If $Z_1, Z_2, Z_3$ are three complex numbers with unit modulus such that $|Z_1-Z_2|^2+|Z_1-Z_3|^2=4$,then $Z_1 \overline{Z_2}+\overline{Z_1} Z_2+Z_1 \overline{Z_3}+\overline{Z_1} Z_3=$

Let $z = 1 + ai$ be a complex number,$a > 0$,such that $z^3$ is a real number. Then the sum $1 + z + z^2 + .... + z^{11}$ is equal to

If $z=\cos 6^{\circ}+i \sin 6^{\circ}$,then $\sum_{n=1}^{20} \operatorname{Im}\left(z^{2 n-1}\right)=$

If $f(x)$ is a polynomial of degree $n$ with rational coefficients and $1+2i, 2-\sqrt{3}$ and $5$ are three roots of $f(x)=0$,then the least value of $n$ is

Let $z$ and $w$ be two complex numbers such that $w = z \bar{z} - 2z + 2$, $\left| \frac{z+i}{z-3i} \right| = 1$ and $\operatorname{Re}(w)$ has a minimum value. Then, the minimum value of $n \in N$ for which $w^n$ is real, is equal to..........

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