2025 Tshwane University Of Technology
Dept. of Mathematics and Statistics
EM115AB - Engineering Mathematics IA
March 17–21, 2025
Worksheet 6
Student #:
Surname & Initials:
1.
a)
Evaluate the following and leave your answers in standard form, that is, a + bi.
5
1
c)
(2 + 5i)(4 − i)
(5 + 6i) + (3 + 2i)
b)
4− i − 9+ i
2
2
d)
(1 − 2i)(8 − 3i)
e)
12 + 7i
f)
g)
1 + 4i
3 + 2i
h)
1
1+i
i)
j)
i3
k)
i100
l)
n)
(2 − 3i)2 (4 − i)
m)
√
√
−3 −12
2i( 21 − i)
3
4 − 3i
√
−16
2.
For each of the complex number below, find the complex conjugate and the modulus of the
number.
a)
12 − 5i
3.
Prove the following property of complex numbers:
√
1 + 2 2i
b)
−4i
c)
zw = z w.
Hint: let z = a + bi and w = c + di.
4.
Solve the following equations:
a)
4x2 + 9 = 0
b)
x4 = 1
d)
z2 + z + 2 = 0
e)
z 2 + 12 z + 14 = 0
5.
Write the following complex numbers in polar form with argument between 0 and 2π.
a)
−3 + 3i
6.
Find polar forms for zw and z/w by first putting z and w in polar form:
a)
b)
c)
√
√
z= √
3 + i,
w = 1 + 3i
z = 2 √3 − 2i,
w = −1 + i
z = 4( 3 + i),
w = −3 − 3i
b)
1−
c)
√
3i
c)
1
x2 + 2x + 5 = 0
3 + 4i
d)
8i
7.
Use De Moivre’s Theorem to find:
a)
(1 + i)20
8.
Use De Moivre’s Theorem with n = 3 to express cos(3θ) and sin(3θ) in terms of cos θ
and sin θ. Hint: use r = 1.
9.
Find the indicated roots. Sketch the roots in the complex plane:
a)
The fifth roots of 32
10.
Write the number in the form a + bi:
a)
e)
eiπ/2
eπ+i
11.
Use Euler’s formula to prove that
(1 −
b)
b)
b)
√
3i)5
The cube roots of i
e2πi
e−iπ
c)
cos x =
√
(2 3 + 2i)5
c)
c)
d)
The cube roots of 1 + i
e2+iπ
eix + e−ix
.
2
EXAM–TYPE QUESTIONS
12.
Find a complex number z such that
z + iz = (2 + i)2 + i(2 − 4i).
13.
If z = 4[cos(π/5) + i sin(π/5)]. Calculate z 3 and z 3 .
14.
a)
b)
√
Consider√the complex number 3 − i.
Express 3 − i in polar form.
√
Find all complex numbers z such that z 4 = 3 − i and represent all your solutions in
the complex plane.
15.
Consider z1 = 5 + 6i and z2 = 3 + 2i. Find z1 + z2 and sketch your solution on the
complex plane.
16.
Simplify
17.
a)
If z1 = 1 + 2i and z2 = −2 + 3i. Find:
z1
z1
b)
z2 +
z2
z2
18.
Determine the values of x and y if (3 + 2i)2 − 3(x + iy) = x + iy.
19.
Determine the square roots of z = 5 + 12i.
Do NOT round off any value in the intermediate steps also work in radians.
i100 − i27
and leave your answer in the form a + bi.
i
c)
2
2
z2
+
z1
3
20.
Let z be a complex number. Then
(A) Re(iz)=−Im(z) (B) Re(iz)=Im(z) (C) Re(iz)=iRe(z) (D) Im(iz)=Im(z) Answer:
21.
Assume that z1 = 2[cos( 5π
) + i sin( 5π
)] and z2 = cos( π2 ) + i sin( π2 ). Find (z3 )6 , leaving your
3
3
answer in standard form, that is, a + bi where z3 = z1 · z2 .
3