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Shvoong Home>Science>Mathematics>Problem: Complex Numbers 04 Summary

# Problem: Complex Numbers 04

( 31 ) If ω ( ≠ 1 ) is a cube root of unity, then
i i ω 1 1
1 i 1 ω 1
1 1 iω ω
2
2 2
- - -
- - -
+
+
equals
( a ) 0 ( b ) 1 ( c ) i ( d ) ω [ IIT 1995 ]
( 32 ) If z and ω be two non-zero complex numbers such that l z l = l ω l and
Arg z+ • Arg ω = π, then z equals
( a ) ω ( b ) - ω ( c ) ω ( d ) - ω [ IIT 1995 ]
( 33 ) If z and w be two complex numbers such that l z l ≤ 1, l w l ≤ 1 and
l z + iw l = l z - iw l = 2, then z equals
( a ) 1 or i ( b ) i or - i ( c ) 1 or - 1 ( d ) i or - 1 [ IIT 1995 ]
( 34 ) The complex numbers sin x + i cos 2x and cos x - i sin 2x are conjugate to each other
for
( a ) x = nπ ( b ) x = 0 ( c ) x = ( n + 1 / 2 ) π ( d ) no value of x [ IIT 1988 ]
02 - COMPLEX NUMBERS Page 6
( Answers at the end of all questions )
( 35 ) If z1 and z2 are two non-zero complex numbers such that l z1 + z2 l = l z1 l + l z2 l ,
then arg z1 - arg z2 is equal to
( a ) - π ( b ) -
2
π ( c ) 0 ( d )
2
π ( e ) π [ IIT 1987 ]
( 36 ) The value of Σ=



6  π π
k 1
7
i cos 2 k
7
sin 2 k - is
( a ) - 1 ( b ) 0 ( c ) - i ( d ) i ( e ) none of these [ IIT 1987 ]
( 37 ) Let z1 and z2 be complex numbers such that z1 ≠ z2 and l z1 l = l z2 l . If z1 has
positive real part and z2 has negative imaginary part, then
1 2
1 2
z z
z z
-
+
may be
( a ) zero ( b ) real and positive ( c ) real and negative
( d ) purely imaginary ( e ) none of these [ IIT 1986 ]
( 38 ) If a, b, c and u, v, w are complex numbers representing the vertices of two triangles
such that c = ( 1 - r ) a + rb and w = ( 1 - r ) u + rv, where r is a complex
number, then the two triangles
( a ) have the same area ( b ) are similar
( c ) are congruent ( d ) none of these [ IIT 1985 ]
( 39 ) If z1 = a + ib and z2 = c + id are complex numbers such that l z1 l = l z2 l = 1 and
Re ( z1 z2 ) = 0, then the pair of complex numbers w1 = a + ic and w2 = b + id
satisfies
( a ) l w1 l = 1 ( b ) l w2 l = 1
( c ) Re ( w1 w2 ) = 0 ( d ) none of these [ IIT 1985 ]
( 40 ) If z = x + iy and w =
z i
1 iz
-
- , then l w l = 1 implies that, in the complex plane,
( a ) z lies on the imaginary axis ( b ) z lies on the real axis
( c ) z lies on the unit circle ( d ) None of these [ IIT 1983 ]

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
c d a a c d b d d b a b c d a,e d a,d b a,b,c b

Published: June 12, 2010
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