IIT-JEE 2008 · Paper 1 · Mathematics Q23

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MathematicsSingle correct+4 / -1Question 23 of 69

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Paragraph for Question Nos. 21 to 23

Let $A$ , $B$ , $C$ be three sets of complex numbers as defined below
$A=\{z:\operatorname{Im}z\ge1\},\qquad B=\{z:|z-2-i|=3\},\qquad C=\{z:\operatorname{Re}((1-i)z)=\sqrt2\}.$

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

−6 and 3

−3 and 6

−6 and 6

−3 and 9

Answer and worked solution

Official answer: B

The point z. $z\approx-0.76+2.18i$ , so $|z|\approx2.30$ .

Range of $|w|$ . $w$ is inside the circle of centre $2+i$ and radius 3, which contains the origin; so $0\le|w|<\sqrt5+3\approx5.24$ .

Range of the expression. $|z|-|w|+3\in(0.07,\ 5.30]$ . This lies between −3 and 6 (B), and also between −6 and 6 (C) and between −3 and 9 (D), but not inside (A)'s range. The official key accepts B, C or D.

(B) (key: B or C or D)

Topics:Modulus and argument Geometry in the complex plane

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IIT-JEE 2008 · Paper 1, Mathematics question 23