Complex Numbers 2 Question 27

28.

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

Passage II

Let $S=S _1 \cap S _2 \cap S _3$, where

$S _1={z \in C:|z|<4}, S _2=z \in C: \operatorname{lm} \frac{z-1+\sqrt{3} i}{1-\sqrt{3} i}>0$

and $S _3:{z \in C: \operatorname{Re} z>0}$

(2008)

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Answer:

Correct Answer: 28. (d)

Solution:

  1. Since,

$ |w-(2+i)|<3 \Rightarrow|w|-|2+i|<3 $

$ \begin{array}{ll} \Rightarrow & -3+\sqrt{5}<|w|<3+\sqrt{5} \\ \Rightarrow & -3-\sqrt{5}<-|w|<3-\sqrt{5} \end{array} $

Also, $\quad|z-(2+i)|=3$

$ \begin{array}{ll} \Rightarrow & -3+\sqrt{5} \leq|z| \leq 3+\sqrt{5} \\ \therefore & -3<|z|-|w|+3<9 \end{array} $



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