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MFE512S - MATHEMATICS FOR ECONOMICS 1B - 2ND OPP - JANUARY 2024 |
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·-- nAml BIA UnlVERSITY
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FacultyofHealth,Natural
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Schoolof Natural andApplied
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Departmentof Mathematics,
StatisticsandActuarialScience
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QUALIFICATION : BACHELOR OF ECONOMICS (07BECO)
QUALIFICATION CODE: 07BECO
COURSE:MATHEMATICS FOR ECONOMICS lB
DATE: JANUARY 2024
DURATION: 3 HOURS
LEVEL:5
COURSECODE: MFE512S
SESSION: 2
MARKS: 100
SECOND OPPORTUNITY: QUESTION PAPER
EXAMINER: Mrs. Hilma Yvonne Nkalle; Mr. Tobias Kaenandunge; Mr. llenikemanya Ndadi
MODERATOR: Ms. Kornelia David
INSTRUCTIONS:
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the
examiner.
4. No books, notes and other additional aids are allowed.
5. Mark all answers clearly with their respective question numbers.
PERMISSIBLE MATERIALS:
l. Non-Programmable Calculator
This paper consists of 5 pages including this front page
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Question 1 (Multiple choice questions, 2 marks each) (20 Marks]
1.1 Which of the following matrices is most likely to have an inverse?
A) A square matrix with determinant equal to 0.
B) A square matrix with all zero entries.
C) A square matrix with determinant not equal to 0.
D) A non-square matrix with all nonzero entries.
1.2 Given the matrix equation AX::;B,where A is a square matrix and X,B are column
matrices, how can the solution for X be obtained?
A) By dividing B by A.
B) By finding the inverse of A and multiplying it with B
C) By finding the determinant of A and dividing it into B.
D) By subtracting B from A.
1.3 If a square matrix A has an inverse, which of the following statements is true.
A) The determinant of A is 1.
B) The determinant of A is 0.
C)The product of A and its inverse is the identity matrix.
D) The transpose of A is its inverse.
1.4 Supposed you have a system of linear equations represented by the matrix equation
AX::;B,where A is a square matrix. Which of the following is the correct expression to solve
for X?
A)X::;AB
B) X::;A1"B
C) X::;BA
D) X::;B1A·
1.5 If a matrix A is given and it has an inverse, which of the following is a correct way to find
the inverse?
A) Compute the transpose of A.
B) Divide each entry of A by its determinant.
C) Swap rows and columns of A.
D) Use Gaussian elimination to row-reduce A to the identity matrix.
Course Name (MFE512S)
2nd Opportunity November 2023
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1.6 For a 2x2 matrix A with a nonzero determinant, what is the formula to calculate its
inverse?
= A) A- 1 1/det(A) x adj(A)
= B) A- 1 1/trace(A) x adj(A)
C)A- 1 = 1/det(A) x A
D) A- 1 = A/ det (A)
1.7 If a square matrix A is invertible, which ofthe equations is true?
= A)A x A- 1 I, where I is the identity matrix.
= B) A+ A- 1 I
C) AX A- 1 = 0
= D) A -A- 1 I
1.8 What is the minimum requirement for a matrix to have an inverse?
A) It must be a square matrix.
B) It must have all positive entries.
C) It must be a non-square matrix.
D) It must have a determinant of 1.
1.9 A square matrix A is a singular (non-invertible), which of the following is true?
A) The matrix A is diagonal.
B) The matrix has no solution.
C) The matrix A has an infinite number of solution.
D) The determinant of matrix A is zero.
1.10 When solving for the inverse of a matrix A, why is it important to check whether the
determinant of A is nonzero?
A) If the determinant is nonzero, the inverse does not exist.
B) If the determinant is zero, the inverse does not exist.
C) The determinant affects the size of the inverse matrix.
D) The determinant determines the number of the rows.
Course Name (MFE512S)
2nd Opportunity November 2023
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Question 2 (true/false questions, 2 marks each) (10 marks]
2.1 A 4x3 matrix has three rows and four columns.
2.2 Every diagonal matrix is an upper triangular matrix.
2.3 A zero matrix is a lower triangular matrix provided it is a square matrix.
2.4 A square matrix is a matrix whose entries are square numbers.
2.5 In a matrix, the entry a23 and the entry a32 represent the same.
Question 3 (2 Marks]
Give an example of a 3x3 lower triangular matrix.
Question 4 (12 Marks]
A company produces three types of products A, Band C. The total annual sales of these
products for the years 1985 and 1986 on the four regions is given below.
For the year 1985:
Products
A
B
C
Khomas
region
15000
5000
8000
Omusati
region
8000
24000
4000
Oshana region Ohangwena region
6000
7000
31000
12000
8000
6000
For the year 1986:
Products
A
B
C
Khomas
region
17000
5000
13000
Omusati
region
10000
22000
6000
Oshana region Ohangwena region
5000
11000
39000
7000
4000
5000
Find the total sales of the three products for two years.
Question 5 [13 Marks]
Find the inverse of the following matdx, A=[~
!n
Course Name (MFE512S)
2nd Opportunity November 2023
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Question 6 [10 Marks]
Use Gaussian elimination method to find the solution (s) of the following system of linear
equations.
4y + 8z = 12
x-y+ 3z = -1
= 3x- 2y + Sz 6
Question 7 [16 Marks]
Given the system of linear equations
= Max P 6x + 8y
Subject to: 30x + 20 y :5 300
Sx + lOy :5 110
X;y 0, find the unknown variables.
Hint: Introduce slack variables; Formulate the initial simplex tableau; Derive the optimum
tableau; Interpret the final tableau.
Question 8 [7; 5; 5 Marks]
(a)
Given A=
2
[6
-1
4
~] , B = [~
!]find A+B.
(b) BC= [} 9 1~], Find (BC)2 .
(c) Given the following matrices, A=r~
!l,B=[-/
~] find AB.
End of 2nd opportunity Exam!
Course Name (MFE512S)
2nd Opportunity November 2023
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