Let z be a complex number satisfying the equality
$$i \cdot z + 1 = 2 ( 1 - \bar { z } )$$
What is the real part of the complex number z?
A) $\frac { 1 } { 6 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 2 }$
D) $\frac { 2 } { 3 }$
E) $\frac { 5 } { 6 }$

$$( 1 + i ) ^ { 4 } \cdot \left( 2 - \frac { 2 } { i } \right) ^ { 2 }$$
What is the result of this operation?
A) $4 i$
B) 16
C) $- 32 i$
D) - 8
E) 12

$\frac { \left( 1 - i ^ { 2 } \right) \cdot \left( 1 - i ^ { 6 } \right) \cdot \left( 1 - i ^ { 10 } \right) } { ( 1 - i ) \cdot \left( 1 - i ^ { 3 } \right) \cdot \left( 1 - i ^ { 5 } \right) }$\ What is the result of this operation?\ A) 1\ B) 2\ C) $2 + 2 i$\ D) $2 + 2 i$\ E) $1 + 2 i$

$4 z - 3 \bar { z } = \frac { 1 - 18 i } { 2 - i }$\ Which of the following is the complex number $z$ that satisfies this equality?\ A) $- 2 + i$\ B) $- 3 + i$\ C) $4 + 2 i$\ D) $3 - 2 i$\ E) $4 - i$

Let a be a real number. In complex numbers,
$$\frac { 1 - a i } { a - i } = i$$
the equality is given.
Accordingly, what is a?
A) 4 B) 3 C) 2 D) 1 E) 0

In the set of complex numbers
$$\frac { ( 4 - 2 i ) \cdot ( 6 + 3 i ) } { ( 1 - i ) \cdot ( 1 + i ) }$$
What is the result of this operation?
A) 15
B) 12
C) 10
D) 9
E) 6

In the set of complex numbers
$$\frac { i \cdot ( 2 - i ) \cdot ( 2 - 4 i ) } { ( 1 - i ) \cdot ( 1 + i ) }$$
what is the result of the operation?
A) 2
B) 5
C) 10
D) $2 i$
E) $5 i$

Let $\bar{z}$ be the conjugate of the complex number $z$,
$$\frac { 6 + 2 i } { z } = \bar { z } + i$$
the sum of the complex numbers $z$ that satisfy the equality is what?
A) $1 + 3 i$
B) $2 + i$
C) $3 + 2 i$
D) $4 + i$