Let $\theta _ { 1 } , \theta _ { 2 } , \ldots , \theta _ { 13 }$ be real numbers and let $A$ be the average of the complex numbers $e ^ { i \theta _ { 1 } } , e ^ { i \theta _ { 2 } } \ldots , e ^ { i \theta _ { 13 } }$, where $i = \sqrt { - 1 }$. As the values of $\theta$'s vary over all 13-tuples of real numbers, find (i) the maximum value attained by $| A |$, (ii) the minimum value attained by $| A |$.

Let $z$ be a complex number such that $\frac{z - i}{z - 1}$ is purely imaginary. Then the minimum value of $|z - (2 + 2i)|$ is
(A) $2\sqrt{2}$
(B) $\sqrt{2}$
(C) $\frac{3}{\sqrt{2}}$
(D) $\frac{1}{\sqrt{2}}$.

Let $A , B , C$ be three sets of complex numbers as defined below $$\begin{aligned} & A = \{ z : \operatorname { Im } z \geq 1 \} \\ & B = \{ z : | z - 2 - i | = 3 \} \\ & C = \{ z : \operatorname { Re } ( ( 1 - i ) z ) = \sqrt { 2 } \} \end{aligned}$$ Let $z$ be any point in $A \cap B \cap C$. Then, $| z + 1 - i | ^ { 2 } + | z - 5 - i | ^ { 2 }$ lies between
(A) 25 and 29
(B) 30 and 34
(C) 35 and 39
(D) 40 and 44

Let $A , B , C$ be three sets of complex numbers as defined below $$\begin{aligned} & A = \{ z : \operatorname { Im } z \geq 1 \} \\ & B = \{ z : | z - 2 - i | = 3 \} \\ & C = \{ z : \operatorname { Re } ( ( 1 - i ) z ) = \sqrt { 2 } \} \end{aligned}$$ Let $z$ be any point in $A \cap B \cap C$ and let $w$ be any point satisfying $| w - 2 - i | < 3$. Then, $| z | - | w | + 3$ lies between
(A) -6 and 3
(B) - 3 and 6
(C) - 6 and 6
(D) - 3 and 9

Let $S = S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$, where $$S _ { 1 } = \{ z \in \mathbb { C } : | \mathrm { z } | < 4 \} , \quad S _ { 2 } = \left\{ z \in \mathbb { C } : \operatorname { Im } \left[ \frac { z - 1 + \sqrt { 3 } i } { 1 - \sqrt { 3 } i } \right] > 0 \right\}$$ and $S _ { 3 } = \{ z \in \mathbb { C } : \operatorname { Re } z > 0 \}$.
$\min _ { z \in S } | 1 - 3 i - z | =$
(A) $\frac { 2 - \sqrt { 3 } } { 2 }$
(B) $\frac { 2 + \sqrt { 3 } } { 2 }$
(C) $\frac { 3 - \sqrt { 3 } } { 2 }$
(D) $\frac { 3 + \sqrt { 3 } } { 2 }$

Let $S$ be the set of all complex numbers $z$ satisfying $| z - 2 + i | \geq \sqrt { 5 }$. If the complex number $z _ { 0 }$ is such that $\frac { 1 } { \left| z _ { 0 } - 1 \right| }$ is the maximum of the set $\left\{ \frac { 1 } { | z - 1 | } : z \in S \right\}$, then the principal argument of $\frac { 4 - z _ { 0 } - \overline { z _ { 0 } } } { z _ { 0 } - \overline { z _ { 0 } } + 2 i }$ is
(A) $- \frac { \pi } { 2 }$
(B) $\frac { \pi } { 4 }$
(C) $\frac { \pi } { 2 }$
(D) $\frac { 3 \pi } { 4 }$

Let $S$ be the set of all complex numbers $z$ satisfying $|z^2 + z + 1| = 1$. Which of the following statements is(are) TRUE?
(A) $\left| z + \frac{1}{2} \right| \leq \frac{1}{2}$ for all $z \in S$
(B) $|z| \leq 2$ for all $z \in S$
(C) $\left| z + \frac{1}{2} \right| \geq \frac{1}{2}$ for all $z \in S$
(D) The set $S$ has exactly four elements

If $| z + 4 | \leq 3$, then the maximum value of $| z + 1 |$ is
(1) 4
(2) 10
(3) 6
(4) 0

If $z$ is a complex number satisfying $| \operatorname { Re } ( z ) | + | \operatorname { Im } ( z ) | = 4$, then $| z |$ cannot be
(1) $\sqrt { \frac { 17 } { 2 } }$
(2) $\sqrt { 10 }$
(3) $\sqrt { 7 }$
(4) $\sqrt { 8 }$

The least value of $| z |$ where $z$ is complex number which satisfies the inequality $e ^ { \left( \frac { ( | z | + 3 ) ( | z | - 1 ) } { | | z | + 1 | } \log _ { \mathrm { e } } 2 \right) } \geq \log _ { \sqrt { 2 } } | 5 \sqrt { 7 } + 9 i | , i = \sqrt { - 1 }$, is equal to :
(1) 3
(2) $\sqrt { 5 }$
(3) 2
(4) 8

If $z$ is a complex number such that $\frac { z - i } { z - 1 }$ is purely imaginary, then the minimum value of $| z - ( 3 + 3i ) |$ is :
(1) $3 \sqrt { 2 }$
(2) $2 \sqrt { 2 }$
(3) $2 \sqrt { 2 } - 1$
(4) $6 \sqrt { 2 }$

A point $z$ moves in the complex plane such that $\arg \left( \frac { z - 2 } { z + 2 } \right) = \frac { \pi } { 4 }$, then the minimum value of $| z - 9 \sqrt { 2 } - 2 i | ^ { 2 }$ is equal to

Let $S = \{ z \in \mathbb { C } : | z - 3 | \leq 1$ and $z ( 4 + 3 i ) + \bar { z } ( 4 - 3 i ) \leq 24 \}$. If $\alpha + i \beta$ is the point in $S$ which is closest to $4 i$, then $25 ( \alpha + \beta )$ is equal to $\_\_\_\_$.

For $z \in \mathbb{C}$ if the minimum value of $(|z - 3\sqrt{2}| + |z - p\sqrt{2}i|)$ is $5\sqrt{2}$, then a value of $p$ is $\_\_\_\_$.
(1) 3
(2) $\frac{7}{2}$
(3) 4
(4) $\frac{9}{2}$

Let $S_1 = \{z_1 \in \mathbb{C} : |z_1 - 3| = \frac{1}{2}\}$ and $S_2 = \{z_2 \in \mathbb{C} : |z_2 - z_2 + 1| = |z_2 + z_2 - 1|\}$. Then, for $z_1 \in S_1$ and $z_2 \in S_2$, the least value of $|z_2 - z_1|$ is
(1) 0
(2) $\frac{1}{2}$
(3) $\frac{3}{2}$
(4) $\frac{3}{2}$

Let the minimum value $v _ { 0 }$ of $v = | z | ^ { 2 } + | z - 3 | ^ { 2 } + | z - 6 i | ^ { 2 } , z \in \mathbb { C }$ is attained at $z = z _ { 0 }$. Then $\left| 2 z _ { 0 } ^ { 2 } - \bar { z } _ { 0 } ^ { 3 } + 3 \right| ^ { 2 } + v _ { 0 } ^ { 2 }$ is equal to
(1) 1000
(2) 1024
(3) 1105
(4) 1196

If the set $\left\{ \operatorname { Re } \left( \frac { z - \bar { z } + z \bar { z } } { 2 - 3 z + 5 \bar { z } } \right) : z \in \mathbb { C } , \operatorname { Re } z = 3 \right\}$ is equal to the interval $( \alpha , \beta ]$, then $24 ( \beta - \alpha )$ is equal to
(1) 36
(2) 27
(3) 30
(4) 42

Let $C$ be the circle in the complex plane with centre $z _ { 0 } = \frac { 1 } { 2 } ( 1 + 3 i )$ and radius $r = 1$. Let $z _ { 1 } = 1 + i$ and the complex number $z _ { 2 }$ be outside circle $C$ such that $\left| z _ { 1 } - z _ { 0 } \right| \left| z _ { 2 } - z _ { 0 } \right| = 1$. If $z _ { 0 } , z _ { 1 }$ and $z _ { 2 }$ are collinear, then the smaller value of $\left| z _ { 2 } \right| ^ { 2 }$ is equal to
(1) $\frac { 5 } { 2 }$
(2) $\frac { 7 } { 2 }$
(3) $\frac { 13 } { 2 }$
(4) $\frac { 3 } { 2 }$

If $z$ is a complex number such that $|z| \leq 1$, then the minimum value of $\left|z + \frac{1}{2}(3 + 4i)\right|$ is:
(1) 2
(2) $\frac{5}{2}$
(3) $\frac{3}{2}$
(4) 3

Let $z$ be a complex number such that the real part of $\frac { z - 2 i } { z + 2 i }$ is zero. Then, the maximum value of $| z - ( 6 + 8 i ) |$ is equal to
(1) 12
(2) 10
(3) 8
(4) $\infty$

Let the curve $z ( 1 + i ) + \bar { z } ( 1 - i ) = 4 , z \in \mathrm { C }$, divide the region $| z - 3 | \leq 1$ into two parts of areas $\alpha$ and $\beta$. Then $| \alpha - \beta |$ equals :
(1) $1 + \frac { \pi } { 2 }$
(2) $1 + \frac { \pi } { 3 }$
(3) $1 + \frac { \pi } { 6 }$
(4) $1 + \frac { \pi } { 4 }$

Let $|z_1 - 8 - 2i| \leq 1$ and $|z_2 - 2 + 6i| \leq 2$, $z_1, z_2 \in \mathbf{C}$. Then the minimum value of $|z_1 - z_2|$ is:
(1) 13
(2) 10
(3) 3
(4) 7

Consider complex numbers $z$ such that
$$z \bar { z } - ( 1 - 2 i ) z - ( 1 + 2 i ) \bar { z } \leqq 15 .$$
(1) On a complex number plane, the figure represented by inequality (1) is the interior and circumference of the circle having the center $\mathbf{L} + \mathbf{M} i$ and the radius $\mathbf{NO}$.
(2) Let us consider all complex numbers $z$ which are on the straight line
$$( 1 - i ) z - ( 1 + i ) \bar { z } = 2 i$$
and satisfy the inequality (1). Of those, let $z _ { 1 }$ be the $z$ such that $| z |$ is maximized and $z _ { 2 }$ be the $z$ such that $| z |$ is minimized. Then we have
$$z _ { 1 } = \sqrt { \mathbf { P Q } } + \mathbf{Q} + ( \sqrt { \mathbf { S T } } + \mathbf { U } ) i ,$$ $$z _ { 2 } = - \frac { \mathbf { U } } { \mathbf { V } } + \frac{\mathbf{W}}{\mathbf{P}} i .$$

Below, line segments $[ A B ]$ and $[ C D ]$ are given in the complex number plane.
For each complex number z taken on these line segments, the number $\mathrm { w } = \mathrm { z } \cdot \overline { \mathrm { z } }$ is defined.
Accordingly, in which of the following are the minimum and maximum values that w can take given respectively?
A) 5 and 20
B) 5 and 25
C) 5 and 30
D) 10 and 20
E) 10 and 25