9000035604
Level:
C
Solve the following equation.
\[
x^{2} - 2\mathrm{i}x + 3 = 0
\]
\(\{ -\mathrm{i};3\mathrm{i}\}\)
\(\{ - 4\mathrm{i};4\mathrm{i}\}\)
\(\{1 - 2\mathrm{i};1 + 2\mathrm{i}\}\)
\(\{ - 3\mathrm{i};\mathrm{i}\}\)
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 \(\root{3}\of{z}\).
\(\begin{aligned}[t] &w_{0} = \root{6}\of{8}\left (\cos \frac{\pi }
{4} + \mathrm{i}\sin \frac{\pi }
{4}\right ) &
\\&w_{1} = \root{6}\of{8}\left (\cos \frac{11\pi }
{12} + \mathrm{i}\sin \frac{11\pi }
{12}\right )
\\&w_{2} = \root{6}\of{8}\left (\cos \frac{19\pi }
{12} + \mathrm{i}\sin \frac{19\pi }
{12}\right )
\\ \end{aligned}\)
\(\begin{aligned}[t] &w_{0} = 2\left (\cos \frac{\pi }
{4} + \mathrm{i}\sin \frac{\pi }
{4}\right ) &
\\&w_{1} = 2\left (\cos \frac{11\pi }
{12} + \mathrm{i}\sin \frac{11\pi }
{12}\right )
\\&w_{2} = 2\left (\cos \frac{19\pi }
{12} + \mathrm{i}\sin \frac{19\pi }
{12}\right )
\\ \end{aligned}\)
\(\root{3}\of{-2} + \root{3}\of{2}\)
\(\begin{aligned}[t] &w_{0} = 2\left (\cos \frac{\pi }
{3} + \mathrm{i}\sin \frac{\pi }
{3}\right )&
\\&w_{1} = 2\left (\cos \pi +\mathrm{i}\sin \pi \right )
\\&w_{2} = 2\left (\cos \frac{5\pi }
{3} + \mathrm{i}\sin \frac{5\pi }
{3}\right )
\\ \end{aligned}\)
9000035607
Level:
C
Identify the quadratic equation with solutions
\(x_{1} = 2\mathrm{i}\),
\(x_{2} = -\mathrm{i}\).
\(x^{2} -\mathrm{i}x + 2 = 0\)
\(x^{2} + \mathrm{i}x + 2 = 0\)
\(x^{2} + \mathrm{i}x - 2 = 0\)
\(x^{2} -\mathrm{i}x - 2 = 0\)
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.
\(7^{\circ }\)
\(3^{\circ }\)
\(45^{\circ }\)
\(83^{\circ }\)
9000033810
Level:
C
Find the parity (odd/even) of the function
\(l\colon y = |\mathop{\mathrm{cotg}}\nolimits x|\).
The function \(l\)
is an even function.
The function \(l\)
is an odd function.
The function \(l\)
is neither an odd nor even function.
9000034303
Level:
C
Find the solution set of the following equation in the set of complex numbers.
\[
x^{3} + \mathrm{i} = 0
\]
\(\{\mathrm{i};\ \frac{\sqrt{3}}
{2} -\frac{1}
{2}\mathrm{i};\ -\frac{\sqrt{3}}
{2} -\frac{1}
{2}\mathrm{i}\}\)
\(\{ - 1;\ -\frac{\sqrt{3}}
{2} + \frac{1}
{2}\mathrm{i};\ -\frac{\sqrt{3}}
{2} -\frac{1}
{2}\mathrm{i}\}\)
\(\{ - 1;\ \frac{\sqrt{3}}
{2} -\frac{1}
{2}\mathrm{i};\ -\frac{\sqrt{3}}
{2} -\frac{1}
{2}\mathrm{i}\}\)
\(\{\mathrm{i};\ -\frac{\sqrt{3}}
{2} + \frac{1}
{2}\mathrm{i};\ -\frac{\sqrt{3}}
{2} -\frac{1}
{2}\mathrm{i}\}\)
9000034307
Level:
C
Let \(x\)
be any of the complex solutions of the following equation.
\[
x^{5} + \sqrt{3} -\mathrm{i} = 0
\]
Find the absolute value of \(x\).
\(\root{5}\of{2}\)
\(2\)
\(\root{5}\of{4}\)
\(\root{10}\of{2}\)
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.
\(x_{3} = \root{6}\of{2}\left (\cos \frac{21}
{12}\pi + \mathrm{i}\sin \frac{21}
{12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{9}
{12}\pi + \mathrm{i}\sin \frac{9}
{12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{17}
{12}\pi + \mathrm{i}\sin \frac{17}
{12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{19}
{12}\pi + \mathrm{i}\sin \frac{19}
{12}\pi \right )\)