If $z_1$ and $z_2$ are any two complex numbers,then $|z_1 + \sqrt{z_1^2 - z_2^2}| + |z_1 - \sqrt{z_1^2 - z_2^2}|$ is equal to

Let $z$ be a complex number such that $|z+2|=1$ and $\operatorname{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $|\operatorname{Re}(\overline{z+2})|$ is:

$\cos (x + iy)$ is equal to

Let $Z_1, Z_2, Z_3$ be three non-zero complex numbers such that $a = |Z_1|, b = |Z_2|, c = |Z_3|$. If the determinant $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = 0$,then:

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	$(4)$ $10$
	$(5)$ $7$

Find the complex number $z$ satisfying the equations $\left| \frac{z - 12}{z - 8i} \right| = \frac{5}{3}$ and $\left| \frac{z - 4}{z - 8} \right| = 1$.