Let $z _ { 1 } = 2 + 3 i$ and $z _ { 2 } = 3 + 4 i$. The set $\mathrm { S } = \left\{ \mathrm { z } \in \mathrm { C } : \left| \mathrm { z } - \mathrm { z } _ { 1 } \right| ^ { 2 } - \left| \mathrm { z } - \mathrm { z } _ { 2 } \right| ^ { 2 } = \left| \mathrm { z } _ { 1 } - \mathrm { z } _ { 2 } \right| ^ { 2 } \right\}$ represents a
(1) straight line with sum of its intercepts on the coordinate axes equals 14
(2) hyperbola with the length of the transverse axis 7
(3) straight line with the sum of its intercepts on the coordinate axes equals $-18$
(4) hyperbola with eccentricity 2

Let z be a complex number such that $\left| \frac { z - 2 i } { z + i } \right| = 2 , z \neq - i$. Then $z$ lies on the circle of radius 2 and centre
(1) $( 2,0 )$
(2) $( 0,2 )$
(3) $( 0,0 )$
(4) $( 0 , - 2 )$

For all $z \in C$ on the curve $C_1 : |z| = 4$, let the locus of the point $z + \frac{1}{z}$ be the curve $C_2$. Then
(1) the curves $C_1$ and $C_2$ intersect at 4 points
(2) the curves $C_1$ lies inside $C_2$
(3) the curves $C_1$ and $C_2$ intersect at 2 points
(4) the curves $C_2$ lies inside $C_1$

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 $\mathrm { S } = \mathrm { z } \in \mathrm { C } : | \mathrm { z } - \mathrm { i } | = | \mathrm { z } + \mathrm { i } | = | \mathrm { z } - 1 |$, then, $\mathrm { n } ( \mathrm { S } )$ is:
(1) 1
(2) 0
(3) 3
(4) 2

If $z = \frac { 1 } { 2 } - 2 i$, is such that $| z + 1 | = \alpha z + \beta ( 1 + i ) , i = \sqrt { - 1 }$ and $\alpha , \beta \in \mathrm { R }$, then $\alpha + \beta$ is equal to
(1) - 4
(2) 3
(3) 2
(4) - 1

The area (in sq. units) of the region $S = \{ z \in \mathbb { C } : | z - 1 | \leq 2 ; ( z + \bar { z } ) + i ( z - \bar { z } ) \leq 2 , \operatorname { Im } ( z ) \geq 0 \}$ is
(1) $\frac { 7 \pi } { 3 }$
(2) $\frac { 7 \pi } { 4 }$
(3) $\frac { 17 \pi } { 8 }$
(4) $\frac { 3 \pi } { 2 }$

Let $S _ { 1 } = \{ z \in C : | z | \leq 5 \} , S _ { 2 } = \left\{ z \in C : \operatorname { Im } \left( \frac { z + 1 - \sqrt { 3 } i } { 1 - \sqrt { 3 } i } \right) \geq 0 \right\}$ and $S _ { 3 } = \{ z \in C : \operatorname { Re } ( z ) \geq 0 \}$. Then the area of the region $S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$ is :
(1) $\frac { 125 \pi } { 12 }$
(2) $\frac { 125 \pi } { 4 }$
(3) $\frac { 125 \pi } { 24 }$
(4) $\frac { 125 \pi } { 6 }$

If $z _ { 1 } , z _ { 2 }$ are two distinct complex number such that $\left| \frac { z _ { 1 } - 2 z _ { 2 } } { \frac { 1 } { 2 } - z _ { 1 } \bar { z } _ { 2 } } \right| = 2$, then
(1) $z _ { 1 }$ lies on a circle of radius $\frac { 1 } { 2 }$ and $z _ { 2 }$ lies on a circle of radius 1.
(2) both $z _ { 1 }$ and $z _ { 2 }$ lie on the same circle.
(3) either $z _ { 1 }$ lies on a circle of radius $\frac { 1 } { 2 }$ or $z _ { 2 }$ lies on a circle of radius 1.
(4) either $z _ { 1 }$ lies on a circle of radius 1 or $z _ { 2 }$ lies on a circle of radius $\frac { 1 } { 2 }$.

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 $| z + 2 | = 1$ and $\operatorname { Im } \left( \frac { z + 1 } { z + 2 } \right) = \frac { 1 } { 5 }$. Then the value of $| \operatorname { Re } ( \overline { z + 2 } ) |$ is
(1) $\frac { 2 \sqrt { 6 } } { 5 }$
(2) $\frac { 24 } { 5 }$
(3) $\frac { 1 + \sqrt { 6 } } { 5 }$
(4) $\frac { \sqrt { 6 } } { 5 }$

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 complex numbers $\alpha$ and $\frac { 1 } { \alpha }$ lie on the circles $\mathrm { z } - \mathrm { z } _ { 0 } { } ^ { 2 } = 4$ and $\mathrm { z } - \mathrm { z } _ { 0 } { } ^ { 2 } = 16$ respectively, where $\mathrm { z } _ { 0 } = 1 + \mathrm { i }$. Then, the value of $100 | \alpha | ^ { 2 }$ is $\_\_\_\_$ .

Let $O$ be the origin, the point $A$ be $z _ { 1 } = \sqrt { 3 } + 2 \sqrt { 2 } i$, the point $B \left( z _ { 2 } \right)$ be such that $\sqrt { 3 } \left| z _ { 2 } \right| = \left| z _ { 1 } \right|$ and $\arg \left( z _ { 2 } \right) = \arg \left( z _ { 1 } \right) + \frac { \pi } { 6 }$. Then
(1) area of triangle ABO is $\frac { 11 } { \sqrt { 3 } }$
(2) ABO is an obtuse angled isosceles triangle
(3) area of triangle ABO is $\frac { 11 } { 4 }$
(4) ABO is a scalene triangle

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

Let $\left| \frac { \bar { z } - i } { 2 \bar { z } + i } \right| = \frac { 1 } { 3 } , z \in C$, be the equation of a circle with center at $C$. If the area of the triangle, whose vertices are at the points $( 0,0 ) , \mathrm { C }$ and $( \alpha , 0 )$ is 11 square units, then $\alpha ^ { 2 }$ equals:
(1) 50
(2) 100
(3) $\frac { 81 } { 25 }$
(4) $\frac { 121 } { 25 }$

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 $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$ be three complex numbers on the circle $| z | = 1$ with $\arg \left( z _ { 1 } \right) = \frac { - \pi } { 4 } , \arg \left( z _ { 2 } \right) = 0$ and $\arg \left( z _ { 3 } \right) = \frac { \pi } { 4 }$. If $\left| z _ { 1 } \bar { z } _ { 2 } + z _ { 2 } \bar { z } _ { 3 } + z _ { 3 } \bar { z } _ { 1 } \right| ^ { 2 } = \alpha + \beta \sqrt { 2 } , \alpha , \beta \in \mathbf { Z }$, then the value of $\alpha ^ { 2 } + \beta ^ { 2 }$ is:
(1) 24
(2) 29
(3) 41
(4) 31

Q61. The area (in sq. units) of the region $S = \{ z \in \mathbb { C } : | z - 1 | \leq 2 ; ( z + \bar { z } ) + i ( z - \bar { z } ) \leq 2 , \operatorname { Im } ( z ) \geq 0 \}$ is
(1) $\frac { 7 \pi } { 3 }$
(2) $\frac { 7 \pi } { 4 }$
(3) $\frac { 17 \pi } { 8 }$
(4) $\frac { 3 \pi } { 2 }$

Q61. Let $S _ { 1 } = \{ z \in C : | z | \leq 5 \} , S _ { 2 } = \left\{ z \in C : \operatorname { Im } \left( \frac { z + 1 - \sqrt { 3 } i } { 1 - \sqrt { 3 } i } \right) \geq 0 \right\}$ and $S _ { 3 } = \{ z \in C : \operatorname { Re } ( z ) \geq 0 \}$. Then the area of the region $S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$ is :
(1) $\frac { 125 \pi } { 12 }$
(2) $\frac { 125 \pi } { 4 }$
(3) $\frac { 125 \pi } { 24 }$
(4) $\frac { 125 \pi } { 6 }$

Q61. If $z _ { 1 } , z _ { 2 }$ are two distinct complex number such that $\left| \frac { z _ { 1 } - 2 z _ { 2 } } { \frac { 1 } { 2 } - z _ { 1 } \bar { z } _ { 2 } } \right| = 2$, then
(1) $z _ { 1 }$ lies on a circle of radius $\frac { 1 } { 2 }$ and $z _ { 2 }$ lies on a
(2) both $z _ { 1 }$ and $z _ { 2 }$ lie on the same circle. both $z _ { 1 }$ and $z _ { 2 }$ lie on the same circle.
(3) either $z _ { 1 }$ lies on a circle of radius $\frac { 1 } { 2 }$ or $z _ { 2 }$ lies on
(4) either $z _ { 1 }$ lies on a circle of radius 1 or $z _ { 2 }$ lies on a a circle of radius 1 . circle of radius $\frac { 1 } { 2 }$.

Q61. The sum of all possible values of $\theta \in [ - \pi , 2 \pi ]$, for which $\frac { 1 + i \cos \theta } { 1 - 2 i \cos \theta }$ is purely imaginary, is equal
(1) $3 \pi$
(2) $2 \pi$
(3) $5 \pi$
(4) $4 \pi$

Q62. 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 $A = \{ Z \in C : | Z - 2 | \leq 4 \}$ and $B = \{ Z \in C : | Z - 2 | + | Z + 2 | \leq 4 \}$ then $\boldsymbol { \operatorname { m a x } } \left\{ \mathrm { Z } _ { 1 } - \mathrm { Z } _ { 2 } \right\} : \mathrm { Z } _ { 1 } \in \mathrm {~A} \text { and } \mathrm { Z } _ { 2 } \in \mathrm {~B} \text { is equal to }$ (A) 8 (B) 6 (C) 4

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
$$z _ { 1 } = \sqrt { \mathbf { P Q } } + \mathbf{Q} + ( \sqrt { \mathbf { S T } } + \mathbf { U } ) i ,$$ $$z _ { 2 } = - \frac { \mathbf { U } } { \mathbf { V } } + \frac{\mathbf{W}}{\mathbf{P}} i .$$