Complex Numbers
1 Introduction
Complex numbers arouse feelings of both awe and suspicion for many when
they come across them for the first time. However those who work with com-
plex numbers simply regard them as a natural extension of the real numbers,
m

2IA Introductory Algebra
14, 15 August 2014
Tutorial 3
Groups
(For all questions show your working.)
1. Recall that the set of all n n real matrices is denoted by Mn (R). A matrix A
Mn (R) is called symmetric if A = AT , and orthogonal if AAT = In , wher

2IA Introductory Algebra
16,17 October 2014
Tutorial 11
Rings and fields
1. For each of the following statements say whether it is true or false. Do this quickly,
but try to get consensus in your group. Then give a reason for each of your answers.
(a) The

Student Number:
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MAM1000W
Final Exam (Paper 2)
7 November 2014
Time : 14h00 16h00
Full marks: 100
This question paper consists of 15 pages (including this on

Student Number:
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Test 3
28 May 2013
Time : 17h00 19h00
Full marks: 100
This question paper consists of 14 pages (including this one).
Answer all questions in the spaces pr

Student Number:
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Class Test 2
22 April 2013
Time : 17h30 19h00
Full marks: 80
This question paper consists of 15 pages (including this one).
Answer all questions in the sp

CALCULUS I
Extras
Paul Dawkins
Calculus I
Table of Contents
Preface . ii
Extras . 2
Introduction . 3
Proof of Various Limit Properties . 4
Proof of Various Derivative Facts/Formulas/Properties . 15
Proof of Trig Limits . 28
Proofs of Derivative Applicatio

Student Number:
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Final Exam (Paper 2)
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2IA Introductory Algebra
7, 8 August 2014
Tutorial 2
Binary operations and error-correcting codes
(For all questions show your working.)
1. Consider the binary operation of subtraction () on the set Z of integers. For each
of the following statements say

MAM 1006H
Resource Book
2015
21nd Semester
Please bring this Book to all
Tutorials
I I I a I I I I I I I I l I - n A n I I I I I I I I I - I I I I I I I l u - 1 I I I I I I I Eiﬁiﬁi 7
Informatioh for Smdents
"Diary fer Terms 3 and 4
Tutorials
Further No

5i
MAM 1006H Taylor and Maclaurin Series / Expansions / Polynomials
H
Introduction:
In this section we use infinite polynomial series (also called power series) to help as approximate
familiar functions. Taylor and Maclaurin series provide us a way of app

A Statement of the Principle of Mathematical Induction:
Let n be a positive integer, and let PM) he a “statement” (or proposition)
about ii.
If (A) we can prove thatP( 1) is true, and
(B) if we assume that P(k) is true for any It 2 1, and
(C) then can pro

is determined if you know the angleﬁ and the distance 7".
Im
Re.
This idea motivates the two deﬁnitions (modulus and argument) of this
section.
4.2 The‘modulus
Deﬁnition The modulus of the complex number 2 = a + 52', denoted by I2]
is deﬁned by
lzl = «a?

,-
(=>) If e; = 6‘” with z =-a + hi and w = c + 032', then by (a) of the same
theorem we have a? 2 e5, and so a = c. Norar
eh1 = e‘IE1 :> cosb+isinb= cosd+isind
so 5 = d + 27m and hence the result.
A last observation for this section: the famous equation

~Tutorial 28
Determinants, Rotation Matrices
l.Evaluate: .
1 2 7 11 2 3 x o 0 0 a 0
(a)—i —21 (b)2 41 (0):; y 0 (d)0 O b
8 1 11 1 213 b c z c 0 O
b
2. Prove the following properties/laws/theorerns for any 2 x 2 matrix A, where A = {a j .
(a) [A] = M1.
(

3?
MAM 1006B Solving Systems of Linear Equations
How can we solve a system of n linear equations in n unknowns, Ax = .5?
Basically there are 4 methods:
By substitution (school method) me very eenﬁistng if there are mere than 3 01“ .4,
equatiens.
b. By us

10. If it is given that 2 + 1' is a solution of the equation 423 —1922 + 322 '15 = 0, ﬁnd the only
real solution to the equation.
1 1. Solve the equation 23 = —1 + $3} a and plot the solutions on a circle in an Argancl diagram.
What is the angle between t

1 4+8i
Sl+i 9 1—21'
e) 2.2 , Where
10. Now that you know how to add and multiply complex numbers you can study the
properties of these Operations. ls addition commutative? Is multiplication associative? Does
the distributive law work for complex numbers

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INVESTIGATION INTO PROPOSED HOUSING
DEVELOPMENT IN ESTUARY POINT TO
DETERMINE MOST SUITABLE DEVELOPMENT
PLAN
PREPARED FOR: Mr Robbie Byrd
Municipal manager
Estuary point
PREEPARED BY: development researching team
Nosihle Gwala
Phathu Siboiboi

INF1002FClassExercise1
2013
INF 1002F Introduction to Excel
Formatting, Formulas, & Cell Referencing
LEARNING OBJECTIVE:
Be able to interpret, Manipulate & Transform data in Excel
Case 1
1. Download the file Excel Class Exercise 1 -2012.xlsx from Vula and

Examples of how to cite according to the Author-date style:
An electronic journal:
Aird, A. 2001. E-commerce in higher education: can we afford to do
nothing? Ariadne. 26. (Online). Available:
http:/www.ariadne.ac.uk/issue26/e-commerce/ (2003, November 25

Question
4 points
1
Save
Each of the following, except one, is a condition that characterizes
a perfectly competitive labor market. Which is the exception?
Workers appear identical to firms.
Workers receive wages that are above their marginal
revenue prod

ECO110H and ECO110S Examination January/February 2005
UNIVERSITY OF CAPE TOWN
SCHOOL OF ECONOMICS
ECO110H and ECO110S
SUPPLEMENTARY EXAMINATION
JANUARY/FEBRUARY 2005
TIME: 3 HOURS
MARKS: 180
SECTIONS
A
TOPIC
MCQs
QUESTIONS
1-30
B
Structured
Questions
1-5

ECO110H and ECO110S Examination November 2004
UNIVERSITY OF CAPE TOWN
SCHOOL OF ECONOMICS
ECO110H and ECO110S
EXAMINATION
NOVEMBER 2004
TIME: 3 HOURS
MARKS: 180
SECTION
A
TOPIC
MCQs
QUESTIONS
1-30
B
Structured
Questions
1-5
MARKS
60
(60 minutes)
120
(120