Question

plss alguém me ajuda, preciso de todas as contas detalhadas até chegar no resultado!

Answer 1

$\displaystyle \text z=\frac{6+6\text i}{1-\text i} \\\\ \underline{\text{Racionalizando}}: \\\\ \text z=\frac{(6+6\text i)}{(1-\text i)}\frac{(1+\text i)}{(1+\text i)}\to \text z= \frac{6+6\text i+6\text i+6\text i^2}{1-\text i^2} \\\\\\ \text z= \frac{6+12 \text i+6(-1)}{1-(-1)} \to \text z=\frac{12\text i}{2} \\\\\\ \boxed{\text z = 6\text i}$

Analisando as alternativas :

Item A) correto

$\text z=6\text i$

item B) correto

$\text z = \text a+\text b\text i \to 6\text i = \text a+\text b\text i \to \text a = 0 \ , \ \text b = 6 \\\\ |\text z| = \sqrt{\text a^2 +\text b^2} \to |\text z| = \sqrt{0^2+6^2} \to |\text z| = 6\ \checkmark$

item C) correto

$\displaystyle \text z = |\text z|.[\ \text{cos}(\theta )+\text i.\text{sen}(\theta) \ ] \\\\ \text z=6\text i\\\\ \text z =6[0+\text i] \\\\ \text z=6[\ \text{cos}(\frac{\pi}{2})+\text i.\text{sen}(\frac{\pi}{2}) \ ]$

item D) incorreto

Justificativa do item C

item E) Correto

$\text z=6\text i \to \text z^2=6^2.\text i^2 \to \text z^2=-36 \to \text{Real}$