MATH10260 Linear Algebra for Engineers
Problem Set 2: Polynomials and Complex Numbers.
1. Check that 2 is a root of the polynomial
P (x) = x4 + 4x3 16x 16 .
Compute the multiplicity of this root. Does P (x) have other real roots?
If yes, nd them all and c
1. A pharmaceutical company produces a drug from two different compounds, A and B.
Each compound contains the same three active ingredients (AI1, AI2 and AI3) but in
different proportions.
The drug requires at least 6 units of AI1, where one gram of compo
Question 1
A small company is involved in the daily production of two simple products but has not up
until now analysed the production output and assessed the best product mix in terms of short
term profitability. At present it manufactures approximately
Question 1
The management of ABC company have decided to deploy a scientific approach to their
decision analysis given the complexities involved in their operations and the potential
for greater efficiencies. One aspect of the companys operations involves
1. A company wishes to estimate the profit from two new products. The estimated
cost of producing the first product is 3 per unit and it is expected to sell for 5
per unit while the second product costs 5 per unit and is expected to sell for 8
per unit. T
1.
(a) Explain what you understand by the terms feasible region and binding constraints in
respect of linear programming.
(3 Marks)
(b) A pharmaceutical company produces two types of pain relief tablets: Paincure and
PaincurePlus. The company wishes to ma
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
B
C
D
E
F
G
H
I
Special Products Co. BreakEven Analysis
Data
Unit Revenue
$2,000
Fixed Cost $10,000,000
Marginal Cost
$1,000
Sales Forecast
30000
Production Quantity
Formula
Total Revenue
Tota
Microsoft Excel 16.0 Feasibility Report
Worksheet: [cafeteria_mine_solver.xlsx]Sheet1
Report Created: 10/14/2016 5:23:29 PM
Constraints Which Make the Problem Infeasible
Cell
Name
Cell Value
Formula
Status
$F$9 vitamin per lb
0 $F$9>=$H$9 Violated
Slack

Unit Rev (in Eur)
Fixed Cost
Cost Per Unit
Sales Forecast
2,000
10,000,000
1,000
30,000
Production Q
10,000
Total Rev
Total Fixed Cost
Total Var Cost
Production Q
BEP
1.5
250,000
2,000
4,500
100
300
Fixed Cost
Variable Cost
Rev / Unit
BEP
Sales Forecast
E
1.
MATH10260 Linear Algebra for Engineers
a) Write the number (1 + 3)4 in the form a + b 3 with rational numbers a, b.
b) Show that the number (3 + 5i)6 + (3 5i)6 is real.
1+i
2
c) Compute
2014
.
d) Express the number (1 +
3i)8 in polar form rei with r 0
MATH10260 Linear Algebra for Engineers
Problem Set 3: Basic Properties of Complex Numbers
1. Express the following complex numbers in the form a + bi where a and b
are real numbers:
a) (1 + 3i)1
b) (1 + i)(1 i)
c) (1 + i)(2 i)
d) (1 i)(2 i)
e) (7 + i)( +
MATH10260 Linear Algebra for Engineers
Problem Set 1: Numbers and Polynomials.
1. Write numbers represented by the following nite and periodic decimal
n
expansions in the form of reduced fraction m , e.g.:
0.0123123123 . . . =
123
10000
k=0
123
1
1
123
41
MATH10260 Linear Algebra for Engineers
Problem Set 5: Linear systems, method of elimination, vectors and matrices
1. Solve the following linear systems by the method of elimination:
2x + 3y = 13
x + 2y
=8
a)
b)
x 2y = 3
3x 4y = 4
5x + 2y = 27
2x 3y + 4z =
MATH10260 Linear Algebra for Engineers
Problem Set 4: Exponential with complex argument, Eulers formula and
trigonometry
1. Express the following complex numbers in the form a + bi where a and b
are real numbers:
a) e2i
e) 2 e
b) ei
5
3 i
f ) 1.5 e
7
8 i
MATH10260 Linear Algebra for Engineers
Problem Set 6: Matrices and linear systems, the inverse of a matrix
1. Bring the following matrices to reduced row echelon form:
1 1
2 5
3
1 2
3 9
2 5
1 9 3
a) 2 1 1 8
b)
2 1 1 3 11
3 0 1 3
1 3 2
7
5
1 2 0 3 1 0 2
SEMESTER 2 EXAMINATION 2013/2014
MATH 10260
Linear Algebra for Engineers
Dr. Patrick Murphy
Dr. Masha Vlasenko
Time Allowed: 2 hours
Instructions for Candidates
All questions have equal weight. Please attempt six out of seven questions.
Show all work!
Ins
MATH10260 Linear Algebra for Engineers
Problem Set 8: Properties of Determinants
1. Consider the matrices
1
D = 1
1
2 x+1
x
3
3
3
and
1
F = 2
4
1
x .
x2
1
3
9
Find all values of x for which
a)
a1
2. Let A = b1
c1
det(D) = 0 ;
b)
det(F ) = 0 ;
b)
a2
b2
c2
MATH10260 Linear Algebra for Engineers
Problem Set 10: Linear Independence, Dimension and Bases
1. a) Construct a basis in the vector space of 2 2 matrices over a eld K.
b) Construct a basis in the vector space of 3 3 matrices over a eld K.
c) What is the
MATH10260 Linear Algebra for Engineers
Problem Set 7: Determinants
1. Find the number of inversions in each of the following permutations of 5 elements
a) 52134
b) 45213
c) 42135
d) 54321
and say which of them are even and which are odd.
2. In each of the
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
B
C
D
E
F
G
H
I
J
K
L
M
Keeping Time BreakEven Analysis
Unit Revenue
Fixed Cost
Marginal Cost
Sales Forecast
Production Quantity
Data
$4,500
$250,000
$2,000
?
100
Total Revenue
Total Fixed Cost
Total Variabl