Complex numbers in algebraic and polar form
Absolute Value and Argument of a Complex Number
Submitted by vladimir.arzt on Sun, 10/13/2024 - 21:102010013113
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
\(z=\frac{\mathrm{i}^{12}+1}
{\mathrm{i}^{11}+1} \).
\(\sqrt{2}\left (\cos \frac{\pi }
{4} + \mathrm{i}\sin \frac{\pi }
{4}\right )\)
\(\cos \frac{\pi }
{2} + \mathrm{i}\sin \frac{\pi }
{2}\)
\(\sqrt{2}\left (\cos \frac{5\pi }
{4} + \mathrm{i}\sin \frac{5\pi }
{4}\right )\)
\(\sqrt{2}\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )\)
2010013110
Level:
B
Let \(z=\frac{1}{\cos \frac{7\pi}{6} + \mathrm{i} \sin\frac{7\pi}{6}}\). Which of the following numbers is not the value of the argument of \(z\)?
\(210^\circ\)
\(150^\circ\)
\(-\frac{7\pi}{6}\)
\(\frac{5\pi}{6}\)
2010013109
Level:
B
Let \(z=\frac{1}{\cos \frac{2\pi}{3} + \mathrm{i} \sin\frac{2\pi}{3}}\). Which of the following numbers is not the value of the argument of \(z\)?
\(120^\circ\)
\(-\frac{2\pi}{3}\)
\(\frac{4\pi}{3}\)
\(240^\circ\)
2010013108
Level:
C
Solve the following equation for \(z\in \mathbb{C}\). By \(\overline{z }\) the complex conjugate of \(z \) is denoted.
\[
2z - 3\overline{z } = 10 - 15\mathrm{i}
\]
\(-10 - 3\mathrm{i}\)
\(-10 + 15\mathrm{i}\)
\(10 + 3\mathrm{i}\)
\(-10 + 3\mathrm{i}\)
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 \(\frac{z_1}{z_2}\).
\(\cos \frac{7\pi}{6} + \mathrm{i} \sin \frac{7\pi}{6}\)
\(\cos \frac{\pi}{6} +\mathrm{i} \sin \frac{\pi}{6}\)
\( \frac{\sqrt{3}}{2} + \frac{\mathrm{i}}{2}\)
\(\cos \left(-\frac{\pi}{6}\right) - \mathrm{i} \sin \left(-\frac{\pi}{6}\right)\)