9000037401
Level:
A
Evaluate \(\mathrm{i}^{50}\).
\(- 1\)
\(\mathrm{i}\)
\(1\)
\(-\mathrm{i}\)
Complex numbers in algebraic and polar form
9000038606
Level:
B
Find the algebraic form of the following complex number.
\[
\cos \frac{\pi }
{4} + \mathrm{i}\sin \frac{\pi }
{4}
\]
\(\frac{\sqrt{2}}
{2} + \mathrm{i}\frac{\sqrt{2}}
{2} \)
\(\frac{\sqrt{2}}
{2} -\mathrm{i}\frac{\sqrt{2}}
{2} \)
\(\frac{\sqrt{3}}
{2} + \mathrm{i}\frac{\sqrt{3}}
{2} \)
\(\frac{\sqrt{3}}
{2} -\mathrm{i}\frac{\sqrt{3}}
{2} \)
9000037402
Level:
A
Evaluate \(\mathrm{i}^{7}\).
\(-\mathrm{i}\)
\(- 1\)
\(\mathrm{i}\)
\(1\)
9000038607
Level:
B
Find the algebraic form of the following complex number.
\[
3\left (\cos \frac{\pi }
{2} + \mathrm{i}\sin \frac{\pi }
{2}\right )
\]
\(3\mathrm{i}\)
\(- 3\mathrm{i}\)
\(3 + 3\mathrm{i}\)
\(3 - 3\mathrm{i}\)
9000037408
Level:
B
Find the polar form of the following complex number
\[z=\frac{1}
{\cos \frac{2\pi }
{3} +\mathrm{i}\sin \frac{2\pi }
{3} }. \]
\(\cos \frac{4\pi }
{3} + \mathrm{i}\sin \frac{4\pi }
{3}\)
\(\cos \left (-\frac{4\pi }
{3}\right ) + \mathrm{i}\sin \left (-\frac{4\pi }
{3}\right )\)
\(\cos \frac{3}
{2\pi } + \mathrm{i}\sin \frac{3}
{2\pi }\)
\(\cos \frac{3}
{2\pi } -\mathrm{i}\sin \frac{3}
{2\pi }\)
9000038608
Level:
B
Find the algebraic form of the following complex number.
\[
8(\cos \pi + \mathrm{i}\sin \pi )
\]
\(- 8\)
\(8\)
\(8 + 8\mathrm{i}\)
\(8 - 8\mathrm{i}\)
9000037409
Level:
B
Find the polar form of the complex number
\[z=\frac{1}
{\cos \frac{7\pi }
{6} +\mathrm{i}\sin \frac{7\pi }
{6} }. \]
\(\cos \frac{5\pi }
{6} + \mathrm{i}\sin \frac{5\pi }
{6}\)
\(\cos \left (-\frac{5\pi }
{6}\right ) + \mathrm{i}\sin \left (-\frac{5\pi }
{6}\right )\)
\(\cos \frac{\pi }{6} + \mathrm{i}\sin \frac{\pi }{6}\)
\(\cos \left (-\frac{\pi }{6}\right ) + \mathrm{i}\sin \left (-\frac{\pi }{6}\right )\)
9000038609
Level:
B
Find the algebraic form of the following complex number.
\[
5\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )
\]
\(-\frac{5\sqrt{2}}
{2} + \mathrm{i}\frac{5\sqrt{2}}
{2} \)
\(\frac{5\sqrt{2}}
{2} -\mathrm{i}\frac{5\sqrt{2}}
{2} \)
\(\frac{5}
{2} + \mathrm{i}\frac{5}
{2}\)
\(\frac{5}
{2} -\mathrm{i}\frac{5}
{2}\)
9000037501
Level:
A
Find the absolute value of the following complex number.
\[
3 + \sqrt{2}\mathrm{i}
\]
\(\sqrt{11}\)
\(\sqrt{13}\)
\(3\)
\(3\sqrt{2}\)
9000038610
Level:
B
Find the algebraic form of the following complex number.
\[
2\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )
\]
\(-\sqrt{2} + \mathrm{i}\sqrt{2}\)
\(\sqrt{2} -\mathrm{i}\sqrt{2}\)
\(2 + 2\mathrm{i}\)
\(2 - 2\mathrm{i}\)