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

Match the statements in Column-I with those in Column-II. [Note: Here $z$ takes values in the complex plane and $\operatorname { Im } z$ and $\operatorname { Re } z$ denote, respectively, the imaginary part and the real part of $z$.]
Column I
A) The set of points $z$ satisfying $| z - i | z \| = | z + i | z \mid$ is contained in or equal to
B) The set of points $z$ satisfying $| z + 4 | + | z - 4 | = 10$ is contained in or equal to
C) If $| w | = 2$, then the set of points $z = w - \frac { 1 } { w }$ is contained in or equal to
D) If $| w | = 1$, then the set of points $z = w + \frac { 1 } { w }$ is contained in or equal to
Column II p) an ellipse with eccentricity $\frac { 4 } { 5 }$ q) the set of points $z$ satisfying $\operatorname { Im } z = 0$ r) the set of points $z$ satisfying $| \operatorname { Im } z | \leq 1$ s) the set of points $z$ satisfying $| \operatorname { Re } z | \leq 2$ t) the set of points $z$ satisfying $| z | \leq 3$

Let complex numbers $\alpha$ and $\frac { 1 } { \bar { \alpha } }$ lie on circles $\left( x - x _ { 0 } \right) ^ { 2 } + \left( y - y _ { 0 } \right) ^ { 2 } = r ^ { 2 }$ and $\left( x - x _ { 0 } \right) ^ { 2 } + \left( y - y _ { 0 } \right) ^ { 2 } = 4 r ^ { 2 }$, respectively. If $z _ { 0 } = x _ { 0 } + i y _ { 0 }$ satisfies the equation $2 \left| z _ { 0 } \right| ^ { 2 } = r ^ { 2 } + 2$, then $| \alpha | =$
(A) $\frac { 1 } { \sqrt { 2 } }$
(B) $\frac { 1 } { 2 }$
(C) $\frac { 1 } { \sqrt { 7 } }$
(D) $\frac { 1 } { 3 }$

Let $w = \frac { \sqrt { 3 } + \mathrm { i } } { 2 }$ and $P = \left\{ w ^ { n } : n = 1,2,3 , \ldots \right\}$. Further $H _ { 1 } = \left\{ \mathrm { z } \in \mathbb { C } : \operatorname { Re } z > \frac { 1 } { 2 } \right\}$ and $H _ { 2 } = \left\{ \mathrm { z } \in \mathbb { C } : \operatorname { Re } z < \frac { - 1 } { 2 } \right\}$, where $\mathbb { C }$ is the set of all complex numbers. If $z _ { 1 } \in P \cap H _ { 1 }$, $z _ { 2 } \in P \cap H _ { 2 }$ and $O$ represents the origin, then $\angle z _ { 1 } O z _ { 2 } =$
(A) $\frac { \pi } { 2 }$
(B) $\frac { \pi } { 6 }$
(C) $\frac { 2 \pi } { 3 }$
(D) $\frac { 5 \pi } { 6 }$

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 $S _ { 3 } = \{ z \in \mathbb { C } : \operatorname { Re } z > 0 \}$.
Area of $S =$
(A) $\frac { 10 \pi } { 3 }$
(B) $\frac { 20 \pi } { 3 }$
(C) $\frac { 16 \pi } { 3 }$
(D) $\frac { 32 \pi } { 3 }$

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 $a , b \in \mathbb { R }$ and $a ^ { 2 } + b ^ { 2 } \neq 0$. Suppose $S = \left\{ z \in \mathbb { C } : z = \frac { 1 } { a + i b t } , t \in \mathbb { R } , t \neq 0 \right\}$, where $i = \sqrt { - 1 }$.
If $z = x + i y$ and $z \in S$, then $( x , y )$ lies on
(A) the circle with radius $\frac { 1 } { 2 a }$ and centre $\left( \frac { 1 } { 2 a } , 0 \right)$ for $a > 0 , b \neq 0$
(B) the circle with radius $- \frac { 1 } { 2 a }$ and centre $\left( - \frac { 1 } { 2 a } , 0 \right)$ for $a < 0 , b \neq 0$
(C) the $x$-axis for $a \neq 0 , b = 0$
(D) the $y$-axis for $a = 0 , b \neq 0$

Let $s , t , r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + i y ( x , y \in \mathbb { R } , i = \sqrt { - 1 } )$ of the equation $s z + t \bar { z } + r = 0$, where $\bar { z } = x - i y$. Then, which of the following statement(s) is (are) TRUE?
(A) If $L$ has exactly one element, then $| s | \neq | t |$
(B) If $| s | = | t |$, then $L$ has infinitely many elements
(C) The number of elements in $L \cap \{ z : | z - 1 + i | = 5 \}$ is at most 2
(D) If $L$ has more than one element, then $L$ has infinitely many elements

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 $\left| z ^ { 2 } + z + 1 \right| = 1$. Then 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

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

Let $\mathbb { R }$ denote the set of all real numbers. Let $z _ { 1 } = 1 + 2 i$ and $z _ { 2 } = 3 i$ be two complex numbers, where $i = \sqrt { - 1 }$. Let
$$S = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : \left| x + i y - z _ { 1 } \right| = 2 \left| x + i y - z _ { 2 } \right| \right\}$$
Then which of the following statements is (are) TRUE?

(A)	$S$ is a circle with centre $\left( - \frac { 1 } { 3 } , \frac { 10 } { 3 } \right)$
(B)	$S$ is a circle with centre $\left( \frac { 1 } { 3 } , \frac { 8 } { 3 } \right)$
(C)	$S$ is a circle with radius $\frac { \sqrt { 2 } } { 3 }$
(D)	$S$ is a circle with radius $\frac { 2 \sqrt { 2 } } { 3 }$

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

The area of the triangle whose vertices are complex numbers $z, iz, z+iz$ in the Argand diagram is
(1) $2|z|^{2}$
(2) $\frac{1}{2}|z|^{2}$
(3) $4|z|^{2}$
(4) $|z|^{2}$

If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg\left(\frac{1+z}{1+\bar{z}}\right)$ equals
(1) $-\theta$
(2) $\frac{\pi}{2}-\theta$
(3) $\theta$
(4) $\pi-\theta$

If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg\left(\frac{1+z}{1+\bar{z}}\right)$ can be equal to (given $z \neq -1$)
(1) $\theta$
(2) $\pi - \theta$
(3) $-\theta$
(4) $\frac{\pi}{2} - \theta$

If $z$ is a complex number such that $| z | \geq 2$, then the minimum value of $\left| z + \frac { 1 } { 2 } \right|$:
(1) Is strictly greater than $\frac { 5 } { 2 }$
(2) Is strictly greater than $\frac { 3 } { 2 }$ but less than $\frac { 5 } { 2 }$
(3) Is equal to $\frac { 5 } { 2 }$
(4) Lies in the interval $( 1,2 )$

A complex number $z$ is said to be unimodular if $|z| = 1$. Suppose $z_1$ and $z_2$ are complex numbers such that $\frac{z_1 - 2z_2}{2 - z_1\bar{z}_2}$ is unimodular and $z_2$ is not unimodular. Then the point $z_1$ lies on a:
(1) straight line parallel to $x$-axis
(2) straight line parallel to $y$-axis
(3) circle of radius 2
(4) circle of radius $\sqrt{2}$

Let $z \in C$, the set of complex numbers. Then the equation, $2 | z + 3 i | - | z - i | = 0$ represents:
(1) A circle with radius $\frac { 8 } { 3 }$
(2) An ellipse with length of minor axis $\frac { 16 } { 9 }$
(3) An ellipse with length of major axis $\frac { 16 } { 3 }$
(4) A circle with diameter $\frac { 10 } { 3 }$

Let $z _ { 1 }$ and $z _ { 2 }$ be two complex numbers satisfying $\left| z _ { 1 } \right| = 9$ and $\left| z _ { 2 } - 3 - 4 i \right| = 4$. Then the minimum value of $\left| z _ { 1 } - z _ { 2 } \right|$ is :
(1) 2
(2) $\sqrt { 2 }$
(3) 0
(4) 1

Let $z_0$ be a root of quadratic equation, $x^2 + x + 1 = 0$. If $z = 3 + 6iz_0^{81} - 3iz_0^{93}$, then $\arg(z)$ is equal to:
(1) 0
(2) $\frac{\pi}{4}$
(3) $\frac{\pi}{6}$
(4) $\frac{\pi}{3}$

Let $z$ be a complex number such that $\left| \frac { z - i } { z + 2 i } \right| = 1$ and $| z | = \frac { 5 } { 2 }$. Then, the value of $| z + 3 i |$ is
(1) $\sqrt { 10 }$
(2) $\frac { 7 } { 2 }$
(3) $\frac { 15 } { 4 }$
(4) $2 \sqrt { 3 }$

If $z _ { 1 } , z _ { 2 }$ are complex numbers such that $\operatorname { Re } \left( z _ { 1 } \right) = \left| z _ { 1 } - 1 \right|$ and $\operatorname { Re } \left( z _ { 2 } \right) = \left| z _ { 2 } - 1 \right|$ and $\arg \left( z _ { 1 } - z _ { 2 } \right) = \frac { \pi } { 6 }$ , then $\operatorname { Im } \left( z _ { 1 } + z _ { 2 } \right)$ is equal to :
(1) $2 \sqrt { 3 }$
(2) $\frac { \sqrt { 3 } } { 2 }$
(3) $\frac { 1 } { \sqrt { 3 } }$
(4) $\frac { 2 } { \sqrt { 3 } }$

If the four complex numbers $z , \bar { z } , \bar { z } - 2 \operatorname { Re } ( \bar { z } )$ and $z - 2 \operatorname { Re } ( z )$ represent the vertices of a square of side 4 units in the Argand plane, then $| z |$ is equal to :
(1) $4 \sqrt { 2 }$
(2) 4
(3) $2 \sqrt { 2 }$
(4) 2