1.Algebra Booster

4.28 Algebra Booster
12. Show that the area of a triangle on the Argand diagram
1 2
formed by the complex numbers z, iz, z + iz is ||.
2 z
[IIT-JEE, 1986]
13. Complex numbers z 1
, z 2
, z 3
are the vertices A, B and C
respectively of an isosceles right-angled triangle with
right angle at C. Show that
(z 1
– z 2
) 2 = 2(z 1
– z 3
)(z 3
– z 2
)
[IIT-JEE, 1986]
14. If z 1
and z 2
be two non-zero complex numbers such that
|z 1
+ z 2
| = |z 1
| + |z 2
|, then Arg(z 1
) – Arg(z 2
) is equal to
p
p
(a) –p (b) - (c) 0 (d)
2
2
[IIT-JEE, 1987]
15. The value of
6
Â Ê Ê2pkˆ Ê2pkˆˆ
ÁsinÁ ˜ - icos
Ë Ë Á ˜
7 ¯ Ë 7 ¯˜
¯
is
k = 1
(a) –1 (b) 0 (c) –i (d) i
[IIT-JEE, 1987]
16. The complex numbers
sin x + i cos 2x and cos x – i sin 2x
are conjugate to each other for
(a) x = np (b) x = 0
Ê 1ˆ
(c) Án
+ ˜p
(d) no values of x
Ë 2¯
[IIT-JEE, 1988]
No questions asked in 1989.
17. Let z 1
= 10 + 6i and z 2
= 4 + 6i. If z be any complex
Ê z - z1
ˆ
number such that the argument of Á
Ë z - z ˜
2 ¯ is p , then
4
prove that | z – 7 –9 i | = 3 2
[IIT-JEE, 1990]
18. The equation not representing a circle is given by
Ê1
+ zˆ (a) ReÁ
= 0
Ë1
- z˜
(b) zz – + iz – iz – + 1 = 0
¯
Ê z - 1ˆ p
z - 1
(c) argÁ
=
Ë z + 1˜
(d) = 1
¯ 2
z + 1
[IIT-JEE, 1991]
19. If z = –1, the principal value of the Arg(z 2/3 ) is equal to
p
2p
(a)
(b) or 0
3
3
10p
(c)
(d) p
3
[IIT-JEE, 1991]
20. If z be a complex number such that z π 0 and Re(z) = 0,
then
(a) Re(z 2 ) = 0 (b) Im(z 2 ) = 0
(c) Re(z 2 ) = Im(z 2 ) (d) none
[IIT-JEE, 1992]
Ê1+
2iˆ
21. The complex number Á
Ë 1 - i ˜ lies in the
¯
(a) 1st quadrant
(c) IIIrd quadrant
(b) IInd quadrant
(d) IVth quadrant.
[IIT-JEE, 1992]
22. If a and b be different complex numbers with
b – a
|b| = 1, then is equal to
1– ab
(a) 0 (b) 1/2 (c) 1 (d) 2
[IIT-JEE, 1992]
23. 1, w, w 2 be the cube roots of unity, the value of
(1 + w) 3 – (1 + w 2 ) 3 is
(a) 2w (b) 2 (c) –2 (d) 0
[IIT-JEE, 1993]
24. If a and b be two non-zero complex numbers and z be a
variable complex number. If the lines az – + a – z +1=0
and bz – + b – z – 1 = 0 are mutually perpendicular, then
(a) ab + a – b – = 0 (b) ab – a – b – = 0
(c) a – b – ab – = 0 (d) ab – + a – b = 0
[IIT-JEE, 1993]
25. If z 1
, z 2
, z 3
, be the vertices of an equilateral triangle inscribed
in the circle |z| = 2 and if z1 = 1+ i 3 , then
(a) z2=- 2, z3= 1 -i
3
(b) z2= 2, z3= 1 -i
3
(c) z2=- 2, z3=-1 -i
3
(d) z2= 1– i 3, z3=-1-i
3
[IIT-JEE, 1994]
26. If w(π1) be a cube root of unity and (1 + w) 7 = A + Bw,
then A and B are respectively the numbers are
(a) 0, 1 (b) 1, 1 (c) 1, 0 (d) –1, 1
[IIT-JEE, 1995]
27. Let z and w be two non-zero complex numbers such
that |z| = |w | and Arg(z) + Arg(w) = p, then z is equal to
(a) w (b) –w (c) w - (d) –w -
[IIT-JEE, 1995]
28. 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 (b) i or –i
(c) 1 or –1 (d) i or –1
[IIT-JEE, 1995]
29. If iz 3 + z 2 – z + i = 0, show that |z| = 1
[IIT-JEE, 1995]
30. If |z| £ 1, |w | £ 1, show that
|z – w| 2 £ (|z| – |w |) 2 + (Arg(z) – Arg(w)) 2 .
[IIT-JEE, 1995]
31. For positive integers n 1
, n 2
, the value of the expression
(1 + i) n1 + (1 + i 3 ) n1 + (1 + i 5 ) n2 + (1 + i 7 ) n2
is a real number if and only if
(a) n 1
= n 2
+ 1 (b) n 1
= n 2
– 1
(c) n 1
= n 2
(d) n 1
> 0, n 2
> 0
[IIT-JEE, 1996]
32. Find all complex numbers z satisfying z – = iz 2 .
[IIT-JEE, 1996]
33. Let b – z + bz – = c, b π 0 be a line in the complex plane,
where b – is the complex conjugate of b. If a point z 1
is
the reflection of a point z 2
through the line, show that
–
z = c. [IIT-JEE, 1997]
1 2