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Mathematics
Mathematics
Problem
Let
w
=
3
+
i
2
w = \frac{\sqrt{3} + i}{2}
w
=
2
3
+
i
and
P
=
{
w
n
:
n
=
1
,
2
,
3
,
…
}
P = \{w^n : n = 1, 2, 3, \dots\}
P
=
{
w
n
:
n
=
1
,
2
,
3
,
…
}
. Further,
H
1
=
{
z
∈
C
:
Re
(
z
)
>
1
2
}
and
H
2
=
{
z
∈
C
:
Re
(
z
)
<
(
−
1
2
)
}
,
H_1 = \left\{ z \in C : \operatorname{Re}(z) > \frac{1}{2} \right\} \text{ and } H_2 = \left\{ z \in C : \operatorname{Re}(z) < \left(-\frac{1}{2}\right) \right\},
H
1
=
{
z
∈
C
:
Re
(
z
)
>
2
1
}
and
H
2
=
{
z
∈
C
:
Re
(
z
)
<
(
−
2
1
)
}
,
where
C
C
C
is the set of all complex numbers. If
z
1
∈
P
∩
H
1
z_1 \in P \cap H_1
z
1
∈
P
∩
H
1
,
z
2
∈
P
∩
H
2
z_2 \in P \cap H_2
z
2
∈
P
∩
H
2
and
O
O
O
represents the origin, then
∠
z
1
O
z
2
\angle z_1 O z_2
∠
z
1
O
z
2
equals to a.
π
2
\frac{\pi}{2}
2
π
b.
π
6
\frac{\pi}{6}
6
π
c.
2
π
3
\frac{2\pi}{3}
3
2
π
d.
5
π
6
\frac{5\pi}{6}
6
5
π
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