b}_2,\vec{b}_3\}$.
\end{zad}


\begin{zad}
Ispitati i grafički predstaviti funkciju 
$\displaystyle{
y=\frac{x^3-2}{2x^2}
}$.
\end{zad}


\begin{zad}
Odrediti površinu figure ograničene parabolom $y=x^2+4x$ i pravom $x-y+4=0$.
\end{zad}


\begin{zad}
Riješiti diferencijalnu jednačinu 
$\displaystyle{
y'^2-xy'+y=0
}$.
\end{zad}



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\section{Grupa C, 05.09.2013, ispit pisati hemiskom olovkom}
\setcounter{zad}{0}



\begin{zad}
Odrediti sve vrijednosti parametra $m$ tako da vektori 
$\vec{a}=\left(
\begin{matrix} 
m-1\\
m-1\\
m-1\\
\end{matrix}
\right)$,
$\vec{b}=\left(
\begin{matrix} 
1\\
m-1\\
1\\
\end{matrix}
\right)$,
$\vec{c}=\left(
\begin{matrix} 
2&
3&
m-1\\
\end{matrix}
\right)^\top$, nisu baza (ne čine bazu) vektorskog prostora $\R^3$. Za najveću dobijenu vrijednost parametra $m$ izraziti vektor $\vec{c}$ kao linearnu kombinaciju vektora $\vec{a}$ i $\vec{b}$.
\end{zad}



\begin{zad}
Ispitati i grafički predstaviti funkciju 
$\displaystyle{
y=\ln\frac{x^2-3x+2}{x^2+1}
}$.
\end{zad}


\begin{zad}
Odrediti površinu figure ograničenu parabolom $4y=8x-x^2$ i pravom $4y=x+6$.
\end{zad}


\begin{zad}
Riješiti diferencijalnu jednačinu 
$\displaystyle{
(y-y'x)^2=1+y'^2
}$.
\end{zad}






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\section{Grupa D, 05.09.2013, ispit pisati hemiskom olovkom}
\setcounter{zad}{0}


\begin{zad}
Ako je $\{\vec{a}_1,\vec{a}_2,\vec{a}_3\}$ jedna baza vektorskog prostora $\R^3$, dokazati da i vektori $\vec{b}_1=\vec{a}_1+2\vec{a}_2+3\vec{a}_3$, $\vec{b}_2=\vec{a}_1+\vec{a}_2+2\vec{a}_3$ i $\vec{b}_3=2\vec{a}_1+\vec{a}_2+4\vec{a}_3$ također čine bazu prostora $\R^3$ i izraziti vektor $\vec{c}=\vec{a}_1+\vec{a}_2+\vec{a}_3$ preko vektora vaze $\{\vec{b}_1,\vec{b}_2,\vec{b}_3\}$.
\end{zad}



\begin{zad}
Ispitati i grafički predstaviti funkciju 
$\displaystyle{
y=\frac{x^2+10}{x^2+4x+4}
}$.
\end{zad}


\begin{zad}
Odrediti površinu figure ograničene hiperbolom $xy=6$ i pravom $y=7-x$.
\end{zad}


\begin{zad}
Riješiti diferencijalnu jednačinu 
$\displaystyle{
y=y'x+\sqrt{4+y'^2}
}$.
\end{zad}






\end{document}