COMPLEX NUMBERS EXAMPLES & SOLUTIONS
COMPLEX NUMBERS EXAMPLES & SOLUTIONS
Department of Mathematics & Philosophy
The Open University of Sri Lanka
Produced by The Open University of Sri Lanka
2015
0
Examples for Complex numbers
COMPLEX NUMBERS:EXAMPLES & SOLUTIONS
Question (01)
(i) Find the real values of x and y such that (1- i)x + 2i + (2 + 3i) y + i = - i
3-i
3+i
(ii) Find the real values of x and y are the complex numbers 3 - ix2 y and -x2 - y - 4i conjugate of each other.
(iii) Find the square roots of 4 + 4i
(iv) Find the complex number Z satisfying the equation Z -12 = 5 and Z - 4 = 1
Z - 8i 3
Z -8
(v) Find real such that 3 + 2i sin is 1- 2i sin
(a) real
(b) imaginary
Solution
(i) (1- i)x + 2i + (2 + 3i) y + i = - i
3-i
3+i
{(1- i)x + 2i}(3 + i) + {(2 + 3i) y + i}(3 - i) = - i
(3 - i)(3 + i)
(3 + i)(1- i)x + 6i + 2i2 + (2 + 3i)(3 - i) y + 3i - i2 9 -i2
=
-i
(3 + i - 3i - i2 )x + 6i + (6 + 9i - 2i - 3i2 ) y + 3i + i2 = - i 9 +1
(4 - 2i)x + (9 - 7i) y + 9i -1 = -10i
Produced by The Open University of Sri Lanka
2015
1
[4x + 9 y -1] + i(19 - 2x - 7 y) = 0
4x + 9 y -1 = 0......................(1)
19 - 2x - 7 y = 0.....................(2)
By solving equations (1) and (2)
x = -82 y = 37
5
5
(ii) 3 - ix2 y = -x2 - y - 4i
3 - ix2 y = -(x2 + y) + 4i
-(x2 + y) = 3
-x2 y = 4
x2 = - 4 y
4y-y=3
y2 +3y - 4 = 0
( y + 4)( y -1) = 0 y = - 4 or y = 1
when y = - 4
x2 = - 4 = 1 x = ?1 -4
y =1
x2
=
-4 1
= -4
(Not real)
x = ?1 and y = - 4
(iii) Let z2 = 4 + 4i and z = (x + iy)
z = 4 + 4i z2 = (x + iy)(x + iy)
= x 2 +2ixy + i2 y2
z2 = (x2 - y2 ) + i2xy
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2015
2
(x2 - y 2 ) + i2xy = 4 + 4i
x2 - y2 = 4 2xy = 4
y = 2x
{ } x2 -
2 x
2
=4
x 4 -4x2 + 4 = 0 (x 2 -2)2 = 0
x2 = 2
x=? 2
y =
2 ?
=? 2
2
z = 2 + 2 i or z = - 2 - 2 i
(iv) z -12 = 5 .................(1) z - 8i 3 z - 4 = 1 ....................(2) z -8
Let z = x + iy
z -12 = (x -12) + iy
z -12 = (x -12)2 + y2
z - 8i = x + i( y - 8)
z - 8i = x2 + ( y - 8)2
z - 4 = (x - 4) + iy
z - 4 = (x - 4)2 + y2
z - 8 = (x - 8) + iy
z - 8 = (x - 8)2 + y2
From (1); z -12 = 5 3 (x -12)2 + y2 = 5 x2 + ( y - 8)2 z - 8i 3
{ } 9 (x -12)2 + y2 = 25 x2 + ( y - 8)2
..............(1)
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2015
3
From (2) z - 4 = 1 z -8
(x - 4)2 + y2 = (x - 8)2 + y2
(x - 4)2 + y2 = (x - 8)2 + y2
x2 - 8x +16 + y2 = x2 -16x + 64 + y2
8x = 48
x =6
form (1); 9 (6 -12)2 + y2 = 25 62 + ( y - 8)2
9 (36 + y2 ) = 25 36 + ( y - 8)2 16 y2 - 400 y + (2500 - 324) = 0
16 y2 - 400 y + 2176 = 0
y2 - 25y +136 = 0
( y -17) ( y - 8) = 0
y = 17 or y = 8
z = 6 +17i or z = 6 + 8i
(v)
z
=
3 + 2i sin 1- 2i sin
=
(3 + 2i sin )(1+ 2i sin ) (1- 2isin )(1+ 2i sin )
=
(3 - 1+
4sin2 4sin2
)
+
8i sin 1+ 4sin2
If
z
is
real
1
8 sin + 4sin2
=0
= 2
= n + (-1)n 2 n
If
z
is imaginary
3 - 4 sin2 1+ 4sin2
=0
Produced by The Open University of Sri Lanka
2015
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