Given that the equation $z^2 + (p + iq)z + r + is = 0$,where $p, q, r, s$ are real and non-zero,has a real root,then:

Let $z_1, z_2 \in \mathbb{C}$ be the distinct solutions of the equation $z^2 + 4z - (1 + 12i) = 0$. Then $|z_1|^2 + |z_2|^2$ is equal to:

Let $A = \left\{ \frac{1967 + 1686 i \sin \theta}{7 - 3 i \cos \theta} : \theta \in R \right\}$. If $A$ contains exactly one positive integer $n$,then the value of $n$ is

Let $A = \{\theta \in (0, 2\pi) : \frac{1+2i \sin \theta}{1-i \sin \theta} \text{ is purely imaginary} \}$. Then the sum of the elements in $A$ is

Let $z$ be a complex number satisfying $|z|^3 + 2z^2 + 4\bar{z} - 8 = 0$,where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be non-zero.
Match each entry in List-$I$ to the correct entries in List-$II$.

List-$I$	List-$II$
$(P)$ $|z|^2$ is equal to	$(1)$ $12$
$(Q)$ $|z-\bar{z}|^2$ is equal to	$(2)$ $4$
$(R)$ $|z|^2+|z+\bar{z}|^2$ is equal to	$(3)$ $8$
$(S)$ $|z+1|^2$ is equal to	$(4)$ $10$
	$(5)$ $7$

$\cosh (\alpha + i\beta ) - \cosh (\alpha - i\beta )$ is equal to