9000037504
Level:
A
Given complex numbers
\[
a = 5 + 2\mathrm{i},\quad b = 3 -\mathrm{i},\quad c = \mathrm{i}\text{,}
\]
find the product \(abc\).
\(- 1 + 17\mathrm{i}\)
\(1 - 17\mathrm{i}\)
\(- 1 - 17\mathrm{i}\)
\(1 + 17\mathrm{i}\)
A
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 \(\frac{a}
{b}\).
\(\frac{1}
{5} + \mathrm{i}\frac{13}
{5} \)
\(\frac{1}
{3} + \mathrm{i}\frac{13}
{3} \)
\(\frac{1}
{5} + \mathrm{i}\frac{7}
{5}\)
\(\frac{1}
{3} + \mathrm{i}\frac{7}
{3}\)
9000035710
Level:
A
Find the complex conjugate of \(z=\frac{3+\mathrm{i}}
{2-\mathrm{i}} + (\mathrm{i} + 1)(2 + \mathrm{i})\).
\(2 - 4\mathrm{i}\)
\(2 + 4\mathrm{i}\)
\(- 2 - 4\mathrm{i}\)
\(- 2 + 4\mathrm{i}\)
9000035807
Level:
A
Given the complex numbers \(a = 2 - 3\mathrm{i}\),
\(b = 1 + 2\mathrm{i}\), find the
quotient \(\frac{a}
{b}\).
\(-\frac{4}
{5} -\frac{7}
{5}\mathrm{i}\)
\(2 -\frac{3}
{2}\mathrm{i}\)
\(\frac{8}
{5} -\frac{7}
{5}\mathrm{i}\)
\(\frac{4}
{3} + \frac{7}
{3}\mathrm{i}\)
9000035706
Level:
A
Find the absolute value of the complex number
\(z = \frac{2+6\mathrm{i}}
{1-2\mathrm{i}}\).
\(2\sqrt{2}\)
\(2\sqrt{5}\)
\(2\)
\(2\sqrt{3}\)
9000035708
Level:
A
Find the imaginary part of the complex number
\(z=1 + 2\mathrm{i}^{12} + 3\mathrm{i}^{19} -\mathrm{i}^{22} + 2\mathrm{i}^{105}\).
\(- 1\)
\(- 5\)
\(1\)
\(4\)
9000035707
Level:
A
Find the real part of the complex number
\(z= 2 + 2\mathrm{i}^{2} + \mathrm{i}^{3} -\mathrm{i}^{4} + 2\mathrm{i}^{5}\).
\(- 1\)
\(1\)
\(5\)
\(- 3\)
9000035701
Level:
A
What is the algebraic form of the complex number \( A \) graphed in the complex
plane (as shown in the picture)?
\( -3 + 2\mathrm{i}\)
\( 2 - 3\mathrm{i}\)
\( 2 + 3\mathrm{i}\)
\( -3 - 2\mathrm{i}\)