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MFE512S - MATHEMATICS FOR ECONOMISTS - 1ST OPP - NOV 2022 |
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nAmlBIA UnlVERSITY
OF SCIEnCE Ano TECHnOLOGY
FACULTY OF HEALTH AND APPLIED SCIENCES
DEPARTMENTOF MATHEMATICS AND STATISTICS
QUALIFICATION:BACHELOROF ECONOMICS
QUALIFICATIONCODE: 07BECO
COURSECODE: MFE512S
SESSION: November 2022
DURATION: 3 HOURS
LEVEL: 5
COURSENAME: MATHEMATICS FOR
ECONOMISTS1B
PAPER:THEORY
MARKS: 100
EXAMINER
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
Mr. T. KAENANDUNGE,Mr. F. Ndinodiva, MS Y. NKALLE
MODERATOR:
Mr. I.D.O. NDADI
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. For section A, write True for statements that are true and False for
statements that are false. For section B, write down the correct letter
of your choice. Show clearly all the steps used in the calculations
when answering section C questions.
3. All written work must be done in blue or black ink and sketches must
be done in pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
ATTACHMENT: Graph paper
THIS QUESTION PAPERCONSISTSOF 5 PAGES{Including this front page)
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MFE512S- MATHEMATICS FOR ECONOMISTS 1B: 1sr OPPORTUNITY QUESTION PAPER-NOVEMBER,
2022
SECTIONA (True or false Questions)
QUESTION 1
State whether each of the following statement is true or false.
[15 marks]
1.1 A 4x 5 matrix has 4 columns and 5 rows.
[1]
1.2 The following system of linear equations is homogeneous.
x+y-z=O
2x + y + 2z -1 = 0
[1]
3x+2y-z=0
1.3 If A is a 3 x 4 matrix and Bis a 4 x 3 matrix, then the product BA is a 4 x 4 matrix.
[1]
1.4 Every square matrix has an inverse.
[1]
1.5 The following system of linear equations has only one solution.
x+2y=0
[1]
2x=y
1.6 For any matrix M, MTM is always possible.
[1]
1.7 A unit or identity matrix is a square matrix whose every entry which is not in the main
diagonal is a 0.
[1]
1.8 Two matrices may only be multiplied if they have the same order.
[1]
1.9 If matrix Mis a singular matrix, then M =MT.
[1]
1.10 Any matrix of any order may be multiplied by a scalar.
[1]
1.11 (: : ) issingular.
[1]
1.12 Every homogeneous system of equations is consistent.
[1]
1.13 Elementary row operations can be used to solve any system of linear equations
[1]
1.14 Jacobians are used to test for second order conditions of the stationary points.
[1]
1.15 Hessian determinants are used to test if a matrix has an inverse.
[1]
SECTION B (Multiple choice Questions)
Question 2
Write down the letter that corresponds to the right answer only.
a,2
2.1
The order
of
A
=
[
a.,
a 21
G22
a3, G32
a,,,]
a 2,, is
a3,,
A. 3x3 B. 3x4 C. nx3 D. 3xn E. None ofthese answers
[10 marks]
[2]
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MFE512S- MATHEMATICS FOR ECONOMISTS 1B: 15r OPPORTUNITY QUESTION PAPER-NOVEMBER,
2022
2.2 Which of the following matrix is a diagonal matrix?
[2]
E. None of these
2.3 Which of the following linear programming problem is a standard maximizing problem? [2]
Maximize P = 3x - 2y
Subject to 2x+3y .:s6;
A.
x+2.:s;I-y
x,y2'.:0
Maximize P =3x- 2y
Subject to 2x+3y .:s6;
B.
x-y.:s;I
x.:s;0,y2'.:0
Maximize P = 3x-2y·
Subject to 2x +3y .:s6;
C.
x-y.:s;I
x,y2'.:0
Minimize P =3x - 2y
Subject to 2x+3y .:s6;
D.
x-y.:s;I
x,y2'.:0
E. None of these
2.4 The solution for the inequality: 3 .:s5; - 2x .:s1; 1is
[2]
A.
B.
C.
D. No solution
2.5 Use the Hessian to test for the nature critical point (2, 1) of the function
f (x,y ) =--x1 2 +2xy+y-+?6x-6y-IO.
[2]
2
A.
It is a local maximum
B.
It is a local minimum
C.
It is a saddle point
D.
Undetermined (test fails)
SECTION C (Structured questions)
Question 3
[75 marks]
3.1
J If B -[ ~1 and A - (1 -2 - 3) determine the products AB and BA if they exist. [S]
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MFE512S- MATHEMATICS FOR ECONOMISTS 18: 15r OPPORTUNITY QUESTION PAPER-NOVEMBER,
2022
3.2 Use Cramer's rule to solve each of the following systems of linear equations.
3.2.1 x+3y+3z = -2
4y+x+4z = -2+ y
x+z+4y+ 2 = -l-2z
for y only.
[SJ
3.2.2 x+2y=3
3x+6y = 9
3.3 Find the inverse of A= [
[2]
:
!]if it exists and use matrix inverse method to solve the
following system of linear equations.
x+2y+3z=2
2x+4y+5z = 3.
[15)
3x+5y+6z =4
3.4 A fruit juice company makes two special drinks by blending apple and pineapple juices. The
first drink uses 30% apple juice and 70% pineapple juice, while the second drink uses 60%
apple juice and 40% pineapple juice. There are 1000 liters of apple juice and 1500 liters of
pineapple juice available. If the profit for the first drink is N$.60 per liter and that for the
second drink is N$.S0, use the simplex method to find the number of liters of each drink that
should be produced in order to maximize the profit.
[15)
3.5 Sky Aviation Industries has two plants, I and II, which produce the "Venag" jet engines used
in their light commercial airplanes. The maximum capacities of these two plants are 110
units and 100 units per month, respectively. The engines are shipped to two of the
company's main assembly plants, A and B. The shipping costs (in dollars) per engine from
plants I and II to the main assembly plants A and Bare as follows:
To
AB
From I [120 70]
II 100 60
In a certain month, assembly plant A needs 70 engines while assembly plant B needs 80
engines. Set up a linear programming model that will determine the number of engines to
be shipped from each plant to each main assembly plant that will minimise the total cost
and use the graphical method to solve this model. Use the scale 5 small squares: 10 units on
both axes.
[15)
3.6
3.6.1
A motor company manufacture and sell cars and motorbikes. The cost of manufacturing x
motorbikes and y cars is given by C(x,y) = 800x2 +400.xy + 2900y2. Each motorbike is
sold for N$36 000-00 and each car is sold for N$180 000-00.
Use Gaussian elimination to determine the number of motorbikes and the number of cars
that should be manufactured and sold for a maximum profit P and determine the
maximum profit F-:nax.
[8]
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MFE512S-MATHEMATICFSORECONOMISTS1B: 15r OPPORTUNITYQUESTIONPAPER-NOVEMBER,
2022
3.6.2 Use the Hessian to confirm that the amounts in 3.6.1 will produce maximum profit.
[S]
3.6.3 Use the Jacobian to test for functional dependence between the cost function and the
revenue function.
[S]
****************************END OF EXAMINATION***************************
GOODLUCK!!!!!!!!!
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