2010004610
Level:
A
Identify the complex number that equals to \((\sqrt{3} - \mathrm{i})^{37} \).
\(2^{36}(\sqrt{3}-\mathrm{i})\)
\(2^{36}(-1+\mathrm{i}\sqrt{3})\)
\(2^{36}(\sqrt{3}+\mathrm{i})\)
\(2^{36}(-1-\mathrm{i}\sqrt{3})\)
Powers and roots of complex numbers
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:
\[
\left (\frac{1}
{2} +\sin \frac{\pi }
{6} + \mathrm{i}\sin \frac{\pi}{2} \right )^{3}
\]
\(- 2 +2\mathrm{i}\)
\(2 -2\mathrm{i}\)
\(-2 - 2\mathrm{i}\)
\(2 + 2\mathrm{i}\)
2010004604
Level:
A
Find the algebraic form of the complex number:
\[
\left (\cos \frac{\pi }
{3} + \mathrm{i}\sin \frac{\pi }
{3}\right )^{5}
\]
\(\frac{1}
{2} - \mathrm{i}\frac{\sqrt{3}}
{2} \)
\(\frac{\sqrt{3}}
{2} +\mathrm{i} \frac{1}
{2} \)
\(\frac{\sqrt{3}}
{2} -\mathrm{i} \frac{1}
{2} \)
\(\frac{1}
{2} + \mathrm{i}\frac{\sqrt{3}}
{2} \)
2010004603
Level:
A
Identify a complex number that is not equal to \(\left (\cos \frac{\pi }{6} + \mathrm{i}\sin \frac{\pi }{6}\right )^{13}\).
\( \frac{\sqrt{3}}{2} - \frac{1}{2}\mathrm{i}\)
\( \frac{\sqrt{3}}{2} + \frac{1}{2}\mathrm{i}\)
\( \cos \frac{\pi }{6} + \mathrm{i}\sin \frac{\pi }{6} \)
\( \cos \frac{25\pi }{6} + \mathrm{i}\sin \frac{25\pi }{6} \)
2010004602
Level:
A
Identify a complex number that is not equal to \(\left (\cos \frac{\pi }{3} + \mathrm{i}\sin \frac{\pi }{3}\right )^{19}\).
\( \frac{1}{2} - \frac{\sqrt{3}}{2}\mathrm{i}\)
\( \cos \frac{\pi }{3} + \mathrm{i}\sin \frac{\pi }{3} \)
\( \frac{1}{2} + \frac{\sqrt{3}}{2}\mathrm{i}\)
\( \cos \left(-\frac{5\pi }{3}\right) + \mathrm{i}\sin \left(-\frac{5\pi }{3}\right) \)
2010004601
Level:
A
Evaluate \(\left (\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )^{40}\) and find the algebraic form of the result.
\(1\)
\(-1\)
\( -\mathrm{i}\)
\( \mathrm{i} \)