9000035801 Level: AFind 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: AGiven 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: AFind the algebraic form of the following complex number. By \(\overline{z }\) the complex conjugate of \(z \) is denoted. \[ \overline{\overline{(2 + \mathrm{i}) }\; \overline{(3 + 2\mathrm{i}) } } \]\(4 + 7\mathrm{i}\)\(8 + 7\mathrm{i}\)\(8 - 7\mathrm{i}\)\(4 - 7\mathrm{i}\)
9000035805 Level: BGiven the complex numbers \[ \text{$a = 2\left (\cos \frac{2\pi } {3} + \mathrm{i}\sin \frac{2\pi } {3}\right )$, $b = \sqrt{2}\left (\cos \frac{3\pi } {4} + \mathrm{i}\sin \frac{3\pi } {4}\right )$,} \] find the product \(ab\).\(2\sqrt{2}\left (\cos \frac{17\pi } {12} + \mathrm{i}\sin \frac{17\pi } {12}\right )\)\(2\sqrt{2}\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )\)\(2\sqrt{2}\left (\cos \frac{5\pi } {7} + \mathrm{i}\sin \frac{5\pi } {7}\right )\)\(2\sqrt{2}\left (\cos \frac{5\pi } {12} + \mathrm{i}\sin \frac{5\pi } {12}\right )\)
9000035806 Level: BGiven the complex numbers \[ \text{ $a = 2\left (\cos \frac{5\pi } {3} + \mathrm{i}\sin \frac{5\pi } {3}\right )$, $b = 3\left (\cos \frac{11\pi } {6} + \mathrm{i}\sin \frac{11\pi } {6} \right )$,} \] find the quotient \(\frac{a} {b}\).\(\frac{2} {3}\left (\cos \frac{11\pi } {6} + \mathrm{i}\sin \frac{11\pi } {6} \right )\)\(\frac{2} {3}\left (\cos \frac{\pi } {6} + \mathrm{i}\sin \frac{\pi } {6}\right )\)\(\frac{2} {3}\left (\cos \frac{5\pi } {6} + \mathrm{i}\sin \frac{5\pi } {6}\right )\)\(\frac{2} {3}\left (\cos \frac{7\pi } {6} + \mathrm{i}\sin \frac{7\pi } {6}\right )\)
9000035710 Level: AFind 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: AGiven 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: AFind 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: AFind 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: AFind the real part of the complex number \(z= 2 + 2\mathrm{i}^{2} + \mathrm{i}^{3} -\mathrm{i}^{4} + 2\mathrm{i}^{5}\).\(- 1\)\(1\)\(5\)\(- 3\)