UNIVERSITY EXAMINATIONS
OCTOBER/NOVEMBER 2025
MAT1581
Engineering Mathematics I
Welcome to the MAT1581 exam.
Date: 13 October 2025
Time: 02:00 pm
Duration: 2 Hours
Examiner name: Mr P.T. Malima
Internal moderator name: Dr A. Dlamini
Duration: 2 Hours 80 Marks
This paper consists of 10 pages (including the cover page)
Instructions:
• The information sheet is attached from page 5 to 9. One table of integrals (page 10).
• You are allowed to use the calculator.
• NO TYPED SOLUTIONS WILL BE ACCEPTED AND THE SCRIPT WILL BE CANCELLED
• You need to declare your honesty regarding writing this paper.
• This exam is IRIS invigilated. You must have your camera on for entire duration of the
exam. Failure to have your camera on for the duration of the exam, will lead to disciplinary
action taken against you.
Additional student instructions
1. Students must upload their answer scripts in a single PDF file (answer scripts must not be
password protected or uploaded as “read only” files)
2. Incorrect file format and uncollated answer scripts will not be considered.
3. NO emailed scripts will be accepted.
4. Students are advised to preview submissions (answer scripts) to ensure legibility and that
the correct answer script file has been uploaded.
5. Incorrect answer scripts and/or submissions made on unofficial examinations platforms
(including the invigilator cell phone application) will not be marked and no opportunity will
be granted for resubmission. Only the last answer file uploaded within the stipulated
submission duration period will be marked.
6. Mark awarded for incomplete submission will be the student’s final mark. No opportunity for
resubmission will be granted.
7. Mark awarded for illegible scanned submission will be the student’s final mark. No
opportunity for resubmission will be granted.
8. Submissions will only be accepted from registered student accounts.
9. Students who have not utilised the proctoring tool will be deemed to have transgressed
Unisa’s examination rules and will have their marks withheld. If a student is found to have
been outside the proctoring tool for a total of 10 minutes during their examination session,
they will be considered to have violated Unisa’s examination rules and their marks will be
withheld. For examinations which use the IRIS invigilator system, IRIS must be recording
throughout the duration of the examination until the submission of the examinations scripts.
10. Students have 48 hours from the date of their examination to upload their invigilator results
from IRIS. Failure to do so will result in students deemed not to have utilized the proctoring
tools.
11. Students suspected of dishonest conduct during the examinations will be
subjected to disciplinary processes. Students may not communicate with any other
person or request assistance from any other person during their examinations.
, Plagiarism is a violation of academic integrity and students who plagiarise, copy
from published work or Artificial Intelligence Software (eg ChatGPT) or online
sources (eg course material), will be in violation of the Policy on Academic Integrity
and the Student Disciplinary Code and may be referred to a disciplinary hearing.
Unisa has a zero tolerance for plagiarism and/or any other forms of academic
dishonesty.
12. Listening to audio (music) and making use of audio-to-text software is strictly
prohibited during your examination session unless such usage of the software is
related to a student’s assistive device which has been so declared. Failure to do so
will be a transgression of Unisa’s examination rules and the student's marks will be
withheld.
13. Students are provided 30 minutes to submit their answer scripts after the official
examination time. Students who experience technical challenges should report the
challenges to the SCSC on 080 000 1870 or their College exam support centres
(refer to the Get help during the examinations by contacting the Student
Communication Service Centre [unisa.ac.za]) within 30 minutes. Queries received
after 30 minutes of the official examination duration time will not be responded to.
Submissions made after the official examination time will be rejected according to
the examination regulations and will not be marked. Only communication received
from your myLife account will be considered.
14. Non-adherence to the processes for uploading examination responses will not
qualify the student for any special concessions or future assessments.
15. Queries that are beyond Unisa’s control include the following: a. Personal network
or service provider issues.
b. Load shedding/limited space on personal computer
c. Crashed computer.
d. Non-functioning cameras or web cameras.
e. Using work computers that block access to the myExams site (employer firewall
challenges).
f. Unlicensed software (eg license expires during exams).
Postgraduate students experiencing the above challenges are advised to apply for
an aegrotat and submit supporting evidence within ten days of the examination
session. Students will not be able to apply for an aegrotat for a third examination
opportunity.
Postgraduate/undergraduate students experiencing the above challenges in their
second examination opportunity will have to reregister for the affected module.
16. Students suspected of dishonest conduct during the examinations will be subjected to
disciplinary processes. UNISA has a zero tolerance for plagiarism and/or any other forms
of academic dishonesty.
17. Students experiencing network or load shedding challenges are advised to apply together
with supporting evidence for an Aegrotat within 3 days of the examination session.
, QUESTION 1
(a) Find only x using Cramer’s rule given the following simultaneous equation. (8)
x +2y +2z =4
3x −y +4z = 25
3x +2y −z = −4
√
(b) Expand 1 − x2 to three terms using the binomial theorem. Simplify the terms. (5)
3x2 + 4x + 4
(c) Resolve the proper fraction completely into partial fractions. (7)
x3 + 4x
[20]
QUESTION 2
(a) Express the complex number z = 3|45o in exponential form. (2)
Z1
(b) If Z1 = 5(cos π2 + jsin π2 ) and Z2 = 4(cos π3 + jsin π3 ). Find (3)
Z2
Leave the answer in polar form.
(c) Find the square roots of the complex number x if x = −3 − 4j. (8)
(d) Write down the radius and the co-ordinates of the center of the circle (3)
(x − 4)2 + (y + 3)2 = 16
(e) Write down the equation of the ellipse with center (0; 0), one focal point at (0; 4) and one vertex at (0; 5). (4)
[20]
QUESTION 3
cos x − 1
(a) Determine lim (3)
x→0 2 sin x
(b) Differentiate the following with respect to x:
(i) y = ln(2x2 + 1). (2)
ex + 1
(ii) y = (4)
ex − 1
(iii) y = x4 tan 3x (2)
q
(c) The period T of a simple pendulum of length l is given by T = 2π gl , (3)
dT
where g is a constant called the acceleration due to gravity. Find .
dl
(d) Find the turning point of the function y = x4 − 8x2 + 16 and determine their nature. (8)
[22]
[TURN OVER]
, QUESTION 4
(a) Determine the following integrals:
4x3 − x
Z
(i) dx (3)
4x4
Z
2
(ii) xe(4−x ) dx (4)
Z
(iii) (cos 2x + sin x) dx (2)
(b) Determine the area enclosed by y = x2 + 2x − 3 and the x-axis between x = 0 and x = 2. (9)
[18]
TOTAL MARKS: [80]
©
UNISA 2025
OCTOBER/NOVEMBER 2025
MAT1581
Engineering Mathematics I
Welcome to the MAT1581 exam.
Date: 13 October 2025
Time: 02:00 pm
Duration: 2 Hours
Examiner name: Mr P.T. Malima
Internal moderator name: Dr A. Dlamini
Duration: 2 Hours 80 Marks
This paper consists of 10 pages (including the cover page)
Instructions:
• The information sheet is attached from page 5 to 9. One table of integrals (page 10).
• You are allowed to use the calculator.
• NO TYPED SOLUTIONS WILL BE ACCEPTED AND THE SCRIPT WILL BE CANCELLED
• You need to declare your honesty regarding writing this paper.
• This exam is IRIS invigilated. You must have your camera on for entire duration of the
exam. Failure to have your camera on for the duration of the exam, will lead to disciplinary
action taken against you.
Additional student instructions
1. Students must upload their answer scripts in a single PDF file (answer scripts must not be
password protected or uploaded as “read only” files)
2. Incorrect file format and uncollated answer scripts will not be considered.
3. NO emailed scripts will be accepted.
4. Students are advised to preview submissions (answer scripts) to ensure legibility and that
the correct answer script file has been uploaded.
5. Incorrect answer scripts and/or submissions made on unofficial examinations platforms
(including the invigilator cell phone application) will not be marked and no opportunity will
be granted for resubmission. Only the last answer file uploaded within the stipulated
submission duration period will be marked.
6. Mark awarded for incomplete submission will be the student’s final mark. No opportunity for
resubmission will be granted.
7. Mark awarded for illegible scanned submission will be the student’s final mark. No
opportunity for resubmission will be granted.
8. Submissions will only be accepted from registered student accounts.
9. Students who have not utilised the proctoring tool will be deemed to have transgressed
Unisa’s examination rules and will have their marks withheld. If a student is found to have
been outside the proctoring tool for a total of 10 minutes during their examination session,
they will be considered to have violated Unisa’s examination rules and their marks will be
withheld. For examinations which use the IRIS invigilator system, IRIS must be recording
throughout the duration of the examination until the submission of the examinations scripts.
10. Students have 48 hours from the date of their examination to upload their invigilator results
from IRIS. Failure to do so will result in students deemed not to have utilized the proctoring
tools.
11. Students suspected of dishonest conduct during the examinations will be
subjected to disciplinary processes. Students may not communicate with any other
person or request assistance from any other person during their examinations.
, Plagiarism is a violation of academic integrity and students who plagiarise, copy
from published work or Artificial Intelligence Software (eg ChatGPT) or online
sources (eg course material), will be in violation of the Policy on Academic Integrity
and the Student Disciplinary Code and may be referred to a disciplinary hearing.
Unisa has a zero tolerance for plagiarism and/or any other forms of academic
dishonesty.
12. Listening to audio (music) and making use of audio-to-text software is strictly
prohibited during your examination session unless such usage of the software is
related to a student’s assistive device which has been so declared. Failure to do so
will be a transgression of Unisa’s examination rules and the student's marks will be
withheld.
13. Students are provided 30 minutes to submit their answer scripts after the official
examination time. Students who experience technical challenges should report the
challenges to the SCSC on 080 000 1870 or their College exam support centres
(refer to the Get help during the examinations by contacting the Student
Communication Service Centre [unisa.ac.za]) within 30 minutes. Queries received
after 30 minutes of the official examination duration time will not be responded to.
Submissions made after the official examination time will be rejected according to
the examination regulations and will not be marked. Only communication received
from your myLife account will be considered.
14. Non-adherence to the processes for uploading examination responses will not
qualify the student for any special concessions or future assessments.
15. Queries that are beyond Unisa’s control include the following: a. Personal network
or service provider issues.
b. Load shedding/limited space on personal computer
c. Crashed computer.
d. Non-functioning cameras or web cameras.
e. Using work computers that block access to the myExams site (employer firewall
challenges).
f. Unlicensed software (eg license expires during exams).
Postgraduate students experiencing the above challenges are advised to apply for
an aegrotat and submit supporting evidence within ten days of the examination
session. Students will not be able to apply for an aegrotat for a third examination
opportunity.
Postgraduate/undergraduate students experiencing the above challenges in their
second examination opportunity will have to reregister for the affected module.
16. Students suspected of dishonest conduct during the examinations will be subjected to
disciplinary processes. UNISA has a zero tolerance for plagiarism and/or any other forms
of academic dishonesty.
17. Students experiencing network or load shedding challenges are advised to apply together
with supporting evidence for an Aegrotat within 3 days of the examination session.
, QUESTION 1
(a) Find only x using Cramer’s rule given the following simultaneous equation. (8)
x +2y +2z =4
3x −y +4z = 25
3x +2y −z = −4
√
(b) Expand 1 − x2 to three terms using the binomial theorem. Simplify the terms. (5)
3x2 + 4x + 4
(c) Resolve the proper fraction completely into partial fractions. (7)
x3 + 4x
[20]
QUESTION 2
(a) Express the complex number z = 3|45o in exponential form. (2)
Z1
(b) If Z1 = 5(cos π2 + jsin π2 ) and Z2 = 4(cos π3 + jsin π3 ). Find (3)
Z2
Leave the answer in polar form.
(c) Find the square roots of the complex number x if x = −3 − 4j. (8)
(d) Write down the radius and the co-ordinates of the center of the circle (3)
(x − 4)2 + (y + 3)2 = 16
(e) Write down the equation of the ellipse with center (0; 0), one focal point at (0; 4) and one vertex at (0; 5). (4)
[20]
QUESTION 3
cos x − 1
(a) Determine lim (3)
x→0 2 sin x
(b) Differentiate the following with respect to x:
(i) y = ln(2x2 + 1). (2)
ex + 1
(ii) y = (4)
ex − 1
(iii) y = x4 tan 3x (2)
q
(c) The period T of a simple pendulum of length l is given by T = 2π gl , (3)
dT
where g is a constant called the acceleration due to gravity. Find .
dl
(d) Find the turning point of the function y = x4 − 8x2 + 16 and determine their nature. (8)
[22]
[TURN OVER]
, QUESTION 4
(a) Determine the following integrals:
4x3 − x
Z
(i) dx (3)
4x4
Z
2
(ii) xe(4−x ) dx (4)
Z
(iii) (cos 2x + sin x) dx (2)
(b) Determine the area enclosed by y = x2 + 2x − 3 and the x-axis between x = 0 and x = 2. (9)
[18]
TOTAL MARKS: [80]
©
UNISA 2025
