1.Algebra Booster

Complex Numbers 4.17
106. Let z = cos q + i sin q, find the value of
15
Â
m=
1
2m-1
Im( z ) at q = 2°.
107. Let a complex number a, a π 1 be a root of an equation
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.
ROATATION
108. If a point P(3, 4) is rotated through an angle of 90°
in anti-clockwise sense about the origin, find the new
position of P.
109. If a point Q(3, 4) is rotated through an angle of 180°
in anti-clockwise sense about the origin, find the new
position of Q.
110. If a point P(3, 4) is rotated through an angle of 30° in
anti-clockwise sense about the point Q(1, 0), find the
new position of P.
111. The complex number 3 + i becomes - 1+ i 3 after
rotating an angle q about the origin in anti-clock-wise
sense, find the angle q.
112. The three vertices of a triangle are represented by the
complex numbers 0, z 1
, z 2
. If the triangle is equilateral
2 2
triangle, prove that z1 + z2 = z1z2
113. If the origin and the roots of z 2 + az + b = 0 form an
equilateral triangle, prove that a 2 = 3b.
114. If the area of a triangle on the complex plane formed by
the complex numbers z, iz and z + iz is 50 sq.u., find |z|.
115. If the area of a triangle on the complex plane formed by
the complex numbers z, wz, z + wz is 16 3 sq.u. , find
the value of (|z| 2 + |z| + 2).
116. Let z 1
and z 2
be the nth roots of unity which subtend a
right angle at the origin, prove that n = 4k, where k ΠN.
117. Suppose z 1
, z 2
, z 3
be the vertices of an equilateral triangle
inscribed in the circle |z| = 2. If z1 = 1+ i 3 , prove
that z 2
= –2 and z3 = 1-i
3.
118. If a and b be real numbers between 0 and 1 such that
the points z 1
= a + i, z 2
= 1 + ib and z 3
= 0 form an equilateral
triangle, prove that a = 2- 3 = b.
119. The adjacent vertices of a regular polygon of n-sides
are the points z and its conjugate z .
If Im( z ) = 2 - 1, find the value of n.
Re( z )
120. The vertices A and C of a square are ABCD are 2 + 3i
and 3 – 2i respectively. Find the vertices B and D.
121. A, B, C are the vertices of an equilateral triangle whose
centre is i. If A represents the complex number –i, find
the vertices B and C.
122. A man walks a distance of 3 units from the origin towards
the north-east (N45°E) direction. From there,
he walks a distance of 4 units towards the north-west
(N45°W) direction to reach a point P. Find the position
of the point P in the Argand plane.
123. A particle P starts from the point z 0
= 1 + 2i, where
i = - 1 . It moves first horizontally away from origin
by 5 units and then vertically away from the origin 3
units to reach a point z 1
. From z 1
the particle moves
2 units in the direction of the vector iˆ+ ˆj
and then
p
through an angle in anti-clockwise direction on a
2
circle with the centre at the origin to reach a point z 2
.
Find the point z 2
.
124. Let z 1
and z 2
be the roots of the equation z 2 + pz + q
= 0, where the co-efficients p and q may be complex
numbers. Let A and B be represent z 1
and z 2
in the complex
plane. If –AOB = a, and OA = OB, where O is the
2 2Êa
ˆ
origin, prove that p = 4q
cos Á .
Ë
˜
2 ¯
LOCI OF A COMPLEX NUMBER
Ê z - 1ˆ p
125. Find the locus of z, if Arg Á = .
Ë z + 1˜
¯ 4
126. Find the locus of z, if |z – 1| + |z + 1| £ 4.
127. Find the locus of z, if |z – 2| + |z + 2| £ 4.
2
128. Find the locus of z, if z = t + 5+ i 4 -t , t ŒR.
Ê
2
z ˆ
129. If Á
Ë z - 1
˜ is always real, find the locus of z.
¯
1
130. If Re Ê Á
ˆ ˜ = cc , π 0 , find the locus of z.
Ë z¯
131. If |z 2 – 1| = |z| 2 + 1, find the locus of z.
LEVEL II
(Mixed Problems)
1. If a complex number z satisfying z + |z| = 1 + 7i, the
value of |z| 2 is
(a) 625 (b) 169 (c) 49 (d) 25
2. If z = (3 + 7i)(p + iq), p, q ΠI Р{0} is purely imaginary
number, the minimum value of |z| 2 is
(a) 0 (b) 58 (c) 3364 (d) 3364/3
3
3. The number of complex numbers z satisfying z = z is
(a) 1 (b) 2 (c) 4 (d) 5
4. The number of real and purely imaginary solution of
the equation z 3 + iz – 1 = 0 is
(a) 0 (a) 1 (c) 2 (d) 3
5. A point z moves on the curve |z – 4 – 3i| = 2 in an argand
plane. The maximum and minimum values of |z| are
(a) 2, 1 (b) 6, 5 (c) 4, 3 (d) 7, 3
6. If z be a complex number satisfying the equation |z + i|
+ |z – i| = 8 on the complex plane, the maximum value
of |z| is
(a) 2 (b) 4 (c) 6 (d) 8
7. Let z be a complex number satisfying the equation
(z 3 + 3) 2 = –16, the value of |z| is
(a) 5 1/2 (b) 5 1/3 (c) 5 2/3 (d) 5