Complex Numbers and Quadratic Equations

Let $z$ be a complex number such that $|z|=1$. If $\displaystyle\frac{2 + k^2 z}{k + \overline{z}} = k z$, $k\in\mathbb{R}$, then the maximum distance of $k + i k^2$ from the circle $|z - (1 + 2i)| = 1$ is:

Let $P_n=\alpha^n+\beta^n,\ n\in\mathbb{N}$. If $P_{10}=123,\ P_9=76,\ P_8=47$ and $P_1=1$, then the quadratic equation having roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is:

If $z_1, z_2, z_3\in\mathbb{C}$ are the vertices of an equilateral triangle whose centroid is $z_0$, then $\displaystyle\sum_{k=1}^3 (z_k - z_0)^2$ is equal to:

Let $\alpha$ and $\beta$ be the roots of $x^2 + \sqrt{3}\,x - 16 = 0$, and $\gamma$ and $\delta$ be the roots of $x^2 + 3x - 1 = 0$. If $P_n = \alpha^n + \beta^n$ and $Q_n = \gamma^n + \delta^n$, then $\frac{P_{25} + \sqrt{3}\,P_{24}}{2P_{23}} \;+\; \frac{Q_{25} - Q_{23}}{Q_{24}}$ is equal to:

Let $z\in\mathbb{C}$ be such that $\frac{z^2 + 3i}{z - 2 + i} = 2 + 3i$. Then the sum of all possible values of $z^2$ is:

Let the product of $\omega_1 = (8+i)\sin\theta + (7+4i)\cos\theta$ and $\omega_2 = (1+8i)\sin\theta + (4+7i)\cos\theta$ be $\alpha + i\beta$, where $i=\sqrt{-1}$. Let $p$ and $q$ be the maximum and minimum values of $\alpha+\beta$, respectively. Then $p+q$ is equal to:

Consider the equation x² + 4x - n = 0, where n \in [20,100] is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to:

If the locus of $z \in \mathbb{C}$, such that $\Re\bigl(\tfrac{z-1}{2z + i}\bigr) + \Re\bigl(\tfrac{\overline{z}-1}{2\overline{z} - i}\bigr) = 2$, is a circle of radius $r$ and center $(a,b)$, then $\tfrac{15ab}{r^2}$ is equal to:

The number of real roots of the equation $x\lvert x - 2\rvert + 3\lvert x - 3\rvert + 1 = 0$ is:

Among the statements: (S1): The set $\{ z \in \mathbb{C} \setminus \{-i\} : |z| = 1 \text{ and } \frac{z - i}{z + i} \text{ is purely real} \}$ contains exactly two elements, and (S2): The set $\{ z \in \mathbb{C} \setminus \{-1\} : |z| = 1 \text{ and } \frac{z - 1}{z + 1} \text{ is purely imaginary} \}$ contains infinitely many elements. Which of the following is correct?