C

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.
\(16\, \mathrm{N}\)
\(15\, \mathrm{N}\)
\(17\, \mathrm{N}\)
\(18\, \mathrm{N}\)

9000036103

Level: 
C
Three forces \(F_{1}\), \(F_{2}\) and \(F_{3}\) 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 \(F_{1} = 8\, \mathrm{N}\) and \(F_{2} = 10\, \mathrm{N}\) and the angle between \(F_{1}\) and \(F_{2}\) is \(55^{\circ }\). Find the angle between \(F_{3}\) and \(F_{1}\). Round your answer to the nearest degrees.
\(149^{\circ }\)
\(125^{\circ }\)
\(55^{\circ }\)
\(30^{\circ }\)

9000036106

Level: 
C
Two straight roads go off from the crossing. The angle between directions of the roads is \(52^{\circ }18'\). A significant tree is on the first road in the distance \(250\, \mathrm{m}\) from the crossing. A rock with a beautiful view is on the second road in the distance \(380\, \mathrm{m}\) from the crossing. Find the direct distance (length of a line segment) from the rock to the tree and round your answer to nearest meters.
\(301\, \mathrm{m}\)
\(411\, \mathrm{m}\)
\(568\, \mathrm{m}\)
\(629\, \mathrm{m}\)

9000035602

Level: 
C
Find the values of the parameter \(m\in \mathbb{C}\) which guarantee that the following quadratic equation has a double solution. \[ mx^{2} - 2x - 1 + \mathrm{i} = 0 \]
\(m = -\frac{1} {2} -\frac{1} {2}\mathrm{i}\)
\(m = -1\)
\(m = -1 + \mathrm{i}\)
\(m = -\frac{1} {2} + \frac{1} {2}\mathrm{i}\)