Let $\mathbb { C }$ denote the set of all complex numbers. Define $$\begin{aligned} & A = \{ ( z , w ) \mid z , w \in \mathbb { C } \text { and } | z | = | w | \} \\ & B = \left\{ ( z , w ) \mid z , w \in \mathbb { C } , \text { and } z ^ { 2 } = w ^ { 2 } \right\} \end{aligned}$$ Then,
(A) $A = B$
(B) $A \subset B$ and $A \neq B$
(C) $B \subset A$ and $B \neq A$
(D) none of the above

For each natural number $k$, choose a complex number $z _ { k }$ with $\left| z _ { k } \right| = 1$ and denote by $a _ { k }$ the area of the triangle formed by $z _ { k } , i z _ { k } , z _ { k } + i z _ { k }$. Then, which of the following is true for the series below?
$$\sum _ { k = 1 } ^ { \infty } \left( a _ { k } \right) ^ { k }$$
(A) It converges only if every $z _ { k }$ lies in the same quadrant.
(B) It always diverges.
(C) It always converges.
(D) none of the above.

Let $i$ be a root of the equation $x^{2} + 1 = 0$ and let $\omega$ be a root of the equation $x^{2} + x + 1 = 0$. Construct a polynomial
$$f(x) = a_{0} + a_{1}x + \ldots + a_{n}x^{n}$$
where $a_{0}, a_{1}, \ldots, a_{n}$ are all integers such that $f(i + \omega) = 0$.

The number of complex roots of the polynomial $z ^ { 5 } - z ^ { 4 } - 1$ which have modulus 1 is
(A) 0
(B) 1
(C) 2
(D) more than 2 .

Find the smallest positive real number $k$ such that the following inequality holds $$\left|z_1 + \ldots + z_n\right| \geq \frac{1}{k}\left(\left|z_1\right| + \ldots + \left|z_n\right|\right)$$ for every positive integer $n \geq 2$ and every choice $z_1, \ldots, z_n$ of complex numbers with non-negative real and imaginary parts. [Hint: First find $k$ that works for $n = 2$. Then show that the same $k$ works for any $n \geq 2$.]

Let $1 , \omega , \omega ^ { 2 }$ be the cube roots of unity. Then the product $$\left( 1 - \omega + \omega ^ { 2 } \right) \left( 1 - \omega ^ { 2 } + \omega ^ { 2 ^ { 2 } } \right) \left( 1 - \omega ^ { 2 ^ { 2 } } + \omega ^ { 2 ^ { 3 } } \right) \cdots \left( 1 - \omega ^ { 2 ^ { 9 } } + \omega ^ { 2 ^ { 10 } } \right)$$ is equal to:
(A) $2 ^ { 10 }$
(B) $3 ^ { 10 }$
(C) $2 ^ { 10 } \omega$
(D) $3 ^ { 10 } \omega ^ { 2 }$

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 $z = x + i y$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation
$$z \bar { z } ^ { 3 } + \bar { z } z ^ { 3 } = 350$$
is
(A) 48
(B) 32
(C) 40
(D) 80

The quadratic equation $p(x) = 0$ with real coefficients has purely imaginary roots. Then the equation
$$p(p(x)) = 0$$
has
(A) only purely imaginary roots
(B) all real roots
(C) two real and two purely imaginary roots
(D) neither real nor purely imaginary roots

Let $z_k = \cos\left(\frac{2k\pi}{10}\right) + i\sin\left(\frac{2k\pi}{10}\right)$; $k = 1,2,\ldots,9$.
List I P. For each $z_k$ there exists a $z_j$ such that $z_k \cdot z_j = 1$ Q. There exists a $k \in \{1,2,\ldots,9\}$ such that $z_1 \cdot z = z_k$ has no solution $z$ in the set of complex numbers. R. $\frac{|1-z_1||1-z_2|\cdots|1-z_9|}{10}$ equals S. $1 - \sum_{k=1}^{9} \cos\left(\frac{2k\pi}{10}\right)$ equals
List II
1. True
2. False
3. 1
4. 2
P Q R S
(A) 1243
(B) 2134
(C) 1234
(D) 2143

For any integer $k$, let $\alpha _ { k } = \cos \left( \frac { k \pi } { 7 } \right) + i \sin \left( \frac { k \pi } { 7 } \right)$, where $i = \sqrt { - 1 }$. The value of the expression $\frac { \sum _ { k = 1 } ^ { 12 } \left| \alpha _ { k + 1 } - \alpha _ { k } \right| } { \sum _ { k = 1 } ^ { 3 } \left| \alpha _ { 4 k - 1 } - \alpha _ { 4 k - 2 } \right| }$ is

Let $z = \frac{-1 + \sqrt{3}\,i}{2}$, where $i = \sqrt{-1}$, and $r, s \in \{1,2,3\}$. Let $P = \left[\begin{array}{cc} (-z)^r & z^{2s} \\ z^{2s} & z^r \end{array}\right]$ and $I$ be the identity matrix of order 2. Then the total number of ordered pairs $(r,s)$ for which $P^2 = -I$ is

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] $1 - \sqrt{1 + y^2}$

Let $s , t , r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + i y ( x , y \in \mathbb { R } , i = \sqrt { - 1 } )$ of the equation $s z + t \bar { z } + r = 0$, where $\bar { z } = x - i y$. Then, which of the following statement(s) is (are) TRUE?
(A) If $L$ has exactly one element, then $| s | \neq | t |$
(B) If $| s | = | t |$, then $L$ has infinitely many elements
(C) The number of elements in $L \cap \{ z : | z - 1 + i | = 5 \}$ is at most 2
(D) If $L$ has more than one element, then $L$ has infinitely many elements

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 $\_\_\_\_$

Let $S = \{ a + b \sqrt { 2 } : a , b \in \mathbb { Z } \} , T _ { 1 } = \left\{ ( - 1 + \sqrt { 2 } ) ^ { n } : n \in \mathbb { N } \right\}$, and $T _ { 2 } = \left\{ ( 1 + \sqrt { 2 } ) ^ { n } : n \in \mathbb { N } \right\}$.
Then which of the following statements is (are) TRUE?
(A) $\mathbb { Z } \bigcup T _ { 1 } \bigcup T _ { 2 } \subset S$
(B) $T _ { 1 } \cap \left( 0 , \frac { 1 } { 2024 } \right) = \phi$, where $\phi$ denotes the empty set.
(C) $T _ { 2 } \cap ( 2024 , \infty ) \neq \phi$
(D) For any given $a , b \in \mathbb { Z } , \cos ( \pi ( a + b \sqrt { 2 } ) ) + i \sin ( \pi ( a + b \sqrt { 2 } ) ) \in \mathbb { Z }$ if and only if $b = 0$, where $i = \sqrt { - 1 }$.

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 $Z _ { 1 }$ 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) $\left| z _ { 1 } \right| ^ { 2 } + \left| z _ { 2 } \right| ^ { 2 }$

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

If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg\left(\frac{1+z}{1+\bar{z}}\right)$ can be equal to (given $z \neq -1$)
(1) $\theta$
(2) $\pi - \theta$
(3) $-\theta$
(4) $\frac{\pi}{2} - \theta$

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) $y ^ { 2 } + 4 x + 2 = 0$

If $z$ is a complex number such that $| z | \geq 2$, then the minimum value of $\left| z + \frac { 1 } { 2 } \right|$:
(1) Is strictly greater than $\frac { 5 } { 2 }$
(2) Is strictly greater than $\frac { 3 } { 2 }$ but less than $\frac { 5 } { 2 }$
(3) Is equal to $\frac { 5 } { 2 }$
(4) Lies in the interval $( 1,2 )$

Let $w ( \operatorname { Im } w \neq 0 )$ be a complex number. Then, the set of all complex numbers $z$ satisfying the equation $w - \bar { w } z = k ( 1 - z )$, for some real number $k$, is
(1) $\{ z : z \neq 1 \}$
(2) $\{ z : | z | = 1 , z \neq 1 \}$
(3) $\{ z : z = \bar { z } \}$
(4) $\{ z : | z | = 1 \}$