IIT-JEE 2010 · Paper 2 · Mathematics Q37

MathematicsMatrix match+8Question 37 of 57

Match the statements in Column-I with those in Column-II.

[Note: Here $z$ takes values in the complex plane and Im $z$ and Re $z$ denote, respectively, the imaginary part and the real part of $z$ .]

Column I

A
The set of points $z$ satisfying $\big|z-i|z|\big|=\big|z+i|z|\big|$ is contained in or equal to
B
The set of points $z$ satisfying $|z+4|+|z-4|=10$ is contained in or equal to
C
If $|w|=2$ , then the set of points $z=w-\dfrac1w$ is contained in or equal to
D
If $|w|=1$ , then the set of points $z=w+\dfrac1w$ is contained in or equal to

Column II

Answer and worked solution

Official answer: A → Q, R; B → P; C → P, S, T; D → Q, R, S, T

A. $z$ is equidistant from $i|z|$ and $-i|z|$ , two points symmetric about the real axis, so it lies on their perpendicular bisector, the real axis: $\operatorname{Im}z=0$ . That set is also inside $|\operatorname{Im}z|\le1$ . → Q, R.
B. Sum of distances to $\pm4$ is 10: an ellipse with $a=5$ , $c=4$ , $e=\frac45$ . (It reaches $|\operatorname{Re}z|=5$ and $|z|=5$ , so S and T fail.) → P.
C. $w=2e^{i\theta}$ : $z=\frac32\cos\theta+\frac52i\sin\theta$ , an ellipse with semi-axes $\frac52$ and $\frac32$ , so $e=\sqrt{1-\frac9{25}}=\frac45$ . Also $|\operatorname{Re}z|\le\frac32\le2$ and $|z|\le\frac52\le3$ . → P, S, T.
D. $w=e^{i\theta}$ : $z=2\cos\theta$ , the real segment $[-2,2]$ . It satisfies Q, R, S and T. → Q, R, S, T.

Answer: A → Q, R; B → P; C → P, S, T; D → Q, R, S, T

IIT-JEE 2010 · Paper 2, Mathematics question 37