lOMoARcPSD|54878315
UNIVERSITY EXAMINATIONS
October/November 2025
STA3710
Mathematical Techniques in Statistics
100 Marks
Duration: 2 Hours
EXAMINERS:
FIRST: Ms MR Mashabela
SECOND: Prof P Ndlovu
EXTERNAL: Prof S Mukeru, UNISA
____________________________________________________________________
This paper consists of 6 pages.
INSTRUCTIONS
1. This is an OPEN BOOK examination. You may use your textbook, your course
material or any other printed material to answer the questions.
2. You must UPLOAD your answers in one PDF file.
3. Answer ALL the questions.
4. Show all relevant computations and steps.
5. Marks denote percentages
6. This examination is IRIS invigilated. If IRIS is not used while you are writing, you will
be subjected to disciplinary processes.
[TURN OVER]
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STA3701
OCT/NOV 2025
ADDITIONAL STUDENT INSTRUCTIONS
1. This is a fully online examination.
2. Students are provided with one submission opportunity for their MCQ/Quiz examinations. No
additional time will be allocated for resubmissions.
3. Students are advised that the examination duration time commences as stipulated on the
official examination timetable. Examination duration is limited to the session duration as
reflected on the official examination timetable.
4. No additional reading time is provided during MCQ/Quiz examinations.
5. DO NOT open your examination in multiple browser windows or tabs at the same time. If you
do so, the system will automatically submit responses on your behalf without your knowledge.
Marks awarded for automatic submissions will be final marks. No additional opportunity for
resubmissions will be given.
6. DO NOT click your browser's back button while taking your examination. Students are at risk
of losing previous responses should they utilise browser navigation buttons. Students are
advised to always use Moodle navigation buttons to move to the next and previous (if
allowed) question, and to "Save" and "Submit".
7. The system will automatically save your responses as you click on the next question.
8. Students who have not utilised invigilation or proctoring tools will be deemed to have
transgressed Unisa’s examination rules and will have their marks withheld. For examinations
that use the IRIS invigilation system, IRIS must be recording throughout the duration of the
examination until the student makes the final submission of the answers.
9. Students must complete the online declaration of their work when submitting their
examination. 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. The use of Telegram,
WhatsApp, or any other instant messaging services with any other person (except for when
asking for technical assistance with via official channels of the SCSC or the Invigilator
WhatsApp line) are strictly prohibited. Plagiarism is a violation of academic integrity and
students who plagiarise, copy from published work or unethical usage use of 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.
10. Listening to audio (music) or making use of audio-to-text software is strictly prohibited during
your examination session unless such usage of the software is related to student’s assisted
devices which have been so declared. Failure to do so will be a transgression of Unisa’s
examination rules and the student's mark will be withheld.
[TURN OVER]
2
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STA3701
OCT/NOV 2025
11. Non-adherence to the processes for submitting MCQ/Quiz responses will not qualify the
student for any special concessions or future assessments.
12. 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 (work firewall challenges)
f. Unlicensed software (eg the license expires during exams)
13. 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.
14. Postgraduate/undergraduate students who experience the above challenges in their second
examination opportunity will have to reregister for the affected module.
Students experiencing technical challenges should immediately contact the SCSC telephonically on
080 000 1870 and choose option 4 for exams or via e-mail at , or refer to
the Get help during the examinations by contacting the Student Communication Service Centre
(unisa.ac.za) resource for the list of additional contact numbers. Only communication received from
your myLife account will be considered.
[TURN OVER]
3
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, lOMoARcPSD|54878315
STA3701
OCT/NOV 2025
QUESTION 1 [10 Marks]
Determine (with evidence) whether the following statements are True or False.
𝑎11 𝑎12
1.1 The characteristic function of 𝐀 = (𝑎 ) is 𝜆2 − (𝑎11 + 𝑎22 )𝜆 + (𝑎11 𝑎22 − 𝑎12 𝑎22 ). (3)
21 𝑎22
4 2
1.2 The eigenvalues of the matrix 𝐀 = ( ) are 7 and 1. (3)
3 5
𝟏
𝟗
2 𝟏
1 𝟏𝟖
1.3 The Moore-Penrose inverse of the vector 𝒂 = ( ) is 𝒂+ = 𝟏 . (3)
3
𝟔
2 𝟏
(𝟗)
1.4 𝐈𝑚 ⊗ 𝐈𝑛 can also be written as 𝐈𝑛𝑚 . (1)
QUESTION 2 [30 Marks]
𝑎11 𝑎21
2.1 Consider the matrix 𝐀 = (𝑎 𝑎22 ). Obtain the expressions for the two eigenvalues of 𝐀 in
21
terms of the elements of 𝐀.
(4)
2.2 Suppose that
cos θ − sin θ
𝐐 = ( ).
sin θ cos θ
2.2.1 Show that 𝐐 is an orthogonal matrix. (4)
2.2.2 Determine the eigenvalues of 𝐐. (4)
2.3 Consider the matrix
0 0 2 0
1 0 1 0
𝐀 = ( ).
0 1 −2 0
0 0 0 1
2.3.1 Find the eigenvalues of 𝐀. (6)
2.3.2 Determine the normalized eigenvector corresponding to each eigenvalue in part (2.3.1).
(10)
2.3.3 Compute 𝑡𝑟(𝐀4 ). (2)
[TURN OVER]
4
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UNIVERSITY EXAMINATIONS
October/November 2025
STA3710
Mathematical Techniques in Statistics
100 Marks
Duration: 2 Hours
EXAMINERS:
FIRST: Ms MR Mashabela
SECOND: Prof P Ndlovu
EXTERNAL: Prof S Mukeru, UNISA
____________________________________________________________________
This paper consists of 6 pages.
INSTRUCTIONS
1. This is an OPEN BOOK examination. You may use your textbook, your course
material or any other printed material to answer the questions.
2. You must UPLOAD your answers in one PDF file.
3. Answer ALL the questions.
4. Show all relevant computations and steps.
5. Marks denote percentages
6. This examination is IRIS invigilated. If IRIS is not used while you are writing, you will
be subjected to disciplinary processes.
[TURN OVER]
Downloaded by Stephen ()
, lOMoARcPSD|54878315
STA3701
OCT/NOV 2025
ADDITIONAL STUDENT INSTRUCTIONS
1. This is a fully online examination.
2. Students are provided with one submission opportunity for their MCQ/Quiz examinations. No
additional time will be allocated for resubmissions.
3. Students are advised that the examination duration time commences as stipulated on the
official examination timetable. Examination duration is limited to the session duration as
reflected on the official examination timetable.
4. No additional reading time is provided during MCQ/Quiz examinations.
5. DO NOT open your examination in multiple browser windows or tabs at the same time. If you
do so, the system will automatically submit responses on your behalf without your knowledge.
Marks awarded for automatic submissions will be final marks. No additional opportunity for
resubmissions will be given.
6. DO NOT click your browser's back button while taking your examination. Students are at risk
of losing previous responses should they utilise browser navigation buttons. Students are
advised to always use Moodle navigation buttons to move to the next and previous (if
allowed) question, and to "Save" and "Submit".
7. The system will automatically save your responses as you click on the next question.
8. Students who have not utilised invigilation or proctoring tools will be deemed to have
transgressed Unisa’s examination rules and will have their marks withheld. For examinations
that use the IRIS invigilation system, IRIS must be recording throughout the duration of the
examination until the student makes the final submission of the answers.
9. Students must complete the online declaration of their work when submitting their
examination. 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. The use of Telegram,
WhatsApp, or any other instant messaging services with any other person (except for when
asking for technical assistance with via official channels of the SCSC or the Invigilator
WhatsApp line) are strictly prohibited. Plagiarism is a violation of academic integrity and
students who plagiarise, copy from published work or unethical usage use of 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.
10. Listening to audio (music) or making use of audio-to-text software is strictly prohibited during
your examination session unless such usage of the software is related to student’s assisted
devices which have been so declared. Failure to do so will be a transgression of Unisa’s
examination rules and the student's mark will be withheld.
[TURN OVER]
2
Downloaded by Stephen ()
, lOMoARcPSD|54878315
STA3701
OCT/NOV 2025
11. Non-adherence to the processes for submitting MCQ/Quiz responses will not qualify the
student for any special concessions or future assessments.
12. 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 (work firewall challenges)
f. Unlicensed software (eg the license expires during exams)
13. 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.
14. Postgraduate/undergraduate students who experience the above challenges in their second
examination opportunity will have to reregister for the affected module.
Students experiencing technical challenges should immediately contact the SCSC telephonically on
080 000 1870 and choose option 4 for exams or via e-mail at , or refer to
the Get help during the examinations by contacting the Student Communication Service Centre
(unisa.ac.za) resource for the list of additional contact numbers. Only communication received from
your myLife account will be considered.
[TURN OVER]
3
Downloaded by Stephen ()
, lOMoARcPSD|54878315
STA3701
OCT/NOV 2025
QUESTION 1 [10 Marks]
Determine (with evidence) whether the following statements are True or False.
𝑎11 𝑎12
1.1 The characteristic function of 𝐀 = (𝑎 ) is 𝜆2 − (𝑎11 + 𝑎22 )𝜆 + (𝑎11 𝑎22 − 𝑎12 𝑎22 ). (3)
21 𝑎22
4 2
1.2 The eigenvalues of the matrix 𝐀 = ( ) are 7 and 1. (3)
3 5
𝟏
𝟗
2 𝟏
1 𝟏𝟖
1.3 The Moore-Penrose inverse of the vector 𝒂 = ( ) is 𝒂+ = 𝟏 . (3)
3
𝟔
2 𝟏
(𝟗)
1.4 𝐈𝑚 ⊗ 𝐈𝑛 can also be written as 𝐈𝑛𝑚 . (1)
QUESTION 2 [30 Marks]
𝑎11 𝑎21
2.1 Consider the matrix 𝐀 = (𝑎 𝑎22 ). Obtain the expressions for the two eigenvalues of 𝐀 in
21
terms of the elements of 𝐀.
(4)
2.2 Suppose that
cos θ − sin θ
𝐐 = ( ).
sin θ cos θ
2.2.1 Show that 𝐐 is an orthogonal matrix. (4)
2.2.2 Determine the eigenvalues of 𝐐. (4)
2.3 Consider the matrix
0 0 2 0
1 0 1 0
𝐀 = ( ).
0 1 −2 0
0 0 0 1
2.3.1 Find the eigenvalues of 𝐀. (6)
2.3.2 Determine the normalized eigenvector corresponding to each eigenvalue in part (2.3.1).
(10)
2.3.3 Compute 𝑡𝑟(𝐀4 ). (2)
[TURN OVER]
4
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