Powers and roots of complex numbers

1003118405

Level: 
C
All the solutions of the equation \( x^6-4\sqrt3+4\mathrm{i} = 0 \) can be displayed as points in the rectangular coordinate system. What is the distance of the two most distant points?
\( 2\sqrt2 \)
\( \sqrt2 \)
\( 2\sqrt[3]4 \)
\( \sqrt[3]4 \)
\( 2\sqrt3 \)
\( \sqrt3 \)

1003118406

Level: 
C
All the solutions of the equation \( x^4+1+\sqrt3\mathrm{i} = 0 \) are complex numbers, whose arguments are from the interval \( [0; 2\pi) \). Find the sum of the arguments of all the solutions of the equation.
\( \frac{13}3\pi \)
\( 4\pi \)
\( \frac{25}6\pi \)
\( \frac92\pi \)

1103118404

Level: 
C
Consider an equation \( x^n+b=0 \), where \( n \) is a positive integer and \( b \) is a complex number. In the picture the points that correspond to the roots of the equation are depicted in black. Find the equation.
\( x^3 + 4\sqrt2 - 4\sqrt2\mathrm{i} = 0 \)
\( x^3 + 4\sqrt2 +4\sqrt2\mathrm{i} = 0 \)
\( x^3 - 4\sqrt2 - 4\sqrt2\mathrm{i} = 0 \)
\( x^3 - 4\sqrt2 +4\sqrt2\mathrm{i} = 0 \)

2010013406

Level: 
C
Find the solution set of the following equation in the set of complex numbers. \[ x^{3} + 8\mathrm{i} = 0 \]
\(\left\{2\mathrm{i};\ \sqrt{3} -\mathrm{i};\ -\sqrt{3}-\mathrm{i}\right\}\)
\(\left\{ -2\mathrm{i};\ \sqrt{3} -\mathrm{i};\ -\sqrt{3}-\mathrm{i}\right\}\)
\(\left\{ -2;\ -\sqrt{3} +\mathrm{i};\ \sqrt{3}+\mathrm{i}\right\}\)
\(\left\{ 2;\ -\sqrt{3} +\mathrm{i};\ \sqrt{3}+\mathrm{i}\right\}\)

2010013407

Level: 
C
Two solutions of the equation \[ x^{3} + 1 - \mathrm{i} = 0 \] are \[ \begin{aligned}x_{1}& = \root{6}\of{2}\left (\cos \frac{\pi} {4} + \mathrm{i}\sin \frac{\pi} {4} \right ),& \\x_{2}& = \root{6}\of{2}\left (\cos \frac{11} {12}\pi + \mathrm{i}\sin \frac{11} {12}\pi \right ). \\ \end{aligned} \] Find the third solution.
\(x_{3} = \root{6}\of{2}\left (\cos \frac{19} {12}\pi + \mathrm{i}\sin \frac{19} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{7} {12}\pi + \mathrm{i}\sin \frac{7} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{5} {12}\pi + \mathrm{i}\sin \frac{5} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{13} {12}\pi + \mathrm{i}\sin \frac{13} {12}\pi \right )\)

2010013408

Level: 
C
Three solutions of the equation \[ x^{4} - 2\mathrm{i} = 0 \] are \[\begin{aligned}x_{1} = \root{4}\of{2}\left (\cos \frac{1}{8}\pi + \mathrm{i}\sin \frac{1}{8}\pi \right ),\\ x_{2} = \root{4}\of{2}\left (\cos \frac{5}{8}\pi + \mathrm{i}\sin \frac{5}{8}\pi \right ),\\ x_{3} = \root{4}\of{2}\left (\cos \frac{9}{8}\pi + \mathrm{i}\sin \frac{9}{8}\pi \right ).\\ \end{aligned}\] Find the fourth solution.
\(x_{4} = \root{4}\of{2}\left (\cos \frac{13}{8}\pi + \mathrm{i}\sin \frac{13}{8}\pi \right )\)
\(x_{4} = \root{4}\of{2}\left (\cos \frac{11}{8}\pi + \mathrm{i}\sin \frac{11}{8}\pi \right )\)
\(x_{4} = \root{4}\of{2}\left (\cos \frac{15}{8}\pi + \mathrm{i}\sin \frac{15}{8}\pi \right )\)
\(x_{4} = \root{4}\of{2}\left (\cos \frac{3}{8}\pi + \mathrm{i}\sin \frac{3}{8}\pi \right )\)