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 a non-zero complex number $z$, let $\arg ( z )$ denote the principal argument with $- \pi < \arg ( z ) \leq \pi$. Then, which of the following statement(s) is (are) FALSE?
(A) $\arg ( - 1 - i ) = \frac { \pi } { 4 }$, where $i = \sqrt { - 1 }$
(B) The function $f : \mathbb { R } \rightarrow ( - \pi , \pi ]$, defined by $f ( t ) = \arg ( - 1 + i t )$ for all $t \in \mathbb { R }$, is continuous at all points of $\mathbb { R }$, where $i = \sqrt { - 1 }$
(C) For any two non-zero complex numbers $z _ { 1 }$ and $z _ { 2 }$, $$\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right) - \arg \left( z _ { 1 } \right) + \arg \left( z _ { 2 } \right)$$ is an integer multiple of $2 \pi$
(D) For any three given distinct complex numbers $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$, the locus of the point $z$ satisfying the condition $$\arg \left( \frac { \left( z - z _ { 1 } \right) \left( z _ { 2 } - z _ { 3 } \right) } { \left( z - z _ { 3 } \right) \left( z _ { 2 } - z _ { 1 } \right) } \right) = \pi$$ lies on a straight line

Let $\omega \neq 1$ be a cube root of unity. Then the minimum of the set $$\left\{ \left| a + b \omega + c \omega ^ { 2 } \right| ^ { 2 } : a , b , c \text { distinct non-zero integers } \right\}$$ equals $\_\_\_\_$

For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^{4} - |z|^{4} = 4iz^{2}$, where $i = \sqrt{-1}$. Then the minimum possible value of $|z_{1} - z_{2}|^{2}$, where $z_{1}, z_{2} \in S$ with $\operatorname{Re}(z_{1}) > 0$ and $\operatorname{Re}(z_{2}) < 0$, is $\_\_\_\_$

Let $\theta_1, \theta_2, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_1 + \theta_2 + \cdots + \theta_{10} = 2\pi$. Define the complex numbers $z_1 = e^{i\theta_1}$, $z_k = z_{k-1} e^{i\theta_k}$ for $k = 2, 3, \ldots, 10$, where $i = \sqrt{-1}$. Consider the statements $P$ and $Q$ given below:
$P: |z_2 - z_1| + |z_3 - z_2| + \cdots + |z_{10} - z_9| + |z_1 - z_{10}| \leq 2\pi$
$Q: |z_2^2 - z_1^2| + |z_3^2 - z_2^2| + \cdots + |z_{10}^2 - z_9^2| + |z_1^2 - z_{10}^2| \leq 4\pi$
Then,
(A) $P$ is TRUE and $Q$ is FALSE
(B) $Q$ is TRUE and $P$ is FALSE
(C) both $P$ and $Q$ are TRUE
(D) both $P$ and $Q$ are FALSE

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 $z$ 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) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 3 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 4 )$

For a non-zero complex number $z$, let $\arg ( z )$ denote the principal argument of $z$, with $- \pi < \arg ( z ) \leq \pi$. Let $\omega$ be the cube root of unity for which $0 < \arg ( \omega ) < \pi$. Let
$$\alpha = \arg \left( \sum _ { n = 1 } ^ { 2025 } ( - \omega ) ^ { n } \right) .$$
Then the value of $\frac { 3 \alpha } { \pi }$ is $\_\_\_\_$.

Let $\alpha, \beta$ be real and $z$ be a complex number. If $z^{2}+\alpha z+\beta=0$ has two distinct roots on the line $\operatorname{Re}z=1$, then it is necessary that
(1) $\beta\in(-1,0)$
(2) $|\beta|=1$
(3) $\beta\in(1,\infty)$
(4) $\beta\in(0,1)$

If $\omega(\neq 1)$ is a cube root of unity, and $(1+\omega)^{7}=A+B\omega$. Then $(A,B)$ equals
(1) $(1,1)$
(2) $(1,0)$
(3) $(-1,1)$
(4) $(0,1)$

Let $p , q , r \in R$ and $r > p > 0$. If the quadratic equation $p x ^ { 2 } + q x + r = 0$ has two complex roots $\alpha$ and $\beta$, then $| \alpha | + | \beta |$ is
(1) equal to 1
(2) less than 2 but not equal to 1
(3) greater than 2
(4) equal to 2

Let $Z _ { 1 }$ and $Z _ { 2 }$ be any two complex number. Statement 1: $\left| Z _ { 1 } - Z _ { 2 } \right| \geq \left| Z _ { 1 } \right| - \left| Z _ { 2 } \right|$ Statement 2: $\left| Z _ { 1 } + Z _ { 2 } \right| \leq \left| Z _ { 1 } \right| + \left| Z _ { 2 } \right|$
(1) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.
(2) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
(3) Statement 1 is true, Statement 2 is false.
(4) Statement 1 is false, Statement 2 is true.

$\left| z _ { 1 } + z _ { 2 } \right| ^ { 2 } + \left| z _ { 1 } - z _ { 2 } \right| ^ { 2 }$ is equal to
(1) $2 \left( \left| z _ { 1 } \right| + \left| z _ { 2 } \right| \right)$
(2) $2 \left( \left| z _ { 1 } \right| ^ { 2 } + \left| z _ { 2 } \right| ^ { 2 } \right)$
(3) $\left| z _ { 1 } \right| \left| z _ { 2 } \right|$
(4) $\left| z _ { 1 } \right| ^ { 2 } + \left| z _ { 2 } \right| ^ { 2 }$

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.

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 $Z _ { 1 } \neq 0$ and $Z _ { 2 }$ be two complex numbers such that $\frac { Z _ { 2 } } { Z _ { 1 } }$ is a purely imaginary number, then $\left| \frac { 2 Z _ { 1 } + 3 Z _ { 2 } } { 2 Z _ { 1 } - 3 Z _ { 2 } } \right|$ is equal to:
(1) 2
(2) 5
(3) 3
(4) 1

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.

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 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

Let $z = 1 + a i$, be a complex number, $a > 0$, such that $z ^ { 3 }$ is a real number. Then, the sum $1 + z + z ^ { 2 } + \ldots + z ^ { 11 }$ is equal to :
(1) $1365 \sqrt { 3 } i$
(2) $- 1365 \sqrt { 3 } i$
(3) $- 1250 \sqrt { 3 } i$
(4) $1250 \sqrt { 3 } i$