Let $a \neq b$ be two non-zero real numbers. Then the number of elements in the set $X = \left\{ z \in C : \operatorname{Re}\left(az^{2} + bz\right) = a$ and $\operatorname{Re}\left(bz^{2} + az\right) = b\right\}$ is equal to
(1) 0
(2) 1
(3) 3
(4) 2

Let $S = \left\{ z = x + iy : \frac { 2z - 3i } { 4z + 2i } \text{ is a real number} \right\}$. Then which of the following is NOT correct?
(1) $y + x ^ { 2 } + y ^ { 2 } \neq - \frac { 1 } { 4 }$
(2) $( x , y ) = \left( 0 , - \frac { 1 } { 2 } \right)$
(3) $x = 0$
(4) $y \in \left( - \infty , - \frac { 1 } { 2 } \right) \cup \left( - \frac { 1 } { 2 } , \infty \right)$

For $\alpha, \beta, z \in \mathbb{C}$ and $\lambda > 1$, if $\sqrt{\lambda - 1}$ is the radius of the circle $|z - \alpha|^{2} + |z - \beta|^{2} = 2\lambda$, then $|\alpha - \beta|$ is equal to $\_\_\_\_$.

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

If the center and radius of the circle $\left|\frac{z - 2}{z - 3}\right| = 2$ are respectively $(\alpha, \beta)$ and $\gamma$, then $3\alpha + \beta + \gamma$ is equal to
(1) 11
(2) 9
(3) 10
(4) 12

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) $( 0 , - 2 )$

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) $\frac { 3 } { 2 }$

Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z = 2 - i \left( 2 \tan \frac { 5 \pi } { 8 } \right)$, then $( r , \theta )$ is equal to
(1) $\left( 2 \sec \frac { 3 \pi } { 8 } , \frac { 3 \pi } { 8 } \right)$
(2) $\left( 2 \sec \frac { 3 \pi } { 8 } , \frac { 5 \pi } { 8 } \right)$
(3) $\left( 2 \sec \frac { 5 \pi } { 8 } , \frac { 3 \pi } { 8 } \right)$
(4) $\left( 2 \sec \frac { 11 \pi } { 8 } , \frac { 11 \pi } { 8 } \right)$

If $z = x + i y , x y \neq 0$, satisfies the equation $z ^ { 2 } + i \bar { z } = 0$, then $\left| z ^ { 2 } \right|$ is equal to:
(1) 9
(2) 1
(3) 4
(4) $\frac { 1 } { 4 }$

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 $z_1$ and $z_2$ be two complex number 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
(3) $15\sqrt{15}$
(4) $25\sqrt{3}$

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 $O$ 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

In a complex number plane, consider the complex numbers $z$ such that $z^3$ is a real number.
(1) Let $C$ be the figure formed by the set of complex numbers $z = x + iy$ satisfying the above condition. Since the arguments of the complex numbers $z$ satisfy $$\arg z = \frac{\pi}{\mathbf{M}}k \quad (k : \text{integer}),$$ figure $C$ consists of three straight lines represented in terms of $x$ and $y$ by the equations $$y = \mathbf{N}, \quad y = \sqrt{\mathbf{O}}\,x, \quad y = -\sqrt{\mathbf{O}}\,x.$$
(2) Suppose that on $C$ there exists only one complex number $z$ satisfying $|z - 1 - i| = r$. Then the value of $r$ is $$r = \frac{\sqrt{\mathbf{Q}} - \square\mathbf{R}}{\square}$$ and the value of $z$ is $$z = \frac{\mathbf{T} + \sqrt{\mathbf{U}}}{\square\mathbf{V}}(1 + \sqrt{\mathbf{W}}\,i).$$

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

Let $\alpha$ and $\beta$ be the two solutions of the quadratic equation $x ^ { 2 } + \sqrt { 3 } x + 1 = 0$, where $0 < \arg \alpha < \arg \beta < 2 \pi$. Consider the complex numbers $z$ satisfying the following three conditions:
$$\begin{cases} \arg \dfrac { \alpha - z } { \beta - z } = \dfrac { \pi } { 2 } & \ldots\ldots\ldots (1)\\ ( 1 + i ) z + ( 1 - i ) \bar { z } + k = 0 & \ldots\ldots\ldots (2)\\ \dfrac { \pi } { 2 } < \arg z < \pi , & \ldots\ldots\ldots (3) \end{cases}$$
where $k$ is a real number.
Let us denote the points on the complex number plane which express $\alpha$, $\beta$ and $z$ by $\mathrm{ A }$, $\mathrm{ B }$ and P.
(1) The arguments of $\alpha$ and $\beta$ are
$$\arg \alpha = \frac { \mathbf { A } } { \mathbf { B } } \pi \quad \text{ and } \quad \arg \beta = \frac { \mathbf { C } } { \mathbf { D } } \pi .$$
(2) For each of $\mathbf { E }$ $\sim$ $\mathbf { Q }$ in the following sentences, choose the correct answer from among (0) $\sim$ (9) below.
Since $\mathbf { E } = \dfrac { \pi } { 2 }$ from (1), the point P is located on the circumference of the circle with the center $-\dfrac{ \sqrt{\mathbf{F}} }{ \mathbf{G} }$ and the radius $\dfrac { \mathbf { H } } { \mathbf { I } }$.
On the other hand, from (2), the point P is on the straight line which has the slope $\mathbf{J}$ and passes through a certain point.
From these, we see that when $n$ is the number of complex numbers $z$ which simultaneously satisfy (1), (2) and (3), the maximum value of $n$ is $\mathbf { M }$, and in this case the range of values of $k$ is
$$\mathbf { N } + \sqrt { \mathbf { O } } < k < \sqrt { \mathbf { P } } + \sqrt { \mathbf { Q } }$$
where $\mathbf { P } < \mathbf { Q }$.
(0) 0
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5 (6) 6 (7) $\angle \mathrm{ PAB }$ (8) $\angle \mathrm{ PBA }$ (9) $\angle \mathrm{ APB }$

Let $z$ be a complex number satisfying $| z | = 2$. In the complex number plane with the origin O, let A and B be the points representing $1 + z$ and $1 - \frac { 1 } { 2 } z$, respectively.
First of all, we can express the complex number $z$ as
$$z = \mathbf { M } ( \cos \theta + i \sin \theta ) \quad ( - \pi \leqq \theta < \pi ) .$$
(1) If $z$ is not a real number, then the area $S$ of the triangle OAB is $S = \mathbf { N }$. For $\mathbf{N}$, choose the correct answer from among (0) $\sim$ (8) below.
Hence, when $\theta = \pm \frac { \mathbf { O } } { \mathbf { P } } \pi$, $S$ is maximized.
(0) $\frac { 1 } { 2 } \left| \sin \left( \theta + \frac { 1 } { 3 } \pi \right) \right|$ (1) $\frac { 1 } { 2 } | \sin \theta |$ (2) $\frac { 1 } { 2 } \left| \sin \left( \theta - \frac { 1 } { 3 } \pi \right) \right|$ (3) $\left| \sin \left( \theta + \frac { 1 } { 3 } \pi \right) \right|$ (4) $| \sin \theta |$ (5) $\left| \sin \left( \theta - \frac { 1 } { 3 } \pi \right) \right|$ (6) $\frac { 3 } { 2 } \left| \sin \left( \theta + \frac { 1 } { 3 } \pi \right) \right|$ (7) $\frac { 3 } { 2 } | \sin \theta |$ (8) $\frac { 3 } { 2 } \left| \sin \left( \theta - \frac { 1 } { 3 } \pi \right) \right|$
(2) When the triangle OAB is an isosceles triangle where $\mathrm { OA } = \mathrm { OB }$, we see that
$$| 1 + z | = \left| 1 - \frac { 1 } { 2 } z \right| = \sqrt { \mathbf { Q } }$$
and
$$\arg ( 1 + z ) = \pm \frac { \mathbf { R } } { \mathbf { S } } \pi , \quad \arg \left( 1 - \frac { 1 } { 2 } z \right) = \mp \frac { \mathbf { T } } { \mathbf{U} } \pi ,$$
where the right-hand sides of the equations are of opposite signs, and where $- \pi \leqq \arg ( 1 + z ) < \pi$ and $- \pi \leqq \arg \left( 1 - \frac { 1 } { 2 } z \right) < \pi$.