Answer the three questions below. To answer (i) and (ii), replace ? with exactly one of the following four options: $<$, $=$, $>$, not enough information to compare.
(i) Suppose $z_1, z_2$ are complex numbers. One of them is in the second quadrant and the other is in the third quadrant. Then $\left|\left|z_1\right| - \left|z_2\right|\right| \quad ? \quad \left|z_1 + z_2\right|$.
(ii) Complex numbers $z_1$, $z_2$ and $0$ form an equilateral triangle. Then $\left|z_1^2 + z_2^2\right| \quad ? \quad \left|z_1 z_2\right|$.
(iii) Let $1, z_1, z_2, z_3, z_4, z_5, z_6, z_7$ be the complex 8th roots of unity. Find the value of $\prod_{i=1,\ldots,7}(1 - z_i)$, where the symbol $\Pi$ denotes product.

Let complex numbers $z _ { 1 } , z _ { 2 }$ satisfy $\left| z _ { 1 } \right| = \left| z _ { 2 } \right| = 2 , z _ { 1 } + z _ { 2 } = \sqrt { 3 } + \mathrm { i }$ , then $\left| z _ { 1 } - z _ { 2 } \right| = $ $\_\_\_\_$.

Let $z$ and $w$ be complex numbers lying on the circles of radii 2 and 3 respectively, with centre ( 0,0 ). If the angle between the corresponding vectors is 60 degrees, then the value of $| z + w | / | z - w |$ is:
(A) $\frac { \sqrt { 19 } } { \sqrt { 7 } }$
(B) $\frac { \sqrt { 7 } } { \sqrt { 19 } }$
(C) $\frac { \sqrt { 12 } } { \sqrt { 7 } }$
(D) $\frac { \sqrt { 7 } } { \sqrt { 12 } }$.

If the points $z_1$ and $z_2$ are on the circles $|z| = 2$ and $|z| = 3$, respectively, and the angle included between these vectors is $60^\circ$, then the value of $\frac{|z_1 + z_2|}{|z_1 - z_2|}$ is
(A) $\sqrt{\frac{19}{7}}$
(B) $\sqrt{19}$
(C) $\sqrt{7}$
(D) $\sqrt{\frac{7}{19}}$

Let $\theta_1, \theta_2, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_1 + \theta_2 + \cdots + \theta_{10} = 2\pi$. Define the complex numbers $z_1 = e^{i\theta_1}$, $z_k = z_{k-1} e^{i\theta_k}$ for $k = 2, 3, \ldots, 10$, where $i = \sqrt{-1}$. Consider the statements $P$ and $Q$ given below:
$P: |z_2 - z_1| + |z_3 - z_2| + \cdots + |z_{10} - z_9| + |z_1 - z_{10}| \leq 2\pi$
$Q: |z_2^2 - z_1^2| + |z_3^2 - z_2^2| + \cdots + |z_{10}^2 - z_9^2| + |z_1^2 - z_{10}^2| \leq 4\pi$
Then,
(A) $P$ is TRUE and $Q$ is FALSE
(B) $Q$ is TRUE and $P$ is FALSE
(C) both $P$ and $Q$ are TRUE
(D) both $P$ and $Q$ are FALSE

Let $\bar { z }$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
$$( \bar { z } ) ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$
are integers, then which of the following is/are possible value(s) of $| z |$ ?
(A) $\left( \frac { 43 + 3 \sqrt { 205 } } { 2 } \right) ^ { \frac { 1 } { 4 } }$
(B) $\left( \frac { 7 + \sqrt { 33 } } { 4 } \right) ^ { \frac { 1 } { 4 } }$
(C) $\left( \frac { 9 + \sqrt { 65 } } { 4 } \right) ^ { \frac { 1 } { 4 } }$
(D) $\left( \frac { 7 + \sqrt { 13 } } { 6 } \right) ^ { \frac { 1 } { 4 } }$