9000035707
Level:
A
Find the real part of the complex number
\(z= 2 + 2\mathrm{i}^{2} + \mathrm{i}^{3} -\mathrm{i}^{4} + 2\mathrm{i}^{5}\).
\(- 1\)
\(1\)
\(5\)
\(- 3\)
Complex Numbers in Algebraic and Polar Form
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.
\(5\)
\(\sqrt{5}\)
\(3\)
\(4\)
9000034803
Level:
A
Find the complex conjugate of \(z = 1 - 3\mathrm{i}\).
\(1 + 3\mathrm{i}\)
\(- 1 - 3\mathrm{i}\)
\(- 1 + 3\mathrm{i}\)
\(1 - 3\mathrm{i}\)
9000034802
Level:
A
Find the opposite number to the complex number
\(z = 3 -\mathrm{i}\).
\(- 3 + \mathrm{i}\)
\(- 3 -\mathrm{i}\)
\(3 + \mathrm{i}\)
\(3 -\mathrm{i}\)
9000034806
Level:
A
Simplify \(\mathrm{i}^{15}\).
\(-\mathrm{i}\)
\(- 1\)
\(1\)
\(\mathrm{i}\)
9000034805
Level:
A
Find the complex number \(z\)
which satisfies \(2z = 2 - 3\mathrm{i}\).
\(1 -\frac{3}
{2}\mathrm{i}\)
\(- 3\mathrm{i}\)
\(4 - 6\mathrm{i}\)
\(- 1 + \frac{3}
{2}\mathrm{i}\)
9000034807
Level:
B
Find the polar form of the complex number
\(z = 2\mathrm{i}\).
\(2\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )\)
\(\sqrt{2}\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )\)
\(\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\)
\(2\left (\cos 0 + \mathrm{i}\sin 0\right )\)
9000034808
Level:
B
Find the algebraic form of the complex number
\(z = 2\left (\cos \pi + \mathrm{i}\sin \pi \right )\).
\(- 2\)
\(2\)
\(- 2\mathrm{i}\)
\(2\mathrm{i}\)