MAT1511/101/3/2020
Tutorial Letter 101/3/2020
PRECALCULUS MATHEMATICS B
MAT1511
Semesters 1 & 2
Department of Mathematical Sciences
IMPORTANT INFORMATION:
This tutorial letter contains important
information about your module.
BAR CODE
Define tomorrow.
university
of south africa
CONTENTS
Page
1
INTRODUCTION.................................................................................................................. 4
2
PURPOSE OF AND OUTCOMES........................................................................................ 4
2.1
Purpose................................................................................................................................ 4
2.2
Outcomes............................................................................................................................. 4
3
LECTURER(S) AND CONTACT DETAILS.......................................................................... 5
3.1
Lecturer(s)............................................................................................................................ 5
3.2
Department........................................................................................................................... 5
3.3
University.............................................................................................................................. 6
4
RESOURCES....................................................................................................................... 6
4.1
Prescribed book(s)............................................................................................................... 6
4.2
Department........................................................................................................................... 6
4.3
University.............................................................................................................................. 6
5
RESOURCES....................................................................................................................... 6
5.1
Prescribed books.................................................................................................................. 6
5.2
Recommended books........................................................................................................... 6
5.3
Electronic Reserves (e-reserves)......................................................................................... 7
5.4
Library services and resources............................................................................................. 7
6
STUDENT SUPPORT SERVICES....................................................................................... 7
7
STUDY PLAN....................................................................................................................... 8
8
PRACTICAL WORK............................................................................................................. 8
9
ASSESSMENT..................................................................................................................... 8
9.1
Assessment criteria.............................................................................................................. 8
9.2
Assessment plan.................................................................................................................. 8
9.3
Assignment numbers............................................................................................................ 9
9.3.1 General assignment numbers............................................................................................... 9
9.3.2 Unique assignment numbers................................................................................................ 9
9.4
Assignment due dates.......................................................................................................... 9
9.5
Submission of assignments.................................................................................................. 9
9.6
The assignments................................................................................................................ 10
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MAT1511/101
9.7
Other assessment methods................................................................................................ 10
9.8
Examination........................................................................................................................ 10
10
FREQUENTLY ASKED QUESTIONS................................................................................ 10
11
SOURCES CONSULTED................................................................................................... 10
12
IN CLOSSING.................................................................................................................... 10
13
ADDENDUM A: SEMESTER 1 STUDENTS...................................................................... 11
14
ADDENDUM B: SEMESTER 2 STUDENTS...................................................................... 22
3
1
INTRODUCTION
Welcome to this module. We trust that you will find it both interesting and rewarding. The university
has decided that this module will now be a semester module, so please start studying as soon as
you receive your study material. Note that if you are registered for Semester 1 (Semester 2), then
all your assignments need to be submitted by 14 February 2020 (28 September 2020).
You will receive a number of tutorial letters during the semester. A tutorial letter is our way of
communicating with you about teaching, learning and assessment.
This tutorial letter contains important information about the scheme of work, resources and assignments for this module as well as exam admission. We urge you to read it carefully and to keep it
at hand when working through the study material, preparing the assignment(s), preparing for the
examination and addressing questions to your lecturers.
In this tutorial letter, you will find the assignments as well as instructions on the preparation and
submission of the assignments. This tutorial letter also provides all the information you need with
regard to the prescribed study material and other resources and how to obtain it. Please study this
information carefully and make sure that you obtain the prescribed material as soon as possible.
We have also included certain general and administrative information about this module. Please
study this section of the tutorial letter carefully.
Right from the start we would like to point out that you must read all the tutorial letters you
receive during the semester immediately and carefully, as they always contain important and,
sometimes urgent information.
We hope that you will enjoy this module and wish you all the best!
2
PURPOSE OF AND OUTCOMES
2.1
Purpose
This module will be useful to students in developing basic skills in algebra which can be applied in
the natural sciences and social sciences. Students credited with this module will have an understanding of basic ideas of algebra and to apply the basic techniques for handling zeros of polynomials, systems of equations, matrices, complex numbers, polar and rectangular coordinates, vectors,
systems of linear inequalities, binomial expansions, mathematical induction. In particular, the focus
is on building strong algebraic and trigonometric skills that will support the development of analytical thinking skills and problem solving in more advanced mathematics and related mathematical
science subjects.
2.2
Outcomes
The broad outcomes of this module are
• To find and describe the zeros of polynomials
• To perform basic operations on complex numbers
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MAT1511/101
• To solve certain system of equations using matrices, inverses of matrices, Cramer’s rule
• To prove mathematical statements using mathematical induction
• To write an expansion of a binomial using the Binomial Theorem
Specific outcomes are listed in the study guide.
3
LECTURER(S) AND CONTACT DETAILS
3.1
Lecturer(s)
The lecturer responsible for this module is Dr Sunday Faleye. You can contact him at:
Dr Sunday Faleye
Tel: (011) 670 9287
Room no: C 6–49
GJ Gerwel Building
e-mail: efaleys1@unisa.ac.za
A notice will be posted on myUnisa if there are any changes and/or an additional lecturer is appointed to this module.
Please do not hesitate to consult your lecturer whenever you experience difficulties with your studies. You may contact your lecturer by phone or through correspondence or by making a personal
visit to his/her office. Please arrange an appointment in advance (by telephone or by e-mail)
to ensure that your lecturer will be available when you arrive.
If you should experience any problems with the exercises in the study guide, your lecturer will
gladly help you with them, provided that you send in your bona fide attempts. When sending in
any queries or problems, please do so separately from your assignments and address them
directly to your lecturer.
3.2
Department
Department of Mathematical Sciences
Fax number: 011 670 9171 (RSA)
Departmental Secretary: 011 670 9147 (RSA)
e-mails: mathsciences@unisa.ac.za or
+27 11 670 9171 (International)
+27 11 670 9147 (International)
swanem@unisa.ac.za
or the mail address
Department of Mathematical Sciences
University of South Africa
PO Box 392
UNISA
0003
5
3.3
University
Consult the brochure Study @ Unisa for information on how to contact the university. Always use
your student number when you contact the University.
4
RESOURCES
4.1
Prescribed book(s)
There is no prescribed textbook for this module.
4.2
Department
Department of Mathematical Sciences
Fax number: 011 670 9171 (RSA)
Departmental Secretary: 011 670 9147 (RSA)
e-mails: mathsciences@unisa.ac.za or
+27 11 670 9171 (International)
+27 11 670 9147 (International)
swanem@unisa.ac.za
or the mail address
Department of Mathematical Sciences
University of South Africa
PO Box 392
UNISA
0003
4.3
University
Consult the brochure Study @ Unisa for information on how to contact the university. Always use
your student number when you contact the University.
5
RESOURCES
5.1
Prescribed books
The prescribed book is
Precalculus, Mathematics for Calculus
by J. Stewart, L. Redlin, S. Watson, 6th edition,
Brooks/Cole Publishers.
The book is also prescribed for MAT1510, so if you are registered for both MAT1510 and MAT1511,
you only have to buy one prescribed book.
You must buy this book since you have to study from it directly – you cannot do the module
without it.
5.2
Recommended books
There is no official recommended books for this module.
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MAT1511/101
5.3
Electronic Reserves (e-reserves)
There are no e-reserves for this module.
E-reserves can be downloaded from the Library catalogue. More information is available at:
http://libguides.unisa.ac.za/request/request
5.4
Library services and resources
The Unisa Library offers a range of information services and resources:
• for detailed Library information go to http://www.unisa.ac.za/sites/corporate/default/Library
• for research support and services (e.g. personal librarians and literature search services) go
to
http://www.unisa.ac.za/sites/corporate/default/Library/Library-services/Research-support
The Library has created numerous Library guides:
http://libguides.unisa.ac.za
Recommended guides:
• Request and download recommended material:
http://libguides.unisa.ac.za/request/request
• Postgraduate information services:
http://libguides.unisa.ac.za/request/postgrad
• Finding and using library resources and tools:
http://libguides.unisa.ac.za/Research_skills
• Frequently asked questions about the Library:
http://libguides.unisa.ac.za/ask
• Services to students living with disabilities:
http://libguides.unisa.ac.za/disability
6
STUDENT SUPPORT SERVICES
The Study @ Unisa brochure is available on myUnisa: www.unisa.ac.za/brochures/studies
This brochure has all the tips and information you need to succeed at distance learning and, specifically, at Unisa.
7
7
STUDY PLAN
As adults you will make your own decisions about when you do your work. Those of you who
are employed or who have other responsibilities will have less time than those who are full-time
students, but it is very important to set aside regular time to study, and to try to stick to your
schedule. The closing dates for the assignments already provide the outline of a schedule for you.
Based on these dates, you should thus try to keep to the following plan. You may progress through
the material faster and submit assignments earlier. In this way you will be safer and provide for
unforeseen circumstances which may occur.
See the brochure Study @ Unisa for general time management and planning skills.
8
PRACTICAL WORK
There are no practicals for this module.
9
ASSESSMENT
9.1
Assessment criteria
There are three assignments and one examination.
Examination admission.
Please note that lecturers are not responsible for examination admission, and ALL enquiries about
examination admission should be directed by e-mail to exams@unisa.ac.za
You will be admitted to the examination if and only if Assignment 01 reaches
the Assignment Section by 14 February 2020 if you are registered for Semester
1, or by 28 August 2020 if you are registered for Semester 2.
9.2
Assessment plan
You will receive the solutions for Assignments 1, 2 and 3 automatically, even if you did not submit
the relevant assignment. These solutions will be posted on myUnisa under Additional Resources
about one week after the closing date of the relevant assignment, so it is important to submit your
assignments so that they reach the Assignment Department at Unisa by the closing date.
Markers will comment on the work that you submit in your assignments. The assignments and the
comments constitute an important part of your learning and should help you to be better prepared
for the examination.
N.B. Please don’t wait for an assignment to be returned to you before starting to work on the next
assignment.
The assignment questions are included in Addendum of this tutorial letter.
ASSIGNMENTS THAT REACH UNISA AFTER THE CLOSING DATE WILL NOT BE MARKED.
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MAT1511/101
9.3
Assignment numbers
9.3.1 General assignment numbers
The assignments are numbered as 01, 02 and 03 for each semester.
9.3.2 Unique assignment numbers
Please note that each assignment has a unique assignment number which must be written on the
cover of your assignment. See the table below for the unique numbers.
9.4
Assignment due dates
The closing dates for submission of the assignments are:
Semester 1
Assignment no
01
02
03
Unique N
560103
632606
537117
Due date
14 February 2020 13 March 2020 03 April 2020
Semester 2
Assignment no
01
02
03
Unique No
813197
504062
578161
Due date
28 August 2020 18 September 2020 07 October 2020
9.5
Submission of assignments
You may submit your assignments either by post or electronically via myUnisa. Assignments may
not be submitted by fax or e–mail. For detailed information and requirements as far as assignments
submissions are concerned, see the brochure Study @ Unisa that you received with your study
material.
Assignments should be addressed to:
The Registrar
P O Box 392
UNISA
0003
To submit an assignment via myUnisa:
•
•
•
•
•
•
Go to myUnisa.
Log in with your student number and password.
Select the module.
Click on "Assignments" in the menu on the left–hand side of the screen.
Click on the assignment number you wish to submit.
Follow the instructions.
9
9.6
The assignments
The assignment questions for Semester 1 are contained in Addendum A and those for Semester 2
in Addendum B. Make sure that you do the correct assignments.
Solutions will be available on myUnisa under Additional Resources before the examination date.
9.7
Other assessment methods
There are no other assessment methods for this module.
9.8
Examination
If you are registered for the first semester, you will write the examination in May/ June 2020 and the
supplementary examination will be written in October/ November 2020. If you are registered for the
second semester you will write the examination in October/ November 2020 and the supplementary
examination will be written in May/ June 2021.
During the relevant semester, the Examination Section will provide you with information regarding
the examination in general, examination venues, examination dates and examination times.
The exam is a two hour exam. The use of a calculator is NOT permissible. The examination
questions will be similar to the questions asked in the study guide and in the assignments.
This is not the type of module that you can master by “cramming” just before the exam.
You will need to work consistently throughout the semester, since you need to thoroughly
understand each unit before studying the next one.
DOING THE ASSIGNMENTS IS THE MOST IMPORTANT PART OF YOUR STUDY PROGRAMME.
Preparing for the exam without having done the assignments would be like training for the Comrades Marathon without ever actually jogging. Just as one cannot get fit by watching other people
exercise, one cannot master mathematics by only studying worked examples.
Consult the brochure Study @ Unisa for general exam guidelines as well as advice on exam preparation.
10
FREQUENTLY ASKED QUESTIONS
The Study @ Unisa brochure contains an A-Z guide of the most relevant study information.
11
SOURCES CONSULTED
The prescribed book.
12
IN CLOSSING
Remember that there are no "short cuts" to studying and understanding mathematics. You need to
be dedicated, work consistently and practise, practise and practise some more! We hope that you
will enjoy studying this module and we wish you success in your studies.
Your MAT1511 lecturer
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MAT1511/101
13
ADDENDUM A: SEMESTER 1 STUDENTS
ASSIGNMENT 01
3.2 – 3.5, 8.1, 8.3, 8.4 and 9.8
FIXED CLOSING DATE: 14 FEBRUARY 2020
UNIQUE NUMBER: 560103
NO EXTENSION CAN BE GRANTED
1. Let P (x) = 2x 4 + 15x 3 + 31x 2 + 20x + 4
1
(a) Determine whether x +
is a factor of P (x) .
2
(2)
(b) Find all the possible rational zeros of P (x) by using the Rational Zeros Theorem.
(2)
(c) Solve P (x) = 0.
(5)
2. Use Descartes’ Rule of Signs to determine the possible number of positive, negative and
imaginary zeros of P (x) .
P (x) = x 6 − 4x 5 − 2x 4 + 10x 3 − 11x 2 + 14x − 8.
(4)
3. Let P (x) = x 4 − 2x 3 + x 2 − 9x + 2.
(a) Use the Upper and Lower Bounds Theorem to show that the real zeros of P (x) lie
between −2 and 3.
(4)
(b) Use the Rational Zeros Theorem and synthetic division or the Factor Theorem to show
that the equation P (x) = 0 has no rational roots.
(5)
4. Decompose
x 5 − 2x 4 + x 3 + x + 5
x 3 − 2x 2 + x − 2
into partial fractions (show all the steps).
5. Write
in the form a + bi, where a, b ∈ R.
(5)
(1 + 2i) (3 + i)
−2 + i
(4)
6. An airplane heads due West at 650 km/h. It experiences a 15 km/h cross wind flowing in the
direction S30◦ W. Find the speed and direction of the airplane. (Leave your answer correct
to 2 decimal places.)
(5)
11
√
7. Let z = −1 − 3i.
(a) Write z in trigonometric (polar form).
(4)
(b) Use De Moivre’s Theorem to evaluate z 6 . (Leave your answer in trigonometric (polar)
form.)
(2)
8. (a) Plot the following points in the same polar coordinates system
π π π
3,
, 3, −
, −3, −
.
4
4
4
13π
(b) Convert into rectangular coordinates: −4,
.
6
(3)
(2)
9. Convert the following rectangular coordinates into polar coordinates (r , θ ) so that r < 0 and
0 ≤ θ ≤ 2π :
√
3 3, −3 .
(3)
[TOTAL: 50]
12
MAT1511/101
ASSIGNMENT 02
9.3 – 9.7, 9.9, and 11.6
FIXED CLOSING DATE: 13 MARCH 2020
UNIQUE NUMBER: 632606
NO EXTENSION CAN BE GRANTED
1. Use the matrix method (together with elementary row transformations) to solve the following:
2x + 3y + z = 1
x + 2y + 2z = 2
2x + 3y + z = 1.
[4]
2. Consider the following system of equations
−x −
2z = 2
2x − y
= 2
3y + 4z = 1.
(a) Solve the system by using the inverse of the (coefficient) matrix.
(4)
(b) Use Cramer’s rule to solve the system.
(4)
[8]
3. (a) Consider the following system of constraints, associated with a linear programming problem:
3x + y
2x + 2y
x + 3y
x
y
Maximize z = 5x + 3y.
≤
≤
≤
≥
≥
24
20
24
0
0.
(5)
(b) An electronics firm manufactures two types of personal computers, a desktop model and
a portable model. The production of a desktop computer requires a capital expenditure
of R4 000 and 40 hours of labour. The production of a portable computer requires a
capital expenditure of R2 500 and 30 hours of labour. The firm has R200 000 capital and
2 160 labour hours available for production of desktop and portable computers.
i. What is the maximum number of computers the company is capable of producing?
(5)
ii. If each desktop computer contributes a profit of R3 200 and each portable contributes a profit of R2 200, how much profit will the company make by producing the
maximum number of computers determined in (i), above?
(5)
13
iii. Is the profit in (ii) above, maximum? If not, what is the maximum profit?
(3)
[18]
4. Use mathematical induction to prove the following:
(a) 1 + 8 + 16 + ... + 8 (n − 1) = (2n − 1)2 ; n > 1.
(b)
n2
> 2n; n ≥ 3.
12
1
5. (a) Find the coefficient of x 3 in the expansion of x 2 + 3 .
8
(b) Find the term that contains y 10 in the expansion of x − y 2 , simplify your answer.
(5)
(5)
(5)
(5)
[10]
[50]
14
MAT1511/101
ASSIGNMENT 03
Based on all the work.
FIXED CLOSING DATE: 03 APRIL 2020
UNIQUE NUMBER: 632606
NO EXTENSION CAN BE GRANTED
This assignment is based on the whole year’s work. Please read the study @ Unisa brochure very
carefully before filling in your mark reading sheet. You will receive 5%, for each correct answer.
Although negative marking will not be applied, please do not guess any answers. Work the answers out on separate sheets of paper, and then fill in your mark reading sheet. Keep a copy of
your attempts so that you can compare them with the solutions which will be posted to all students
shortly after the closing date.
1. Express
7 + 3i
in the form a + bi, where a, b ∈ R.
−4i
(i)
3 7
+ i
4 4
(ii)
2 7
− i
3 8
(iii)
3 7
− i
4 4
(iv)
2 7
+ i
3 8
3 7
(v) − + i
4 4
2. Which of the following sets contains only possible rational zeros of P (x) = 3x 3 − 5x 2 − 16x +
12.
1
1
16
(i) ± ; ± ; ± ; ±4
3
5
3
(ii) {±1; ±2; ±6; ±12}
3
3
1
(iii) ±3; ± ; ±1; ± ; ±
2
4
4
1
1
1
1
3
3
(iv) ± ; ± ; ± ; ± ; ± ; ± , ± 3, ± 1
12
4
2
6
4
2
1
1
(v) ± ; ± ; ±2; ±3; ±12
12
3
15
3. If −2 is a zero of P (x) = x 3 − 7x − 6, then all the other zeros of P (x) are
(i) 2, 3 and 6
(ii) 3 and 6
(iii) 3 and −1
(iv) 2, 3 and −1
(v) −3 and 1
4. Use Descartes’ Rule of Signs to find the possible number of positive real zeros of
P (x) = 2x 6 − 3x 5 − 13x 4 + 29x 3 − 27x 2 + 32x − 6
(i) 7 or 1
(ii) 6 or 3 or 2 or 1
(iii) 4 or 2
(iv) 5 or 3 or 1
(v) 5 or 3
5. A polynomial of degree 3, with integer coefficients that has zeros
√
3i and 0 is
(i) x 3 − 2x 2 + 4x
√
(ii) x 3 − x 2 + x 3i
(iii) 3x 3 − 3x 2 + xi
(iv) x 3 + 3x
√
(v) x 3 − x 2 + x 3i
6. A jet is flying in a direction N30o W with a speed of 688 km/h. The horizontal and vertical
components of the velocity of the jet is, respectively
(i) −344 km/h and 595, 8 km/h
(ii) 595, 8 km/h and 344 km/h
(iii) −595, 8 km/h and 344 km/h
(iv) 344 km/u and 595, 8 km/h
(v) −344 km/h and −595, 8 km/h.
16
MAT1511/101
7. Consider the decomposition
x 2 + 3x − 5
(x + 1) (x 2 + 2)
into partial fractions. The correct answer is
(i)
8
7
7
−
−
x + 1 x + 2 (x + 1)2
(ii) −
7
10x − 1
−
3 (x + 1) 3 (x 2 + 2)
1
3
+ 2
x +1 x +2
7
10x − 1
(iv) −
+
3 (x + 1) 3 (x 2 + 2)
(iii)
(v)
10
1
7
+
−
3 (x + 1) x + 2 3 (x + 1)
1+i
into polar (trigonometric) form. The correct form is:
2
π
π
2 cos + i sin
4
4
5π
5π
+ i sin
2 cos
4
4
√
7π
7π
+ i sin
2 cos
4
4
√ 2
π
π
cos + i sin
2
4
4
√ 7π
7π
2
cos
+ i sin
2
4
4
8. Convert
(i)
(ii)
(iii)
(iv)
(v)
9. A solution for
√
z 3 + 2 + 2 3i = 0,
by using the nth root formula for complex numbers, is
4π
4π
2
(i) 2 3 cos
+ i sin
9
9
11π
11π
1
(ii) 4 4 cos
+ i sin
12
12
17π
17π
2
(iii) 2 3 cos
+ i sin
12
12
17
11π
11π
(iv) 2 cos
+ i sin
12
12
7π
7π
1
+ i sin
(v) 4 3 cos
6
6
2
3
√ 10. Convert −3, 3 3 into polar coordinates (r , θ ) so that r ≥ 0 and 0 ≤ θ < 2π.
(i)
(ii)
(iii)
(iv)
(v)
5π
6,
3
2π
6,
3
5π
6,
6
π
6,
6
7π
6,
6
11. Which of the following is equal to AB, for matrices A and B below?


1 −2
−2 1 5
4 
A=
B= 3
3 1 1
0 −1
(i)
−5
1
12 −4
−5
12
(ii)
3 −18
−5
3
(iii)
15 −4
(iv) AB is impossible to compute.
(v) None of the preceding (above).
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MAT1511/101
12. If
(i)
−2
4
6 −8
w x
y z
=
1 0
0 1
, then z = ...
1
2
(ii) 0
(iii) −
1
4
(iv) 1
(v) None of the preceding (above).
13. Evaluate:
1 −2 3
−1
0 1
−2
3 4
(i) 4
(ii) −16
(iii) −18
(iv) 0
(v) −4
14. Which one of the following is the second row of the inverse of matrix A shown below?


1 0
2
A =  0 2 −1 
2 5
2
(i) (−2, −2, 1)
(ii) (−4, −5, 2)
(iii) (0, 0, 1)
(iv) (2, 5, 2)
(v) (0, −2, 1)
19
15. What is the maximum value of z = 3x + 4y for the region given by the system:
x ≥ 0
y ≥ 0
1
x + y ≤ 11
2
3x + 2y ≤ 30.
(i) 48
(ii) 30
(iii) 31
(iv) 11
(v) None of the above
16. Which of the following is one of the three numbers in the solution of this system?
2x + y
= 3
4x
+ 5z = 6
−2y + 5z = −4
(i) 3
(ii) −3
(iii) −2
(iv) 2
(v) None of the preceding.
17. What is the 20th term of the sequence:
1,
(i)
17
15
(ii)
17
495
(iii) 15
20
(iv)
1
15
(v)
17
295
n+2
1
, ... ,
, ... ?
3
2n2 + 3n + 2
MAT1511/101
18. Consider the following statement:
2n > n2 , where n ∈ N.
Which of the following is true?
(i) 2n > n2 for all n ∈ N
(ii) 2n > n2 only for n = 8, 9, ... , 20
(iii) 2n > n2 for all n ≥ 5
(iv) 2n < n2 for all n ∈ N
(v) None of the preceding (above).
19. The middle term in the expansion of
(i)
35
x4
(ii)
70
x4
1 √
+ x
x
8
is:
(iii) 70x 2
(iv) 35x 5
(v) None of the preceding.
20. The term that contains x 5 in the expansion of (2x + y)6 is:
(i) 12x 5 y
(ii) 32x 5 y 4
(iii) 192x 5 y
(iv) 576x 5 y
(v) None of the above.
[TOTAL: 100]
21
14
ADDENDUM B: SEMESTER 2 STUDENTS
SEMESTER 02
ASSIGNMENT 01
3.2 – 3.5, 8.1, 8.3, 8.4 and 9.8
FIXED CLOSING DATE: 28 August 2020
UNIQUE NUMBER: 813197
NO EXTENSION CAN BE GRANTED
NOTE: All references refer to the 5th edition of Steward, Redlin and Watson.
1. Use Descartes’ Rule of Signs to describe all possibilities for the number of positive, negative
and imaginary zeros of
P (x) = x 4 + x 3 + x 2 + x + 12
(Summarize your answer in the form of a table like the example on p. 297).
[3]
2. P (x) = x 4 − 2x 3 − 2x 2 − 2x − 3
(a) Use the Upper and Lower Bounds Theorem to show that all zero of P (x) are bounded
below by −1 and above by 3.
(3)
(b) Find all the possible rational zeros of P (x) by using the Rational Zero Theorem.
(1)
(c) Solve P (x) = 0 (i.e. find all the solutions of P (x) = 0.)
(4)
[8]
3. P (x) = 2x 3 − 5x 2 − x + 8
(a) Use the Factor Theorem to prove that x − 12 is not a factor of P (x) .
(2)
(b) Show that P (x) has no rational zeros by using the Rational Zeros Theorem.
(3)
[5]
4. (a) Find a polynomial of degree 4 that has zeros 3, 0, 1, − 5.
(2)
(b) Write
5 + 3i
2 (2 + i)
in the form a + bi, where a, b ∈ R.
(2)
[4]
22
MAT1511/101
5. Decompose
4x 2 − 14x + 2
4x 2 − 1
into partial fractions (show all the steps).
[4]
6. (a) Find the
(i) magnitude, and
(ii) direction (in degrees)
of the vector h−6, 3i correct to one decimal place.
(2)
(b) A jet is flying through a wind that is blowing at a speed of 88 km/h in the direction N60o E.
The jet has a speed of 1 224 km/h relative to the air, and the pilot heads the jet in the
direction N60o W.
(i) Express the velocity of the wind as a vector in terms of the unit vectors i and j. (1)
(ii) Express the velocity of the jet relative to the air as a vector in terms of the unit
(1)
vectors i and j.
(iii) Find the true velocity of the jet.
(1)
(iv) Find the true speed of the jet, correct to one decimal place.
(1)
(v) Find the true direction of the jet, correct to one decimal place.
(1)
√
7. Let z = −1 − 3i.
(a) Write z in polar (trigonometric) form with argument θ between 0 and 2π.
(4)
(b) Use your answer to (a) above to find z 2 . (Leave your answer in polar (trigonometric)
form.)
(2)
√
(c) Use your answer to (a) above to find z. (Leave your answer in polar (trigonometric)
form.)
(3)
[9]
8. Let
π
π
7π
7π
z1 = 3 cos + i sin
and z2 = 2 cos
+ i sin
.
6
6
4
4
Find the product z1 z2 in polar (trigonometric) form.
[1]
9. (a) Plot the following points on the same polar coordinate system (indicate or label the points
clearly).
π π π π
2,
,
−2,
,
2, −
,
−2, −
(2)
3
3
3
3
π
(b) Give two other polar coordinate representations of the point 3, −
, one with r > 0
3
and one with r < 0, where −2π ≤ θ ≤ 2π.
(2)
23
[4]
5π
10 (a) Find the rectangular coordinates for the point with polar coordinates 3,
.
(2)
6
√
√ (b) Convert the rectangular coordinates −3 2, −3 2 of a point to its polar coordinates
(r , θ ) with r < 0 and 0 ≤ θ < 2π.
(3)
[5]
TOTAL: [50]
24
MAT1511/101
ASSIGNMENT 02
9.3 – 9.7, 9.9, and 11.6
FIXED CLOSING DATE: 18 SEPTEMBER 2020
UNIQUE NUMBER: 504062
NO EXTENSION CAN BE GRANTED
1. Use matrices and elementary row operations to solve the following system:
5x − 3y + 2z = 13
2x − y − 3z = 1
4x − 2y + 4z = 12
[4]
2. Solve the linear system by using the inverse of the coefficient matrix:
x
+ 2z = −1
2x − y
= 2
3y + 4z = 1
[4]
3. Use Cramer’s Rule to solve the following system:
5x − 3y + z = 6
2y − 3z = 11
7x + 10y
= −13.
[4]
4. (a) Consider the following system of constraints, associated with a linear programming problem:
2x + 3y
3x + 2y
x +y
x, y
≥
≥
≤
≥
30
30
15
0
Maximize z = x + 3y subject to the constraints above.
(5)
(b) The officers of a high school senior class are planning to rent busses and vans for a
class trip. Each bus can transport 40 students, requires 3 chaperones and costs R1200
to rent. Each van can transport 8 students, requires 1 chaperone and costs R100 to rent.
The officers want to be able to accommodate at least 400 students with no more than
36 chaperones. How many vehicles of each type should they rent in order to minimize
the transport costs? What are the minimal transporation costs?
(8)
[13]
25
5. (a) Use mathematical induction to prove that the formula
13 + 33 + 53 + · · · + (2n − 1)3 = n2 2n2 − 1 is true for all natural n.
(b) Show that n3 − n + 3 is divisible by 3 for all natural numbers n.
2
(c) Use mathematical induction to prove that (n + 1)
< 2n2
(5)
(5)
for all natural numbers n ≥ 3.(4)
[14]
6. (a) Use the Binomial Theorem to expand the expression:
(3x + y)5 , and simplify.
(5)
(b) Find the middle term in the expansion of
1 √
+ x
x
4
, and simplify your answer.
(c) Determine the coefficient of x 11 in the expansion of x 2 + x1
10
(3)
, simplify your answer. (3)
[11
TOTAL: [50]
26
MAT1511/101
ASSIGNMENT 03
Based on all the work.
FIXED CLOSING DATE: 07 OCTOBER 2020
UNIQUE NUMBER: 578161
NO EXTENSION CAN BE GRANTED
This assignment is based on the whole year’s work. Please read the study @ Unisa brochure very
carefully before filling in your mark reading sheet. You will receive 5%, for each correct answer.
Although negative marking will not be applied, please do not guess any answers. Work the answers out on separate sheets of paper, and then fill in your mark reading sheet. Keep a copy of
your attempts so that you can compare them with the solutions which will be posted to all students
shortly after the closing date.
1. Express
2 − 3i
in the form a + bi, where a, b ∈ R.
1 + 2i
(i)
13
5
(ii)
8 1
+ i
5 5
(iii) 3
7
(iv) −1 − i
5
3
(v) 1 − i
2
2. Which of the following sets contains only possible rational zeros of P (x) = 2x 4 − x 3 + 6x 2 −
5x + 8.
1
1
(i) ± ; ± ; ±1; ±2
4
2
1
1
1
(ii) ± ; ± ; ± ; ±1; ±2
8
4
2
1
(iii) ± ; ±1; ±2; ±4; ±8
8
1
(iv) ± ; ±2; ±8
2
(v) ±1; ±2; ±4; ±8; ±16
27
3. Use Descartes’s Rule of Signs to find the possible number of imaginary (non real)
P (x) = 2x 5 + x 4 + x 3 − 4x 2 − x − 3
(i) 1, 0
(ii) 5, 3
(iii) 0, 2, 4
(iv) 6, 2
(v) None of the above.
4. The decomposition of
x 4 − x 3 − 2x 2 + 4x + 1
x (x − 1)2
into partial fractions gives
(i) x + 1 +
1
2
3
−
+
x x − 1 (x − 1)2
(ii) x 2 + x +
(iii) x + 1 −
2
3
1
+
−
x x − 1 (x − 1)2
1
2
3
+
= 2
x x −1 x −1
(iv) x 3 − x 2 − 2x +
1
2
−
x x −1
(v) None of the above.
5. A sailboat under auxiliary power is proceeding on a bearing 25◦ north of west at 6,25 km/h
in still water. Then a tail wind blowing 15 km/h in the direction 35◦ south of west, alters the
course of the sailboat. The resultant speed and direction of the sailboat is: (Give answers
correct to two decimal places)
(i) 21, 18 km/h, 32, 06o North of West
(ii) 18, 92 km/h, N 18, 38 W
(iii) 21, 18 km/h, 32, 06o South of West
(iv) 18, 92 km/h, 18, 38o South of West
(v) None of the above.
28
MAT1511/101
√
6. Convert 4 3 − 4i into trigonometric form.
√
5π
5π
(i) 4 2 cos
+ i sin
3
3
11π
11π
+ i sin
(ii) 8 cos
6
6
√
−π
−π
(iii) 4 2 cos
+ i sin
6
6
π
π
(iv) 8 cos + i sin
6
6
5π
5π
(v) 8 cos
+ i sin
6
6
q
11π
7. Which of the following is NOT 8 cos 11π
+
i
sin
6
6 ?
(i)
(ii)
(iii)
(iv)
(v)
11π
11π
+ sin
2 cos
12
12
π
π
3
2 2 − cos + i sin
12
12
23π
23π
1
+ i sin
8 2 cos
12
12
π
π
1
8 2 cos − i sin
12
12
23π
23π
3
2 2 cos
− i sin
12
12
3
2
1
8. If z 3 = 2 cos π9 + i sin π9 , then z =
(i)
1
2
23
+
√
3
2
23
π
π
3
(ii) 2 cos + i sin
27
27
√
(iii) 4 + 4 3i
(iv) 8
π π (v) 8 cos3
+ i sin3
9
9
29
9. All the complex roots of x 3 + 8 = 0 are:
√
√
(i) −2; 1 − 3i; 1 + 3i
(ii) −2
(iii) −2; +2; −2
√
√
(iv) −2; −1 − 3i; −1 + 3i
√
√
(v) −2; 1 + 6i; 1 − 6i
√
10. Convert the rectangular coordinates − 3, −1 into polar coordinates (r , θ ) so that r < 0 and
0 ≤ θ < 2π.
√ π 2,
(i)
6
π
(ii) −2,
3
4π
(iii) 2,
3
7π
(iv) 2,
6
4π
(v) −2,
3
11. Suppose a linear system of three equations in three variables is being solved by making use
of elementary row operations and we arrive at the following matrix:


1 2 1 0
 0 1 1 2 
0 0 0 0
What does it mean?
(i) That (t, t, 0) , t ∈ R is a solution.
(ii) That (t, t, r ) , t, r ∈ R is a solution.
(iii) That (t − 2, 1 − t, t, ) t ∈ R is a solution.
(iv) The original system of equation has no solution.
(v) None of the above.
30
MAT1511/101
12. The solution of the following linear system
x
+ 2z = −2
2x − y
= 2
3y + 4z = 1
is
(i) (−2, 1, 1)
11 7 19
, ,−
(ii)
4 2
8
11 7 19
(iii) − , − ,
4
2 8
(iv) (−2, −1, −1)
(v) None of the above.
13. The 1555th term of the sequence
(−1)n+2 1 − 12i 3n+1
bn =
i n+2
where i 2 = −1, is
(i) 13
(ii) −13
(iii) 13 + i
(iv) −13i
(v) 13i
14. The general term of the expansion (x + y)8 is:
8
(i)
x r y 8−r
r
n
(ii)
x r y r −8
8−r
8
(iii)
x r y r −8
8−r
8
(iv)
x n y 8−r
n−r
(v) None of the above.
31
15. A manufacturer produces two models of a certain product: model A and model B. There is
a R3 profit on model A and an R8 profit on model B. Three machines M1 , M2 and M3 are used
jointly to manufacture these models. The number of hours that each machine operates to
produce 1 unit of each model is given in the table:
Machine M1
Machine M2
Machine M3
Model A Model B
1 12
1
3
1 12
4
1 13
1 13
No machine is in operation more than 12 hours per day.
Now let x be the number of model A made per day and y be the be the number of model B
made per day. Then x and y satisfies the following constrains:
3
3
3
4
4
(i) x ≥ 0, y ≥ 0, x + y ≤ 12, x + y ≤ 12, x + y ≥ 0
2
4
2
3
3
(ii) x ≥ 0, y ≤ 0,
3
3
3
4
4
x + y ≥ 12, x + y ≤ 12, x + y ≤ 0
2
4
2
3
3
(iii) x ≥ 0, y ≥ 0,
3
3
3
4
4
x + y ≤ 12, x + y ≤ 12, x + y ≤ 12
2
4
2
3
3
(iv) x ≥ 0, y ≥ 0,
3
3
4
4
3
x − y ≤ 12, x + y ≤ 12, x + y ≤ 12
2
4
2
3
3
(v) None of the above.
16. For question 15 above if we express the daily profit P in terms of x and y, then
(i) P = 3x + 8y
(ii) P = 8x + 5y
(iii) P = x + y
(iv) P = 8x + 8y
(v) None of the above.
17. For question 15 and 16 above, what is the maximum profit in rands?
(i) 40
(ii) 88
(iii) 54
(iv) 66
(v) None of the above.
32
MAT1511/101
18. Find the first four terms in the expansion (xy + 2i)12 :
(i) x 12 y 12 + 12x 11 y 11 + 66ix 10 y 10 + 1760ix 9 y 9
(ii) x 12 y 12 + 24ix 11 y 11 + 264x 10 y 10 + 1760ix 9 y 9
(iii) x 12 y 12 − 24x 11 y 11 − 264x 10 y 10 − 1760x 9 y 9
(iv) x 12 y 12 + 24ix 11 y 1 − 264x 10 y 10 − 1760ix 9 y 9
(v) None of the above.
19. The statement (n + 1)2 < 3n2
(i) is true for all integers n ≥ 1.
(ii) is true for all integers n ≥ 0.
(iii) is true for all integers n ≥ 3.
(iv) is waar vir alle heelgetalle 0 < n ≤ 6.
(v) None of the above.
20. Find the number R such that the complex numbers i, R, −i is a geometric sequence..
(i) −1 + i
(ii) i
(iii) −i
(iv) 1
(v) None of the above.
[TOTAL: 100]
33