9000037402
Level:
A
Evaluate \(\mathrm{i}^{7}\).
\(-\mathrm{i}\)
\(- 1\)
\(\mathrm{i}\)
\(1\)
Complex numbers in algebraic and polar form
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.
\[
2\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )
\]
\(-\sqrt{2} + \mathrm{i}\sqrt{2}\)
\(\sqrt{2} -\mathrm{i}\sqrt{2}\)
\(2 + 2\mathrm{i}\)
\(2 - 2\mathrm{i}\)
9000037502
Level:
A
Find the total sum of the complex numbers
\(a\),
\(b\) and
\(c\).
\[
a = 3 + \sqrt{2}\mathrm{i},\quad b = 1 - 4\mathrm{i},\quad c = \sqrt{3} - 3\mathrm{i}
\]
\(4 + \sqrt{3} + \mathrm{i}(\sqrt{2} - 7)\)
\(4 + \mathrm{i}\sqrt{3}\)
\(4 + \sqrt{2} + \mathrm{i}(\sqrt{3} - 3)\)
\(4 + \sqrt{3} -\mathrm{i}(\sqrt{2} - 7)\)
9000037506
Level:
A
Given complex numbers
\[
a = 3 + 5\mathrm{i}\text{, }\quad b = 2 -\mathrm{i}\text{, }
\]
find the quotient \(\frac{a}
{b}\).
\(\frac{1}
{5} + \mathrm{i}\frac{13}
{5} \)
\(\frac{1}
{3} + \mathrm{i}\frac{13}
{3} \)
\(\frac{1}
{5} + \mathrm{i}\frac{7}
{5}\)
\(\frac{1}
{3} + \mathrm{i}\frac{7}
{3}\)