Download Complex Numbers 1. Ifthenis equal to (a) (b) (c) (d) (e) 2. The value

Complex Numbers
1.
If ( x  iy)1 3  2  3i, then 3x  2 y is equal to
(a)
2.
The value of
(a)
3.
60
120
(c)
(d)
60
(e)
(d)
1  3i
2
156
cos30  i sin 30
is equal to
cos60  i sin 60
(b)
i
(b)
1
If a  cos θ  i sin θ, then
(a)
5.
(b)
i
1  3i
2
(c)
If i  1 and n is a positive integer, then i n  i n1  i n 2  i n3 is equal to
(a)
4.
20
Let z 
(a)
i cot
θ
2
i
in
(c)
(d)
0
(c)
i cos
1 a
is equal to
1 a
(b)
i tan
θ
2
θ
2
(d)
i cos ec
θ
2
11  3i
. If α is a real number such that z  iα is real, then the value of α is
1 i
7
7
3
(b)
(c)
(d)
(e)
4
4
13
6.
The value of sum  (i n  i n 1 ) , where i  1 , equals
n 1
(a)
7.
9.
(b)
i 1
i
(d)
0
1
5  3cosθ
(c)
1
3  5cosθ
(c)
The real part of (1  cos θ  2i sin θ )1 is
(a)
8.
i
1
3  5cosθ
(b)
(d)
1
5  3cosθ
 x y
If z  x  iy and z1 3  p  iq , then    ( p 2  q 2 ) is equal to
 p q
(a)
(b)
(c)
(d)
2
1
1
2
The values of x and y satisfying the equation
(a)
x  1, y  3
(b)
(1  i) x  2i (2  3i) y  i

 i are
3i
3i
x  3, y  1
1
(c)
x  0, y  1
(d)
x  1, y  0
10.
3  3  4i 
 1



 is equal to
1

2
i
1

i  2  4i 

(a)
11.
π
3
(b)
nπ 
π
3
π
3
(d)
None of these
(d)
π 2
(c)
nπ 
45
(b)
15
(c)
10
(d)
6
π
(b)
π
(c)
π 2
1
(b)
2
(c)
x2  y 2  1
(b)
(d)
9
4
(e)
3
z  2i
 1, where z  x  iy, is
2z  i
x2  y 2  1
(c)
x2  y 2  1
(d)
2 x2  3 y 2  1
3
 2i
2
(b)
3
 2i
2
(c)
3  2i
(d)
None of these
0
(b)
2
(d)
1
110
(d)
70
(c)
1
The principal amplitude of (sin 40  i cos40)5 is
(a)
19.
1 9
 i
4 4
For any complex number z, the minimum value of | z |  | z  1| is
(a)
18.
(d)
The solution of equation | z |  z  1  2i is
(a)
17.
2nπ 
The locus of z satisfying the inequality
(a)
16.
1 9
 i
4 4
The modulus of the complex number z such that z  3  i  1 and arg( z )  π is equal to
(a)
15.
(c)
π
If π  arg( z)   then arg( z )  arg( z ) is
2
(a)
14.
1 9
 i
2 2
If x  3  i , then x3  3x 2  8x  15 is equal to
(a)
13.
(b)
3  2i sin θ
will be purely imaginary, if θ is equal to
1  2i sin θ
(a)
12.
1 9
 i
2 2
If
70
(b)
110
(c)
z  25
 5, find the value of | z |
z 1
2
(a)
20.
x  nπ (b)
π
4
π θ
1

x   n  π
2

(b)
6, 6
(b)
x2  y 2  2 x  1 (b)
0
(c)
θ
(d)
θ
x0
(c)
(d)
No value of x
π
2
(c)
3π
4
(d)
5π
4
(c)
6,0
x2  y 2  2 x  1 (c)
(b)
3/ 2
(c)
1/ 2
(d)
(d)
7,2
0, 1
z 1 π
 , then
z 1 4
x2  y 2  2 y  1 (d)
x2  y 2  2 x  1
β α
is
1  αβ
1
(e)
2
The number of solutions for the equations | z  1|  | z  2 |  | z  i | is
one solution
(b)
3 solutions
(c)
2 solutions
(d)
no solution

π
 π 
The real part of 1  cos    i sin    is
5
 5 

(a) 1 (b)
28.
(b)
If α and β are different complex numbers with | β | 1, then
(a)
27.
θ π
If z  x  iy is a variable complex number such that arg
(a)
26.
6
If | z  4 |  3, then the greatest and the least value of | z  1| are respectively
(a)
25.
(d)
Let z, w be complex numbers such that z  iw  0 and arg( zw)  π. Then, arg( z ) equals
(a)
24.
5
The complex numbers sin x  i cos 2 x and cos x  i sin 2 x are conjugate to each other for
(a)
23.
(c)

(a)
22.
4
If arg( z)  θ, then arg z is equal to
(a)
21.
(b)
3
1
(c)
2
1
π 
cos  
2
 10 
1
π
cos  
2
5
(d)
(e)
1 π
sec  
2  10 
Let z1 be a complex number with | z1 |  1 and z 2 be any complex number, then
(a)
0
(b)
(c)
1
3
1
(d)
z1  z2
is equal to
1  z1 z2
2
29.
π
If the amplitude of z  2  3i is , then the locus of z  x  iy, is
4
(a)
x  y 1  0
(b)
x  y 1  0
(c)
x  y 1  0
(d)
x  y 1  0
n
30.
31.
π
π

1  sin 8  i cos 8 
The smallest positive integral value of n such that 
 is purely imaginary,
1  sin π  i cos π 
8
8

3
8
is equal to
(a)
(b)
(c)
(d)
4
2
If n is an integer which leaves remainder one when divided by three, then (1  3i)n  (1  3i)n ,
equals
(a)
32.
If
2n1
(c)
(2)n
(d)
2n
(b)
0
(c)
1
1
(d)
2
  x

 x
sin  2   cos  2   i tan( x) 
 
 
 is real, then the set of all possible value of x is
If the expression 

 x 
1  2i sin  2  
 

(a)
34.
(b)
3
 a  ib, then [(a  2)2  b2 ] is equal to
2  cos θ  i sin θ
(a)
33.
2n1
nπ  α
(b)
(c)
2nπ
nπ
α
2
(d)
None of these
(d)
24
If z 2  z  1  0, where z is a complex number, then the value of
2
2
2
2
12
(c)
1  2 1   3 1 

 6 1
 z     z  2    z  3   ...   z  6  is
z 
z  
z 
z 


(a)
35.
18
30
(b)
(c)
32
2
(d)
None of these
If 1, ω, ω2 are the cube roots of unity, then (1  ω)(1  ω2 )(1  ω4 )(1  ω8 ) is equal to
(a)
37.
(b)
If ω is a cube root of unity, then the value of (1  ω  ω2 )5  (1  ω  ω2 )5 is
(a)
36.
6
1
(b)
(c)
0
ω2
(d)
π

If ω is a complex cube root of unity, then the value of sin (ω10  ω23 )π   is
6

4
ω
1
(a)
2
(b)
3
2
(c)
1

(d)
2
3
2

(e)
1
2
(d)
1
n
38.
 3 i
A value of n such that 
   1 is
 2 2
(a)
39.
(b)
12
(c)
3
2
The value of the expression
1 
1  
1 
1 
1 
1 


2 1  1  2   3 2   2  2   ...  (n  1)  n   n  2  is
ω 
ω 
ω 
ω 
 ω  ω  

 n(n  1) 
 2 


(a)
40.
(b)
 n(n  1) 
 2  n


2
(c)
(d)
None of these
π
π 
π
π 
π
π

The value of  cos  i sin  cos  i sin  cos  i sin  ... is
2
2 
4
4 
8
8

(a)
41.
 n(n  1) 
 2  n


2
2
(b)
1
1
(c)
0
(d)
None of these
If ω( 1) be a cube root of unity and (1  ω2 )n  (1  ω4 )n , then the least positive value of n is
(a)
(b)
2
(c)
3
(d)
5
6
42. If 1, a1 , a2 ,..., an 1 are the n roots of unity, then the value of (1  a1 )(1  a2 )(1  a3 )...(1  an 1 ) is equal to
(a)
43.
44.
i
Let a  e
1
2
(c)
(d)
n
0
. Then, the equation whose roots are a  a 2 and a 2  a 4 is
x2  2 x  4  0
(b)
x2  x  1  0
(d)
x2  2 x  4  0
(e)
x2  2 x  4  0
(c)
x2  x  4  0
6
  2kπ 
 2kπ  
Value of  sin 
  i cos 
  is equal to
 7 
 7 
k 1 
1
(b)
(c)
1
0
(d)
None of these
Let x  α  β, y  αω  βω2 , z  αω2  βω, ω is an imaginary cube rot of unity. The value of xyz is
(a)
46.
2π
3
(b)
(a)
(a)
45.
3
α2  β 2
(b)
α2  β 2
(c)
α3  β 3
If 1, ω, ω2 are the cube roots of unity, then (3  ω2  ω4 )6 is equal to
5
(d)
α3  β 3
(a)
47.
(c)
729
(d)
2
0
The value of (2  ω)(2  ω2 )(2  ω10 )(2  ω11 ), where ω is the complex cube root of unity, is
(a)
48.
(b)
64
49
(b)
50
(c)
(d)
48
47
(e)
The value of the expression 1. (2  ω)(2  ω2 )  2(3  ω)(3  ω2 ) 
64

 (n  1)(n  ω)(n  ω2 )
where ω is an imaginary cube root of unity is
49.
(a)
(n  1)n(n2  3n  4) / 4
(b)
(n  1)n(n2  3n  4) / 2
(c)
(n  1)n(n2  3n  4) / 2
(d)
None of these
If ω is an imaginary cube root of unity, n is a positive integer but not a multiple of 3, then the
value of 1  ωn  ω2n is
(a)
50.
(b)
ω2
(c)
0
(d)
ω2  1
(c)
cos9θ  i sin 9θ
(cos θ  i sin θ ) 4
is equal to
(sin θ  i cos θ )5
(a)
(d)
51
3
cos θ  i sin θ
sin 9θ  i cos9θ
(b)
(e)
sin θ  i cos θ
cos θ  i sin θ
If P is the point in the A grand diagram corresponding to the complex number 3  i and OPQ
is an isosceles right angled triangle, right angled at ' O ' , then Q represents the complex number
(a)
52.
53.
1  i 3 or 1  i 3
(b)
1 i 3
3  i or 1  i 3
(c)
(d) 1  i 3
The centre of a regular hexagon is at the point z  i. If one of its vertices is at 2  i, then the
adjacent vertices of 2  i are at the points
(a)
1  2i
(e)
1  i(1  3)
(b)
i 1 3
2  i(1  3)
(c)
(d)
1  i(1  3)
For all complex numbers z1 , z2 satisfying | z1 | 12 and | z2  3  4i | 5, the minimum value of
| z1  z2 | is
(a)
54.
4
(b)
(c)
3
2
(d)
1
Let z1 and z 2 be roots of the equation z 2  pz  q  0 where p, q are real. The points represented
by z1 , z2 and the origin form an equilateral triangle, if
(a)
p2  3q
(b)
p2  3q
(c)
6
p2  3q
(d)
p 2  2q
55.
The region of the Argand diagram defined by z  1  z  1  4 is
(a)
(c)
56.
exterior of a circle
None of the above
a straight line
(b)
a circle
(c)
a parabola
(d)
None of these
(c)
a straight line
(d)
a circle
 z 1 
If Im 
  4, then locus of z is
 2z  1 
(a)
58.
(b)
(d)
 z 1  π
The locus of the points z which satisfy the condition arg 
  is
 z 1 3
(a)
57.
interior of an ellipse
interior and boundary of an ellipse
an ellipse
(b)
a parabola
The point (4, 1) undergoes the following three transformations successively
(i)
reflection about the line y  x
(ii)
translation through a distance of 2 unit along the positive direction of x-axis
(iii)
rotation through an angle of
π
about the origin in the anti-clockwise direction
4
The final position of the point is
(a)
 1 7 
,


 2 2
(b)
 1 7 
,


2 2

(c)
( 2,7 2)
(d)
( 2,7 2)
ANSWERS
1.
c
2.
a
3.
d
4.
a
5.
d
6.
b
7.
d
8.
d
9.
b
10.
d
11.
c
12.
b
13.
a
14.
e
15.
c
16.
b
17.
b
18.
b
19.
c
20.
d
21.
d
22.
c
23.
b
24.
c
25.
d
26.
a
27.
b
28.
b
29.
d
30.
a
31.
c
32.
b
33.
b
34.
b
35.
b
36.
a
37.
e
38.
a
39.
c
40.
c
41.
b
42.
43.
e
44.
d
45.
c
46.
a
47.
a
48.
7
c
a
49.
c
50.
d
51
a
52.
d
55.
c
56.
b
57.
d
58.
b
53.
d
54.
a
IIT ADVANCED LATEST
1.
Let complex numbers  and
 x  x0    y  y0 
2
2
1
2
2
lie on circles  x  x0    y  y0   r 2 and

[2013]
2
 4r 2 , respectively. If z0  x0  iy0 satisfies the equation 2 z0  r 2  2 ,
then  =
1
2
(a)
2.
Let w 
(b)
1
2
(c)
1
7
(d)
3 i

and P  {wn : n  1, 2,3,...} . Further H1  z  C : Re z 
2

1 

H 2  z  C : Re z   , where C is the set of all complex numbers.
2

1
3
1
 and
2
[2013]
If z1  P  H1 , z2  P  H 2 and O represents the origin, then z1Oz2 
(a)

2
(b)

6
2
3
(c)
Paragraph for Questions 3 and 4
(d)
5
6
Let S  S1  S2  S3 , where
[2013]


 z  1  3i 


S1   z  C : z  4 , S2   z  C : lm 
  0  and S3  z  C : Re z  0


 1  3i 


3.
min 1  3i  z 
zS
(a)
2 3
2
(b)
4.
Area of S 
5.
Matching List Type
(a)
10
3
2 3
2
(c)
20
3
(b)
3 3
2
(c)
 2kπ 
 2kπ 
Let zk  cos 
  i sin 
 ; k  1,2,...,9.
 10 
 10 
List I
List II
8
(d)
3 3
2
16
3
(d)
[2014]
32
3
P.
For each z k there exists a z j such that zk  z j  1
Q.
There exists a k {1,2,...,9} such that z1  z  zk has
1.
True
no solution z in the set of complex numbers
2.
False
R.
|1  z1 ||1  z2 | ...|1  z9 |
equals
10
3.
1
S.
9
 2kπ 
1   k 1 cos 
 equals
 10 
4.
2
Answers :
1.
c
2.
c
3.
c
4.
b
5. P  (1), Q  (2), R  (3), S  (4)
AIEEE/IITMAINS
1.
2.
3.
If z  1 and
2
z
is real, then the point represented by the complex number z lies :
z 1
(a)
either on the axis or on a circle not passing through origin.(b) on the imaginary axis.
(c)
either on the real axis or on a circle passing through the origin.
(d)
on a circle with centre at the origin.
 1 z 
If z is a complex number of unit modulus and argument θ , then arg 
 equals
1 z 
π
θ
π  θ [2013 ]
θ
(a)
(b)
(d)
 θ (c)
2
1
If z is a complex number such that | z |  2, then the minimum value of z  :
2
5
2
(a)
is equal to
(b)
(c)
is strictly greater than
5
2
lies in the interval 1,2 
[2013 ]
3
5
is strictly greater than but less than
2
2
(d)
ANSWERS
1.
[2012]
c
2.
c
3.
D
PAST YEARS IIT SUBJECTIVE EXERCISE
9
1.
Find the real values of x and y for which the following equation is satisfied
(1  i ) x  2i (2  3i ) y  i

i.
3i
3i
2.
[1980]
Let the complex numbers z1 , z2 and z3 be the vertices of an equilateral triangle. Let z0 be the
circumcentre of the triangle. Then prove that z12  z22  z32  3z02
3.
For complex number z1  x1  iy1 and z2  x2  iy2 , we write z1  z2 , if x1  x2 and y1  y2 .
Then for all complex numbers z with 1  z , prove that
4.
1 z
0
1 z
A relation R on the set of complex numbers is defined by z1 R z2 if and only if
Show that R is an equivalence relation.
5.
[1983]
If the complex numbers, z1 , z2 and z3 represent the vertices of an equilateral triangle such that
[1984]
Show that the area of the triangle on the argand diagram formed by the complex numbers z, iz
and z  iz is
8.
z1  z2
is real.
z1  z2
Prove that the complex numbers z1 , z2 , and the origin from an equilateral triangle only if
z1  z2  z3 then z1  z2  z3  0 .
7.
[1981-FCTS]
[1982]
z12  z22  z1 z2  0 .
6.
[1981]
1 2
z .
2
[1986]
Complex numbers z1 , z2 , z3 are the vertices right angled triangle with right angle at C . Show
that ( z1  z2 ) 2  2( z1  z3 )( z3  z2 ) .
[1986]
9.
Prove that the cube roots of the unity when represented on argand diagram from the vertices of
an equilateral triangle.
[1986]
10.
For any two complex numbers z1 , z2 and real numbers a and b . Prove that

az1  bz2  bz1  az2   a 2  b2  z1  z2
2
11.
2
2
2

If α , β , γ are the cube roots of p, p  0 , then for any x, y and z , prove that
xα  yβ  zγ
 w2
xβ  yγ  zα
12.
[1988]
[1989]
If a and b are real numbers between 0 and 1 such that the points z1  a  i, z2  1  bi and
z3  0 from an equilateral triangle, then find a and b .
10
[1989]
13.
Let z1  10  6i and z2  4  6i . If z is any complex number such that the argument of
is
π
, then prove that z  7  9i  3 2 .
4
z  z1
z  z2
[1990]
14.
ABCD is a rhombus. Its diagonals AC and BD intersected at the point M and satisfy
BD  2 AC . If the points D and M represent the complex numbers 1  i and 2  i
respectively, then find the complex numbers representing A .
[1993]
15.
Suppose z1 , z2 , z3 are the vertices of an equilateral triangle inscribed in the circle z  2 . If
z1  1  i 3 then find z2 and z3 .
16.
[1994]
If A1 , A2 ,....., An be vertices of an n  sided polygon such that
1
1
1


. Show that n  7 .
A1 A2 A1 A3 A1 A4
[1994]
17.
If iz 3  z 2  z  i  0 then show that z  1
[1995]
18.
If z  1, w  1, show that z  w  ( z  w ) 2  ( Arg z  Arg w) 2 .
[1995]
19.
Find all non-zero complex numbers z satisfying z  iz 2 .
20.
Show that 1.(2  ω)(2  ω2 )  2.(3  ω)(3  ω2 )  ....  (n  1)(n  ω)(n  ω2 ) 
2
1 2
n [(n  1) 2  n] where ω is an imaginary cube root of unity.
4
22.
[1996]
Let bz  bz  c, b  0, be a line in the complex plane, where b is the complex conjugate of b .
If a point z1 is reflection of a point z2 through the line, then show that c  z1b  z2b . [1997]
23.
Let z1 and z2 be roots of the equation z 2  pz  q  0 where co-efficient p and q may be
complex numbers. Let A and B represent z1 and z2 in the complex plane. If AOB  α  0
α
.
2
and OA  OB, where O is the origin, prove that p 2  4q cos 2 
n 1
24.
Prove that
 (n  k ) cos
k 1
25.
2kπ
n
  , where n  3 is on integer.
n
2
[1997]
[1997]
For complex numbers z and w , prove that
z w  w z  z  w if and only if z  w or zw  1 .
2
2
11
[1999]
26.
Let a complex number α, α  1 , to be root of the equation z p  q  z p  z q  1  0, where p, q
are distinct primes. Show that either 1  α  α 2  ...  α p 1  0 or 1  α  α 2  ...  α q 1  0 but
not both together.
[2000]
27.
If z1 and z2 are two complex numbers such that z1  1  z2 then prove that
1  z1 z2
1.
z1  z2
[2003]
28.
Prove that there exists no complex number z such the z 
1
and
3
n
a z
r 1
r
r
 1 where ar  2 .
[2003]
29.
If one vertices of square circumscribing the circle z  1  2 is 2  3i . Find the affixes of
other vertices of the square.
30.
[2005]
If cos α  cos β  cos γ  0  sin α  sin β  sin γ . Then prove that
cos 3α  cos 3 β  cos 3γ  3cos(α  β  γ) .
31.
[Roorkee-1985]
What is the mistake in computation of 1  1  (1)(1)  1  1  i  i  1
[Roorkee-1987]
32.
Let A & B be two complex numbers such that
A B
  1, then prove that the origin and the
B A
two points represented by A & B from vertices of an equilateral triangle.
33.
For every real value of a  0 , determine the complex numbers which will satisfy the equation :
z  2iz  2a (1  i )  0 .
2
34.
[Roorkee-1990]
Find the range of real number α for which the equation z  α z  1  2i  0; z  x  iy has a
solution. Find the solution.
35.
[Roorkee-1991]
Find the equations in complex variables of all the circles which are orthogonal to :
z  1 and z  4 .
36.
[Roorkee-1989]
[Roorkee-1992]
Find the complex numbers z which simultaneously satisfy the equations :
z  12 5
z4

1.
and
z  8i 3
z 8
[Roorkee-1993]
37.
Use Demoivre’s theorem to solve equation 2 2 x 4  ( 3  1)  i( 3  1)
38.
 & z  3  i  3 . [Roorkee-1995]
Find the complex numbers z for which arg 
 2 z  8  6i  4
12
 3z  6  3i 
π
[Roorkee-1994]
39.
Find all complex numbers satisfying the equation 2 z  z 2  5  i 3  0
40.
 k
2qπ
2qπ 
Evaluate :  (3 p  2)   sin
 i cos

11
11 
p 1
 q 1
41.
If α  e2 π / 7 and f ( x)  A0 
2
32
20
A
k
k 1
p
[Roorkee-1997]
x k then find the value of f ( x)  f (αx)  ....  f (αx)
independent of α .
42.
[Roorkee-1996]
[Roorkee-1999]
 2π 
 2π 
  i sin 
 , n a positive integer, find the equation whose roots
 2n  1 
 2n  1 
Given z  cos 
are α  z  z 3  ....  z 2 n 1 and β  z 2  z 4  ....  z 2 n
[Roorkee-2000]
ANSWERS
1.
x  3, y  1
12.
2  3, 2  3
15.
z2  2, z3  1  i 3
19.

31.
The formula
33.
a  (1  1  a 2  2a )i, 0  a  2  1

z   ( 2  1)  i
,
a  2 1

,
a  2 1
 nosolution
34.
No solution for α 
3 1
 i, i
4 2
21.
14.
i
3i
3  ,1 
2
2
7
29.
(1  3), ( 3  1)  i
a b  ab is applicable only when at least one a & b is positive.
 5

a 2  α 2 (5  4α 2 )
5
,z 
 2i, α  
, 1 ,
2
2
α 1
 2

α 2  α 2 (5  4α 2 )
 2i, α  (0,1),
α2 1
z
α 2  α 2 (5  4α 2 )
 2i, α  (1, 0),
α2 1
z
 5
5
 2i if α  1 and no solution for α  1,
 .
2
 2 
35.
z  7  iβ  (48  β 2 ), β  R
37.
x  cos
z
36.
rπ
rπ
 i sin , r  5, 29,53, 77 .
48
48
13
6  8i or 6  17i
38.
4  
2 
4  
2 


4
  i 1 
 and  4 
  i 1 

5 
5
5 
5


39.
 6 1   1 3 
 

i ;  
 i
2   6 2 
 2
40.
48(1  i )
41.
7( A0  A7 x 7  A14 x14 )
42.
1
 π 
z 2  z  sec 2 

4
 2n  1 
14