Download The number of real values of satisfying the equation is (a) Zero (b

1.
The number of real values of a satisfying the equation a 2  2a sin x  1  0 is
(a) Zero
(b) One
(c) Two
(d) Infinite
2.
For positive integers n1 , n 2 the value of the expression (1  i)n1  (1  i 3 )n1  (1  i 5 )n2  (1  i7 )n2 where i   1 is a
real number if and only if
3.
4.
(a) n1  n 2  1
(b) n1  n 2  1
(c) n1  n 2
(d) n1  0, n 2  0
Given that the equation z 2  ( p  iq)z  r  i s  0, where p, q, r, s are real and non-zero has a real root, then
(a) pqr  r 2  p 2 s
(b) prs  q 2  r 2 p
(c) qrs  p 2  s 2q
(d) pqs  s 2  q 2r
If x  5  2  4 , then the value of the expression x 4  9 x 3  35 x 2  x  4 is
(b) 160
(d) 60
(a) 160
(c) 60
5.
If
(a)
b
d 
3  i  (a  ib)(c  id ) , then tan 1    tan 1   has the value
c
a

3
 2n  , n  I
(c) n  
6.

3
,n I

3
,n  I
b c a
   1, then cos(    )  cos(   )  cos(   ) is equal to
c a b
(a) 3/2
(b) – 3/2
(c) 0
(d) 1
If (1  i)(1  2i)(1  3i).....( 1  ni)  a  ib , then
a2  b 2
(c)
9.
6
If a  cos   i sin , b  cos   i sin  ,
(a) a 2  b 2
8.

(d) 2n  
,n  I
c  cos   i sin  and
7.
(b) n  
2.5.10.... (1  n 2 ) is equal to
(b) a 2  b 2
(d)
a2  b 2
If z is a complex number, then the minimum value of | z |  | z  1| is
(a) 1
(b) 0
(c) 1/2
(d) None of these
For any two complex numbers z1 and z 2 and any real numbers a and b; | (az1  bz 2 )| 2  | (bz 1  az 2 )| 2 
(a) (a 2  b 2 )(| z1 |  | z 2 |)
(b) (a 2  b 2 )(| z1 | 2  | z 2 | 2 )
(c) (a 2  b 2 )(| z1 | 2  | z 2 | 2 ) (d) None of these
10.
11.
The locus of z satisfying the inequality log 1 / 3 | z  1|  log1 / 3 | z  1| is
(a) R (z)  0
(b) R (z)  0
(c) I (z)  0
(d) None of these
If z1  a  ib and z 2  c  id are complex numbers such that | z1 | | z 2 |  1 and R(z1 z 2 )  0, then the pair of
complex numbers w1  a  ic and w2  b  id satisfies
12.
(a) | w1 |  1
(b) | w2 |  1
(c) R(w1 w2 )  0,
(d) All the above
Let z and w be two complex numbers such that | z |  1, | w |  1 and | z  iw | | z  iw |  2 . Then z is equal to
(a) 1 or i
(c) 1 or – 1
13.
(b) i or i
(d) i or –1
The maximum distance from the origin of coordinates to the point z satisfying the equation z 
(a)
1
( a 2  1  a)
2
1
 a is
z
1
( a 2  2  a)
2
1
( a 2  4  a)
(c)
2
(d) None of these
(b)
14.
Find the complex number z satisfying the equations
(b) 6  8i
(d) None of these
(a) 6
(c) 6  8 i, 6  17 i
15.
If z 1 , z 2 , z 3 are complex numbers such that | z1 | | z 2 |  | z 3 | 
(a) Equal to 1
(c) Greater than 3
16.
1
1
1


 1 , then | z 1  z 2  z 3 | is
z1 z 2 z 3
(b) Less than 1
(d) Equal to 3
 z  z1  
  , then the value of | z  7  9i| is
If z1  10  6i, z 2  4  6i and z is a complex number such that amp 

 z  z2  4
equal to
(a)
17.
z  12
5 z4
 ,
1
z  8i
3 z 8
(b) 2 2
2
(c) 3 2
(d) 2 3
If z 1 , z 2 , z 3 be three non-zero complex number, such that z 2  z 1 , a | z 1 |, b | z 2 | and c | z 3 | suppose that
a b c
z
b c a  0 , then arg 3
 z2
c a b
 z  z1
(a) arg 2
 z 3  z1




2

 is equal to


 z  z1
(b) arg 2
 z 3  z1




2
18.
 z  z1 
 z  z1 


(c) arg 3
(d) arg 3


z

z
1 
 z 2  z1 
 2
Let z and w be the two non-zero complex numbers such that | z | | w | and arg z  arg w   . Then z is equal
to
(a) w
19.
(b) w
(c) w
(d)  w
If | z  25 i |  15 , then | max .amp (z)  min .amp (z)| 
3
(a) cos 1  
5
(c)
20.
21.

2
3
 cos 1  
5

z
  arg 2

z

 3
Let z, w be complex numbers such that z  iw  0 and arg zw   . Then arg z equals
(b)  / 2
(d)  / 4
If (1  x )n  C 0  C 1 x  C 2 x 2  .....  C n x n , then the value of C0  C2  C4  C6  ..... is
(c) 2 n sin
n
2
n
n/2
(d) 2 cos
4
(b) 2 n cos
(a) 2 n
23.
3
3
(d) sin 1    cos 1  
5
5
z
If z 1 , z 2 and z 3 , z 4 are two pairs of conjugate complex numbers, then arg 1
 z4

(a) 0
(b)
2
3
(c)
(d) 
2
(a) 5 / 4
(c) 3 / 4
22.
3
(b)   2 cos 1  
5
n
2
If x  cos   i sin  and y  cos   i sin  , then x m y n  x m y n is equal to
(a) cos(m   n  )

 equals


(b) cos(m   n  )
(c) 2 cos(m   n  )
(d) 2 cos(m   n  )
8
  sin
2r
2r 
 i cos
 is
9
9 
24.
The value of
25.
(a) 1
(b) 1
(c) i
(d) i
If a, b, c and u, v, w are complex numbers representing the vertices of two triangles such that c  (1  r)a  rb
r 1
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
26.
Suppose z1 , z 2 , z 3 are the vertices of an equilateral triangle inscribed in the circle | z |  2 . If z1  1  i 3 , then
values of z 3 and z 2 are respectively
27.
28.
(a)  2, 1  i 3
(b) 2, 1  i 3
(c) 1  i 3 ,2
(d) None of these
[IIT 1994]
If the complex number z1 , z 2 the origin form an equilateral triangle then z12  z 22 
(a) z1 z 2
(b) z1 z 2
(c) z 2 z1
(d) | z1 | 2 | z 2 | 2
If at least one value of the complex number z  x  iy satisfy the condition | z  2 |  a 2  3a  2 and the
inequality | z  i 2 |  a 2 , then
29.
(a) a  2
(b) a  2
(c) a  2
(d) None of these
If z, iz and z  iz are the vertices of a triangle whose area is 2 units, then the value of | z | is
(a) – 2
(c) 4
(b) 2
(d) 8
30.
If z 2  z | z |  | z | 2  0 , then the locus of z is
31.
(a) A circle
(b) A straight line
(c) A pair of straight lines (d)
None of these
If cos   cos   cos   sin   sin   sin   0 then cos 3  cos 3   cos 3 equals to
32.
33.
(a) 0
(b) cos(     )
(c) 3 cos(     )
(d) 3 sin(     )
If z r  cos
r
n2
 i sin
r
n2
, where r = 1, 2, 3,….,n, then lim z 1 z 2 z 3 ... z n is equal to
n 
(a) cos   i sin 
(b) cos(/2)  i sin(/2)
(c) e i / 2
(d)
3
e i
If the cube roots of unity be 1, ,  2 , then the roots of the equation (x  1)3  8  0 are
(a)  1, 1  2 , 1  2 2
(b)  1, 1  2 , 1  2 2
(c) 1,  1,  1
(d) None of these
34.
35.
If 1, ,  2 ,  3 .......,  n 1 are the n, n th roots of unity, then (1   )(1   2 ).....( 1   n 1 ) equals
(a) 0
(b) 1
(c) n
(d) n 2
The value of the expression 1.(2   )(2   2 )  2.(3   )(3   2 )  .......
....  (n  1).(n   )(n   2 ),
where  is an imaginary cube root of unity, is
36.
37.
(a)
1
(n  1)n(n 2  3 n  4 )
2
(b)
1
(n  1)n(n 2  3 n  4 )
4
(c)
1
(n  1)n(n 2  3 n  4 )
2
(d)
1
(n  1)n(n 2  3 n  4 )
4
 1 i 3

If i   1 , then 4  5  
 2
2 

334
 1 i 3

 3  
 2
2 

(a) 1  i 3
(b)  1  i 3
(c) i 3
(d)  i 3
365
is equal to
If a  cos(2 / 7)  i sin(2 / 7), then the quadratic equation whose roots are   a  a 2  a 4 and   a 3  a 5  a 6 is
(a) x 2  x  2  0
(b) x 2  x  2  0
38.
(c) x 2  x  2  0
(d) x 2  x  2  0
th
Let z 1 and z 2 be n roots of unity which are ends of a line segment that subtend a right angle at the origin.
39.
Then n must be of the form
(a) 4k + 1
(b) 4k + 2
(c) 4k + 3
(d) 4k
Let  is an imaginary cube roots of unity then the value of
2(  1)( 2  1)  3(2  1)(2 2  1)  .....
 (n  1)(n  1)(n 2  1) is
2
 n(n  1) 
(a) 
 n
 2 
 n(n  1) 
(b) 

 2 
2
2
 n(n  1) 
(c) 
 n
 2 
40.
(d) None of these
 is an imaginary cube root of unity. If (1   2 )m  (1   4 )m , then least positive integral value of m is
(a) 6
(c) 4
(b) 5
(d) 3