9000037505
Level:
A
Find the complex conjugate of the following complex number.
\[
-2\sqrt{3} -\mathrm{i}
\]
\(- 2\sqrt{3} + \mathrm{i}\)
\(2\sqrt{3} -\mathrm{i}\)
\(11\)
\(10\mathrm{i}\)
Complex Numbers in Algebraic and Polar Form
9000037507
Level:
A
Given complex numbers
\[
a = \sqrt{3} + 2\mathrm{i}\text{, }\quad b = \sqrt{2} -\mathrm{i}\text{, }
\]
find the quotient \(\frac{a}
{b}\).
\(\frac{\sqrt{6}-2}
{3} + \mathrm{i}\frac{2\sqrt{2}+\sqrt{3}}
{3} \)
\(\frac{\sqrt{6}-2}
{3} -\mathrm{i}\frac{2\sqrt{2}+\sqrt{3}}
{3} \)
\(\frac{\sqrt{6}-3}
{2} + \mathrm{i}\frac{2\sqrt{2}+\sqrt{3}}
{2} \)
\(\frac{\sqrt{6}-2}
{2} -\mathrm{i}\frac{2\sqrt{2}+\sqrt{3}}
{2} \)
9000037508
Level:
B
Find the absolute value of the following complex number.
\[
\sqrt{2}\left (\cos \frac{\pi }
{3} + \mathrm{i}\sin \frac{\pi }
{3}\right )
\]
\(\sqrt{2}\)
\(\sqrt{2} + 2\)
\(2\)
\(\sqrt{2} - 2\)
9000037509
Level:
B
Given complex numbers
\[
a = 3\left (\cos \frac{\pi }
{3} + \mathrm{i}\sin \frac{\pi }
{3}\right ),\quad b = \sqrt{2}\left (\cos \frac{2\pi }
{3} + \mathrm{i}\sin \frac{2\pi }
{3}\right )
\]
find the product \(ab\).
\(- 3\sqrt{2}\)
\(3\sqrt{2}\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )\)
\(3\sqrt{2}\left (\cos \frac{\pi }{2} -\mathrm{i}\sin \frac{\pi }{2}\right )\)
\(- 3\sqrt{2}\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )\)
9000037510
Level:
B
Given complex numbers
\[
a = \left (\cos \frac{\pi }
{3} + \mathrm{i}\sin \frac{\pi }
{3}\right ),\quad b = \sqrt{2}\left (\cos \frac{2\pi }
{3} + \mathrm{i}\sin \frac{2\pi }
{3}\right )
\]
find the quotient \(\frac{a}
{b}\).
\(\frac{\sqrt{2}}
{2} \left (\cos \left (-\frac{\pi }
{3}\right ) + \mathrm{i}\sin \left (-\frac{\pi }
{3}\right )\right )\)
\(\frac{\sqrt{2}}
{2} \left (\cos \left (-\frac{\pi }
{3}\right ) -\mathrm{i}\sin \left (-\frac{\pi }
{3}\right )\right )\)
\(-\frac{\sqrt{2}}
{2} \left (\cos \left (-\frac{\pi }
{3}\right ) -\mathrm{i}\sin \left (-\frac{\pi }
{3}\right )\right )\)
\(-\frac{\sqrt{2}}
{2} \left (\cos \left (-\frac{\pi }
{3}\right ) + \mathrm{i}\sin \left (-\frac{\pi }
{3}\right )\right )\)
9000038601
Level:
B
Find the polar form of the following complex number.
\[
-\frac{1}
{2} + \mathrm{i}\frac{\sqrt{3}}
{2}
\]
\(\cos \frac{2\pi }
{3} + \mathrm{i}\sin \frac{2\pi }
{3}\)
\(\cos \frac{\pi }
{3} + \mathrm{i}\sin \frac{\pi }
{3}\)
\(\cos \left (-\frac{\pi }{3}\right ) + \mathrm{i}\sin \left (-\frac{\pi }{3}\right )\)
\(\cos \frac{3\pi }
{2} + \mathrm{i}\sin \frac{3\pi }
{2}\)
9000038602
Level:
B
Find the polar form of the following complex number.
\[
\frac{1}
{2} + \mathrm{i}\frac{\sqrt{3}}
{2}
\]
\(\cos \frac{\pi }{3} + \mathrm{i}\sin \frac{\pi }{3}\)
\(\cos \frac{2\pi }
{3} + \mathrm{i}\sin \frac{2\pi }
{3}\)
\(\cos \frac{3\pi }
{2} + \mathrm{i}\sin \frac{3\pi }
{2}\)
\(\cos \left (-\frac{\pi }{3}\right ) + \mathrm{i}\sin \left (-\frac{\pi }{3}\right )\)
9000038603
Level:
B
Find the polar form of the following complex number.
\[
\frac{\sqrt{2}}
{2} + \mathrm{i}\frac{\sqrt{6}}
{2}
\]
\(\sqrt{2}\left (\cos \frac{\pi }{3} + \mathrm{i}\sin \frac{\pi }{3}\right )\)
\(\sqrt{2}\left (\cos \frac{2\pi }
{3} + \mathrm{i}\sin \frac{2\pi }
{3}\right )\)
\(2\left (\cos \frac{4\pi }
{3} + \mathrm{i}\sin \frac{4\pi }
{3}\right )\)
\(2\left (\cos \frac{3\pi }
{2} + \mathrm{i}\sin \frac{3\pi }
{2}\right )\)
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.
\[
3z - 2\overline{z } = 8 - 10\mathrm{i}
\]
\(8 - 2\mathrm{i}\)
\(1 + 5\mathrm{i}\)
\(8 - 10\mathrm{i}\)
\(2 + 2\mathrm{i}\)