Let $\omega$ denote a cube root of unity which is not equal to 1. Then the number of distinct elements in the set $$\left\{ \left( 1 + \omega + \omega ^ { 2 } + \cdots + \omega ^ { n } \right) ^ { m } \quad : \quad m , n = 1, 2, 3, \cdots \right\}$$ is
(A) 4
(B) 5
(C) 7
(D) infinite

Let $\omega$ denote a cube root of unity which is not equal to 1. Then the number of distinct elements in the set $$\left\{ \left( 1 + \omega + \omega ^ { 2 } + \cdots + \omega ^ { n } \right) ^ { m } \quad : \quad m , n = 1, 2, 3, \cdots \right\}$$ is
(A) 4
(B) 5
(C) 7
(D) infinite

Let $S$ be the set of all points $z$ in the complex plane such that $$\left(1 + \frac{1}{z}\right)^4 = 1$$ Then, the points of $S$ are
(A) vertices of a rectangle
(B) vertices of a right-angled triangle
(C) vertices of an equilateral triangle
(D) collinear

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 $f(i + \omega) = 0$.

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) $3 ^ { 10 } \omega ^ { 2 }$

Suppose $z \in \mathbb { C }$ is such that the imaginary part of $z$ is non-zero and $z ^ { 25 } = 1$. Then $$\sum _ { k = 0 } ^ { 2023 } z ^ { k }$$ equals
(A) 0.
(B) 1.
(C) $- 1 - z ^ { 24 }$.
(D) $- z ^ { 24 }$.

Consider the function $f : \mathbb { C } \rightarrow \mathbb { C }$ defined by $$f ( a + i b ) = e ^ { a } ( \cos b + i \sin b ) , a , b \in \mathbb { R }$$ where $i$ is a square root of $-1$. Then
(A) $f$ is one-to-one and onto.
(B) $f$ is one-to-one but not onto.
(C) $f$ is onto but not one-to-one.
(D) $f$ is neither one-to-one nor onto.

If the points $z_1$ and $z_2$ are on the circles $|z| = 2$ and $|z| = 3$, respectively, and the angle included between these vectors is $60^\circ$, then the value of $\frac{|z_1 + z_2|}{|z_1 - z_2|}$ is
(A) $\sqrt{\frac{19}{7}}$
(B) $\sqrt{19}$
(C) $\sqrt{7}$
(D) $\sqrt{\frac{7}{19}}$

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 $S ^ { 1 } = \{ z \in \mathbb { C } | | z \mid = 1 \}$ be the unit circle in the complex plane. Let $f : S ^ { 1 } \rightarrow S ^ { 1 }$ be the map given by $f ( z ) = z ^ { 2 }$. We define $f ^ { ( 1 ) } : = f$ and $f ^ { ( k + 1 ) } : = f \circ f ^ { ( k ) }$ for $k \geq 1$. The smallest positive integer $n$ such that $f ^ { ( n ) } ( z ) = z$ is called the period of $z$. Determine the total number of points in $S ^ { 1 }$ of period 2025. (Hint: $2025 = 3 ^ { 4 } \times 5 ^ { 2 }$)

11. The value of the $\operatorname { sum } \Sigma n = 113 ( i n + i n + 1 )$, where $i = \sqrt { } ( - 1 )$, equals:
(A) i
(B) $\mathrm { i } - 1$
(C) - i
(D) 0

30. If $\mathrm { b } > \mathrm { a }$, then the equation $( \mathrm { x } - \mathrm { a } ) ( \mathrm { x } - \mathrm { b } ) - 1 = 0$ has:
(A) both roots in (a, b)
(B) both roots in ( $- ¥$, a)
(C) both roots in $( b , + ¥ )$
(D) one root in ( $- ¥$, a) and the other in ( $b , + \neq$ )

31. If $\mathrm { z } 1 , \mathrm { z } 2$ and z 3 are complex numbers such that $| z 1 | | z 2 | = | z 3 | = | 1 / z 1 + 1 / z 2 + 1 / z 3 | = 1$, then $| z 1 + z 2 + z 3 |$ is :
(A) equal to 1
(B) less than1
(C) greater than 3
(D) equal to 3

If $z _ { 1 }$ and $z _ { 2 }$ are two complex numbers such that $\left| \mathrm { z } _ { 1 } \right| < 1 < \left| \mathrm { z } _ { 1 } \right|$ then prove that $$\left| \frac { 1 - z _ { 1 } \bar { z } _ { 2 } } { z _ { 1 } - z _ { 2 } } \right| < 1 .$$

16. The minimum value of $\left| a + b w + c w ^ { 2 } \right|$, where $a , b$ and $c$ are all not equal integers and $w \left( \begin{array} { l l } 1 & 1 \end{array} \right)$ is a cube root of unity, is:
(a) $\sqrt { } 3$
(b) $1 / 3$
(c) 1
(d) 0

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1. If $P = \left[ \begin{array} { c c } \sqrt { 3 } / 2 & 1 / 2 \\ - 1 / 2 & \sqrt { 3 } / 2 \end{array} \right] , A = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$ and $\mathrm { Q } = \mathrm { PAP } ^ { \top }$, then $\mathrm { P } ^ { \top } \mathrm { Q } ^ { 2005 } \mathrm { P }$ is:
   (a) $\left[ \begin{array} { c c } 1 & 2005 \\ 0 & 1 \end{array} \right]$
   (b) $\left[ \begin{array} { c c } 1 & 2005 \\ 2005 & 1 \end{array} \right]$
   (c) $\left[ \begin{array} { c c } 1 & 0 \\ 2005 & 1 \end{array} \right] ^ { 1 }$
   (d) $\left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]$
2. The shaded region, where $P \equiv ( - 1,0 ) , Q \equiv ( - 1 + \sqrt { } 2 , \sqrt { } 2 )$ $R \equiv ( - 1 + \sqrt { } 2 , - \sqrt { } 2 ) , S \equiv ( 1,0 )$ is represented by:
   (a) $| z + 1 | > 2 , | \arg ( z + 1 ) | < n / 4$
   (b) $| z + 1 | < 2 , | \arg ( z + 1 ) | < n / 2$
   (c) $| z - 1 | > 2 , | \arg ( z + 1 ) | > \pi / 4$
   (d) $| z - 1 | < 2 , | \arg ( z + 1 ) | > n / 2$
3. The number of ordered pairs ( $a , \beta$ ), where $a , \beta \hat { I } ( - \Pi , \Pi )$ satisfying $\cos ( a - \beta ) = 1$ and $\cos ( a + \beta ) = 1 / e$ is :
   (a) 0
   (b) 1
   (c) 2
   (d) 4
4. Let $f ( x ) = | x | - 1$, then points where $f ( x )$ is not differentiable is/(are):
   (a) $0 , + 1$
   (b) + 1
   (c) 0
   (d) 1
5. The second degree polynomial $f ( x )$, satisfying $f ( 0 ) = 0 , f ( 1 ) = 1 , f ^ { \prime } ( x ) > 0$ for all $x \hat { I }$ ( 0 , 1) :
   (a) $f ( x ) = f$
   (b) $f ( x ) = a x + ( 1 - a ) x ^ { 2 } ; \forall a \hat { I } ( 0 , ¥ )$
   (c) $f ( x ) = a x + ( 1 - a ) x ^ { 2 } ; \forall a \hat { I } ( 0,2 )$
   (d) no such polynomial
6. If $f$ is a differentiable function satisfying $f ( 1 / n ) = 0$ for all $n > 1 , n \hat { I } I$, then :
   (a) $f ( x ) = 0 , x \hat { I } ( 0,1 ]$
   (b) $f ^ { \prime } ( 0 ) = 0 = f ( 0 )$

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(c) $f ( 0 ) = 0$ but $f ^ { \prime } ( 0 )$ not necessarily zero
(d) $| f ( x ) | < 1 , x \hat { I } ( 0,1 ]$

46. If $| z | = 1$ and $z \neq \pm 1$, then all the values of $\frac { z } { 1 - z ^ { 2 } }$ lie on
(A) a line not passing through the origin
(B) $| z | = \sqrt { 2 }$
(C) the $x$-axis
(D) the $y$-axis

Answer

[Figure]
(A)
[Figure]
(B)
[Figure]
(C)
[Figure]
(D)

1. Let $E ^ { c }$ denote the complement of an event $E$. Let $E , F , G$ be pairwise independent events with $P ( G ) > 0$ and $P ( E \cap F \cap G ) = 0$. Then $P \left( E ^ { c } \cap F ^ { c } \mid G \right)$ equals
   (A) $P \left( E ^ { c } \right) + P \left( F ^ { c } \right)$
   (B) $P \left( E ^ { c } \right) - P \left( F ^ { c } \right)$
   (C) $P \left( E ^ { c } \right) - P ( F )$
   (D) $P ( E ) - P \left( F ^ { c } \right)$

Answer ◯
(A)
[Figure]
(B)
[Figure]
(C)
◯
(D)

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

Let $p$ and $q$ be real numbers such that $p \neq 0 , p ^ { 3 } \neq q$ and $p ^ { 3 } \neq - q$. If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha + \beta = - p$ and $\alpha ^ { 3 } + \beta ^ { 3 } = q$, then a quadratic equation having $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$ as its roots is
A) $\left( p ^ { 3 } + q \right) x ^ { 2 } - \left( p ^ { 3 } + 2 q \right) x + \left( p ^ { 3 } + q \right) = 0$
B) $\left( p ^ { 3 } + q \right) x ^ { 2 } - \left( p ^ { 3 } - 2 q \right) x + \left( p ^ { 3 } + q \right) = 0$
C) $\left( \mathrm { p } ^ { 3 } - \mathrm { q } \right) \mathrm { x } ^ { 2 } - \left( 5 \mathrm { p } ^ { 3 } - 2 \mathrm { q } \right) \mathrm { x } + \left( \mathrm { p } ^ { 3 } - \mathrm { q } \right) = 0$
D) $\left( p ^ { 3 } - q \right) x ^ { 2 } - \left( 5 p ^ { 3 } + 2 q \right) x + \left( p ^ { 3 } - q \right) = 0$

Let $z _ { 1 }$ and $z _ { 2 }$ be two distinct complex numbers and let $z = ( 1 - t ) z _ { 1 } + t z _ { 2 }$ for some real number $t$ with $0 < t < 1$. If $\operatorname { Arg } ( w )$ denotes the principal argument of a nonzero complex number $w$, then
A) $\left| z - z _ { 1 } \right| + \left| z - z _ { 2 } \right| = \left| z _ { 1 } - z _ { 2 } \right|$
B) $\operatorname { Arg } \left( z - z _ { 1 } \right) = \operatorname { Arg } \left( z - z _ { 2 } \right)$
C) $\left| \begin{array} { c c } \mathrm { z } - \mathrm { z } _ { 1 } & \overline { \mathrm { z } } - \overline { \mathrm { z } } _ { 1 } \\ \mathrm { z } _ { 2 } - \mathrm { z } _ { 1 } & \overline { \mathrm { z } } _ { 2 } - \overline { \mathrm { z } } _ { 1 } \end{array} \right| = 0$
D) $\operatorname { Arg } \left( z - z _ { 1 } \right) = \operatorname { Arg } \left( z _ { 2 } - z _ { 1 } \right)$

Let $\omega$ be the complex number $\cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 }$. Then the number of distinct complex numbers $z$ satisfying $\left| \begin{array} { c c c } z + 1 & \omega & \omega ^ { 2 } \\ \omega & z + \omega ^ { 2 } & 1 \\ \omega ^ { 2 } & 1 & z + \omega \end{array} \right| = 0$ is equal to

A value of $b$ for which the equations $$\begin{aligned} & x ^ { 2 } + b x - 1 = 0 \\ & x ^ { 2 } + x + b = 0 \end{aligned}$$ have one root in common is
(A) $- \sqrt { 2 }$
(B) $- i \sqrt { 3 }$
(C) $i \sqrt { 5 }$
(D) $\sqrt { 2 }$

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