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 \(\frac{a} {b}\).
\(\frac{\sqrt{6}-2} {3} + \mathrm{i}\frac{2\sqrt{2}+\sqrt{3}} {3} \)
\(\frac{\sqrt{6}-2} {3} -\mathrm{i}\frac{2\sqrt{2}+\sqrt{3}} {3} \)
\(\frac{\sqrt{6}-3} {2} + \mathrm{i}\frac{2\sqrt{2}+\sqrt{3}} {2} \)
\(\frac{\sqrt{6}-2} {2} -\mathrm{i}\frac{2\sqrt{2}+\sqrt{3}} {2} \)

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. \[ \frac{\sqrt{2}} {2} + \mathrm{i}\frac{\sqrt{6}} {2} \]
\(\sqrt{2}\left (\cos \frac{\pi }{3} + \mathrm{i}\sin \frac{\pi }{3}\right )\)
\(\sqrt{2}\left (\cos \frac{2\pi } {3} + \mathrm{i}\sin \frac{2\pi } {3}\right )\)
\(2\left (\cos \frac{4\pi } {3} + \mathrm{i}\sin \frac{4\pi } {3}\right )\)
\(2\left (\cos \frac{3\pi } {2} + \mathrm{i}\sin \frac{3\pi } {2}\right )\)

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. \[ -\frac{\sqrt{5}} {2} + \mathrm{i}\frac{\sqrt{15}} {2} \]
\(\sqrt{5}\left (\cos \frac{2\pi } {3} + \mathrm{i}\sin \frac{2\pi } {3}\right )\)
\(\sqrt{5}\left (\cos \frac{\pi }{3} + \mathrm{i}\sin \frac{\pi }{3}\right )\)
\(\sqrt{5}\left (\cos \frac{2\pi } {5} + \mathrm{i}\sin \frac{2\pi } {5}\right )\)
\(\sqrt{5}\left (\cos \frac{3\pi } {2} + \mathrm{i}\sin \frac{3\pi } {2}\right )\)