Let $( 1 + i ) x = 1 + y i$, where $x , y$ are real numbers, then $| x + y i | =$
(A) 1
(B) $\sqrt { 2 }$
(C) $\sqrt { 3 }$
(D) 2

Let $( 1 + 2 \mathrm { i } ) a + b = 2 \mathrm { i }$ , where $a , b$ are real numbers, then
A. $a = 1 , b = - 1$
B. $a = 1 , b = 1$
C. $a = - 1 , b = 1$
D. $a = - 1 , b = - 1$

Given $z = 1 - 2i$, and $z + a\bar{z} + b = 0$, where $a, b$ are real numbers, then
A. $a = 1, b = -2$
B. $a = -1, b = 2$
C. $a = 1, b = 2$
D. $a = -1, b = -2$

If the complex number $( a + i )( 1 - ai ) = 2$ , then $a =$
A. $-1$
B. $0$
C. $1$
D. $2$

Let $z = x + i y$ be a complex number, which satisfies the equation $( z + \bar { z } ) z = 2 + 4 i$. Then
(A) $y = \pm 2$.
(B) $x = \pm 2$.
(C) $x = \pm 3$.
(D) $y = \pm 1$.

Let $a, b, x$ and $y$ be real numbers such that $a - b = 1$ and $y \neq 0$. If the complex number $z = x + iy$ satisfies $\operatorname{Im}\left(\frac{az + b}{z + 1}\right) = y$, then which of the following is(are) possible value(s) of $x$?
[A] $-1 + \sqrt{1 - y^2}$
[B] $-1 - \sqrt{1 - y^2}$
[C] $1 + \sqrt{1 + y^2}$
[D] $1 - \sqrt{1 + y^2}$

For all complex numbers $z$ of the form $1 + i \alpha , \alpha \in R$, if $z ^ { 2 } = x + i y$, then
(1) $y ^ { 2 } - 4 x + 4 = 0$
(2) $y ^ { 2 } + 4 x - 4 = 0$
(3) $y ^ { 2 } - 4 x + 2 = 0$
(4) $y ^ { 2 } + 4 x + 2 = 0$

A value of $\theta$ for which $\frac{2+3i\sin\theta}{1-2i\sin\theta}$ is purely imaginary, is: (1) $\frac{\pi}{3}$ (2) $\frac{\pi}{6}$ (3) $\sin^{-1}\left(\frac{\sqrt{3}}{4}\right)$ (4) $\sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Let $z \in C$ with $\operatorname { Im } ( z ) = 10$ and it satisfies $\frac { 2 z - n } { 2 z + n } = 2 i - 1$ for some natural number $n$. Then
(1) $n = 20$ and $\operatorname { Re } ( z ) = 10$
(2) $n = 40$ and $\operatorname { Re } ( z ) = 10$
(3) $n = 20$ and $\operatorname { Re } ( z ) = - 10$
(4) $n = 40$ and $\operatorname { Re } ( z ) = - 10$

Let $\mathrm{z}=\mathrm{x}+\mathrm{iy}$ be a non-zero complex number such that $\mathrm{z}^{2}=\mathrm{i}|\mathrm{z}|^{2}$, where $\mathrm{i}=\sqrt{-1}$, then z lies on the:
(1) line, $y=-x$
(2) imaginary axis
(3) line, $y=x$
(4) real axis

If $\alpha , \beta \in R$ are such that $1 - 2 i$ (here $i ^ { 2 } = - 1$ ) is a root of $z ^ { 2 } + \alpha z + \beta = 0$, then ( $\alpha - \beta$ ) is equal to:
(1) - 7
(2) 7
(3) - 3
(4) 3

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) $2 - i$

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 $\_\_\_\_$

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 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) $\frac{1}{3}$

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

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 $b$ and $c$ be real numbers. One root of the polynomial $P(x) = x^{2} + bx + c$ is the complex number $3 - 2i$.
Accordingly, what is $P(-1)$?
A) 5
B) 10
C) 20
D) 25
E) 30

$$\frac { | z | ^ { 2 } + z } { \bar { z } } = z + i$$
Which of the following is the set of complex numbers z that satisfy this equality? (R is the set of real numbers.)
A) $\{ a + a i \mid a \in R , a \neq 0 \}$
B) $\{ a - a i \mid a \in R , a \neq 0 \}$
C) $\{ a + 2 a i \mid a \in R , a \neq 0 \}$
D) $\{ a - 2 a i \mid a \in R , a \neq 0 \}$
E) $\{ 2 a - a i \mid a \in R , a \neq 0 \}$

The functions $\mathrm { f } ( \mathrm { x } ) = \mathrm { x } + \mathrm { xi }$ and $\mathrm { g } ( \mathrm { x } ) = 2 \mathrm { x } - \mathrm { xi }$ are defined from the set of real numbers to the set of complex numbers and satisfy
$$f ( a ) + g ( b ) = 4 + 2 i$$
Accordingly, what is the sum $\mathbf { a } + \mathbf { b }$?
A) $\frac { 7 } { 2 }$
B) $\frac { 9 } { 2 }$
C) $\frac { 10 } { 3 }$
D) $\frac { 13 } { 3 }$
E) $\frac { 15 } { 4 }$