1.Algebra Booster

Complex Numbers 4.27
5. Match the following lists.
(P)
(Q)
(R)
(S)
List I
Let z be a root of x 5 – 1 = 0, the
value of z 15 + z 16 + z 17 + … + z 50
is
If z = x + iy, z 1/3 = a – ib and
x y 2 2
- = ka ( - b),
the value
a b
of k is
The number of common roots of
the equations z 3 + 2z 2 + 2z + 1 =
0 and z 2015 + z 2014 + 1 = 0 is
Ê2rpˆ Ê2rpˆ
If zr
= cosÁ + isin ,
Ë
˜ Á ˜
5 ¯ Ë 5 ¯
where r = 1, 2, 3, 4, 5, then the
value of (z 1
z 2
z 3
z 4
z 5
+ 2) is
Codes:
P Q R S
(A) 3 4 2 1
(B) 3 1 2 4
(C) 1 4 3 1
(D) 3 1 4 2
6. Match the following lists.
(P)
(Q)
(R)
(S)
List I
The number of solution of
the equation 2z = |z| + 2i is
The sum of the non-real
complex values of x in
(x – 1) 4 – 16 = 0 is
z1-
3z2
If = 1 and |z
3 - zz
2
| π 1,
1 2
the value of |z 1
| is
If x = a + b, y = aw + bw 2 and
z = aw 2 + bw such that
x 3 + y 3 + z 3 = l(a 3 + b 3 ),
the value of (l + 1) is
Codes
P Q R S
(A) 2 3 1 4
(B) 3 2 1 4
(C) 2 3 4 1
(D) 3 2 4 2
List II
(1) 4
(2) 2
(3) 1
(4) 3
List II
(1) 4
(2) 1
(3) 2
(4) 3
Questions asked in Previous Years’
JEE-Advanced Examinations
1. If x = a + b, y = aa + bb and z = ab + ba where a, b are
complex cube roots of unity, prove that xyz = a 3 + b 3 .
[IIT-JEE, 1978]
2. If the cube roots of unity are 1, w, w 2 , the roots of
(x –1) 3 + 8 = 0 are
(a) –1, 1 + 2w, 1 + 2w 2 (b) –1, 1 – 2w, 1 – 2w 2
(c) –1, –1, –1
(d) none
[IIT-JEE, 1979]
n
Ê1
+ iˆ 3. The smallest positive integer n for which Á = 1
Ë1
- i˜
is
¯
(a) n = 8 (b) n = 16 (c) n = 12 (d) none
[IIT-JEE, 1980]
4. The complex number z = x + iy which satisfies the
z - 5i
equation = 1 lie on
z + 5i
(a) the x-axis
(b) the straight line y = 5
(c) a circle passing through the origin
(d) none [IIT-JEE, 1981]
5. The inequality |z + 2| < |z – 2| represents the region
given by
(a) Re(z) ≥ 0 (b) Re (z) > 0
(c) Re (z) < 0 (d) none
[IIT-JEE, 1982]
5 5
Ê 3 iˆ Ê 3 iˆ
6. If z = Á + + -
Ë
˜
2 2¯ Á
Ë
˜
2 2¯ , then
(a) Re(z) = 0
(b) Im(z) = 0
(c) Re (z) > 0, Im (z) > 0
(d) Re (z) > 0, Im (z) < 0 [IIT-JEE, 1982]
Ê1
- izˆ
7. If z = x + iy and w = Á
Ë z - i ˜ , then |w| = 1 implies that
¯
in the complex plane
(a) z lies on the imaginary axis
(b) z lies on the real axis
(c) z lies on the unit circle
(d) none
[IIT-JEE, 1983]
8. The points z 1
, z 2
, z 3
, z 4
in the complex plane are the
vertices of a parallelogram taken in order if and only if
(a) z 1
+ z 4
= z 2
+ z 3
(b) z 1
+ z 3
= z 2
+ z 4
(c) z 1
+ z 2
= z 3
+ z 4
(d) none
[IIT-JEE, 1983]
9. If the complex numbers z 1
, z 2
, z 3
represent the vertices
of an equilateral triangle such that |z 1
| = |z 2
| = |z 3
|, then
z 1
+ z 2
+ z 3
= 0. Is it true or false? [IIT-JEE, 1984]
10. If three complex numbers are in AP, they lie on a circle
in the complex plane. Is it true or false?
[IIT-JEE, 1985]
11. If a, b, c and u, v, w be complex numbers representing
the vertices of two triangles such that c = (1 – r)a + rb
and w = (1 – r)u + rv, where r is a complex number, the
two triangles
(a) have the same area (b) are similar
(c) are congruent (d) none
[IIT-JEE, 1985]