The correct option is
A 18OABC (anti-clockwise) is a rhombus where
O is the origin,
A,B,C are represented by the complex numbers .
z1,z2 and
z3.z1 and
z3 lie on
|z|=2. z2 lies on
|z|=3.
Further arg(z1−z2)=π3 as OABC is a rhombus all sides are equal.
i.e |Z1|=|Z3|=|Z1−Z2|=2 as Z1 and Z3 lie on |Z|=2
∠AOB=θ
in ΔAOB OA=|Z1|=2,AB=|Z1−Z2|=2,OB=|Z2|=3
Using cosine rule
cosθ=OA2+OB2−AB22⋅OA⋅OB=4+9−412
⇒cosθ=34
∴cos2θ=2cos2θ−1=18
Hence, option A.