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1

### AIEEE 2010

The number of complex numbers z such that $$\left| {z - 1} \right| = \left| {z + 1} \right| = \left| {z - i} \right|$$ equals
A
1
B
2
C
$$\infty$$
D
0

## Explanation

Let $$z=x+iy$$

$$\left| {z - 1} \right| = \left| {z + 1} \right|{\left( {x - 1} \right)^2} + {y^2}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\left( {x + 1} \right)^2} + {y^2}$$

$$\Rightarrow {\mathop{\rm Re}\nolimits} \,z = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow x = 0$$

$$\left| {z - 1} \right| = \left| {z - i} \right|{\left( {x - 1} \right)^2} + {y^2}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {x^2} + {\left( {y - 1} \right)^2}$$

$$\Rightarrow x = y$$

$$\left| {z + 1} \right| = \left| {z - i} \right|{\left( {x + 1} \right)^2} + {y^2}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {x^2} + {\left( {y - 1} \right)^2}$$

Only $$(0,0)$$ will satisfy all conditions.

$$\Rightarrow$$ Number of complex number $$z=1$$
2

### AIEEE 2008

Let R be the real line. Consider the following subsets of the plane $$R \times R$$ :
$$S = \left\{ {(x,y):y = x + 1\,\,and\,\,0 < x < 2} \right\}$$
$$T = \left\{ {(x,y): x - y\,\,\,is\,\,an\,\,{\mathop{\rm int}} eger\,} \right\}$$,

Which one of the following is true ?

A
Neither S nor T is an equivalence relation on R
B
Both S and T are equivalence relation on R
C
S is an equivalence relation on R but T is not
D
T is an equivalence relation on R but S is not

## Explanation

Given $$S = \left\{ {\left( {x,y} \right):y = x + 1\,\,} \right.\,$$

and $$\,\,\,\left. {0 < x < 2} \right\}$$

As $$\,\,\,\,x \ne x + 1\,\,\,$$

for any $$\,\,\,x \in \left( {0,2} \right) \Rightarrow \left( {x,x} \right) \notin S$$

$$\therefore$$ $$S$$ is not reflexive.

Hence $$S$$ in not an equivalence relation.

Also $$\,\,\,T = \left\{ {x,\left. y \right)} \right.:x - y$$ is an integer $$\left. {} \right\}$$

as $$x - x = 0$$ is an integer $$\forall x \in R$$

$$\therefore$$ $$T$$ is reflexive.

If $$x-y$$ is an integer then $$y-x$$ is also an integer

$$\therefore$$ $$T$$ is symmetric

If $$x-y$$ is an integer and $$y - z$$ is an integer then

$$(x-y)+(y-z)=x-z$$ is also an integer.

$$\therefore$$ $$T$$ is transitive

Hence $$T$$ is an equivalence relation
3

### AIEEE 2008

The conjugate of a complex number is $${1 \over {i - 1}}$$ then that complex number is
A
$${{ - 1} \over {i - 1}}$$
B
$${1 \over {i + 1}}\,$$
C
$${{ - 1} \over {i + 1}}$$
D
$${1 \over {i - 1}}$$

## Explanation

$$\left( {{1 \over {i - 1}}} \right) = {1 \over { - i - 1}} = {{ - 1} \over {i + 1}}$$
4

### AIEEE 2007

If $$\,\left| {z + 4} \right|\,\, \le \,\,3\,$$, then the maximum value of $$\left| {z + 1} \right|$$ is
A
6
B
0
C
4
D
10

## Explanation

$$z$$ lies on or inside the circle with center $$(-4,0)$$ and radius $$3$$ units. From the Argand diagram maximum value of $$\left| {z + 1} \right|$$ is $$6$$

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