9000035808
Level:
A
Evaluate \((1 -\mathrm{i})^{10}\).
\(- 32\mathrm{i}\)
\(32\)
\(32\mathrm{i}\)
\(- 32\)
Powers and roots of complex numbers
9000035810
Level:
C
Given the complex number \(z = -2 + 2\mathrm{i}\),
find all the roots of \(\root{3}\of{z}\).
\(\begin{aligned}[t] &w_{0} = \root{6}\of{8}\left (\cos \frac{\pi }
{4} + \mathrm{i}\sin \frac{\pi }
{4}\right ) &
\\&w_{1} = \root{6}\of{8}\left (\cos \frac{11\pi }
{12} + \mathrm{i}\sin \frac{11\pi }
{12}\right )
\\&w_{2} = \root{6}\of{8}\left (\cos \frac{19\pi }
{12} + \mathrm{i}\sin \frac{19\pi }
{12}\right )
\\ \end{aligned}\)
\(\begin{aligned}[t] &w_{0} = 2\left (\cos \frac{\pi }
{4} + \mathrm{i}\sin \frac{\pi }
{4}\right ) &
\\&w_{1} = 2\left (\cos \frac{11\pi }
{12} + \mathrm{i}\sin \frac{11\pi }
{12}\right )
\\&w_{2} = 2\left (\cos \frac{19\pi }
{12} + \mathrm{i}\sin \frac{19\pi }
{12}\right )
\\ \end{aligned}\)
\(\root{3}\of{-2} + \root{3}\of{2}\)
\(\begin{aligned}[t] &w_{0} = 2\left (\cos \frac{\pi }
{3} + \mathrm{i}\sin \frac{\pi }
{3}\right )&
\\&w_{1} = 2\left (\cos \pi +\mathrm{i}\sin \pi \right )
\\&w_{2} = 2\left (\cos \frac{5\pi }
{3} + \mathrm{i}\sin \frac{5\pi }
{3}\right )
\\ \end{aligned}\)
9000034306
Level:
B
Solve the following equation in the set of complex numbers.
\[
x^{6} - 64 = 0
\]
\(x_{1, 2} =\pm 2,\ x_{3, 4} = 1\pm \mathrm{i}\sqrt{3},\ x_{5, 6} = -1\pm \mathrm{i}\sqrt{3}\)
\(x_{1, 2} =\pm 2,\ x_{3, 4} = \frac{1}
{2}\pm \mathrm{i}\frac{\sqrt{3}}
{2} ,\ x_{5, 6} = -\frac{1}
{2}\pm \mathrm{i}\frac{\sqrt{3}}
{2} \)
\(x_{1, 2} =\pm 4,\ x_{3, 4} = 1\pm \mathrm{i}\sqrt{3},\ x_{5, 6} = -1\pm \mathrm{i}\sqrt{3}\)
\(x_{1, 2} =\pm 8,\ x_{3, 4} = 2\pm 2\mathrm{i}\sqrt{3},\ x_{5, 6} = -2\pm 2\mathrm{i}\sqrt{3}\)
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.
\(x_{3} = \root{6}\of{2}\left (\cos \frac{21}
{12}\pi + \mathrm{i}\sin \frac{21}
{12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{9}
{12}\pi + \mathrm{i}\sin \frac{9}
{12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{17}
{12}\pi + \mathrm{i}\sin \frac{17}
{12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{19}
{12}\pi + \mathrm{i}\sin \frac{19}
{12}\pi \right )\)
9000034309
Level:
C
Find the angle \(\varphi \)
such that the angles in the polar form of any two solutions of the equation
\[
x^{5} - 1 + \mathrm{i}\sqrt{3} = 0
\]
differ by an integer multiple of \(\varphi \).
\(\varphi = \frac{2}
{5}\pi \)
\(\varphi = \frac{3}
{5}\pi \)
\(\varphi = \frac{4}
{5}\pi \)
\(\varphi =\pi \)
9000034301
Level:
B
Find the solution set of the following equation in the set of complex numbers.
\[
x^{3} - 1 = 0
\]
\(\{1;\ -\frac{1}
{2} + \mathrm{i}\frac{\sqrt{3}}
{2} ;\ -\frac{1}
{2} -\mathrm{i}\frac{\sqrt{3}}
{2} \}\)
\(\{1;\ -1 + \mathrm{i}\sqrt{3};\ -1 -\mathrm{i}\sqrt{3}\}\)
\(\{1;\ -\frac{1}
{2} + \mathrm{i}\frac{\sqrt{3}}
{2} \}\)
\(\{1;\ -\frac{1}
{2} -\mathrm{i}\frac{\sqrt{3}}
{2} \}\)
9000034302
Level:
B
Find the solution set of the following equation in the set of complex numbers.
\[
x^{3} + 8 = 0
\]
\(\{ - 2;\ 1 + \mathrm{i}\sqrt{3};\ 1 -\mathrm{i}\sqrt{3}\}\)
\(\{ - 2;\ -1 + \mathrm{i}\sqrt{3};\ -1 -\mathrm{i}\sqrt{3}\}\)
\(\{ - 2;\ \frac{1}
{2} + \mathrm{i}\frac{\sqrt{3}}
{2} ;\ \frac{1}
{2} -\mathrm{i}\frac{\sqrt{3}}
{2} \}\)
\(\{ - 2;\ -\frac{1}
{2} + \mathrm{i}\frac{\sqrt{3}}
{2} ;\ -\frac{1}
{2} -\mathrm{i}\frac{\sqrt{3}}
{2} \}\)
9000034303
Level:
C
Find the solution set of the following equation in the set of complex numbers.
\[
x^{3} + \mathrm{i} = 0
\]
\(\{\mathrm{i};\ \frac{\sqrt{3}}
{2} -\frac{1}
{2}\mathrm{i};\ -\frac{\sqrt{3}}
{2} -\frac{1}
{2}\mathrm{i}\}\)
\(\{ - 1;\ -\frac{\sqrt{3}}
{2} + \frac{1}
{2}\mathrm{i};\ -\frac{\sqrt{3}}
{2} -\frac{1}
{2}\mathrm{i}\}\)
\(\{ - 1;\ \frac{\sqrt{3}}
{2} -\frac{1}
{2}\mathrm{i};\ -\frac{\sqrt{3}}
{2} -\frac{1}
{2}\mathrm{i}\}\)
\(\{\mathrm{i};\ -\frac{\sqrt{3}}
{2} + \frac{1}
{2}\mathrm{i};\ -\frac{\sqrt{3}}
{2} -\frac{1}
{2}\mathrm{i}\}\)
9000034304
Level:
B
Find the solution set of the following equation in the set of complex numbers.
\[
x^{4} - 1 = 0
\]
\(\{1;\ -1;\ \mathrm{i};\ -\mathrm{i}\}\)
\(\{1 -\mathrm{i};\ -1 -\mathrm{i}\}\)
\(\{1 + \mathrm{i};\ -1 + \mathrm{i}\}\)
\(\{1 + \mathrm{i};\ 1 -\mathrm{i};\ -1 + \mathrm{i};\ -1 -\mathrm{i}\}\)