1
IIT-JEE 2009 Paper 1 Offline
+3
-1
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
$${1 \over {4\sin \,{2^ \circ }}}$$
2
IIT-JEE 2009 Paper 1 Offline
+3
-1

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 $$\overline z {z^3} + z{\overline z ^3} = 350$$ is

A
48
B
32
C
40
D
80
3
IIT-JEE 2008 Paper 2 Offline
+3
-1
A particle P stats from the point $${z_0}$$ = 1 +2i, where $$i = \sqrt { - 1}$$. It moves horizontally away from origin by 5 unit and then vertically away from origin by 3 units to reach a point $${z_1}$$. From $${z_1}$$ the particle moves $$\sqrt 2$$ units in the direction of the vector $$\hat i + \hat j$$ and then it moves through an angle $${\pi \over 2}$$ in anticlockwise direction on a circle with centre at origin, to reach a point $${z_2}$$. The point $${z_2}$$ is given by
A
6 + 7i
B
-7 + 6i
C
7 + 6i
D
- 6 + 7i
4
IIT-JEE 2008 Paper 1 Offline
+3
-1

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

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

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

$$C = \left\{ {z:\,{\mathop{\rm Re}\nolimits} (1 - i)z) = \sqrt 2 \,} \right\}$$

The number of elements in the set $$A \cap B \cap C$$ is

A
0
B
1
C
2
D
$$\infty$$
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