9000070103
Level:
A
Evaluate the following complex number.
\[
\left (\cos \pi + \mathrm{i}\sin \pi \right )^{9}
\]
\(- 1\)
\(1\)
\(\mathrm{i}\)
\(-\mathrm{i}\)
Powers and roots of complex numbers
9000070104
Level:
A
Evaluate the following complex number.
\[
\left (\sin 2\pi + \mathrm{i}\cos 2\pi \right )^{11}
\]
\(-\mathrm{i}\)
\(- 1\)
\(1\)
\(\mathrm{i}\)
9000070105
Level:
A
Evaluate the following complex number.
\[
\mathrm{i}^{13}
\]
\(\cos \frac{\pi }
{2} + \mathrm{i}\sin \frac{\pi }
{2}\)
\(\cos \frac{\pi }
{2} -\mathrm{i}\sin \frac{\pi }
{2}\)
\(\sin \frac{\pi }
{2} + \mathrm{i}\cos \frac{\pi }
{2}\)
\(\cos \frac{3\pi }
{2} + \mathrm{i}\sin \frac{3\pi }
{2}\)
9000070106
Level:
A
Evaluate the following complex number.
\[
(1 -\mathrm{i})^{8}
\]
\(16\)
\(- 16\mathrm{i}\)
\(16\mathrm{i}\)
\(- 16\)
9000070107
Level:
A
Find the algebraic form of the complex number:
\[
\left (\frac{1}
{2} +\cos \frac{\pi }
{3} + \mathrm{i}\cos 2\pi \right )^{5}
\]
\(- 4 - 4\mathrm{i}\)
\(- 4 + 4\mathrm{i}\)
\(4 - 4\mathrm{i}\)
\(4 + 4\mathrm{i}\)
9000070108
Level:
A
Evaluate the following complex number.
\[
\left (\frac{1}
{2} + \frac{\sqrt{3}}
{2} \mathrm{i}\right )^{6}
\]
\(1\)
\(- 1\)
\(\mathrm{i}\)
\(-\mathrm{i}\)
9000070109
Level:
A
Evaluate the following complex number.
\[
\left (\sqrt{3} -\mathrm{i}\right )^{3}
\]
\(- 8\mathrm{i}\)
\(8\)
\(- 8\)
\(8\mathrm{i}\)
1003109301
Level:
B
Consider the equation \( x^4 + 81 = 0 \). Find the sum of all its roots in the set of complex numbers.
\( 0 \)
\( 3 \)
\( -3 \)
\( 81 \)
\( - 81 \)
1003109302
Level:
B
Consider the equation \( x^4 - 81 = 0 \). Find the product of all its roots in the set of complex numbers.
\( -81 \)
\( 81 \)
\( -3 \)
\( 3 \)
\( 0 \)
1003109305
Level:
B
Solve the following equation in the set of complex numbers. (Solve the equation by substitution.)
\[ (2x + 3)^4 - 256 = 0 \]
\( \left\{-\frac72;\frac12;-\frac32\pm2\mathrm{i} \right\} \)
\( \left\{-\frac72;\frac12;\frac32\pm2\mathrm{i} \right\} \)
\( \left\{\frac72;-\frac12;\frac32\pm2\mathrm{i} \right\} \)
\( \left\{\frac72;-\frac12;-\frac32\pm2\mathrm{i} \right\} \)