1.Algebra Booster

Complex Numbers 4.31
2
z + 1 w w
w
2
z + w 1 = 0 is equal to …
2
w 1 z + w
[IIT-JEE, 2010]
65. If z be any complex number satisfying |z – 3 – 2i| £ 2,
then the minimum value of |2z – 6 + 5i| is......
[IIT-JEE, 2011]
66. Comprehension
Let a, b and c be three real numbers satisfying
È1 9 7˘
[ abc] Í
Í
8 2 7
˙=
˙
[000]
ÍÎ7 3 7˙˚
…(E)
(i) If the point P(a, b, c) with reference to (E) lies on
the plane 2x + y + z = 1, the value of 7a + b + c is
(a) 0 (b) 12 (c) 7 (d) 6
(ii) Let w be a solution of x 3 – 1 = 0 with Im(w) > 0.
If a = 2 with b and c satisfying (E), the value of
3 1 3
a b c
w w w
is....
(a) –2 (b) 2 (c) 3 (d) –3
[IIT-JEE, 2011]
67. Let z be a complex number such that the imaginary part
of z is non-zero and a = z 2 + z + 1 is real. Then a cannot
take the value
(a) –1
(b)
1
3
(c)
1
2
3
(d)
4
[IIT-JEE, 2012]
68. Let the complex numbers a and 1 lie on circles
a
(x – x 0
) 2 + (y – y 0
) 2 = r 2 and (x – x 0
) 2 + (y – y 0
) 2 = 4r 2
respectively. If z 0
= x 0
+ iy 0
satisfies the equation
2|z 0
| 2 = r 2 + 2,
then |a| is
(a)
1
2
(b)
1
2
(c)
1
7
3 + i
69. Let w = and P = {w n : n = 1, 2, 3, …}
2
Ï
1 ¸
Further H1
= ÌzŒ C:Re( z)
> ˝
Ó
2˛
1
(d)
3
[IIT-JEE, 2013]
Ï
1 ¸
and H2
= ÌzŒ C:Re( z)
0˝
ÔÓ
Ë 1– i 3 ¯ Ô˛
and S 3
= {z Πc: Re(z) > 0}
(i) Area of S
10p
20p
16p
(a) (b) (c)
3 3 3
(ii) min|1 -3 i - z|
is
71. Let
zŒS
(a)
(c)
2-
3
2
3– 3
2
(b)
(d)
Ê2kpˆ Ê2kpˆ
zk
= cosÁ + isin ,
Ë
˜ Á ˜
10 ¯ Ë 10 ¯
where k = 1, 2, 3, …, 9.
List I
(P) For each z k
, there exist z j
such
that z k
◊z j
=1.
(Q) There exist a k Œ {1, 2, … , 9}
such that z k
◊z j
= 1. has no solution
z in the set of complex
numbers
(R)
|1 -z1| |1 - z2| º |1 -z9|
equals
10
(S) 9
Ê2kpˆ
1- Â cosÁ Ë
˜ equals
10 ¯
k = 1
(d)
32p
3
2+
3
2
3+
3
2
[IIT-JEE, 2013]
List II
(1) True
(2) False
(3) 1
(4) 2
[IIT-JEE, 2014]
ANSWERS
LEVEL II
1. (a) 2. (c) 3. (d) 4. (a) 5. (d)
6. (b) 7. (b) 8. (a) 9. (c) 10. (c)
11. (c) 12. (a) 13. (b) 14. (d) 15. (a)
16. (a) 17. (b) 18. (b) 19. (a) 20. (c)
21. (d) 22. (a) 23. (b) 24. (a) 25. (a)
26. (a) 27. (d) 28. (b) 29. (a) 30. (b)
31. (a) 32. (d) 33. (a) 34. (b) 35. (c)
36. (b) 37. (d) 38. (c) 39. (d) 40. (b)