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JEE Advance - Mathematics (2008 - Paper 1 Offline - No. 12)

Let A, B, C be three sets of complex numbers as defined below

$$A = \left\{ {z:\,{\mathop{\rm Im}\nolimits} \,\,z\,\, \ge \,1} \right\}$$

$$B = \left\{ {z:\,\,\left| {z - 2 - i} \right| = 3} \right\}$$

$$C = \left\{ {z:\,{\mathop{\rm Re}\nolimits} (1 - i)z) = \sqrt 2 \,} \right\}$$
Let A, B, C be three sets of complex numbers as defined below

$$A = \left\{ {z:\,{\mathop{\rm Im}\nolimits} \,\,z\,\, \ge \,1} \right\}$$

$$B = \left\{ {z:\,\,\left| {z - 2 - i} \right| = 3} \right\}$$

$$C = \left\{ {z:\,{\mathop{\rm Re}\nolimits} (1 - i)z) = \sqrt 2 \,} \right\}$$
Let A, B, C be three sets of complex numbers as defined below

$$A = \left\{ {z:\,{\mathop{\rm Im}\nolimits} \,\,z\,\, \ge \,1} \right\}$$

$$B = \left\{ {z:\,\,\left| {z - 2 - i} \right| = 3} \right\}$$

$$C = \left\{ {z:\,{\mathop{\rm Re}\nolimits} (1 - i)z) = \sqrt 2 \,} \right\}$$
Let z be any point $$A \cap B \cap C$$ and let w be any point satisfying $$\left| {w - 2 - i} \right| < 3\,$$. Then, $$\left| z \right| - \left| w \right| + 3$$ lies between :
- 6 and 3
- 3 and 6
- 6 and 6
- 3 and 9

Explanation

Since, $$|w - (2 + i)| < 3$$

$$|w| - |2 + i| < 3$$

$$ - 3 + \sqrt 5 < |w| < 3 + \sqrt 5 $$

$$ - 3 - \sqrt 5 < |w| < 3 - \sqrt 5 $$

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

$$ - 3 + \sqrt 5 \le |z| \le 3 + \sqrt 5 $$

$$\therefore$$ $$ - 3 < |z| - |w| + 3 < 9$$

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