Question

(UEL) O conjugado do número complexo
$z = \frac{1}{i} + \frac{1 - i}{1 + i}$


a) 1-i.

b) 1 + i.

c) i.

d) 2i.

e) -2i.
​

Answer 1

Resposta:

$\boxed{\mathtt{D}}$

Explicação passo-a-passo:

Desenvolvendo o número...

$\\ \displaystyle \mathsf{z = \frac{1}{i} + \frac{1 - i}{1 + i}} \\\\\\ \mathsf{z = \frac{1 \cdot (1 + i) + i \cdot (1 - i)}{i(1 + i)}} \\\\\\ \mathsf{z = \frac{1 + i + i - i^2}{i(1 + i)}} \\\\\\ \mathsf{z = \frac{1 + 2i - (- 1)}{i(1 + i)}} \\\\\\ \mathsf{z = \frac{2 + 2i}{i(1 + i)}}$

$\\ \displaystyle \mathsf{z = \frac{2(1 + i)}{i(1 + i)}} \\\\\\ \mathsf{z = \frac{2}{i}}$

Por conseguinte,

$\\ \displaystyle \mathsf{z = \frac{2}{i} \times \frac{i}{i}} \\\\\\ \mathsf{z = \frac{2i}{i^2}} \\\\\\ \mathsf{z = \frac{2i}{- 1}} \\\\ \boxed{\mathsf{z = - 2i}}$

Daí, concluímos que $\displaystyle \boxed{\boxed{\mathsf{\overline{z} = 2i}}}$.