9000035604 Časť: CVyriešte danú rovnicu. \[ x^{2} - 2\mathrm{i}x + 3 = 0 \]\(\{ -\mathrm{i};3\mathrm{i}\}\)\(\{ - 4\mathrm{i};4\mathrm{i}\}\)\(\{1 - 2\mathrm{i};1 + 2\mathrm{i}\}\)\(\{ - 3\mathrm{i};\mathrm{i}\}\)
9000035609 Časť: CRovnica \[ x^{2} + px - 11 = 0 \] s parametrom \(p\in \mathbb{C}\) má jeden koreň \(x_{1} = 3 -\mathrm{i}\sqrt{2}\). Nájdite druhý koreň \(x_{2}\) a parameter \(p\).\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 2\mathrm{i}\sqrt{2}\)\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = 6\)\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 6\)\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\)\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\sqrt{2}\)
9000035810 Časť: CJe dané komplexné číslo \(z = -2 + 2\mathrm{i}\). Všetky navzájom rôzne hodnoty \(\root{3}\of{z}\) sú:\(\begin{aligned}[t] &w_{0} = \root{6}\of{8}\left (\cos \frac{\pi } {4} + \mathrm{i}\sin \frac{\pi } {4}\right ) & \\&w_{1} = \root{6}\of{8}\left (\cos \frac{11\pi } {12} + \mathrm{i}\sin \frac{11\pi } {12}\right ) \\&w_{2} = \root{6}\of{8}\left (\cos \frac{19\pi } {12} + \mathrm{i}\sin \frac{19\pi } {12}\right ) \\ \end{aligned}\)\(\begin{aligned}[t] &w_{0} = 2\left (\cos \frac{\pi } {4} + \mathrm{i}\sin \frac{\pi } {4}\right ) & \\&w_{1} = 2\left (\cos \frac{11\pi } {12} + \mathrm{i}\sin \frac{11\pi } {12}\right ) \\&w_{2} = 2\left (\cos \frac{19\pi } {12} + \mathrm{i}\sin \frac{19\pi } {12}\right ) \\ \end{aligned}\)\(\root{3}\of{-2} + \root{3}\of{2}\)\(\begin{aligned}[t] &w_{0} = 2\left (\cos \frac{\pi } {3} + \mathrm{i}\sin \frac{\pi } {3}\right )& \\&w_{1} = 2\left (\cos \pi +\mathrm{i}\sin \pi \right ) \\&w_{2} = 2\left (\cos \frac{5\pi } {3} + \mathrm{i}\sin \frac{5\pi } {3}\right ) \\ \end{aligned}\)
9000035607 Časť: CUrčte kvadratickú rovnicu, ktorej korene sú čísla \(x_{1} = 2\mathrm{i}\), \(x_{2} = -\mathrm{i}\).\(x^{2} -\mathrm{i}x + 2 = 0\)\(x^{2} + \mathrm{i}x + 2 = 0\)\(x^{2} + \mathrm{i}x - 2 = 0\)\(x^{2} -\mathrm{i}x - 2 = 0\)
9000035608 Časť: CRovnica \[ x^{2} - 2\mathrm{i}x + q = 0 \] s parametrom \(q\in \mathbb{C}\) má jeden koreň \(x_{1} = 1 + 2\mathrm{i}\). Nájdite druhý koreň \(x_{2}\) a parameter \(q\).\(x_{2} = -1,\ q = -1 - 2\mathrm{i}\)\(x_{2} = -1 - 4\mathrm{i},\ q = 9 - 6\mathrm{i}\)\(x_{2} = 1 - 4\mathrm{i},\ q = 7 - 4\mathrm{i}\)\(x_{2} = 1,\ q = -1 - 2\mathrm{i}\)\(x_{2} = -1,\ q = 1 + 2\mathrm{i}\)
9000036101 Časť: CV akom zornom uhle sa javí pozorovateľovi tyč dlhá \(3\, \mathrm{m}\), ak je od jedného jeho konca vzdialený \(20\, \mathrm{m}\) a od druhého konca \(18\, \mathrm{m}\)? Výsledok zaokrúhlite na celé stupne.\(7^{\circ }\)\(3^{\circ }\)\(45^{\circ }\)\(83^{\circ }\)
9000033810 Časť: CRozhodnite o párnosti alebo nepárnosti funkcie \(l\colon y = |\mathop{\mathrm{cotg}}\nolimits x|\).Funkcia \(l\) je párna.Funkcia \(l\) je nepárna.Funkcia \(l\) nie je ani párna, ani nepárna.
9000034303 Časť: CNájdite množinu všetkých riešení danej rovnice v množine komplexných čísel. \[ x^{3} + \mathrm{i} = 0 \]\(\{\mathrm{i};\ \frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i};\ -\frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i}\}\)\(\{ - 1;\ -\frac{\sqrt{3}} {2} + \frac{1} {2}\mathrm{i};\ -\frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i}\}\)\(\{ - 1;\ \frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i};\ -\frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i}\}\)\(\{\mathrm{i};\ -\frac{\sqrt{3}} {2} + \frac{1} {2}\mathrm{i};\ -\frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i}\}\)
9000034307 Časť: CNech \(x\) je jedno z komplexných riešení danej rovnice. \[ x^{5} + \sqrt{3} -\mathrm{i} = 0 \] Nájdite absolútnu hodnotu \(x\).\(\root{5}\of{2}\)\(2\)\(\root{5}\of{4}\)\(\root{10}\of{2}\)
9000034308 Časť: CDve riešenia rovnice \[ x^{3} + 1 + \mathrm{i} = 0 \] sú \[ \begin{aligned}x_{1}& = \root{6}\of{2}\left (\cos \frac{5} {12}\pi + \mathrm{i}\sin \frac{5} {12}\pi \right ),& \\x_{2}& = \root{6}\of{2}\left (\cos \frac{13} {12}\pi + \mathrm{i}\sin \frac{13} {12}\pi \right ). \\ \end{aligned} \] Nájdite tretie riešenie.\(x_{3} = \root{6}\of{2}\left (\cos \frac{21} {12}\pi + \mathrm{i}\sin \frac{21} {12}\pi \right )\)\(x_{3} = \root{6}\of{2}\left (\cos \frac{9} {12}\pi + \mathrm{i}\sin \frac{9} {12}\pi \right )\)\(x_{3} = \root{6}\of{2}\left (\cos \frac{17} {12}\pi + \mathrm{i}\sin \frac{17} {12}\pi \right )\)\(x_{3} = \root{6}\of{2}\left (\cos \frac{19} {12}\pi + \mathrm{i}\sin \frac{19} {12}\pi \right )\)