Mathematics Complex Number MCQ

----------------------- Page 1----------------------02 - COMPLEX

NUMBERS

Page 1 ( Answers at the end of all questions ) 2 ( 1 ) If the cube roots of unity are 1, ω, ω , then the roots of the equation 3 ( x - 1 ) + 8 = 0 are 2 ( a ) - 1, - 1 + 2ω, - 1 - 2ω 2 ( c ) - 1, 1 - 2ω, 1 - 2ω [ AIEEE 2005 ]

( b ) - 1, - 1, - 1, 2 ( d ) - 1, 1 + 2ω, 1 + 2ω

m ( 2 ) If z and z are two non-zero complex numbers such that l z + l = l z l + l z l, 1 2 1 2 1 2 then arg z1 - arg z2 is equal to o π ( a ) ( b ) - π [ AIEEE 2005 ] 2 ( 3 ) If w = z and 1 z - i 3

π ( c ) 0

c

( d ) -

2 l w l = 1, then z lies on

. e

c

( a ) an ellipse ( b ) a circle ( d ) a straig t line ( d ) a parabola [ AIEEE 2005 ] a r ( 4 ) Let z, w be complex numbers such that zw = π. Then arg z z i w + equals m π π ( a ) ( b ) [ AIEEE 2004 ]

3 π ( c )

= 0 and arg

5 π ( d )

a 4

2

4

4

x e

x 1

-

y +

. 3

p

q

o ( 5 ) If z = x y

and

z

= p + iq, then 2 p

2 q

+

is equal t

( a ) 1 ( b ) - 1 [ AIEEE 2004 ] w

( c ) 2

( d ) - 2

w 2 2 6 ) If l z - 1 l = l z l + 1, then z lies on ( a ) the real axis ( b ) the imaginary axis w ( c ) a circle [ AIEEE 2004 ]

( c ) an ellipse

2 ( 7 ) Let z1 and z2 be two roots of the equation z + az + b = 0, z being complex. Further assume that the origin, z1 and z2 form an equilateral triangle. Then 2 ( a ) a = b [ AIEEE 2003 ]

b

2 ( b ) a = 2b

2 ( c ) a = 3b

2 ( d ) a = 4

----------------------- Page 2----------------------02 - COMPLEX

NUMBERS

Page 2 ( Answers at the end of all questions ) ( 8 ) If z and w are two non-zero complex numbers such that l zw l = 1 and π Arg ( z ) - Arg ( w ) = , then z w is equal to 2 ( a ) 1 [ AIEEE 2003 ]

( b ) - 1

( c ) i

( d ) - i

m x 1 ( 9 ) If i n by 1

i + = 1, then the value of smallest positive integer n is g i -

o

( a ) x = 4n ( b ) x = 2n ( c ) x = 4n + 1 ( d ) x = 2n + 1 [ AIEE E 2003 ] c . 2n e 1 ω

ω 2

n

2n ω, ( 10 ) If 1, 1 is

ω

∆ ω ω are the cube roots of unity, then the alue of = c

2n

n a 1

ω

ω

( a ) 1 [ AIEEE 2003 ]

( b ) 0

( c ) ω

2 ( d ) ω

r c i + m 2 2 ( 11 ) If = a ib, where a, b, c are real, then the value of a + b is c i a ( a ) 1 [ AIEEE 2002 ]

( b ) 1

2 ( c ) c

2 ( d ) - c

x e ( 12 ) If z = x + y, then l 3z - 1 l = 3 l z - 2 l represents . ( a ) x xis ( b ) y-axis ( c ) a circle ( d ) line parallel to y-axis [ AIEEE 2002 ] w w 13

2 1 + ω 3 If the cube roots of unity are 1, ω and ω , then the value of

i

s 2 ω w ( a ) 1 [ AIEEE 2002 ]

( b ) - 1

( c ) ω

2 ( d ) ω 1

1 α α β β ab is ( 14 ) If a = cos + i sin and b = cos + i sin , then the value of

+

2 ab ( a ) sin ( α + β ) ( b ) cos ( α + β ) ( c ) sin ( α - β ) ( d ) cos ( α [ AIEEE 2002 ] ----------------------- Page 3----------------------02 - COMPLEX

NUMBERS

Page 3 ( Answers at the end of all questions ) 3n + 1 3n + 5 ( 15 ) If α + α is

is cube root of unity, then for n ∈ N, the value of α

( a ) - 1 ( b ) 0 [ AIEEE 2002 ]

( c ) 1

( d ) 3

( 16 ) Four points P ( - 1, 0 ), Q ( 1, 0 ), R ( 2 - 1, 2 ) and m S ( 2 - 1, - 2 ) are given on a complex plane, equation of the locus of the shaded region excluding the boundaries is given by o π ( a ) l z + 1 l > 2 and l arg ( z + 1 ) l < 4

c .

π ( b ) l z + 1 l > 2 and l arg ( z + 1 ) l < 2 e π ( c ) l z - 1 l > 2 and l arg ( z - 1 ) l < 4 c π ( d ) l z - 1 l > 2 and l arg ( z - 1 ) l < [ IIT 2005 ] 2 a r ( 17 ) If ω is cube root of unity ( ω ≠ 1 ), then the least value of n where n is a positive 2 n 4 n integer such that ( 1 + ω ) = ( 1 + ω ) is m ( a ) 2 ( b ) 3 ( c ) 5 ( d ) 6 [ IIT 2004 ] a x z

1 -

( 18 ) The complex number z is such that l z l = 1, z ≠ 1 , then real part of ω is e z 1 + . 1 - 1 2 ( a ) ( b ) ( c ) 0 [ IIT 2003 ] 2 2 2 z 1 + z 1 + l z 1l + l l l l w

and

ω =

( d )

1 1

1

w 1 3 2 ( 19 ) Let ω = - i + . Then the value of the determinant 1- ω ω is 2 2 2

2 w

1 -

4 1 ω

ω

( a ) 3 ω [ IIT 2002 ]

( 20 ) For all complex numbers l z - 3 - 4i l = 5, the

d

2 ( c ) 3ω

( b ) 3 ω ( ω - 1 )

z , z

( d ) 3 ω ( 1 - ω )

1

satisfying 2

l z l = 12

an

1

2 minimum value of

l z1 - z2 l is

( a ) 0 ( b ) 2 [ IIT 2002 ]

( c ) 7

( d ) 17

----------------------- Page 4----------------------02 - COMPLEX

NUMBERS

Page 4 ( Answers at the end of all questions ) z 1 z 3 - 1 i 3 ( 21 ) The complex are the vertices

numbers

z , z and z 1

2

satisfying 3 z 2

z 3 -

2 of a triangle which is ( a ) of area zero ( c ) equilateral [ IIT 2001 ]

( b ) right-angled isosceles ( d ) obtuse-angled isosceles

m ( 22 ) If z1 and z2 be nth roots of unity which subtend a right ngle at the origin, then n o must be of the form c ( a ) 4k + 1 ( b ) 4k + 2 ( c ) 4k + 3 ( d ) 4k . [ IIT 2001 ] e ( 23 ) If arg ( z ) < 0, then arg ( - z ) - arg ( z ) = c π ( a ) π [ IIT 2000 ]

( b ) - π

π

( c ) -

( d )

2

2 a r

( 24 ) If z , z and z are complex 1 2 3 1

1

m l z l = l z l = l z l = + + is

umbers such that 1 = 1, then l z + z + z l

1

2

3

1 z2

3

2

3

z

a ( a ) 1 ( b ) < 1 ( c ) > 3 [ IIT 2000 ] x e 1 i 3 ( 25 ) If i = th n 4 + 5 - +

( d ) 3 334 + 3

1 i 3 - +

365 is equal

to . ( a ) 1 - i 3 [ IIT 1999 ] w

2

2

( b ) - 1 + i 3

2 ( c ) i 3

2 ( d ) - i 3

w 2 7 ( 26 ) If ω is an imaginary cube root of unity, then ( 1 + ω - ω ) equ als ( a ) 128 ω [ IIT 1998 ]

2 ( c ) 128 ω

( b ) - 128 ω

2 ( d ) - 128 ω

w 13 n n 1 + ( 27 ) The value of the sum ∑ ( i i + ) , where i = 1- , equals n 1 ( a ) i ( b ) i - 1 [ IIT 1998 ]

( c ) - i

( d ) 0

----------------------- Page 5----------------------02 MPLEX

NUMBERS

- CO

Page 5 ( Answers at

the end of all questions ) 6i

3i

1

( 28 ) If

-

= x + iy

, then 4 20

3i 3

1 i

( a ) x = 3, y = 1

( b ) x

( c ) x = 0, y = 3 [ IIT 1998 ]

( d ) x

= 1, y = 3 = 0, y = 0 m ( 29 ) For positive integers n , n , the value of the expression 1

1

( 5 )n2

(

7 )n2 + +

+

+

+

+ + is a real number 1 i ,

1

i

1

)n1

(

3 )n1 (

o

i

1 i wherei 1

if

and only if c . n - 1 ( c ) n

( a ) n = n + 1 ( b ) n [ IIT 1996 ] 1 2 1

= n ( d ) n > , n > 0

1

2

= 2

e2 7

c ( 30 ) If ω ( ≠ 1 ) is a cube root of ( 1 + ω ) = A + Bω, then A and B are respectively the numbers

unity an a c ) , 0

( a ) 0, 1 ( b ) 1, 1 [ IIT 1995 ]

( d ) - 1, 1

(

r 2

2 m

1

1

i + ω

ω 2 ω

-

-

-

of unity, then 1 i

1

equals ( 31 ) If

≠ (

1 ) is a cube root

ω 1 a

- i i - + ω1-

1 ( a ) 0 ( b ) 1 [ IIT 1995 ]

( d ) ω

( c ) i x

e . 32 ) If z and ω be two non-z l z l = l ω l + Arg z · Arg ω = π, then z equal (

ero complex

numbers

such that

s w a ) ω [ IIT 1995 ]

( d ) - ω

( b ) - ω

( c ) ω

w (

33

) If z and w

be two comp

lex

numbers

such that

l z l ≤ 1, l w l ≤ 1 and l z + iw l

= l z - iw l

=

2, then z equals w i

( c ) 1 or - 1

( a ) 1 or i [ IIT 1995 ]

( d ) i or - 1

( b ) i or -

( 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 ] ----------------------- Page 6----------------------02 - COMPLEX

NUMBERS

Page 6 ( Answers at the end of all questions ) ( 35 ) If z and z are two non-zero complex numbers such that l z + z l = l z l + l z l , 1 2 1 2 1 2 then arg z1 - arg z2 is equal to π ( a ) - π [ II 1987 ]

π

( b ) -

( c ) 0

( d )

2

( e ) π 2

6 ( 36 ) The value of sin ∑ k 1 ( a ) - 1 [ IIT 1987 ]

( b ) 0

m 2 k π i 7

2 k π cos 7

is o

( c ) - i

( d ) i

( e ) none of these c e. l

=

l z ( 37 ) Let z1 and z2 be complex numbers 2 l . If z1 has

such that z1 ≠ z2 and l z1

z 1 z 2 + positive real part and z2 has negative imagina y part, then may be c a z1

z 2 -

( a ) zero ( b ) real and positive ( c ) real and negative ( d ) purely imaginary ( e ) none of hese [ IIT 1986 ] r ( 38 ) If a, b, c and u, v, w re complex numbers representing the ver tices of two triangles

m such that c = ( 1 - r ) a + b and w = ( 1 - r ) u + rv, where r is a complex number, then the two riangles a ( a ) have the same area x ( c ) are congruent [ IIT 1985 ]

( b ) are similar ( d ) none of these

e . ( 39 ) If z1 = a + b and z2 = c + id are complex numbers such that l z 1 l = l z2 l = 1 and Re ( z1 z 2 ) = 0, then the pair of complex numbers w1 = a + ic a nd w2 = b + id w s tisfies ( a ) l w1 l = 1

( b ) l w2 l = 1

w ( c ) Re ( w 1 w 2 [ IIT 1985 ]

) = 0 ( d ) none of these

w 1 iz( 40 ) If z = x + iy and w = , then l w l = 1 implies that, in the complex plane, z i ( 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 ] ----------------------- Page 7----------------------02 - COMPLEX

NUMBERS

Page 7 ( Answers at the end of all questions ) ( 41 ) The points z , z , z , z in the complex plane are the vertices of a parallelogram 1 2 3 4 taken in order if and only if ( a ) z1 + z4 = z2 + z3 ( c ) z1 + z2 = z3 + z4 [ IIT 1983 ]

( b ) z1 + z3 = z2 + z4 ( d ) None of these

m ( 42 ) The inequality l z - 4 l < l z - 2 l represents the region given by o ( a ) Re ( z ) > 0 ( b ) Re ( z ) < 0 ( c ) Re ( z ) > 2 ( d ) none of these 1982 ]

c [ IIT

. 3 ( 43 ) If z =

i

5 +

+ 2

3

i

5 , then

-

2

2

e

2 c

( a ) Re ( z ) = 0

( b ) Im ( z ) = 0 a ( c ) Re ( z ) > 0, Im ( z ) > 0 ( d ) Re ( z ) > 0, Im ( z ) < 0 [ IIT 1982 ] r 2 2 ( 44 ) If the cube roots of unity are 1, ω ω , then the roots of the equation ( x - 1 ) + 8 = 0 m are 2 a ( a ) - 1, 1 + 2ω, 1 ω ( c ) - 1, - 1, - 1 x ( d ) [ IIT 1979 ] e w. w 1 18 19 c d b

2 20 c b

2 ( b ) - 1, 1 - 2ω, 1 - 2 ω none of these

Answers

3

4

5

6

7

8

9

10 11

12 13 14

15 16

17

c

c

d

b

c

d

a

b

d

b

b

a

b

b

a

w 21 39 c a,b,c

8 b

22 23 40 d a b

41 42 58 59 60 b d

24 25 26

27 28 29 30

31 32 33

34 35

a

b

a

d

c

43 44 45 b

b

d

46 47

d

d 48 49

b

b

50 51 52

c

53 54

36 37 3

a,e d 55 56

a,d 57