1
IIT-JEE 2003 Screening
+2
-0.5
If $$\,\left| z \right| = 1$$ and $$\omega = {{z - 1} \over {z + 1}}$$ (where $$z \ne - 1$$), then $${\mathop{\rm Re}\nolimits} \left( \omega \right)$$ is
A
0
B
$$- {1 \over {{{\left| {z + 1} \right|}^2}}}$$
C
$$\left| {{z \over {z + 1}}} \right|.{1 \over {{{\left| {z + 1} \right|}^2}}}$$
D
$$\,{{\sqrt 2 } \over {{{\left| {z + 1} \right|}^2}}}$$
2
IIT-JEE 2002
+2
-0.5
Let $$\omega$$ $$= - {1 \over 2} + i{{\sqrt 3 } \over 2},$$ then the value of the det.
$$\,\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - 1 - {\omega ^2}} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^4}} \cr } } \right|$$ is
A
$$3\omega$$
B
$$3\omega \left( {\omega - 1} \right)$$
C
$$3{\omega ^2}$$
D
$$3\omega \left( {1 - \omega } \right)$$
3
IIT-JEE 2002 Screening
+2
-0.5
For all complex numbers $${z_1},\,{z_2}$$ satisfying $$\left| {{z_1}} \right| = 12$$ and $$\left| {{z_2} - 3 - 4i} \right| = 5,$$
the minimum value of $$\left| {{z_1} - {z_2}} \right|$$ is
A
0
B
2
C
7
D
17
4
IIT-JEE 2001 Screening
+2
-0.5
The complex numbers $${z_1},\,{z_2}$$ and $${z_3}$$ satisfying $${{{z_1} - {z_3}} \over {{z_2} - {z_3}}} = {{1 - i\sqrt 3 } \over 2}\,$$ are the vertices of a triangle which is
A
of area zero
B
right-angled isosceles
C
equilateral
D
obtuse-angled isosceles
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