C

2010013412

Level: 
C
Which of the given numbers does not belong to the solution set of the following equation? \[x^{4}-1+\mathrm{i}=0\]
\(\root{8}\of{2}\left (\cos \frac{3\pi}{16} + \mathrm{i}\sin \frac{3\pi}{16}\right )\)
\(-\root{4}\of{1-\mathrm{i}}\)
\(-\mathrm{i}\root{4}\of{1-\mathrm{i}}\)
\(\root{8}\of{2}\left (\cos \left(-\frac{\pi}{16}\right) + \mathrm{i}\sin \left(-\frac{\pi}{16}\right)\right )\)

2010013409

Level: 
C
Three solutions of the equation \[ x^{4} + 8\mathrm{i} = 0 \] are \[\begin{aligned}x_{1} = \root{4}\of{8}\left (\cos \frac{3}{8}\pi + \mathrm{i}\sin \frac{3}{8}\pi \right ), \\ x_{2} = \root{4}\of{8}\left (\cos \frac{7}{8}\pi + \mathrm{i}\sin \frac{7}{8}\pi \right ),\\ x_{3} = \root{4}\of{8}\left (\cos \frac{15}{8}\pi + \mathrm{i}\sin \frac{15}{8}\pi \right ).\\ \end{aligned}\] Find the fourth solution.
\(x_{4} = \root{4}\of{8}\left (\cos \frac{11}{8}\pi + \mathrm{i}\sin \frac{11}{8}\pi \right )\)
\(x_{4} = \root{4}\of{8}\left (\cos \frac{9}{8}\pi + \mathrm{i}\sin \frac{9}{8}\pi \right )\)
\(x_{4} = \root{4}\of{8}\left (\cos \frac{5}{8}\pi + \mathrm{i}\sin \frac{5}{8}\pi \right )\)
\(x_{4} = \root{4}\of{8}\left (\cos \frac{1}{8}\pi + \mathrm{i}\sin \frac{1}{8}\pi \right )\)

2010013408

Level: 
C
Three solutions of the equation \[ x^{4} - 2\mathrm{i} = 0 \] are \[\begin{aligned}x_{1} = \root{4}\of{2}\left (\cos \frac{1}{8}\pi + \mathrm{i}\sin \frac{1}{8}\pi \right ),\\ x_{2} = \root{4}\of{2}\left (\cos \frac{5}{8}\pi + \mathrm{i}\sin \frac{5}{8}\pi \right ),\\ x_{3} = \root{4}\of{2}\left (\cos \frac{9}{8}\pi + \mathrm{i}\sin \frac{9}{8}\pi \right ).\\ \end{aligned}\] Find the fourth solution.
\(x_{4} = \root{4}\of{2}\left (\cos \frac{13}{8}\pi + \mathrm{i}\sin \frac{13}{8}\pi \right )\)
\(x_{4} = \root{4}\of{2}\left (\cos \frac{11}{8}\pi + \mathrm{i}\sin \frac{11}{8}\pi \right )\)
\(x_{4} = \root{4}\of{2}\left (\cos \frac{15}{8}\pi + \mathrm{i}\sin \frac{15}{8}\pi \right )\)
\(x_{4} = \root{4}\of{2}\left (\cos \frac{3}{8}\pi + \mathrm{i}\sin \frac{3}{8}\pi \right )\)

2010013407

Level: 
C
Two solutions of the equation \[ x^{3} + 1 - \mathrm{i} = 0 \] are \[ \begin{aligned}x_{1}& = \root{6}\of{2}\left (\cos \frac{\pi} {4} + \mathrm{i}\sin \frac{\pi} {4} \right ),& \\x_{2}& = \root{6}\of{2}\left (\cos \frac{11} {12}\pi + \mathrm{i}\sin \frac{11} {12}\pi \right ). \\ \end{aligned} \] Find the third solution.
\(x_{3} = \root{6}\of{2}\left (\cos \frac{19} {12}\pi + \mathrm{i}\sin \frac{19} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{7} {12}\pi + \mathrm{i}\sin \frac{7} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{5} {12}\pi + \mathrm{i}\sin \frac{5} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{13} {12}\pi + \mathrm{i}\sin \frac{13} {12}\pi \right )\)

2010013406

Level: 
C
Find the solution set of the following equation in the set of complex numbers. \[ x^{3} + 8\mathrm{i} = 0 \]
\(\left\{2\mathrm{i};\ \sqrt{3} -\mathrm{i};\ -\sqrt{3}-\mathrm{i}\right\}\)
\(\left\{ -2\mathrm{i};\ \sqrt{3} -\mathrm{i};\ -\sqrt{3}-\mathrm{i}\right\}\)
\(\left\{ -2;\ -\sqrt{3} +\mathrm{i};\ \sqrt{3}+\mathrm{i}\right\}\)
\(\left\{ 2;\ -\sqrt{3} +\mathrm{i};\ \sqrt{3}+\mathrm{i}\right\}\)

2010018105

Level: 
C
The values of variables \( x \) and \( y \) are listed in the following table and visualized in the next graph. Calculate the correlation coefficient of \( x \) and \( y \) and round it to four decimal places. \[ \begin{array}{|c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4& 4.5 \\\hline y & 6 & 4 &5 & 3 & 3.5 \\\hline \end{array} \]
\(-0.8120\)
\(-0.8211\)
\(-0.8305\)
\(-0.8021\)

2010018004

Level: 
C
A rectangle-shaped land has dimensions \(5 \times 8\,\mathrm{cm}\) on a map with scale \(1:500\). The owner increased the size of his land by buying some land from his neighbor. The new land has dimensions \(7\times 9\,\mathrm{cm}\) on the map. Find the actual increase of the perimeter of the land (i.e. find the increase in the length of the fence required to enclose the whole land). Give your answer in meters.
\(30\,\mathrm{m}\)
\(15\,\mathrm{m}\)
\(40\,\mathrm{m}\)
\(60\,\mathrm{m}\)