Let $z_{1},z_{2}$ and $z_{3}$ be three complex numbers on the circle $|z|=1$ with $\arg(z_{1})=-\dfrac{\pi}{4}$, $\arg(z_{2})=0$ and $\arg(z_{3})=\dfrac{\pi}{4}$. If $\bigl|z_{1}\overline{z_{2}}+z_{2}\overline{z_{3}}+z_{3}\overline{z_{1}}\bigr|^{2}=\alpha+\beta\sqrt{2}$, $\alpha,\beta\in\mathbb{Z}$, then the value of $\alpha^{2}+\beta^{2}$ is:
Let $\alpha,\beta$ be the roots of the equation $x^{2}-ax-b=0$ with $\operatorname{Im}(\alpha)<\operatorname{Im}(\beta)$. Let $P_{n}=\alpha^{n}-\beta^{n}$. If $P_{3}=-5\sqrt{7}\,i$, $P_{4}=-3\sqrt{7}\,i$, $P_{5}=11\sqrt{7}\,i$ and $P_{6}=45\sqrt{7}\,i$, then $\bigl|\alpha^{4}+\beta^{4}\bigr|$ is equal to \rule{2cm}{0.4pt}\,.
Let integers $a, b \in [-3,3]$ be such that $a+b\neq0$. Then the number of all possible ordered pairs $(a,b)$ for which $\left|\dfrac{z-a}{z+b}\right|=1$ and $\begin{vmatrix} z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{vmatrix}=1$, $z\in\mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2+x+1=0$, is equal to
Let $\alpha$ and $\beta$ be the roots of $x^2-ax-b=0$ with $\text{Im}(\alpha)<\text{Im}(\beta)$. Let $P_n=\alpha^n-\beta^n$. If $P_3=-5\sqrt{7}\,i$, $P_4=-3\sqrt{7}\,i$, $P_5=11\sqrt{7}\,i$, and $P_6=45\sqrt{7}\,i$, then $|\alpha^4+\beta^4|$ is equal to
For $a\in\mathbb{C}$, let $A=\{z\in\mathbb{C}:\ \text{Re}(a+\bar{z})>\text{Im}(\bar{a}+z)\}$ and $B=\{z\in\mathbb{C}:\ \text{Re}(a+\bar{z})<\text{Im}(\bar{a}+z)\}$. Then among the two statements: (S1): If $\text{Re}(a),\text{Im}(a)>0$, then $A$ contains all real numbers. (S2): If $\text{Re}(a),\text{Im}(a)<0$, then $B$ contains all real numbers.
Let integers $a,b\in[-3,3]$ be such that $a+b\ne 0$. Then the number of all possible ordered pairs $(a,b)$, for which $\left|\dfrac{z-a}{z+b}\right|=1$ and $\left|\,\begin{matrix} z+1 & \omega & \omega^{2}\\ \omega & z+\omega^{2} & 1\\ \omega^{2} & 1 & z+\omega\end{matrix}\,\right|=1$, where $\omega$ and $\omega^{2}$ are the complex cube roots of unity, is equal to \rule{2cm}{0.4pt}.