1.Algebra Booster

Complex Numbers 4.29
34. Let z 1
and z 2
be the roots of z 2 + pz + q = 0, where the coefficients
p and q may be complex numbers. Let A and
B represent z 1
and z 2
in the complex plane. If –AOB
= a and OA = OB, where O is the origin, prove that
2 2Êa
ˆ
p = 4q
cos Á .
Ë
˜
2 ¯
[IIT-JEE, 1997]
n-1
Ê2kpˆ
n
35. Prove that  ( n- k)cosÁ
˜ =- where n ≥ 3 is
Ë
k = 1
n ¯ 2
an integer. [IIT-JEE, 1997]
36. If w be an imaginary cube root of unity, then
(1 + w – w 2 ) 7 equals
(a) 128 w
(b) –128 w
(c) 128 w 2 (d) –128 w 2
37. The sum of
13
n n 1
 i + i + is
n=
1
[IIT-JEE, 1998]
(a) i (b) i – 1 (c) –i (d) 0
[IIT-JEE, 1998]
38. If i = - 1 , the value of
334 365
1 3 1 3
4+ 5 Ê Á- + i
ˆ ˜ + 3
Ê Á- + i
ˆ
˜ is
Ë 2 2 ¯ Ë 2 2 ¯
(a) 1- i 3
(b) - 1+i
3
(c) i 3
(d) – i 3
[IIT-JEE, 1999]
39. For complex numbers z and w, prove that |z 2 |w – |w 2 |z =
z – w if z = w or zw – = 1. [IIT-JEE, 1999]
40. Find all the roots of (3z – 1) 4 + (z – 2) 4 = 0.
[IIT-JEE, 1999]
41. If arg (z) < 0, then Arg(–z) – Arg(z) =
(a) p (b) –p (c)
p p
- (d)
2 2
[IIT-JEE, 2000]
42. If z 1
, z 2
, z 3
be three complex numbers such that
| z1| = | z2| = | z3| =
1 1 1
+ +
z1 z2 z3
= 1, then |z 1
+ z 2
+ z 3
| is
(a) 1 (b) 3 (d) 3
[IIT-JEE, 2000]
43. Let z 1
and z 2
be the nth roots of unity which subtend a
right-angle at the origin, then n must be of the form
(a) 4k +1 (b) 4k +2
(c) 4k +3 (d) 4 k [IIT-JEE, 2001]
44. The complex numbers z 1
, z 2
and z 3
satisfying
Ê z1- z2ˆ 1-
i 3
Á =
Ë z2-
z ˜
3¯
2
are the vertices of the triangle
which is
(a) of area zero
(c) equilateral
(b) rt angled
(d) obtuse angled
[IIT-JEE, 2001]
1 3
45. Let w =- + i . Then the value of
2 2
1 1 1
1
2
-1- w
2
w is
1
2
w
4
w
(a) 3w (b) 3w(w – 1)
(c) 3w 2 (d) –3w(w – 1)
[IIT-JEE, 2002]
46. For all complex numbers z 1
, z 2
satisfying |z 1
| = 12 and
|z 2
– 3 – 4i| = 5, find the minimum value of |z 1
– z 2
| is
(a) 0 (b) 2 (c) 7 (d) 17
[IIT-JEE, 2002]
47. Let a complex number a, a π 1 be a root of
z p+q – z p – z q + 1 = 0,
where p and q are distinct primes. Show that either
1 + a + a 2 + … + a p–1 = 0
or 1 + a + a 2 + … + a q–1 = 0
but not both together. [IIT-JEE, 2002]
z - 1
48. If |z| = 1 and w = , where z π –1, then Re(w) is
z + 1
1
(a) 0 (b) –
| z + 1|
2
1
(c)
2
| z + 1|
2
(d)
2
| z + 1|
[IIT-JEE, 2003]
49. If z 1
and z 2
be two complex numbers such that
1 - zz 1 2
|z 1
| < 1, |z 2
| > 1, prove that < 1
z - z
1 2
[IIT-JEE, 2003]
50. Prove that there exist no complex number z such that
n
1
r
|| z < and  ( az r ) = 1, where |a r
| < 2
3 r = 1
[IIT-JEE-2003]
51. If w(π1) be a cube root of unity and (1 + w 2 ) n = (1 + w 4 ) n ,
the least value of n is
(a) 2 (b) 3 (c) 5 (d) 6
[IIT-JEE, 2004]
52. Find the centre and the radius of the circle given by
z - a
= k , where k π 1
z - b
z = x + iy, a = a 1
+ ia 2
, b = b 1
+ ib 2
[IIT-JEE, 2004]
53. Let | z - 1| = 2 is a circle inscribed in a square whose
one vertex is 2+ i 3 . Find the remaining vertices.
[IIT-JEE, 2005]
54. PQ and PR are two infinite rays and QAR is an arc.
Point lying in the shaded region excluding the boundary
satisfies