1.Algebra Booster

Complex Numbers 4.23
2. Find the area of a triangle on the Argand plane formed
by the complex numbers –z, iz, z– iz.
3. If a complex number z lies on a circle of radius 1/2,
prove that the complex number (–1 + 4z) lies on a circle
of radius 2.
4. If |z 1
| = 1, |z 2
| = 2, |z 3
| = 3 and |9z 1
z 2
+ 4z 1
z 3
+ z 2
z 3
| = 12,
find the value of |z 1
+ z 2
+ z 3
|.
5. If z 1
, z 2
, z 3
be the complex numbers representing the
2 1 1
points A, B, C such that = + , prove that a
z1 z2 z3
circle through A, B, C passes through the origin.
6. Let a, b, c be three distinct complex numbers such that
a b c
= = = k , find the value of k.
1-b 1-c 1-a
7. If a and b are positive integers such that N = (a + ib) 3 –
107i is a positive integer, find N.
8. Find the set of points on the Argand plane for which the
real part of the complex number (1 + i)z 2 is positive,
where z = x + iy, x, y ΠR and i = -1.
9. C is the complex number and f : C Æ R is defined by
f(z) = |z 3 – z + 2|. What is the maximum value of f on the
unit circle |z| = 1.
10. Let z = x + iy be a complex number, where x and y are
real numbers. Let A and B be two sets defined as A =
{z :|z| £ 2} and B= { z:(1 - i) z + (1 + i) z £ 4} . Find
the area of the region A « B.
11. Show that
2
n
2 2 2
È 1 i È 1 i ˘È
1 i
˘È
1 i
˘
Ê + ˆ˘ Ê + ˆ Ê + ˆ Ê + ˆ
Í1+ Á ˜ 1 1 1
2
˙Í + Á ˜ ˙Í + Á ˜
˙Í + Á ˜ ˙
Î Ë ¯˚Î Ë 2 ¯ ˚ÎÍ Ë 2 ¯ ˚˙Î Í Ë 2 ¯ ˙˚
Ê 1 ˆ
= Á1 - (1+ i), n ≥ 2
n
2
˜
Ë 2 ¯
12. Show that the locus formed by z in the equation
z 3 + iz = 1 never crosses the co-ordinate axes in the
Argand plane. Further show that
- Im( z)
| z|
=
2Re( z)Im( z) + 1
1
13. For all real numbers x, let the function f()
x = x - 1
,
where i = - 1 . If there exist real number a, b, c and d
for which f(a), f(b), f(c) and f(d) form a square on the
complex plane, find the area of the square.
14. If
i 2
e p
n
a= and
=
20
0 +Â k
k
find the value of
k = 1
f() x A A x ,
f(x) + f(ax) + f(a 2 x) + … + f(a k x), which is independent
of a
15. Solve the equation z 7 – 1 = 0 and deduce that
Ê2 4 6 1
cos p ˆ Ê
cos p ˆ Ê
cos p ˆ
Á + + = - .
Ë
˜
7 ¯ Á
Ë
˜ Á ˜
7 ¯ Ë 7 ¯ 2
16. Solve the equation z 7 + 1 = 0 and deduce that
Ê 3 5 1
cos p ˆ Ê
cos p ˆ Ê
cos p ˆ
Á ◊ ◊ = - .
Ë
˜
7¯ Á
Ë
˜ Á ˜
7 ¯ Ë 7 ¯ 8
17. If w be the nth root of unity and z 1
, z 2
be any two complex
numbers, prove that
n –1
Â
k = 0
k 2 2 2
1+ w 2 = 1 + 2
| z z | n(|z | |z | ).
18. Given that |z – 1| = 1, where z is a point on the complex
plane, show that
z - 2 = itan (Arg ( z))
z
Ê 2p
ˆ Ê 2p
ˆ
19. Given z = cosÁ + isin
Ë2n+ 1˜ ¯
Á
Ë2n+
1˜
, where n is a
¯
positive integer, find the equation whose roots are
a = z + z 3 + … + z 2n – 1 and b = z 2 + z 4 + … + z 2n .
1
20. If a and b be the roots of z + = 2(cos q+ isin q)
,
z
where 0 < q < p, prove that |a – i| = |b – i|.
21. If a complex number z lies on the curve |z – (–1 + i)| = 1,
then find the locus of
z + i
w = , i = -1.
z - i
22. Consider a triangle formed by the points
ip
5
2 2
i
i p
Ê ˆ Ê - p
ˆ Ê 2 - ˆ
A e
2
, B e
6
, C e
6 . Let P(z) be
Á
Ë
˜ Á ˜ Á ˜
3 ¯ Ë 3 ¯ Ë 3 ¯
any point on its circle, prove that AP 2 + BP 2 + CP 2 = 5.
23. Find the common tangents of the curves
Re(z) = |z – 2a| and |z – 4a| = 3a.
24. Show that the complex numbers whose real and
imaginary parts are integers and satisfy the equation
3 3
zz + zz = 350 form a rectangle in the Argand plane
with the length of the diagonal having an integral number
of units.
25. If b π 1 be any nth root of unity, prove that the value of
2n
1 + 3b + 5b 2 + … to n terms is
b - 1
.
26. The equation x 3 = 9 + 46i, where i = - 1 has a solution
of the form a + ib, where a and b are integers. Find
the value of (a 3 + b 3 ).
27. If w be the ffth root of 2 and x = w + w 2 , prove that
x 5 = 10x 2 + 10x + 6.
28. Let Z is a complex number satisfying the equation z 2
Р(3 + i)z + m + 2i = 0, where m ΠR. Suppose the equation
has a real root, find the value of m.
29. a, b, c are real numbers in the polynomial
P(Z) = 2Z 4 + aZ 3 + bZ 2 + cZ + 3.
If two roots of P(Z) = 0 are 2 and i (iota), find the value
of a.