1.Algebra Booster

Complex Numbers 4.19
30. The number 15th roots of unity which are also 25th
roots of unity is
(a) 3 (b) 5 (c) 10 (d) 13.
31. The complex number z satisfies the equation
25
z - = 24. The maximum distance from the origin
z
to z is
(a) 25 (b) 30 (c) 32 (d) 40
32. If the area of a triangle formed by the points z, iz and z
+ iz is 200, then |z| is
(a) 5 (b) 10 (c) 15 (d) 20
33. Let z be a root of z 5 – 1 = 0 with z π 1. The value of
z 15 + z 16 + … + z 50 is
(a) 1 (b) –1 (c) 0 (d) 5
34. The set of all real x satisfying the inequality
|4i – 1 – log 2
x| ≥ 5 is
Ê 1 ˘
Ê 1 ˘
(a) Á0, Ë 16 ˙
(b) Á0, [4, )
˚
Ë 16˙
˚
Ê 1 ˘
(c) [4, ) (d) Á ,4
Ë16
˙
˚
Ê x + 1ˆ 35. The number of roots of Á = 1
Ë x -1˜
, where n ΠR and
x ΠR, is
¯
(a) n (b) 1 (c) n – 1 (d) n – 2
36. If w is an imaginary fifth root of 2 and x = w + w 2 , the
value of x 5 – 10x 2 – 10x is
(a) 4 (b) 6 (c) 20 (d) 12
37. Suppose A is complex number and n ΠN, such that
A n = (A + 1) n = 1, then the least value of n is
(a) 3 (b) 6 (c) 9 (d) 12
38. If z 3 – iz 2 – 2iz – 2 = 0, then z can be
(a) 1 – i (b) 1 (c) 1+ i (d) –1 –i
100 99
39. If ( 3 + i) = 2 ( a+ ib)
, the value of a 2 + b 2 is
(a) 1 (b) 2 (c) 3 (d) 4
40. The points of intersection of the two curves |z – 3| = 2
and |z| = 2 in an argand plane are
1 1
(a) (7 ± i 3) (b) (3 ± i 3)
2
2
Ê3 7ˆ
Ê7 3ˆ
(c) Á ± i
Ë
˜ (d) i
2 2¯
Á ±
Ë
˜
2 2¯
41. Let a and b be the roots of x 2 + x + 1 = 0. The equation
whose roots are a 19 , b 7 is
(a) x 2 – x – 1 = 0 (b) x 2 – x + 1 = 0
(c) x 2 + x – 1 = 0 (d) x 2 + x + 1 = 0
42. If a, b, c, p, q, r are complex numbers such that
p q r
1 i
a b c
a b c
+ + =
p q r
0 , the value of
2 2 2
p q r
+ + is
2 2 2
a b c
(a) 0 (b) –1 (c) 2i (d) –2i
n
43. Let |z 1
| = c = |z 2
|, the value of |z 1
+ z 2
| 2 + |z 1
– z 2
| 2 is
c
(a) c 2 (b) 4c 2 (c) –c 2 (d)
2
44. The adjacent vertices of a regular polygon of n sides are
the points z and its conjugate z . If Im( z ) 2 1,
Re( z )
= -
the value of n is
(a) 4 (b) 8 (c) 6 (d) 10
45. The vertices A and C of a square ABCD are 2 + 3i and
3 – 2i respectively. The vertices B and D are
(a) B = (0, 0), D = (5, 1)
(b) B = (0, 0), D = (–5, 1)
(c) B = (1, 0), D = (–5, –1)
(d) B = (1, 1), D = (–5, –1)
46. A, B, C are the vertices of an equilateral triangle whose
centre is i. If A represents the complex number –i, the
vertices of B and D are
(a) B= (2i - 3), C = (2i
+ 3)
(b) B= (2i + 3), C = (2i
+ 3)
(c) B= (2i - 3), C = (2 i – 3)
(d) B= (2i - 3), C =- (2i
+ 3)
47. Let A and B represents the complex number a + i and
3 + bi and O be the origin. If a triangle OAB forms an
isosceles triangle with right angle at B, the value of a
and b are
(a) a = 7, b = 4 (b) a = 4, b = 4
(c) a = 4, b = 7 (d) a = 7, b = 7
48. The complex number 3 + i becomes - 1+ i 3 after
rotating by an angle about the origin in anti-clockwise
direction. Then the angle q is
(a) p/2 (b) p/4 (c) –p/4 (d) p/6
49. ABCD is a rhombus. Its diagonal AC and BD intersect
at a point M and satisfy BD = 2AC. If the points D and
M are 1 + i and 2 – i respectively, the possible value of
A is
(a)
i 3i
i 3i
A = 3– ,1- (b) A = 3– ,3-
2 2
2 2
3i
3i
i 3i
(c) A = 1– ,1- (d) A = 1– ,1-
2 2
2 2
50. If z 1
, z 2
, z 3
be the vertices of an equilateral triangle in
the argand plane such that
2 2 2
1 + 2 + 3 = 1 2+ 2 3+
3 1
( z z z ) k( z z z z z z ),
the value of k is
(a) 1 (b) 2 (c) 0 (d) –2
51. For all complex numbers z 1
, z 2
satisfying |z 1
| = 12 and
|z 2
– 3 – 4i| = 5, the minimum value of |z 1
– z 2
| is
(a) 0 (b) 2 (c) 12 (d) 10.
10
52. The value of Â Ê Ê2pkˆ Ê2pkˆˆ
ÁsinÁ ˜ - cosÁ ˜
k = 1Ë
Ë 11 ¯ Ë 11 ¯˜
¯
is
(a) –1 (b) 0 (c) –1 (d) i
2