9000037406
Level:
A
Evaluate \(\left (\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )^{40}\).
\(1\)
\(1 + \mathrm{i}\)
\(\mathrm{i}\)
\(1 -\mathrm{i}\)
A
9000037407
Level:
A
Evaluate \(\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )^{13}\) and find the algebraic form of the result.
\(\mathrm{i}\)
\(1 + 2\mathrm{i}\)
\(1 -\mathrm{i}\)
\(1\)
9000037410
Level:
A
Evaluate \(\left (1 -\mathrm{i}\right )^{3}\).
\(- 2 - 2\mathrm{i}\)
\(2 + 2\mathrm{i}\)
\(1 + \mathrm{i}\)
\(\mathrm{i}\)
9000037401
Level:
A
Evaluate \(\mathrm{i}^{50}\).
\(- 1\)
\(\mathrm{i}\)
\(1\)
\(-\mathrm{i}\)
9000035801
Level:
A
Find the complex conjugate of the following complex number.
\[
\mathrm{i} + 3\mathrm{i}(2 -\mathrm{i})^{2} - 4(1 -\mathrm{i})^{3}
\]
\(20 - 18\mathrm{i}\)
\(20 - 24\mathrm{i}\)
\(20 + 18\mathrm{i}\)
\(- 8 + 26\mathrm{i}\)
9000035803
Level:
A
Given the complex number \(z = -1 + 2\mathrm{i}\),
find the imaginary part of the complex number
\(\frac{1}
{z}\).
\(-\frac{2}
{5}\)
\(\frac{1}
{2}\)
\(\frac{2}
{5}\)
\(-\frac{1}
{2}\)
9000035804
Level:
A
Find the algebraic form of the following complex number. By \(\overline{z }\) the complex conjugate of \(z \) is denoted.
\[
\overline{\overline{(2 + \mathrm{i}) }\; \overline{(3 + 2\mathrm{i}) } }
\]
\(4 + 7\mathrm{i}\)
\(8 + 7\mathrm{i}\)
\(8 - 7\mathrm{i}\)
\(4 - 7\mathrm{i}\)
9000035709
Level:
A
Simplify \((1 -\mathrm{i})^{-3}\).
\(-\frac{1}
{4} + \frac{1}
{4}\mathrm{i}\)
\(1 + 3\mathrm{i}\)
\(- 2 - 2\mathrm{i}\)
\(\frac{1}
{2} + \frac{1}
{2}\mathrm{i}\)
9000035809
Level:
A
Given the complex number \(z = -1 + \mathrm{i}\),
find the angle in the polar form of the number
\(z^{6}\).
\(\frac{\pi }
{2}\)
\(\frac{3\pi }
{2}\)
\(\frac{3\pi }
{4}\)
\(\frac{7\pi }
{4}\)
9000035603
Level:
A
Find the solution set of the following equation.
\[
4x^{2} + 9 = 0
\]
\(\left \{-\frac{3}
{2}\mathrm{i}; \frac{3}
{2}\mathrm{i}\right \}\)
\(\left \{-\frac{2}
{3}\mathrm{i}; \frac{2}
{3}\mathrm{i}\right \}\)
\(\left \{-\frac{9}
{4}\mathrm{i}; \frac{9}
{4}\mathrm{i}\right \}\)
\(\left \{-\frac{3}
{2}; \frac{3}
{2}\right \}\)