Complex numbers in algebraic and polar form

2010013113

Level: 
B
Find the polar form of the opposite number of \(z=-\frac{\sqrt5}{2} + \mathrm{i}\frac{\sqrt{15}}{2}\).
\(\sqrt{5}\left (\cos \left(-\frac{\pi} {3}\right) + \mathrm{i}\sin \left(-\frac{\pi} {3}\right )\right)\)
\(\sqrt{5}\left (\cos \frac{2\pi } {3} + \mathrm{i}\sin \frac{2\pi } {3}\right )\)
\(\sqrt{10}\left (\cos \frac{5\pi } {3} + \mathrm{i}\sin \frac{5\pi } {3}\right )\)
\(\sqrt{5}\left (\cos \frac{\pi } {3} + \mathrm{i}\sin \frac{\pi } {3}\right )\)

2010013112

Level: 
B
Find the polar form of the complex conjugate of \(z=-\frac{\sqrt5}{2} + \mathrm{i}\frac{\sqrt{15}}{2}\).
\(\sqrt{5}\left (\cos \left(-\frac{2\pi} {3}\right) + \mathrm{i}\sin \left(-\frac{2\pi} {3}\right )\right)\)
\(\sqrt{5}\left (\cos \frac{2\pi } {3} + \mathrm{i}\sin \frac{2\pi } {3}\right )\)
\(\sqrt{10}\left (\cos \frac{4\pi } {3} + \mathrm{i}\sin \frac{4\pi } {3}\right )\)
\(\sqrt{5}\left (\cos \frac{\pi } {3} + \mathrm{i}\sin \frac{\pi } {3}\right )\)

2010013111

Level: 
B
Find the polar form of the complex number \(z=\frac{\mathrm{i}^{12}+1} {\mathrm{i}^{11}+1} \).
\(\sqrt{2}\left (\cos \frac{\pi } {4} + \mathrm{i}\sin \frac{\pi } {4}\right )\)
\(\cos \frac{\pi } {2} + \mathrm{i}\sin \frac{\pi } {2}\)
\(\sqrt{2}\left (\cos \frac{5\pi } {4} + \mathrm{i}\sin \frac{5\pi } {4}\right )\)
\(\sqrt{2}\left (\cos \frac{3\pi } {4} + \mathrm{i}\sin \frac{3\pi } {4}\right )\)

2010013107

Level: 
C
Let \( z_1 = x^2 + 9y\,\mathrm{i}-10\,\mathrm{i} \) and \( z_2 = 8x-15+ y^2\,\mathrm{i} \). Find all \( [x;y] \in \mathbb{R}\times\mathbb{R} \) such that \( z_1= \overline{z_2} \).
\( [x;y]\in\left\{[3;-10], [3;1], [5;-10], [5;1]\right\} \)
\( [x;y]\in\left\{[-10;3], [1;3], [-10;5], [1;5]\right\} \)
\( [x;y]\in\left\{[3;10], [3;-1], [5;10], [5;-1]\right\} \)
\( [x;y]\in\left\{[-3;-10], [-3;1], [-5;-10], [-5;1]\right\} \)

2010013106

Level: 
B
Let \(z_1 = 1 + \mathrm{i}\sqrt{3}\), \(z_2=\sqrt3 + \mathrm{i}\). Identify a complex number that is not equal to \(\frac{z_1}{z_2}\).
\(\cos \frac{7\pi}{6} + \mathrm{i} \sin \frac{7\pi}{6}\)
\(\cos \frac{\pi}{6} +\mathrm{i} \sin \frac{\pi}{6}\)
\( \frac{\sqrt{3}}{2} + \frac{\mathrm{i}}{2}\)
\(\cos \left(-\frac{\pi}{6}\right) - \mathrm{i} \sin \left(-\frac{\pi}{6}\right)\)

2010013105

Level: 
B
Let \(z_1 = \sqrt3 + \mathrm{i}\), \(z_2=1 + \mathrm{i}\sqrt3\). Identify a complex number that is not equal to \(\frac{z_1}{z_2}\).
\(\cos \frac{5\pi}{6} + \mathrm{i} \sin \frac{5\pi}{6}\)
\(\cos \left(-\frac{\pi}{6}\right) + \mathrm{i} \sin \left(-\frac{\pi}{6}\right)\)
\( \frac{\sqrt{3}}{2} - \frac{\mathrm{i}}{2}\)
\(\cos \frac{\pi}{6} - \mathrm{i} \sin \frac{\pi}{6}\)

2010013104

Level: 
B
Let \( [x;y]\in\mathbb{R}\times\mathbb{R} \), \( z_1 = -2 + xy\,\mathrm{i} \) and \( z_2 = x + y + 8\,\mathrm{i}\). Find all \( [x;y] \) such that \( z_1 \) and \( z_2 \) are the opposite numbers.
\( [x;y] \in\left\{[4;-2],[-2;4]\right\} \)
\( [x;y] \in \left\{[4;-2]\right\} \)
\( [x;y] \in\left\{[-4;2],[2;-4]\right\} \)
\( [x;y] \in \left\{[-2;4]\right\} \)