C

9000035609

Level: 
C
One of the roots of the equation \( x^{2} + px - 11 = 0\) with the parameter \(p\in \mathbb{C}\) is \(x_{1} = 3 -\mathrm{i}\sqrt{2}\). Find the second root \(x_{2}\) and the corresponding value of the parameter \(p\).
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 2\mathrm{i}\sqrt{2}\)
\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = 6\)
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 6\)
\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\)
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\sqrt{2}\)

9000035810

Level: 
C
Given the complex number \(z = -2 + 2\mathrm{i}\), find all the roots of \(\root{3}\of{z}\).
\(\begin{aligned}[t] &w_{0} = \root{6}\of{8}\left (\cos \frac{\pi } {4} + \mathrm{i}\sin \frac{\pi } {4}\right ) & \\&w_{1} = \root{6}\of{8}\left (\cos \frac{11\pi } {12} + \mathrm{i}\sin \frac{11\pi } {12}\right ) \\&w_{2} = \root{6}\of{8}\left (\cos \frac{19\pi } {12} + \mathrm{i}\sin \frac{19\pi } {12}\right ) \\ \end{aligned}\)
\(\begin{aligned}[t] &w_{0} = 2\left (\cos \frac{\pi } {4} + \mathrm{i}\sin \frac{\pi } {4}\right ) & \\&w_{1} = 2\left (\cos \frac{11\pi } {12} + \mathrm{i}\sin \frac{11\pi } {12}\right ) \\&w_{2} = 2\left (\cos \frac{19\pi } {12} + \mathrm{i}\sin \frac{19\pi } {12}\right ) \\ \end{aligned}\)
\(\root{3}\of{-2} + \root{3}\of{2}\)
\(\begin{aligned}[t] &w_{0} = 2\left (\cos \frac{\pi } {3} + \mathrm{i}\sin \frac{\pi } {3}\right )& \\&w_{1} = 2\left (\cos \pi +\mathrm{i}\sin \pi \right ) \\&w_{2} = 2\left (\cos \frac{5\pi } {3} + \mathrm{i}\sin \frac{5\pi } {3}\right ) \\ \end{aligned}\)

9000035608

Level: 
C
The equation \[ x^{2} - 2\mathrm{i}x + q = 0 \] with a parameter \(q\in \mathbb{C}\) has a solution \(x_{1} = 1 + 2\mathrm{i}\). Find the second solution \(x_{2}\) and the parameter \(q\).
\(x_{2} = -1,\ q = -1 - 2\mathrm{i}\)
\(x_{2} = -1 - 4\mathrm{i},\ q = 9 - 6\mathrm{i}\)
\(x_{2} = 1 - 4\mathrm{i},\ q = 7 - 4\mathrm{i}\)
\(x_{2} = 1,\ q = -1 - 2\mathrm{i}\)
\(x_{2} = -1,\ q = 1 + 2\mathrm{i}\)

9000036101

Level: 
C
A \(3\, \mathrm{m}\) long rod is in a slant position with respect to the observer's eye: one end is in the distance \(20\, \mathrm{m}\) and the other one \(18\, \mathrm{m}\). Find the visual angle of the rod (the angle between the lines which connect the observer's eye and the ends of the rod) and round to the nearest degrees.
\(7^{\circ }\)
\(3^{\circ }\)
\(45^{\circ }\)
\(83^{\circ }\)

9000036102

Level: 
C
Three forces act on the same body in the same point and the total force on the body is zero (the forces cancel). The first two forces are \(8\, \mathrm{N}\) and \(10\, \mathrm{N}\) and the angle between these forces is \(55^{\circ }\). Find the third force.
\(16\, \mathrm{N}\)
\(15\, \mathrm{N}\)
\(17\, \mathrm{N}\)
\(18\, \mathrm{N}\)

9000034303

Level: 
C
Find the solution set of the following equation in the set of complex numbers. \[ x^{3} + \mathrm{i} = 0 \]
\(\{\mathrm{i};\ \frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i};\ -\frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i}\}\)
\(\{ - 1;\ -\frac{\sqrt{3}} {2} + \frac{1} {2}\mathrm{i};\ -\frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i}\}\)
\(\{ - 1;\ \frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i};\ -\frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i}\}\)
\(\{\mathrm{i};\ -\frac{\sqrt{3}} {2} + \frac{1} {2}\mathrm{i};\ -\frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i}\}\)

9000034308

Level: 
C
Two solutions of the equation \[ x^{3} + 1 + \mathrm{i} = 0 \] are \[ \begin{aligned}x_{1}& = \root{6}\of{2}\left (\cos \frac{5} {12}\pi + \mathrm{i}\sin \frac{5} {12}\pi \right ),& \\x_{2}& = \root{6}\of{2}\left (\cos \frac{13} {12}\pi + \mathrm{i}\sin \frac{13} {12}\pi \right ). \\ \end{aligned} \] Find the third solution.
\(x_{3} = \root{6}\of{2}\left (\cos \frac{21} {12}\pi + \mathrm{i}\sin \frac{21} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{9} {12}\pi + \mathrm{i}\sin \frac{9} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{17} {12}\pi + \mathrm{i}\sin \frac{17} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{19} {12}\pi + \mathrm{i}\sin \frac{19} {12}\pi \right )\)