C

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.
\(30\,\mathrm{m}\)
\(15\,\mathrm{m}\)
\(40\,\mathrm{m}\)
\(60\,\mathrm{m}\)

2010013308

Level: 
C
Find the solution set of the following quadratic equation in the complex plane. \[ 2x^2-(2-4\mathrm{i})x + 3 - 2\mathrm{i}= 0 \]
\( \left\{ \frac12+\frac12\mathrm{i}; \frac12-\frac52\mathrm{i} \right\} \)
\( \left\{ \frac12+\frac12\mathrm{i}; \frac12-\frac12\mathrm{i} \right\} \)
\( \emptyset \)
\( \left\{ -\frac12-\frac12\mathrm{i}; -\frac12+\frac52\mathrm{i} \right\} \)
\( \left\{ 1+\mathrm{i}; 1-5\mathrm{i} \right\} \)

2010013208

Level: 
C
Which of the given quadratic equations has the roots \( x_1 = -1 - 3\mathrm{i} \) and \( x_2 = 1 +3\mathrm{i} \)?
\( x^2 + (3 - 3\mathrm{i}) + 5 - 3\mathrm{i} = 0 \)
\( x^2 - (3 - 3\mathrm{i}) + 5 + 3\mathrm{i} = 0 \)
\( x^2 + (3 - 3\mathrm{i}) + 5 + 3\mathrm{i} = 0 \)
\( x^2 - (3 - 3\mathrm{i}) + 5 - 3\mathrm{i} = 0 \)

2010013207

Level: 
C
Find the complex roots of the following quadratic equation. \[ 2x^2 + 5\mathrm{i} = 0 \]
\( x_1=-\frac{\sqrt5}2+\frac{\sqrt5}2\mathrm{i},\ \ x_2=\frac{\sqrt5}2-\frac{\sqrt5}2\mathrm{i} \)
\( x_1=\frac{\sqrt5}2+\frac{\sqrt5}2\mathrm{i},\ \ x_2=-\frac{\sqrt5}2-\frac{\sqrt5}2\mathrm{i} \)
\( x_1=\frac{\sqrt5}2\mathrm{i},\ \ x_2=-\frac{\sqrt5}2\mathrm{i} \)
\( x_1=-\frac{\sqrt5}2,\ \ x_2=\frac{\sqrt5}2\)

2010013206

Level: 
C
Find the complex roots of the following quadratic equation. \[ 3\mathrm{i}x^2 + 2 = 0 \]
\( x_1=\frac{\sqrt3}3+\frac{\sqrt3}3\mathrm{i},\ \ x_2=-\frac{\sqrt3}3-\frac{\sqrt3}3\mathrm{i} \)
\( x_1=-\frac{\sqrt3}3+\frac{\sqrt3}3\mathrm{i},\ \ x_2=\frac{\sqrt3}3-\frac{\sqrt3}3\mathrm{i} \)
\( x_1=-\frac{\sqrt3}6+\frac{\sqrt3}6\mathrm{i},\ \ x_2=-\frac{\sqrt3}6-\frac{\sqrt3}6\mathrm{i} \)
\( x_1=\frac{\sqrt3}6+\frac{\sqrt3}6\mathrm{i},\ \ x_2=-\frac{\sqrt3}6-\frac{\sqrt3}6\mathrm{i} \)