Polar and Euler form

Let $S = \left\{a + b\sqrt{2} : a, b \in \mathbb{Z}\right\}$, $T_1 = \left\{\left(-1+\sqrt{2}\right)^n : n \in \mathbb{N}\right\}$, and $T_2 = \left\{\left(1+\sqrt{2}\right)^n : n \in \mathbb{N}\right\}$. Then which of the following statements is (are) TRUE?

The area of the polygon, whose vertices are the non-real roots of the equation $\bar z=iz^2$ is :

Let for some real numbers $\alpha$ and $\beta$, $a=\alpha-i\beta$. If the system of equations $4ix+(1+i)y=0$ and $8\left(\cos\dfrac{2\pi}3+i\sin\dfrac{2\pi}3\right)x+\bar ay=0$ has more than one solution, then $\dfrac\alpha\beta$ is equal to

Let $\theta_1, \theta_2, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_1 + \theta_2 + \cdots + \theta_{10} = 2\pi$. Define the complex numbers $z_1 = e^{i\theta_1}$, $z_k = z_{k-1}e^{i\theta_k}$ for $k = 2, 3, \ldots, 10$, where $i = \sqrt{-1}$. Consider the statements $P$ and $Q$ given below:
$$P: |z_2 - z_1| + |z_3 - z_2| + \cdots + |z_{10} - z_9| + |z_1 - z_{10}| \le 2\pi$$
$$Q: |z_2^2 - z_1^2| + |z_3^2 - z_2^2| + \cdots + |z_{10}^2 - z_9^2| + |z_1^2 - z_{10}^2| \le 4\pi$$
Then,

For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^4 - |z|^4 = 4iz^2$, where $i = \sqrt{-1}$. Then the minimum possible value of $|z_1 - z_2|^2$, where $z_1, z_2 \in S$ with $\operatorname{Re}(z_1) > 0$ and $\operatorname{Re}(z_2) < 0$, is ______

Let $w=\dfrac{\sqrt3+i}2$ and $P=\{w^n:n=1,2,3,\ldots\}$. Further $H_1=\left\{z\in\mathbb C:\operatorname{Re}z>\frac12\right\}$ and $H_2=\left\{z\in\mathbb C:\operatorname{Re}z<\frac{-1}2\right\}$, where $\mathbb C$ is the set of all complex numbers. If $z_1\in P\cap H_1$, $z_2\in P\cap H_2$ and $O$ represents the origin, then $\angle z_1Oz_2=$

A man walks a distance of 3 units from the origin towards the north-east (N 45° E) direction. From there, he walks a distance of 4 units towards the north-west (N 45° W) direction to reach a point $P$. Then the position of $P$ in the Argand plane is