Let $z$ be a complex number satisfying $|z|^3 + 2z^2 + 4\bar{z} - 8 = 0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Which of the following statements is TRUE?
(A) $|z|^2 = 2$
(B) $|z|^2 = 4$
(C) $|z|^2 = 8$
(D) $|z|^2 = 16$

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 $z$ be a complex number with non-zero imaginary part. If
$$\frac { 2 + 3 z + 4 z ^ { 2 } } { 2 - 3 z + 4 z ^ { 2 } }$$
is a real number, then the value of $| z | ^ { 2 }$ is $\_\_\_\_$.

Let $\bar { z }$ denote the complex conjugate of a complex number $z$ and let $i = \sqrt { - 1 }$. In the set of complex numbers, the number of distinct roots of the equation
$$\bar { z } - z ^ { 2 } = i \left( \bar { z } + z ^ { 2 } \right)$$
is $\_\_\_\_$.

Let $\bar { z }$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
$$( \bar { z } ) ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$
are integers, then which of the following is/are possible value(s) of $| z |$ ?
(A) $\left( \frac { 43 + 3 \sqrt { 205 } } { 2 } \right) ^ { \frac { 1 } { 4 } }$
(B) $\left( \frac { 7 + \sqrt { 33 } } { 4 } \right) ^ { \frac { 1 } { 4 } }$
(C) $\left( \frac { 9 + \sqrt { 65 } } { 4 } \right) ^ { \frac { 1 } { 4 } }$
(D) $\left( \frac { 7 + \sqrt { 13 } } { 6 } \right) ^ { \frac { 1 } { 4 } }$

Let $A = \left\{ \frac { 1967 + 1686 i \sin \theta } { 7 - 3 i \cos \theta } : \theta \in \mathbb { R } \right\}$. If $A$ contains exactly one positive integer $n$, then the value of $n$ is

Let $z$ be a complex number satisfying $| z | ^ { 3 } + 2 z ^ { 2 } + 4 \bar { z } - 8 = 0$, where $\bar { z }$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-I to the correct entries in List-II.
List-I
(P) $| z | ^ { 2 }$ is equal to
(Q) $| z - \bar { z } | ^ { 2 }$ is equal to
(R) $| z | ^ { 2 } + | z + \bar { z } | ^ { 2 }$ is equal to
(S) $| z + 1 | ^ { 2 }$ is equal to
List-II
(1) 12
(2) 4
(3) 8
(4) 10
(5) 7
The correct option is:
(A) $( P ) \rightarrow ( 1 ) \quad ( Q ) \rightarrow ( 3 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 4 )$
(B) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 1 ) \quad ( R ) \rightarrow ( 3 ) \quad ( S ) \rightarrow ( 5 )$
(C) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 4 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 1 )$
(D) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 3 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 4 )$

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

Let $Z$ and $W$ be complex numbers such that $| Z | = | W |$, and $\arg Z$ denotes the principal argument of $Z$. Statement 1: If $\arg Z + \arg W = \pi$, then $Z = - \bar { W }$. Statement 2: $| Z | = | W |$, implies $\arg Z - \arg \bar { W } = \pi$.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
(4) Statement 1 is false, Statement 2 is true.

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 $z \neq 1$ and $\frac{z^{2}}{z-1}$ is real, then the point represented by the complex number $z$ lies
(1) either on the real axis or on a circle passing through the origin
(2) on a circle with centre at the origin
(3) either on the real axis or on a circle not passing through the origin
(4) on the imaginary axis

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) $\pi-\theta$

Let $a = \operatorname { Im } \left( \frac { 1 + z ^ { 2 } } { 2 i z } \right)$, where $z$ is any non-zero complex number. The set $\mathrm { A } = \{ a : | z | = 1$ and $z \neq \pm 1 \}$ is equal to:
(1) $( - 1,1 )$
(2) $[ - 1,1 ]$
(3) $[ 0,1 )$
(4) $( - 1,0 ]$

Let $z$ satisfy $| z | = 1$ and $z = 1 - \bar { z }$. Statement $1 : z$ is a real number. Statement 2 : Principal argument of z is $\frac { \pi } { 3 }$
(1) Statement 1 is true Statement 2 is true; Statement 2 is a correct explanation for Statement 1.
(2) Statement 1 is false; Statement 2 is true.
(3) Statement 1 is true, Statement 2 is false.
(4) Statement 1 is true; Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.

A complex number $z$ is said to be unimodular if $| z | = 1$. Let $z _ { 1 }$ and $z _ { 2 }$ are complex numbers such that $\frac { z _ { 1 } - 2 z _ { 2 } } { 2 - z _ { 1 } \bar { z } _ { 2 } }$ is unimodular and $z _ { 2 }$ is not unimodular, then the point $z _ { 1 }$ lies on a
(1) circle of radius $\sqrt { 2 }$
(2) straight line parallel to $x$-axis
(3) straight line parallel to $y$-axis
(4) circle of radius 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 $\frac { 10 } { 3 }$

If $\frac { z - \alpha } { z + \alpha } ( \alpha \in R )$ is a purely imaginary number and $| z | = 2$, then a value of $\alpha$ is :
(1) 1
(2) $\frac { 1 } { 2 }$
(3) $\sqrt { 2 }$
(4) 2

All the points in the set $S = \left\{ \frac { \alpha + i } { \alpha - i } , \alpha \in R \right\} , i = \sqrt { - 1 }$ lie on a
(1) straight line whose slope is - 1
(2) circle whose radius is $\sqrt { 2 }$
(3) circle whose radius is 1
(4) straight line whose slope is 1

If the equation $x ^ { 2 } + b x + 45 = 0 , b \in R$ has conjugate complex roots and they satisfy $| z + 1 | = 2 \sqrt { 10 }$, then
(1) $b ^ { 2 } - b = 30$
(2) $b ^ { 2 } + b = 72$
(3) $b ^ { 2 } - b = 42$
(4) $b ^ { 2 } + b = 12$

If $\operatorname { Re } \left( \frac { z - 1 } { 2 z + i } \right) = 1$, where $z = x + i y$, then the point $(x, y)$ lies on a
(1) circle whose centre is at $\left( - \frac { 1 } { 2 } , - \frac { 3 } { 2 } \right)$
(2) straight line whose slope is $- \frac { 2 } { 3 }$
(3) straight line whose slope is $\frac { 3 } { 2 }$
(4) circle whose diameter is $\frac { \sqrt { 5 } } { 2 }$

If $\frac { 3 + i \sin \theta } { 4 - i \cos \theta } , \theta \in [ 0,2 \pi ]$, is a real number, then an argument of $\sin \theta + i \cos \theta$ is
(1) $\pi - \tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)$
(2) $\pi - \tan ^ { - 1 } \left( \frac { 3 } { 4 } \right)$
(3) $- \tan ^ { - 1 } \left( \frac { 3 } { 4 } \right)$
(4) $\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)$

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