Let A , B , C be three sets of complex numbers as defined below A = { z : Im ( z ) 1 } B = { z : | z - 2 - i | = 3 } C = { z : Re ( ( 1 - i ) z ) = 2 } Q. The number of elements in the set A B C is [2008]

Question

Let $A, B, C$ be three sets of complex numbers as defined below [2008]

$A = {z : Im (z) \geq 1}$

$B = {z : | z - 2 - i | = 3}$

$C = {z : Re ((1 - i) z) = \sqrt{2}}$

Q. The number of elements in the set $A \cap B \cap C$ is

Smart Achievers · Accepted Answer

(2)

$Given : A = {z : Im (z) \geq 1} = {(x, y) : y \geq 1}$

Clearly A is the set of all points lying on or above the line $y = 1$ in cartesian plane.

$B = {z : | z - 2 - i | = 3} = {(x, y) : {(x - 2)}^{2} + {(y - 1)}^{2} = 9}$

$\Rightarrow$ B is the set of all points lying on the boundary of the circle with centre (2,1) and radius 3.

$C = {z : R e [(1 - i) z] = \sqrt{2}} = {(x, y) : x + y = \sqrt{2}}$

$\Rightarrow$ C is the set of all points lying on the straight line represented by $x + y = \sqrt{2}$ .

Graphically, the three sets are represented as shown below :

From graph $A \cap B \cap C$ consists of only one point P [the common point of the region $y \geq 1, {(x - 2)}^{2} + {(y - 1)}^{2} = 9$ and $x + y = \sqrt{2}$

$∴$ $n (A \cap B \cap C) = 1$

Solution

Q.
Let $A, B, C$ be three sets of complex numbers as defined below [2008]

$A = {z : Im (z) \geq 1}$

$B = {z : | z - 2 - i | = 3}$

$C = {z : Re ((1 - i) z) = \sqrt{2}}$

Q. The number of elements in the set $A \cap B \cap C$ is

1 0

2 1

3 2

4 $\infty$

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Solution

Q. Let A,B,C be three sets of complex numbers as defined below [2008] A={z:Im(z)≥1} B={z:|z-2-i|=3} C={z:Re((1-i)z)=2} Q. The number of elements in the set A∩B∩C is

1 0

2 1

3 2

4 ∞

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Q.
Let $A, B, C$ be three sets of complex numbers as defined below [2008]

$A = {z : Im (z) \geq 1}$

$B = {z : | z - 2 - i | = 3}$

$C = {z : Re ((1 - i) z) = \sqrt{2}}$

Q. The number of elements in the set $A \cap B \cap C$ is

4 $\infty$