1

### IIT-JEE 2010 Paper 2 Offline

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

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

Column I

(A) The set of points z satisfying $$\left| {z - i} \right|\left. {z\,} \right\|\,\, = \left| {z + i} \right|\left. {\,z} \right\|$$ is contained in or equal to
(B) The set of points z satisfying $$\left| {z + 4} \right| + \,\left| {z - 4} \right| = 10$$ is contained in or equal to
(C) If $$\left| w \right|$$= 2, then the set of points $$z = w - {1 \over w}$$ is contained in or equal to
(D) If $$\left| w \right|$$ = 1, then the set of points $$z = w + {1 \over w}$$ is contained in or equal to.

Column II

(p) an ellipse with eccentricity $${4 \over 5}$$
(q) the set of points z satisfying Im z = 0
(r) the set of points z satisfying $$\left| {{\rm{Im }}\,{\rm{z }}} \right| \le 1$$
(s) the set of points z satisfying $$\,\left| {{\mathop{\rm Re}\nolimits} \,\,z} \right| < 2$$
(t) the set of points z satisfying $$\left| {\,z} \right| \le 3$$
A
(A) - q, s ; (B) - p ; (C) - p, t ; (D) - q, r, s, t
B
(A) - q, r ; (B) - p ; (C) - p, s, t ; (D) - q, r, s, t
C
(A) - p, r ; (B) - p ; (C) - p, t ; (D) -q, r, s, t
D
(A) - p ; (B) - q ; (C) - r, s ; (D) -q, r, s, t
2

### IIT-JEE 2009

Let $$z = \,\cos \,\theta \, + i\,\sin \,\theta$$ . Then the value of $$\sum\limits_{m = 1}^{15} {{\mathop{\rm Im}\nolimits} } ({z^{2m - 1}})\,at\,\theta \, = {2^ \circ }$$ is
A
$${1 \over {\sin \,{2^ \circ }}}$$
B
$${1 \over {3\sin \,{2^ \circ }}}$$
C
$${1 \over {2\sin \,{2^ \circ }}}$$
D
$${1 \over {4\sin \,{2^ \circ }}}$$
3

### IIT-JEE 2009

Let z = x + iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation : $$z\,{z^{ - 3}}\, + \,\overline z \,{z^3}\, = 350$$ is
A
48
B
32
C
40
D
80
4

### IIT-JEE 2008

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

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

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