9000039102
Level:
B
Identify a complex number with absolute value different from
\(1\).
\(1 + \mathrm{i}\)
\(\frac{1}
{2} -\frac{\sqrt{3}}
{2} \mathrm{i}\)
\(-\frac{3}
{5} -\frac{4}
{5}\mathrm{i}\)
\(-\mathrm{i}\)
Complex numbers in algebraic and polar form
9000039101
Level:
B
Find the polar form of the complex number
\(z=\frac{\mathrm{i}^{14}-1}
{\mathrm{i}^{9}+1} \).
\(\sqrt{2}\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )\)
\(\sqrt{2}\left (\cos \frac{5\pi }
{4} + \mathrm{i}\sin \frac{5\pi }
{4}\right )\)
\(\sqrt{2}\left (\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )\)
\(\sqrt{2}\left (\cos \frac{7\pi }
{4} + \mathrm{i}\sin \frac{7\pi }
{4}\right )\)
9000039103
Level:
C
Assuming \(x\in \mathbb{R}\),
\(y\in \mathbb{R}\), solve
the following equation.
\[
(2 + 5\mathrm{i})x + (1 -\mathrm{i})y = 13\mathrm{i} + 8
\]
\(x = 3,\ y = 2\)
\(x = 13,\ y = 8\)
\(x = 8,\ y = 13\)
\(x = 2,\ y = 3\)
9000039104
Level:
C
Assuming \(x\in \mathbb{R}\),
\(y\in \mathbb{R}\), solve
the following equation.
\[
(3 - 2\mathrm{i})x + (5 - 7\mathrm{i})y = 1 + 3\mathrm{i}
\]
\(x = 2,\ y = -1\)
\(x = -1,\ y = 2\)
\(x = -2,\ y = -1\)
\(x = -1,\ y = -2\)
9000039109
Level:
C
Assuming \(z\in \mathbb{C}\),
solve the following equation.
\[
2z -\overline{iz} = 1 -\mathrm{i}
\]
\(z = 1 -\mathrm{i}\)
\(z = 1 + \mathrm{i}\)
\(z = \frac{1}
{3} -\frac{1}
{3}\mathrm{i}\)
\(z = -\frac{1}
{3} + \frac{1}
{3}\mathrm{i}\)
9000039110
Level:
C
Assuming \(z\in \mathbb{C}\),
solve the following equation.
\[
\left (1 + \mathrm{i}\sqrt{3}\right )z = 1 -\mathrm{i}\sqrt{3}
\]
\(z = -\frac{1}
{2} -\frac{\sqrt{3}}
{2} \mathrm{i}\)
\(z = \frac{\sqrt{3}}
{2} + \frac{1}
{2}\mathrm{i}\)
\(z = -\frac{1}
{2} + \frac{\sqrt{3}}
{2} \mathrm{i}\)
\(z = -\frac{\sqrt{3}}
{2} + \frac{1}
{2}\mathrm{i}\)
9000039108
Level:
C
Assuming \(z\in \mathbb{C}\),
solve the following equation. By \(\overline{z }\) the complex conjugate of \(z \) is denoted.
\[
2z -\mathrm{i}\, \overline{z} = 1 -\mathrm{i}
\]
\(z = \frac{1}
{3} -\frac{1}
{3}\mathrm{i}\)
\(z = 1 + \mathrm{i}\)
\(z = -\frac{3}
{5} + \frac{6}
{5}\mathrm{i}\)
\(z = -\frac{1}
{5} -\frac{3}
{5}\mathrm{i}\)
9000038603
Level:
B
Find the polar form of the following complex number.
\[
\frac{\sqrt{2}}
{2} + \mathrm{i}\frac{\sqrt{6}}
{2}
\]
\(\sqrt{2}\left (\cos \frac{\pi }{3} + \mathrm{i}\sin \frac{\pi }{3}\right )\)
\(\sqrt{2}\left (\cos \frac{2\pi }
{3} + \mathrm{i}\sin \frac{2\pi }
{3}\right )\)
\(2\left (\cos \frac{4\pi }
{3} + \mathrm{i}\sin \frac{4\pi }
{3}\right )\)
\(2\left (\cos \frac{3\pi }
{2} + \mathrm{i}\sin \frac{3\pi }
{2}\right )\)
9000038604
Level:
B
Find the polar form of the following complex number.
\[
\frac{\sqrt{3}}
{\sqrt{2}} + \mathrm{i}\frac{\sqrt{3}}
{\sqrt{2}}
\]
\(\sqrt{3}\left (\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )\)
\(\sqrt{3}\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )\)
\(\sqrt{2}\left (\cos \frac{\pi }{3} + \mathrm{i}\sin \frac{\pi }{3}\right )\)
\(\sqrt{2}\left (\cos \frac{2\pi }
{3} + \mathrm{i}\sin \frac{2\pi }
{3}\right )\)
9000038605
Level:
B
Find the polar form of the following complex number.
\[
-\frac{\sqrt{5}}
{2} + \mathrm{i}\frac{\sqrt{15}}
{2}
\]
\(\sqrt{5}\left (\cos \frac{2\pi }
{3} + \mathrm{i}\sin \frac{2\pi }
{3}\right )\)
\(\sqrt{5}\left (\cos \frac{\pi }{3} + \mathrm{i}\sin \frac{\pi }{3}\right )\)
\(\sqrt{5}\left (\cos \frac{2\pi }
{5} + \mathrm{i}\sin \frac{2\pi }
{5}\right )\)
\(\sqrt{5}\left (\cos \frac{3\pi }
{2} + \mathrm{i}\sin \frac{3\pi }
{2}\right )\)