Let $\alpha$ and $\beta$ be the roots of the equation $x ^ { 2 } + ( 2 i - 1 ) = 0$. Then, the value of $\left| \alpha ^ { 8 } + \beta ^ { 8 } \right|$ is equal to
(1) 50
(2) 250
(3) 1250
(4) 1550

If $z = 2 + 3 i$, then $z ^ { 5 } + ( \bar { z } ) ^ { 5 }$ is equal to:
(1) 244
(2) 224
(3) 245
(4) 265

If $z \neq 0$ be a complex number such that $z - \frac{1}{z} = 2$, then the maximum value of $|z|$ is
(1) $\sqrt{2}$
(2) 1
(3) $\sqrt{2} - 1$
(4) $\sqrt{2} + 1$

Let for some real numbers $\alpha$ and $\beta , a = \alpha - i \beta$. If the system of equations $4 i x + ( 1 + i ) y = 0$ and $8 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) x + \bar { a } y = 0$ has more than one solution then $\frac { \alpha } { \beta }$ is equal to
(1) $2 - \sqrt { 3 }$
(2) $2 + \sqrt { 3 }$
(3) $- 2 + \sqrt { 3 }$
(4) $- 2 - \sqrt { 3 }$

Let $S$ be the set of all $( \alpha , \beta ) , \pi < \alpha , \beta < 2 \pi$, for which the complex number $\frac { 1 - i \sin \alpha } { 1 + 2 i \sin \alpha }$ is purely imaginary and $\frac { 1 + i \cos \beta } { 1 - 2 i \cos \beta }$ is purely real. Let $Z _ { \alpha \beta } = \sin 2 \alpha + i \cos 2 \beta , ( \alpha , \beta ) \in S$. Then $\sum _ { ( \alpha , \beta ) \in S } \left( i Z _ { \alpha \beta } + \frac { 1 } { i \bar { Z } _ { \alpha \beta } } \right)$ is equal to
(1) 3
(2) $3 i$
(3) 1
(4) $2 - i$

Let $S = \left\{ z \in \mathbb { C } : z ^ { 2 } + \bar { z } = 0 \right\}$. Then $\sum _ { z \in S } ( \operatorname { Re } ( z ) + \operatorname { Im } ( z ) )$ is equal to $\_\_\_\_$ .

Let $\mathrm { z } = \mathrm { a } + i b , \mathrm { b } \neq 0$ be complex numbers satisfying $\mathrm { z } ^ { 2 } = \overline { \mathrm { z } } \cdot 2 ^ { 1 - | z | }$. Then the least value of $n \in N$, such that $z ^ { n } = ( z + 1 ) ^ { n }$, is equal to $\_\_\_\_$.

Let $p , \quad q \in \mathbb { R }$ and $( 1 - \sqrt { 3 } i ) ^ { 200 } = 2 ^ { 199 } ( p + i q ) , i = \sqrt { - 1 }$. Then, $p + q + q ^ { 2 }$ and $p - q + q ^ { 2 }$ are roots of the equation.
(1) $x ^ { 2 } + 4 x - 1 = 0$
(2) $x ^ { 2 } - 4 x + 1 = 0$
(3) $x ^ { 2 } + 4 x + 1 = 0$
(4) $x ^ { 2 } - 4 x - 1 = 0$

Let the complex number $z = x + iy$ be such that $\frac{2z - 3i}{2z + i}$ is purely imaginary. If $x + y^2 = 0$, then $y^4 + y^2 - y$ is equal to
(1) $\frac{2}{3}$
(2) $\frac{3}{2}$
(3) $\frac{3}{4}$
(4) $\frac{1}{3}$

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

For two non-zero complex number $z _ { 1 }$ and $z _ { 2 }$, if $\operatorname { Re } \left( z _ { 1 } z _ { 2 } \right) = 0$ and $\operatorname { Re } \left( z _ { 1 } + z _ { 2 } \right) = 0$, then which of the following are possible?
(A) $\operatorname { Im } \left( \mathrm { z } _ { 1 } \right) > 0$ and $\operatorname { Im } \left( \mathrm { z } _ { 2 } \right) > 0$
(B) $\operatorname { Im } \left( \mathrm { z } _ { 1 } \right) < 0$ and $\operatorname { Im } \left( \mathrm { z } _ { 2 } \right) > 0$
(C) $\operatorname { Im } \left( \mathrm { z } _ { 1 } \right) > 0$ and $\operatorname { Im } \left( \mathrm { z } _ { 2 } \right) < 0$
(D) $\operatorname { Im } \left( \mathrm { z } _ { 1 } \right) < 0$ and $\operatorname { Im } \left( \mathrm { z } _ { 2 } \right) < 0$ Choose the correct answer from the options given below:
(1) B and D
(2) B and C
(3) A and B
(4) A and C

Let $\alpha = 8 - 14 \mathrm { i } , \mathrm { A } = \left\{ \mathrm { z } \in \mathbb { C } : \frac { \alpha \mathrm { z } - \bar { \alpha } \overline { \mathrm { z } } } { \mathrm { z } ^ { 2 } - ( \overline { \mathrm { z } } ) ^ { 2 } - 112 \mathrm { i } } = 1 \right\}$ and $B = \{ z \in \mathbb { C } : | z + 3 i | = 4 \}$
Then, $\sum _ { z \in A \cap B } ( \operatorname { Re } z - \operatorname { Im } z )$ is equal to $\_\_\_\_$

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 $C_1$

If for $z = \alpha + i \beta , | z + 2 | = z + 4 ( 1 + i )$, then $\alpha + \beta$ and $\alpha \beta$ are the roots of the equation
(1) $x ^ { 2 } + 3 x - 4 = 0$
(2) $x ^ { 2 } + 7 x + 12 = 0$
(3) $x ^ { 2 } + x - 12 = 0$
(4) $x ^ { 2 } + 2 x - 3 = 0$

Let $z_1$ and $z_2$ be two complex numbers such that $z_1 + z_2 = 5$ and $z_1^3 + z_2^3 = 20 + 15i$. Then $|z_1^4 + z_2^4|$ equals
(1) $30\sqrt{3}$
(2) $75\sqrt{2}$
(3) $15\sqrt{15}$
(4) $25\sqrt{3}$

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 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) $4 \pi$

If the set $R = \{ ( a , b ) : a + 5 b = 42 , a , b \in \mathbb { N } \}$ has $m$ elements and $\sum _ { n = 1 } ^ { m } \left( 1 - i ^ { n ! } \right) = x + i y$, where $i = \sqrt { - 1 }$, then the value of $m + x + y$ is
(1) 12
(2) 4
(3) 8
(4) 5

Given that the inverse trigonometric function assumes principal values only. Let $x , y$ be any two real numbers in $[ - 1,1 ]$ such that $\cos ^ { - 1 } x - \sin ^ { - 1 } y = \alpha , \frac { - \pi } { 2 } \leq \alpha \leq \pi$. Then, the minimum value of $x ^ { 2 } + y ^ { 2 } + 2 x y \sin \alpha$ is
(1) 0
(2) - 1
(3) $\frac { 1 } { 2 }$
(4) $- \frac { 1 } { 2 }$

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - x + 2 = 0$ with $\operatorname { Im } ( \alpha ) > \operatorname { Im } ( \beta )$. Then $\alpha ^ { 6 } + \alpha ^ { 4 } + \beta ^ { 4 } - 5 \alpha ^ { 2 }$ is equal to

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 $\_\_\_\_$ .

The number of complex numbers $z$, satisfying $| z | = 1$ and $\left| \frac { z } { \bar { z } } + \frac { \bar { z } } { z } \right| = 1$, is :
(1) 4
(2) 8
(3) 10
(4) 6

If $\alpha + i \beta$ and $\gamma + i \delta$ are the roots of $x ^ { 2 } - ( 3 - 2 i ) x - ( 2 i - 2 ) = 0 , i = \sqrt { - 1 }$, then $\alpha \gamma + \beta \delta$ is equal to :
(1) $-2$
(2) 6
(3) $-6$
(4) 2

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - a x - b = 0$ with $\operatorname { Im } ( \alpha ) < \operatorname { Im } ( \beta )$. Let $P _ { n } = \alpha ^ { n } - \beta ^ { n }$. If $\mathrm { P } _ { 3 } = - 5 \sqrt { 7 } i , \mathrm { P } _ { 4 } = - 3 \sqrt { 7 } i , \mathrm { P } _ { 5 } = 11 \sqrt { 7 } i$ and $\mathrm { P } _ { 6 } = 45 \sqrt { 7 } i$, then $\left| \alpha ^ { 4 } + \beta ^ { 4 } \right|$ is equal to