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Shvoong Home>Science>Mathematics>Problem: Complex Numbers 02 Summary

# Problem: Complex Numbers 02

( 11 ) If
c i
c i
-
+ = a ib, where a, b, c are real, then the value of a2 + b2 is
( a ) 1 ( b ) - 1 ( c ) c2 ( d ) - c2 [ AIEEE 2002 ]
( 12 ) If z = x + iy, then l 3z - 1 l = 3 l z - 2 l represents
( a ) x-axis ( b ) y-axis ( c ) a circle ( d ) line parallel to y-axis [ AIEEE 2002 ]
( 13 ) If the cube roots of unity are 1, ω and ω2, then the value of
3
ω2
ω 1  

 

 + is
( a ) 1 ( b ) - 1 ( c ) ω ( d ) ω2 [ AIEEE 2002 ]
( 14 ) If a = cos α + i sin α and b = cos β + i sin β, then the value of 


 +
ab
ab 1
2
1 is
( a ) sin ( α + β ) ( b ) cos ( α + β ) ( c ) sin ( α - β ) ( d ) cos ( α - β ) [ AIEEE 2002 ]
02 - COMPLEX NUMBERS Page 3
( Answers at the end of all questions )
( 15 ) If α is cube root of unity, then for n ∈ N, the value of α3n + 1 + α3n + 5 is
( a ) - 1 ( b ) 0 ( c ) 1 ( d ) 3 [ AIEEE 2002 ]
( 16 ) Four points P ( - 1, 0 ), Q ( 1, 0 ), R ( 2 - 1, 2 ) and
S ( 2 - 1, - 2 ) are given on a complex plane, equation
of the locus of the shaded region excluding the
boundaries is given by
( a ) l z + 1 l > 2 and l arg ( z + 1 ) l <
4
π
( b ) l z + 1 l > 2 and l arg ( z + 1 ) l <
2
π
( c ) l z - 1 l > 2 and l arg ( z - 1 ) l <
4
π
( d ) l z - 1 l > 2 and l arg ( z - 1 ) l <
2
π [ IIT 2005 ]
( 17 ) If ω is cube root of unity ( ω ≠ 1 ), then the least value of n where n is a positive
integer such that ( 1 + ω2 )n = ( 1 + ω4 )n is
( a ) 2 ( b ) 3 ( c ) 5 ( d ) 6 [ IIT 2004 ]
( 18 ) The complex number z is such that l z l = 1, z ≠ - 1 and ω =
z 1
z 1
+
- , then real part
of ω is
( a ) z 1 2
1
l + l
( b ) z 1 2
1
l + l
- ( c ) z 1 2
2
l + l
( d ) 0 [ IIT 2003 ]
( 19 ) Let ω =
2
i 3
2
- 1 + . Then the value of the determinant
2 4
2 2
1 ω ω
1 1 ω ω
1 1 1
- - is
( a ) 3 ω ( b ) 3 ω ( ω - 1 ) ( c ) 3ω2 ( d ) 3 ω ( 1 - ω ) [ IIT 2002 ]
( 20 ) For all complex numbers z1, z2 satisfying l z1 l = 12 and l z2 - 3 - 4i l = 5, the
minimum value of l z1 - z2 l is
( a ) 0 ( b ) 2 ( c ) 7 ( d ) 17 [ IIT 2002 ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
c c c c d b c d a b a d b b b a b d b b

Published: June 12, 2010
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