Fórmula binómica y trigonométrica de números complejos

2010013102

Parte: 
B
Dados los números complejos \( a=2\left(\cos⁡ \frac{\pi}{3}+\mathrm{i}\sin⁡ \frac{\pi}{3}\right) \), \( b=\sqrt{2}\left(\cos⁡ \frac{5\pi}{4}+\mathrm{i}\sin \frac{5\pi}{4}\right) \) y \( c=2\sqrt{2}\left(\cos \left(-\frac{\pi}{6}\right)+\mathrm{i}\sin \left(-\frac{\pi}{6}\right)\right) \), calcula \( a\cdot b\cdot c \).
\(8\left(\cos \frac{17\pi}{12}+\mathrm{i}\sin \frac{17\pi}{12} \right) \)
\(8\left(\cos \frac{17\pi}{12}-\mathrm{i}\sin \frac{17\pi}{12} \right) \)
\(8\left(\cos \frac{7\pi}{4}+\mathrm{i}\sin \frac{7\pi}{4} \right) \)
\(4\sqrt{2}\left(\cos \frac{17\pi}{12}+\mathrm{i}\sin \frac{17\pi}{12} \right) \)

2010013103

Parte: 
B
Dados los números complejos \( a=3\sqrt{2}\left(\cos120^{\circ}+\mathrm{i}\sin120^{\circ}\right) \), \( b=2\left(\cos\frac{4\pi}3+\mathrm{i}\sin\frac{4\pi}{3}\right) \) y \( c=\frac{\sqrt{2}}{2}\left(\cos\frac{\pi}4+\mathrm{i}\sin\frac{\pi}{4}\right) \), calcula \( \frac{a\cdot b}{c} \).
\( 12\left(\cos\frac{7\pi}4+\mathrm{i}\sin\frac{7\pi}4\right) \)
\( 12\left(\sin\frac{7\pi}4+\mathrm{i}\cos\frac{7\pi}4\right) \)
\( 6\left(\cos\frac{7\pi}4+\mathrm{i}\sin\frac{7\pi}4\right) \)
\( 12\left(\cos\frac{9\pi}4+\mathrm{i}\sin\frac{9\pi}4\right) \)

2010013104

Parte: 
B
Sean \( [x;y]\in\mathbb{R}\times\mathbb{R} \), \( z_1 = -2 + xy\,\mathrm{i} \) y \( z_2 = x + y + 8\,\mathrm{i}\). Halla todos los \( [x;y] \) tales que \( z_1 \) y \( z_2 \) sean números opuestos.
\( [x;y] \in\left\{[4;-2],[-2;4]\right\} \)
\( [x;y] \in \left\{[4;-2]\right\} \)
\( [x;y] \in\left\{[-4;2],[2;-4]\right\} \)
\( [x;y] \in \left\{[-2;4]\right\} \)

2010013105

Parte: 
B
Sean \(z_1 = \sqrt3 + \mathrm{i}\), \(z_2=1 + \mathrm{i}\sqrt3\). Identifica el número complejo que es distinto a \(\frac{z_1}{z_2}\).
\(\cos \frac{5\pi}{6} + \mathrm{i} \sin \frac{5\pi}{6}\)
\(\cos \left(-\frac{\pi}{6}\right) + \mathrm{i} \sin \left(-\frac{\pi}{6}\right)\)
\( \frac{\sqrt{3}}{2} - \frac{\mathrm{i}}{2}\)
\(\cos \frac{\pi}{6} - \mathrm{i} \sin \frac{\pi}{6}\)

2010013106

Parte: 
B
Sean \(z_1 = 1 + \mathrm{i}\sqrt{3}\), \(z_2=\sqrt3 + \mathrm{i}\). Identifica el número complejo que es distinto a \(\frac{z_1}{z_2}\).
\(\cos \frac{7\pi}{6} + \mathrm{i} \sin \frac{7\pi}{6}\)
\(\cos \frac{\pi}{6} +\mathrm{i} \sin \frac{\pi}{6}\)
\( \frac{\sqrt{3}}{2} + \frac{\mathrm{i}}{2}\)
\(\cos \left(-\frac{\pi}{6}\right) - \mathrm{i} \sin \left(-\frac{\pi}{6}\right)\)

2010013111

Parte: 
B
Halla la forma polar del número complejo \(z=\frac{\mathrm{i}^{12}+1} {\mathrm{i}^{11}+1} \).
\(\sqrt{2}\left (\cos \frac{\pi } {4} + \mathrm{i}\sin \frac{\pi } {4}\right )\)
\(\cos \frac{\pi } {2} + \mathrm{i}\sin \frac{\pi } {2}\)
\(\sqrt{2}\left (\cos \frac{5\pi } {4} + \mathrm{i}\sin \frac{5\pi } {4}\right )\)
\(\sqrt{2}\left (\cos \frac{3\pi } {4} + \mathrm{i}\sin \frac{3\pi } {4}\right )\)

2010013112

Parte: 
B
Halla la forma polar del conjugado del número complejo \(z=-\frac{\sqrt5}{2} + \mathrm{i}\frac{\sqrt{15}}{2}\).
\(\sqrt{5}\left (\cos \left(-\frac{2\pi} {3}\right) + \mathrm{i}\sin \left(-\frac{2\pi} {3}\right )\right)\)
\(\sqrt{5}\left (\cos \frac{2\pi } {3} + \mathrm{i}\sin \frac{2\pi } {3}\right )\)
\(\sqrt{10}\left (\cos \frac{4\pi } {3} + \mathrm{i}\sin \frac{4\pi } {3}\right )\)
\(\sqrt{5}\left (\cos \frac{\pi } {3} + \mathrm{i}\sin \frac{\pi } {3}\right )\)

2010013113

Parte: 
B
Halla la forma polar del opuesto del número complejo \(z=-\frac{\sqrt5}{2} + \mathrm{i}\frac{\sqrt{15}}{2}\).
\(\sqrt{5}\left (\cos \left(-\frac{\pi} {3}\right) + \mathrm{i}\sin \left(-\frac{\pi} {3}\right )\right)\)
\(\sqrt{5}\left (\cos \frac{2\pi } {3} + \mathrm{i}\sin \frac{2\pi } {3}\right )\)
\(\sqrt{10}\left (\cos \frac{5\pi } {3} + \mathrm{i}\sin \frac{5\pi } {3}\right )\)
\(\sqrt{5}\left (\cos \frac{\pi } {3} + \mathrm{i}\sin \frac{\pi } {3}\right )\)