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\hspace*{160mm} 30.08.2012.
\centerline {Pismeni ispit iz predmeta \textbf{Matematika I}}


\vspace*{4mm}
\vspace*{2mm}\textbf{1. } a) 
Metodom matemati\v cke indukcije dokazati da\\ 
\vspace*{2mm}
\hspace*{15mm}
$
1+2+2^2+...+2^n=2^{n+1}-1
$\\
\vspace*{2mm}
vrijedi za sve prirodne brojeve $n\in \mathbb N.$\\
\hspace*{5mm}
b) Neka je 
$
A=\left[
\begin{matrix} 
0 & 0 & 1 \\
1 & 0 & 0 \\
1 & 1 & 0 \\
\end{matrix}
\right]$. Izračunati $A^7.$



\vspace*{2mm}\textbf{2. }
Riješiti linearni sistem jednačina:
\vspace*{-2mm}
\begin{alignat*}{7}
\,\,x_1\,\,   & \,\,+\,\, &  \,\,x_2\,\,  & \,\,+\,\, & \,\,x_3\,\, & \,\,+\,\, &  \,\,x_4\,\,  &= \,\,1\,\,\\
\,\,x_1\,\,   & \,\,+\,\, &  \,\,x_2\,\,  & \,\,+\,\, & \,\,3x_3\,\, & \,\,+\,\, &  \,\,3x_4\,\,  &= \,\,3\,\,\\
\,\,x_1\,\,   & \,\,+\,\, &  \,\,x_2\,\,  & \,\,+\,\, & \,\,2x_3\,\, & \,\,+\,\, &  \,\,3x_4\,\,  &= \,\,3\,\,\\
\,\,x_1\,\,   & \,\,+\,\, &  \,\,3x_2\,\,  & \,\,+\,\, & \,\,3x_3\,\, & \,\,+\,\, &  \,\,3x_4\,\,  &= \,\,4\,\,\,\, .\\
\end{alignat*}




\vspace*{-4mm}\textbf{3. }
Napisati jedna\v cinu ravni koja prolazi kroz ta\v cku $M_1(2,0,-1)$ i normalna je na ravnima $2x-y-3=0$ i $x+y-z+1=0$.\\




\vspace*{1mm}\textbf{4. }
Ispitati i grafi\v cki predstaviti funkciju  
$\displaystyle{
y={x^2\over 1+x}.
}$\\






\vspace*{25mm}





\hspace*{160mm} 30.08.2012.
\centerline {Pismeni ispit iz predmeta \textbf{Matematika I}}


\vspace*{4mm}
\vspace*{2mm}\textbf{1. } a) 
Metodom matemati\v cke indukcije dokazati da\\ 
\vspace*{2mm}
\hspace*{15mm}
$
1+2+2^2+...+2^n=2^{n+1}-1
$\\
\vspace*{2mm}
vrijedi za sve prirodne brojeve $n\in \mathbb N.$\\
\hspace*{5mm}
b) Neka je 
$
A=\left[
\begin{matrix} 
0 & 0 & 1 \\
1 & 0 & 0 \\
1 & 1 & 0 \\
\end{matrix}
\right]$. Izračunati $A^7.$



\vspace*{2mm}\textbf{2. }
Riješiti linearni sistem jednačina:
\vspace*{-2mm}
\begin{alignat*}{7}
\,\,x_1\,\,   & \,\,+\,\, &  \,\,x_2\,\,  & \,\,+\,\, & \,\,x_3\,\, & \,\,+\,\, &  \,\,x_4\,\,  &= \,\,1\,\,\\
\,\,x_1\,\,   & \,\,+\,\, &  \,\,x_2\,\,  & \,\,+\,\, & \,\,3x_3\,\, & \,\,+\,\, &  \,\,3x_4\,\,  &= \,\,3\,\,\\
\,\,x_1\,\,   & \,\,+\,\, &  \,\,x_2\,\,  & \,\,+\,\, & \,\,2x_3\,\, & \,\,+\,\, &  \,\,3x_4\,\,  &= \,\,3\,\,\\
\,\,x_1\,\,   & \,\,+\,\, &  \,\,3x_2\,\,  & \,\,+\,\, & \,\,3x_3\,\, & \,\,+\,\, &  \,\,3x_4\,\,  &= \,\,4\,\,\,\, .\\
\end{alignat*}




\vspace*{-4mm}\textbf{3. }
Napisati jedna\v cinu ravni koja prolazi kroz ta\v cku $M_1(2,0,-1)$ i normalna je na ravnima $2x-y-3=0$ i $x+y-z+1=0$.\\




\vspace*{1mm}\textbf{4. }
Ispitati i grafi\v cki predstaviti funkciju  
$\displaystyle{
y={x^2\over 1+x}.
}$\\









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