9000035801
Level:
A
Find the complex conjugate of the following complex number.
\[
\mathrm{i} + 3\mathrm{i}(2 -\mathrm{i})^{2} - 4(1 -\mathrm{i})^{3}
\]
\(20 - 18\mathrm{i}\)
\(20 - 24\mathrm{i}\)
\(20 + 18\mathrm{i}\)
\(- 8 + 26\mathrm{i}\)
Complex numbers in algebraic and polar form
9000035803
Level:
A
Given the complex number \(z = -1 + 2\mathrm{i}\),
find the imaginary part of the complex number
\(\frac{1}
{z}\).
\(-\frac{2}
{5}\)
\(\frac{1}
{2}\)
\(\frac{2}
{5}\)
\(-\frac{1}
{2}\)
9000035804
Level:
A
Find the algebraic form of the following complex number. By \(\overline{z }\) the complex conjugate of \(z \) is denoted.
\[
\overline{\overline{(2 + \mathrm{i}) }\; \overline{(3 + 2\mathrm{i}) } }
\]
\(4 + 7\mathrm{i}\)
\(8 + 7\mathrm{i}\)
\(8 - 7\mathrm{i}\)
\(4 - 7\mathrm{i}\)
9000035805
Level:
B
Given the complex numbers
\[
\text{$a = 2\left (\cos \frac{2\pi }
{3} + \mathrm{i}\sin \frac{2\pi }
{3}\right )$, $b = \sqrt{2}\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )$,}
\]
find the product \(ab\).
\(2\sqrt{2}\left (\cos \frac{17\pi }
{12} + \mathrm{i}\sin \frac{17\pi }
{12}\right )\)
\(2\sqrt{2}\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )\)
\(2\sqrt{2}\left (\cos \frac{5\pi }
{7} + \mathrm{i}\sin \frac{5\pi }
{7}\right )\)
\(2\sqrt{2}\left (\cos \frac{5\pi }
{12} + \mathrm{i}\sin \frac{5\pi }
{12}\right )\)
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
\(z = 2\left (\cos \pi + \mathrm{i}\sin \pi \right )\).
\(- 2\)
\(2\)
\(- 2\mathrm{i}\)
\(2\mathrm{i}\)
9000034804
Level:
A
Find the absolute value of the complex number
\(z = 3 -\mathrm{i}\).
\(\sqrt{10}\)
\(2\)
\(2\sqrt{2}\)
\(\sqrt{2}\)
9000034809
Level:
B
Given complex numbers \(z_{1} = 2\left (\cos \frac{\pi }{6} + \mathrm{i}\sin \frac{\pi }{6}\right )\)
and \(z_{2} = \sqrt{3}\left (\cos \frac{4\pi }
{3} + \mathrm{i}\sin \frac{4\pi }
{3}\right )\), find the angle in the
polar form of the product \(z_{1}z_{2}\).
\(\frac{3\pi }
{2}\)
\(\frac{2}
{9}\pi \)
\(\frac{5}
{9}\pi \)
\(3\pi \)
9000034810
Level:
B
Given complex numbers \(z_{1} = 2\left (\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )\)
and \(z_{2} = \sqrt{2}\left (\cos \frac{7\pi }
{4} + \mathrm{i}\sin \frac{7\pi }
{4}\right )\),
find the angle in the polar form of the quotient
\(\frac{z_{1}}
{z_{2}} \).
\(\frac{\pi }
{2}\)
\(- \frac{\pi } {2}\)
\(-\frac{3}
{2}\pi \)
\(\frac{3}
{2}\pi \)