Fie $\varepsilon$ o radacina a ecuatiei $x^2+x+1=0$.
a). Calculati: $S=1+\varepsilon+{\varepsilon}^2+{\varepsilon}^3+\dots+{\varepsilon}^{2011}$.
b). Aratati ca: (i). $(z-1)+\varepsilon.(z-\varepsilon)+{\varepsilon}^2.(z-{\varepsilon}^2)=0;\, (\forall)z\in\mathbb{C}$;
(ii). $|z-1|^2+|z-\varepsilon|^2+|z-{\varepsilon}^2|^2=3.(1+|z|^2),\, (\forall)z\in\mathbb{C}$.
P4, locala Bihor(2011).
-
mihai miculita
- Mesaje: 1493
- Membru din: Mar Oct 26, 2010 9:21 pm
- Localitate: ORADEA
Re: P4, locala Bihor(2011).
a) Dand factor comun $(e^{3})^{k}$, unde $k = \overline{1,2,...,670}$, obtinem
$S = 1 + e$, unde inlocuind $e$ obtinem $S$.
b) i) Desfacand parantezele obtinem
$z-1 + e(z-e) + e^{2}(z-e^{2}) = z - 1 + ez - e^{2} + e^{2}z - e^{4}$.
Stim ca $e^{3} = 1$, deoarece $e^{3} - 1 = 0$.
Atunci, avem:
$z + ez + e^{2}z - (1 + e + e^{2}) = (1 + e + e^{2})(z-1) = 0$, adica concluzia.
ii) Folosind $|z|^{2} = z \overline{z}$, avem:
$|z-1|^{2} + |z-e|^{2} + |z-e^{2}|^{2} = 3(1 + |z|^{2}), \forall z \in \mathbb{C} \Leftrightarrow$
$(z-1)(\overline{z} - 1) + (z-e)(\overline{z} - \overline{e}) + (z-e^{2})(\overline{z} - \overline{e}^{2}) = 3 + 3 z \overline{z} \Leftrightarrow$
$z\overline{z} - z - \overline{z} + 1 + z \overline{z} - z \overline{e} - \overline{z} e + |e|^{2} + z \overline{z} - z {\overline{e}}^{2} - \overline{z} \cdot e^{2} + | e |^{4}$ egal cu $3 + 3 z \overline{z}$
$3 + 3 z \overline{z} - (z + z \overline{e} + z \overline{e}^{2}) - (\overline{z} + \overline{z} e + \overline{z} e^{2})$ egal cu $3 + 3z\overline{z} - z(\overline{1+e+e^{2}}) - \overline{z}(1 + e + e^{2}) = 3 + 3z\overline{z}$, ceea ce trebuia demonstrat.
$S = 1 + e$, unde inlocuind $e$ obtinem $S$.
b) i) Desfacand parantezele obtinem
$z-1 + e(z-e) + e^{2}(z-e^{2}) = z - 1 + ez - e^{2} + e^{2}z - e^{4}$.
Stim ca $e^{3} = 1$, deoarece $e^{3} - 1 = 0$.
Atunci, avem:
$z + ez + e^{2}z - (1 + e + e^{2}) = (1 + e + e^{2})(z-1) = 0$, adica concluzia.
ii) Folosind $|z|^{2} = z \overline{z}$, avem:
$|z-1|^{2} + |z-e|^{2} + |z-e^{2}|^{2} = 3(1 + |z|^{2}), \forall z \in \mathbb{C} \Leftrightarrow$
$(z-1)(\overline{z} - 1) + (z-e)(\overline{z} - \overline{e}) + (z-e^{2})(\overline{z} - \overline{e}^{2}) = 3 + 3 z \overline{z} \Leftrightarrow$
$z\overline{z} - z - \overline{z} + 1 + z \overline{z} - z \overline{e} - \overline{z} e + |e|^{2} + z \overline{z} - z {\overline{e}}^{2} - \overline{z} \cdot e^{2} + | e |^{4}$ egal cu $3 + 3 z \overline{z}$
$3 + 3 z \overline{z} - (z + z \overline{e} + z \overline{e}^{2}) - (\overline{z} + \overline{z} e + \overline{z} e^{2})$ egal cu $3 + 3z\overline{z} - z(\overline{1+e+e^{2}}) - \overline{z}(1 + e + e^{2}) = 3 + 3z\overline{z}$, ceea ce trebuia demonstrat.