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

If $z = x + i y$ is a complex number such that $\left| \frac { z - i } { z + i } \right| < 1$, then we must have
(A) $x > 0$
(B) $x < 0$
(C) $y > 0$
(D) $y < 0$.

Let $\alpha , \beta , \gamma$ be complex numbers which are the vertices of an equilateral triangle. Then, we must have:
(A) $\alpha + \beta + \gamma = 0$
(B) $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 0$
(C) $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \alpha \beta + \beta \gamma + \gamma \alpha = 0$
(D) $( \alpha - \beta ) ^ { 2 } + ( \beta - \gamma ) ^ { 2 } + ( \gamma - \alpha ) ^ { 2 } = 0$

Let $\Omega = \{ z = x + iy \in \mathbb{C} : |y| \leq 1 \}$. If $f(z) = z^{2} + 2$, then draw a sketch of $$f(\Omega) = \{ f(z) : z \in \Omega \}.$$ Justify your answer.

Let $z$ and $w$ be complex numbers lying on the circles of radii 2 and 3 respectively, with centre ( 0,0 ). If the angle between the corresponding vectors is 60 degrees, then the value of $| z + w | / | z - w |$ is:
(A) $\frac { \sqrt { 19 } } { \sqrt { 7 } }$
(B) $\frac { \sqrt { 7 } } { \sqrt { 19 } }$
(C) $\frac { \sqrt { 12 } } { \sqrt { 7 } }$
(D) $\frac { \sqrt { 7 } } { \sqrt { 12 } }$.

Consider the following two subsets of $\mathbb { C }$ : $$A = \left\{ \frac { 1 } { z } : | z | = 2 \right\} \text { and } B = \left\{ \frac { 1 } { z } : | z - 1 | = 2 \right\} .$$ Then
(A) $A$ is a circle, but $B$ is not a circle.
(B) $B$ is a circle, but $A$ is not a circle.
(C) $A$ and $B$ are both circles.
(D) Neither $A$ nor $B$ is a circle.

The locus of points $z$ in the complex plane satisfying $z ^ { 2 } + | z | ^ { 2 } = 0$ is
(A) a straight line
(B) a pair of straight lines
(C) a circle
(D) a parabola

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 $a, b, c$ be three complex numbers. The equation $$az + b\bar{z} + c = 0$$ represents a straight line on the complex plane if and only if
(A) $a = b$
(B) $\bar{a}c = b\bar{c}$
(C) $|a| = |b| \neq 0$
(D) $|a| = |b| \neq 0$ and $\bar{a}c = b\bar{c}$

The set of complex numbers $z$ satisfying the equation $$( 3 + 7 i ) z + ( 10 - 2 i ) \bar { z } + 100 = 0$$ represents, in the Argand plane,
(a) a straight line. (B) a pair of intersecting straight lines. (C) a pair of distinct parallel straight lines. (D) a point.

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)

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

A particle $P$ starts from the point $z _ { 0 } = 1 + 2 i$, where $i = \sqrt { - 1 }$. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z _ { 1 }$. From $z _ { 1 }$ the particle moves $\sqrt { 2 }$ units in the direction of the vector $\hat { i } + \hat { j }$ and then it moves through an angle $\frac { \pi } { 2 }$ in anticlockwise direction on a circle with centre at origin, to reach a point $z _ { 2 }$. The point $z _ { 2 }$ is given by
(A) $6 + 7 i$
(B) $- 7 + 6 i$
(C) $7 + 6 i$
(D) $- 6 + 7 i$

Let $A , B , C$ be three sets of complex numbers as defined below $$\begin{aligned} & A = \{ z : \operatorname { Im } z \geq 1 \} \\ & B = \{ z : | z - 2 - i | = 3 \} \\ & C = \{ z : \operatorname { Re } ( ( 1 - i ) z ) = \sqrt { 2 } \} \end{aligned}$$ The number of elements in the set $A \cap B \cap C$ is
(A) 0
(B) 1
(C) 2
(D) $\infty$

Let $A , B , C$ be three sets of complex numbers as defined below $$\begin{aligned} & A = \{ z : \operatorname { Im } z \geq 1 \} \\ & B = \{ z : | z - 2 - i | = 3 \} \\ & C = \{ z : \operatorname { Re } ( ( 1 - i ) z ) = \sqrt { 2 } \} \end{aligned}$$ Let $z$ be any point in $A \cap B \cap C$. Then, $| z + 1 - i | ^ { 2 } + | z - 5 - i | ^ { 2 }$ lies between
(A) 25 and 29
(B) 30 and 34
(C) 35 and 39
(D) 40 and 44

Let $A , B , C$ be three sets of complex numbers as defined below $$\begin{aligned} & A = \{ z : \operatorname { Im } z \geq 1 \} \\ & B = \{ z : | z - 2 - i | = 3 \} \\ & C = \{ z : \operatorname { Re } ( ( 1 - i ) z ) = \sqrt { 2 } \} \end{aligned}$$ Let $z$ be any point in $A \cap B \cap C$ and let $w$ be any point satisfying $| w - 2 - i | < 3$. Then, $| z | - | w | + 3$ lies between
(A) -6 and 3
(B) - 3 and 6
(C) - 6 and 6
(D) - 3 and 9

Match the statements in Column-I with those in Column-II. [Note: Here $z$ takes values in the complex plane and $\operatorname { Im } z$ and $\operatorname { Re } z$ denote, respectively, the imaginary part and the real part of $z$.]
Column I
A) The set of points $z$ satisfying $| z - i | z \| = | z + i | z \mid$ is contained in or equal to
B) The set of points $z$ satisfying $| z + 4 | + | z - 4 | = 10$ is contained in or equal to
C) If $| w | = 2$, then the set of points $z = w - \frac { 1 } { w }$ is contained in or equal to
D) If $| w | = 1$, then the set of points $z = w + \frac { 1 } { w }$ is contained in or equal to
Column II p) an ellipse with eccentricity $\frac { 4 } { 5 }$ q) the set of points $z$ satisfying $\operatorname { Im } z = 0$ r) the set of points $z$ satisfying $| \operatorname { Im } z | \leq 1$ s) the set of points $z$ satisfying $| \operatorname { Re } z | \leq 2$ t) the set of points $z$ satisfying $| z | \leq 3$

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 complex numbers $\alpha$ and $\frac { 1 } { \bar { \alpha } }$ lie on circles $\left( x - x _ { 0 } \right) ^ { 2 } + \left( y - y _ { 0 } \right) ^ { 2 } = r ^ { 2 }$ and $\left( x - x _ { 0 } \right) ^ { 2 } + \left( y - y _ { 0 } \right) ^ { 2 } = 4 r ^ { 2 }$, respectively. If $z _ { 0 } = x _ { 0 } + i y _ { 0 }$ satisfies the equation $2 \left| z _ { 0 } \right| ^ { 2 } = r ^ { 2 } + 2$, then $| \alpha | =$
(A) $\frac { 1 } { \sqrt { 2 } }$
(B) $\frac { 1 } { 2 }$
(C) $\frac { 1 } { \sqrt { 7 } }$
(D) $\frac { 1 } { 3 }$

Let $S = S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$, where $$S _ { 1 } = \{ z \in \mathbb { C } : | \mathrm { z } | < 4 \} , \quad S _ { 2 } = \left\{ z \in \mathbb { C } : \operatorname { Im } \left[ \frac { z - 1 + \sqrt { 3 } i } { 1 - \sqrt { 3 } i } \right] > 0 \right\}$$ and $S _ { 3 } = \{ z \in \mathbb { C } : \operatorname { Re } z > 0 \}$.
Area of $S =$
(A) $\frac { 10 \pi } { 3 }$
(B) $\frac { 20 \pi } { 3 }$
(C) $\frac { 16 \pi } { 3 }$
(D) $\frac { 32 \pi } { 3 }$

Let $S = S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$, where $$S _ { 1 } = \{ z \in \mathbb { C } : | \mathrm { z } | < 4 \} , \quad S _ { 2 } = \left\{ z \in \mathbb { C } : \operatorname { Im } \left[ \frac { z - 1 + \sqrt { 3 } i } { 1 - \sqrt { 3 } i } \right] > 0 \right\}$$ and $S _ { 3 } = \{ z \in \mathbb { C } : \operatorname { Re } z > 0 \}$.
$\min _ { z \in S } | 1 - 3 i - z | =$
(A) $\frac { 2 - \sqrt { 3 } } { 2 }$
(B) $\frac { 2 + \sqrt { 3 } } { 2 }$
(C) $\frac { 3 - \sqrt { 3 } } { 2 }$
(D) $\frac { 3 + \sqrt { 3 } } { 2 }$

Let $a , b \in \mathbb { R }$ and $a ^ { 2 } + b ^ { 2 } \neq 0$. Suppose $S = \left\{ z \in \mathbb { C } : z = \frac { 1 } { a + i b t } , t \in \mathbb { R } , t \neq 0 \right\}$, where $i = \sqrt { - 1 }$.
If $z = x + i y$ and $z \in S$, then $( x , y )$ lies on
(A) the circle with radius $\frac { 1 } { 2 a }$ and centre $\left( \frac { 1 } { 2 a } , 0 \right)$ for $a > 0 , b \neq 0$
(B) the circle with radius $- \frac { 1 } { 2 a }$ and centre $\left( - \frac { 1 } { 2 a } , 0 \right)$ for $a < 0 , b \neq 0$
(C) the $x$-axis for $a \neq 0 , b = 0$
(D) the $y$-axis for $a = 0 , b \neq 0$

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 $S$ be the set of all complex numbers $z$ satisfying $| z - 2 + i | \geq \sqrt { 5 }$. If the complex number $z _ { 0 }$ is such that $\frac { 1 } { \left| z _ { 0 } - 1 \right| }$ is the maximum of the set $\left\{ \frac { 1 } { | z - 1 | } : z \in S \right\}$, then the principal argument of $\frac { 4 - z _ { 0 } - \overline { z _ { 0 } } } { z _ { 0 } - \overline { z _ { 0 } } + 2 i }$ is
(A) $- \frac { \pi } { 2 }$
(B) $\frac { \pi } { 4 }$
(C) $\frac { \pi } { 2 }$
(D) $\frac { 3 \pi } { 4 }$

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