Paragraph: Let A, B, C be three sets of complex numbers as defined below. A=\ z ; lm z ≥ 1\ ; B=\ z:|z-2-i|=3\ ; C=\ z: Re((1-i) z)= 2 \ . Question: Let z be any point in A B C and let w be any point satisfying |w-2-i|

Correct answer: D. -3 and 9 array ll & |w-(2+i)| < 3 & || w|-| 2+i|| < 3 & -3+ 5 < |w| < 3+ 5 & -3- 5 < -|w| < 3- 5 array Also, |z-(2+i)|=3 aligned -3+ 5 & ≤|z| ≤ 3+ 5 -3 & < |z|-|w|+3 < 9 aligned

JEE Advanced 2008Paper 1

Exam: JEE Advanced
Year: 2008
Paper: Paper 1
Subject: Mathematics
Chapter: Complex Number

JEE Advanced 2008 · Paper 1 · Mathematics · Complex Number

Paragraph: Let A, B, C be three sets of complex numbers as defined...

Paragraph: Let

A, B, C

be three sets of complex numbers as defined below.

A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ; C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}

.Question: 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

Options

A $-6$ and 3
B $-3$ and 6
C $-6$ and 6
D $-3$ and 9

Correct answer

D. $-3$ and 9

Step-by-step solution

\begin{array}{ll}& |w-(2+i)| < 3 \\ \Rightarrow & \quad|| w|-| 2+i|| < 3 \\ \Rightarrow & -3+\sqrt{5}< |w| < 3+\sqrt{5}\\ \Rightarrow & -3-\sqrt{5}< -|w| < 3-\sqrt{5} \end{array}

Also,

|z-(2+i)|=3

\begin{aligned} \Rightarrow \quad-3+\sqrt{5}& \leq|z| \leq 3+\sqrt{5}\\ -3 & < |z|-|w|+3 < 9 \end{aligned}

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