Let $z$ be a non-zero complex number such that $\frac { z } { 1 + z }$ is purely imaginary. Then
(a) $z$ is neither real nor purely imaginary
(b) $z$ is real
(c) $z$ is purely imaginary
(d) none of the above.

$z _ { 1 } , z _ { 2 }$ are two complex numbers with $z _ { 2 } \neq 0$ and $z _ { 1 } \neq z _ { 2 }$ and satisfying $\left| \frac { z _ { 1 } + z _ { 2 } } { z _ { 1 } - z _ { 2 } } \right| = 1$. Then $\frac { z _ { 1 } } { z _ { 2 } }$ is
(A) real and negative
(B) real and positive
(C) purely imaginary
(D) none of the above need to be true always

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

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

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

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

Let $z$ be a complex number such that $\frac{z - i}{z - 1}$ is purely imaginary. Then the minimum value of $|z - (2 + 2i)|$ is
(A) $2\sqrt{2}$
(B) $\sqrt{2}$
(C) $\frac{3}{\sqrt{2}}$
(D) $\frac{1}{\sqrt{2}}$.

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

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.

17. If $\arg ( \mathrm { z } ) < 0$, then $\arg ( - \mathrm { z } ) - \arg ( \mathrm { z } ) =$
(A) п
(B) - п
(C) $- \pi / 2$
(D) $\pi / 2$

21. The complex numbers $\mathrm { z } 1 , \mathrm { z } 2$, and z 3 ,satisfying ( $\mathrm { z } 1 - \mathrm { z } 3$ )/ ( $\mathrm { z } 2 - \mathrm { z } 3$ ) $= ( 1 - \mathrm { i } \sqrt { } 3 ) / 2$ are the vertices of a triangle which is :
(A) Of area zero
(B) Right-angled isosceles
(C) Equilateral
(D) Obtuse-angled isosceles

23. Let z 1 and z 2 be nth roots of unity which subtend a right angle at the origin. Then n must be of the form:
(A) $4 \mathrm { k } + 1$
(B) $4 \mathrm { k } + 2$
(C) $4 \mathrm { k } + 3$
(D) 4 k

2. For all complex numbers $z _ { 1 } , z _ { 2 }$ satisfying $\left| z _ { 1 } \right| = 12$ and $\left| z _ { 2 } - 3 - 4 i \right| = 5$, the minimum value of $\left| z _ { 1 } - z _ { 2 } \right|$ is:
(A) 0
(B) 2
(C) 7
(D) 17

18. If $1 / 2 z 1 / 2 = 1$ and $\omega = z - 1 / z + 1$ (where $z \neq - 1$ ), then $\operatorname { Re } ( w )$ is:
(a) 0
(b) $\quad 1 / | z + 1 | ^ { 2 }$
(c) $\quad ( | 1 / ( z + 1 ) | ) \left( 1 / [ z + 1 ] ^ { 2 } \right)$
(d) $\quad \sqrt { } 2 / | z + 1 | ^ { 2 }$

14. If one of the vertices of the square circumscribing the circle $| z - 1 | = \sqrt { } 2$ is $2 + \sqrt { } 3 \mathrm { i }$. Find the other vertices of square.

5. If $\mathrm { w } = \alpha + \mathrm { i } \beta$, where $\beta \neq 0$ and $\mathrm { z } \neq 1$, satisfies the condition that $\left( \frac { \mathrm { w } - \overline { \mathrm { w } } \mathrm { z } } { 1 - \mathrm { z } } \right)$ is purely real, then the set of values of z is
(A) $\{ \mathrm { z } : | \mathrm { z } | = 1 \}$
(B) $\{ \mathrm { z } : \mathrm { z } = \overline { \mathrm { z } } \}$
(C) $\{ z : z \neq 1 \}$
(D) $\{ \mathrm { z } : | \mathrm { z } | = 1 , \mathrm { z } \neq 1 \}$
Sol. (D)
$$\begin{aligned} & \frac { \mathrm { w } - \overline { \mathrm { w } } \mathrm { z } } { 1 - \mathrm { z } } = \frac { \overline { \mathrm { w } } - \mathrm { w } \overline { \mathrm { z } } } { 1 - \overline { \mathrm { z } } } \\ & \Rightarrow \quad ( \mathrm { z } \overline { \mathrm { z } } - 1 ) ( \overline { \mathrm { w } } - \mathrm { w } ) = 0 \\ & \Rightarrow \quad \mathrm { z } \overline { \mathrm { z } } = 1 \Rightarrow | \mathrm { z } | ^ { 2 } = 1 \Rightarrow | \mathrm { z } | = 1 \end{aligned}$$

1. Let $\mathrm { a } , \mathrm { b } , \mathrm { c }$ be the sides of a triangle. No two of them are equal and $\lambda \in \mathrm { R }$. If the roots of the equation $\mathrm { x } ^ { 2 } + 2 ( \mathrm { a } + \mathrm { b } + \mathrm { c } ) \mathrm { x } + 3 \lambda ( a b + b c + c a ) = 0$ are real, then
   (A) $\lambda < \frac { 4 } { 3 }$
   (B) $\lambda > \frac { 5 } { 3 }$
   (C) $\lambda \in \left( \frac { 1 } { 3 } , \frac { 5 } { 3 } \right)$
   (D) $\lambda \in \left( \frac { 4 } { 3 } , \frac { 5 } { 3 } \right)$

Sol. (A) $\mathrm { D } \geq 0$ $\Rightarrow \quad 4 ( \mathrm { a } + \mathrm { b } + \mathrm { c } ) ^ { 2 } - 12 \lambda ( \mathrm { ab } + \mathrm { bc } + \mathrm { ca } ) \geq 0$ $\Rightarrow \lambda \leq \frac { \mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 } + \mathrm { c } ^ { 2 } } { 3 ( \mathrm { ab } + \mathrm { bc } + \mathrm { ca } ) } + \frac { 2 } { 3 }$ Since $| \mathrm { a } - \mathrm { b } | < \mathrm { c } \Rightarrow \mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 } - 2 \mathrm { ab } < \mathrm { c } ^ { 2 }$
$$\begin{aligned} & | b - c | < a \Rightarrow b ^ { 2 } + c ^ { 2 } - 2 b c < a ^ { 2 } \\ & | c - a | < b \Rightarrow c ^ { 2 } + a ^ { 2 } - 2 a c < b ^ { 2 } \end{aligned}$$
From (1), (2) and (3), we get $\frac { a ^ { 2 } + b ^ { 2 } + c ^ { 2 } } { a b + b c + c a } < 2$. Hence $\lambda < \frac { 2 } { 3 } + \frac { 2 } { 3 } \Rightarrow \lambda < \frac { 4 } { 3 }$.

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) $\infty$