9000037410
Level:
A
Evaluate \(\left (1 -\mathrm{i}\right )^{3}\).
\(- 2 - 2\mathrm{i}\)
\(2 + 2\mathrm{i}\)
\(1 + \mathrm{i}\)
\(\mathrm{i}\)
A
9000037401
Level:
A
Evaluate \(\mathrm{i}^{50}\).
\(- 1\)
\(\mathrm{i}\)
\(1\)
\(-\mathrm{i}\)
9000037402
Level:
A
Evaluate \(\mathrm{i}^{7}\).
\(-\mathrm{i}\)
\(- 1\)
\(\mathrm{i}\)
\(1\)
9000037501
Level:
A
Find the absolute value of the following complex number.
\[
3 + \sqrt{2}\mathrm{i}
\]
\(\sqrt{11}\)
\(\sqrt{13}\)
\(3\)
\(3\sqrt{2}\)
9000037502
Level:
A
Find the total sum of the complex numbers
\(a\),
\(b\) and
\(c\).
\[
a = 3 + \sqrt{2}\mathrm{i},\quad b = 1 - 4\mathrm{i},\quad c = \sqrt{3} - 3\mathrm{i}
\]
\(4 + \sqrt{3} + \mathrm{i}(\sqrt{2} - 7)\)
\(4 + \mathrm{i}\sqrt{3}\)
\(4 + \sqrt{2} + \mathrm{i}(\sqrt{3} - 3)\)
\(4 + \sqrt{3} -\mathrm{i}(\sqrt{2} - 7)\)
9000037504
Level:
A
Given complex numbers
\[
a = 5 + 2\mathrm{i},\quad b = 3 -\mathrm{i},\quad c = \mathrm{i}\text{,}
\]
find the product \(abc\).
\(- 1 + 17\mathrm{i}\)
\(1 - 17\mathrm{i}\)
\(- 1 - 17\mathrm{i}\)
\(1 + 17\mathrm{i}\)
9000037505
Level:
A
Find the complex conjugate of the following complex number.
\[
-2\sqrt{3} -\mathrm{i}
\]
\(- 2\sqrt{3} + \mathrm{i}\)
\(2\sqrt{3} -\mathrm{i}\)
\(11\)
\(10\mathrm{i}\)
9000037404
Level:
A
Given \(z = \sqrt{2}\left (\cos \frac{\pi }{3} -\mathrm{i}\sin \frac{\pi }{3}\right )\),
find \(z^{2}\).
\(- 1 -\mathrm{i}\sqrt{3}\)
\(1 + \mathrm{i}\sqrt{3}\)
\(- 2 -\mathrm{i}\sqrt{2}\)
\(2 + \mathrm{i}\sqrt{2}\)
9000037506
Level:
A
Given complex numbers
\[
a = 3 + 5\mathrm{i}\text{, }\quad b = 2 -\mathrm{i}\text{, }
\]
find the quotient \(\frac{a}
{b}\).
\(\frac{1}
{5} + \mathrm{i}\frac{13}
{5} \)
\(\frac{1}
{3} + \mathrm{i}\frac{13}
{3} \)
\(\frac{1}
{5} + \mathrm{i}\frac{7}
{5}\)
\(\frac{1}
{3} + \mathrm{i}\frac{7}
{3}\)
9000037503
Level:
A
Given complex numbers
\[
a = \sqrt{2} + \sqrt{3}\mathrm{i},\quad b = \sqrt{2} -\sqrt{3}\mathrm{i},
\]
find the product \(ab\).
\(5\)
\(2\)
\(\sqrt{2} + \mathrm{i}\sqrt{3}\)
\(\sqrt{2} -\mathrm{i}\sqrt{3}\)