Algebrický a goniometrický tvar komplexného čísla

2010013105

Časť: 
B
Nech \(z_1 = \sqrt3 + \mathrm{i}\), \(z_2=1 + \mathrm{i}\sqrt3\). Identifikujte komplexné číslo, ktoré sa nerovná \(\frac{z_1}{z_2}\).
\(\cos \frac{5\pi}{6} + \mathrm{i} \sin \frac{5\pi}{6}\)
\(\cos \left(-\frac{\pi}{6}\right) + \mathrm{i} \sin \left(-\frac{\pi}{6}\right)\)
\( \frac{\sqrt{3}}{2} - \frac{\mathrm{i}}{2}\)
\(\cos \frac{\pi}{6} - \mathrm{i} \sin \frac{\pi}{6}\)

2010013104

Časť: 
B
Ak \( [x;y]\in\mathbb{R}\times\mathbb{R} \), \( z_1 = -2 + xy\,\mathrm{i} \) a \( z_2 = x + y + 8\,\mathrm{i}\). Nájdite všetky \( [x;y] \) tak že \( z_1 \) and \( z_2 \) sú opačné čísla.
\( [x;y] \in\left\{[4;-2],[-2;4]\right\} \)
\( [x;y] \in \left\{[4;-2]\right\} \)
\( [x;y] \in\left\{[-4;2],[2;-4]\right\} \)
\( [x;y] \in \left\{[-2;4]\right\} \)

2010013103

Časť: 
B
Vzhľadom na komplexné čísla \( a=3\sqrt{2}\left(\cos120^{\circ}+\mathrm{i}\sin120^{\circ}\right) \), \( b=2\left (\cos\frac{4\pi}3+\mathrm{i}\sin\frac{4\pi}{3}\right) \) a \( c=\frac{\sqrt{2}}{2 }\left(\cos\frac{\pi}4+\mathrm{i}\sin\frac{\pi}{4}\right) \), zistite \( \frac{a\cdot b}{c} \).
\( 12\left(\cos\frac{7\pi}4+\mathrm{i}\sin\frac{7\pi}4\right) \)
\( 12\left(\sin\frac{7\pi}4+\mathrm{i}\cos\frac{7\pi}4\right) \)
\( 6\left(\cos\frac{7\pi}4+\mathrm{i}\sin\frac{7\pi}4\right) \)
\( 12\left(\cos\frac{9\pi}4+\mathrm{i}\sin\frac{9\pi}4\right) \)

2010013102

Časť: 
B
Vzhľadom na komplexné čísla \( a=2\left(\cos⁡ \frac{\pi}{3}+\mathrm{i}\sin⁡ \frac{\pi}{3}\right) \), \( b=\sqrt{2}\left(\cos⁡ \frac{5\pi}{4}+\mathrm{i}\sin \frac{5\pi}{4}\right) \) a \( c =2\sqrt{2}\left(\cos \left(-\frac{\pi}{6}\right)+\mathrm{i}\sin \left(-\frac{\pi}{6}\right)\right) \), zistite \( a\cdot b\cdot c \).
\(8\left(\cos \frac{17\pi}{12}+\mathrm{i}\sin \frac{17\pi}{12} \right) \)
\(8\left(\cos \frac{17\pi}{12}-\mathrm{i}\sin \frac{17\pi}{12} \right) \)
\(8\left(\cos \frac{7\pi}{4}+\mathrm{i}\sin \frac{7\pi}{4} \right) \)
\(4\sqrt{2}\left(\cos \frac{17\pi}{12}+\mathrm{i}\sin \frac{17\pi}{12} \right) \)

2010013101

Časť: 
B
Nech \(z_{1} = 4\left (\cos \frac{7\pi} {3} + \mathrm{i}\sin \frac{7\pi} {3} \right )\) a \(z_{2} = \frac12\left (\cos \frac{\pi} {6} + \mathrm{i}\sin \frac{\pi} {6} \right )\), vypočítajte \(z_{1}\cdot z_{2} \).
\( 2\mathrm{i}\)
\(- 2\mathrm{i}\)
\(2\)
\(-2\)

2010010403

Časť: 
A
Sú dané komplexné čísla \[ a = \sqrt{2} + \mathrm{i},\ \quad b = {2} -\sqrt{3}\mathrm{i},\ \] nájdite \(\frac{a} {b}\).
\(\frac{2\sqrt{2}-\sqrt{3}} {7} + \mathrm{i}\frac{\sqrt{6}+2} {7} \)
\(\frac{2\sqrt{2}-\sqrt{3}} {7} - \mathrm{i}\frac{\sqrt{6}+2} {7} \)
\(\frac{2\sqrt{2}+\sqrt{3}} {4} + \mathrm{i}\frac{\sqrt{6}-2} {4} \)
\(\frac{\sqrt{2}-\sqrt{6}} {3} + \mathrm{i}\frac{\sqrt{3}+2} {3} \)