2010004614
Level:
A
Evaluate \( (1-\mathrm{i})^{100} \) and write the result in algebraic form.
\( -2^{50}\)
\( 2^{50}\)
\( -2^{50}\mathrm{i} \)
\( 2^{50}\mathrm{i} \)
A
2010004613
Level:
A
Find the absolute value and the value of the argument of \( z=(2+\mathrm{i}\sqrt{12})^5\).
\( |z|=1024\); \(\varphi = \frac{5}{3}\pi\)
\( |z|=512\); \(\varphi = \frac{5}{3}\pi\)
\( |z|=1024\); \(\varphi = \frac{\pi}{3}\)
\( |z|=4\); \(\varphi = \frac{5}{3}\pi\)
2010004612
Level:
A
Find the absolute value and the value of the argument of \( z=(1-\mathrm{i}\sqrt{3})^4\).
\( |z|=16\); \(\varphi = \frac{2}{3}\pi\)
\( |z|=8\); \(\varphi = \frac{\pi}{3}\)
\( |z|=256\); \(\varphi = \frac{2}{3}\pi\)
\( |z|=2\); \(\varphi = \frac{2}{3}\pi\)
2010004611
Level:
A
Identify the complex number that equals to \(( -1+ \mathrm{i}\sqrt{3})^{67} \).
\(2^{66}(-1+\mathrm{i}\sqrt{3})\)
\(2^{66}(1-\mathrm{i}\sqrt{3})\)
\(2^{66}(\sqrt{3}+\mathrm{i})\)
\(2^{66}(\sqrt{3}-\mathrm{i})\)
2010004610
Level:
A
Identify the complex number that equals to \((\sqrt{3} - \mathrm{i})^{37} \).
\(2^{36}(\sqrt{3}-\mathrm{i})\)
\(2^{36}(-1+\mathrm{i}\sqrt{3})\)
\(2^{36}(\sqrt{3}+\mathrm{i})\)
\(2^{36}(-1-\mathrm{i}\sqrt{3})\)
2010004609
Level:
A
Evaluate \( (2+2\mathrm{i})^{100} \) and write the result in algebraic form.
\( -2^{150} \)
\( -2^{150}\mathrm{i} \)
\( 2^{150}\mathrm{i} \)
\( 2^{150}\)
2010004608
Level:
A
Evaluate \( (1+\mathrm{i})^{22} \) and write the result in algebraic form.
\( -2^{11}\mathrm{i} \)
\( 2^{11}\mathrm{i} \)
\( 2^{22}\mathrm{i} \)
\( -2^{22}\mathrm{i} \)
2010004607
Level:
A
Evaluate \( (\cos 27^{\circ} + \mathrm{i}\sin 27^{\circ})^{5} \) and find the algebraic form of the result.
\( -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}\mathrm{i}\)
\( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}\mathrm{i}\)
\( -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}\mathrm{i}\)
\( \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}\mathrm{i}\)
2010004606
Level:
A
Evaluate \( (\cos 33^{\circ} + \mathrm{i}\sin 33^{\circ})^{10} \) and find the algebraic form of the result.
\( \frac{\sqrt{3}}{2} - \frac{1}{2}\mathrm{i}\)
\( \frac{\sqrt{3}}{2} + \frac{1}{2}\mathrm{i}\)
\( \frac{1}{2} + \frac{\sqrt{3}}{2}\mathrm{i}\)
\( \frac{1}{2} - \frac{\sqrt{3}}{2}\mathrm{i}\)
2010004605
Level:
A
Find the algebraic form of the complex number:
\[
\left (\frac{1}
{2} +\sin \frac{\pi }
{6} + \mathrm{i}\sin \frac{\pi}{2} \right )^{3}
\]
\(- 2 +2\mathrm{i}\)
\(2 -2\mathrm{i}\)
\(-2 - 2\mathrm{i}\)
\(2 + 2\mathrm{i}\)