9000035706
Level:
A
Find the absolute value of the complex number
\(z = \frac{2+6\mathrm{i}}
{1-2\mathrm{i}}\).
\(2\sqrt{2}\)
\(2\sqrt{5}\)
\(2\)
\(2\sqrt{3}\)
Complex Numbers in Algebraic and Polar Form
9000035708
Level:
A
Find the imaginary part of the complex number
\(z=1 + 2\mathrm{i}^{12} + 3\mathrm{i}^{19} -\mathrm{i}^{22} + 2\mathrm{i}^{105}\).
\(- 1\)
\(- 5\)
\(1\)
\(4\)
9000035707
Level:
A
Find the real part of the complex number
\(z= 2 + 2\mathrm{i}^{2} + \mathrm{i}^{3} -\mathrm{i}^{4} + 2\mathrm{i}^{5}\).
\(- 1\)
\(1\)
\(5\)
\(- 3\)
9000035802
Level:
C
Solve the following equation for \(z\in \mathbb{C}\). By \(\overline{z }\) the complex conjugate of \(z \) is denoted.
\[
3z - 2\overline{z } = 8 - 10\mathrm{i}
\]
\(8 - 2\mathrm{i}\)
\(1 + 5\mathrm{i}\)
\(8 - 10\mathrm{i}\)
\(2 + 2\mathrm{i}\)
9000035701
Level:
A
What is the algebraic form of the complex number \( A \) graphed in the complex
plane (as shown in the picture)?
\( -3 + 2\mathrm{i}\)
\( 2 - 3\mathrm{i}\)
\( 2 + 3\mathrm{i}\)
\( -3 - 2\mathrm{i}\)
9000035702
Level:
A
Find the absolute value of the complex number \( A \) graphed in the complex
plane as shown in the picture.
\(5\)
\(\sqrt{5}\)
\(3\)
\(4\)
9000035703
Level:
A
Find the absolute value of the complex number \( A \) graphed in the complex
plane as shown in the picture.
\(2\sqrt{5}\)
\(2\sqrt{3}\)
\(4\)
\(\sqrt{6}\)
9000035704
Level:
B
Find the polar form of the complex number \( A \) graphed in the complex plane as shown in the picture.
\(z = 2\sqrt{2}\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )\)
\(z = 2\sqrt{2}\left (\cos \frac{\pi }{4} -\mathrm{i}\sin \frac{\pi }{4}\right )\)
\(z = 2\sqrt{2}\left (-\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )\)
\(z = 2\sqrt{2}\left (\cos \frac{5\pi }
{4} + \mathrm{i}\sin \frac{5\pi }
{4}\right )\)
9000035705
Level:
A
Find the absolute value of the complex number
\(z = (1 - 2\mathrm{i})(2 + \mathrm{i})\).
\(5\)
\(3\)
\(\sqrt{10}\)
\(2\sqrt{2}\)
9000035801
Level:
A
Find the complex conjugate of the following complex number.
\[
\mathrm{i} + 3\mathrm{i}(2 -\mathrm{i})^{2} - 4(1 -\mathrm{i})^{3}
\]
\(20 - 18\mathrm{i}\)
\(20 - 24\mathrm{i}\)
\(20 + 18\mathrm{i}\)
\(- 8 + 26\mathrm{i}\)