MODULE MATHEMATICS

Entry Test: Mathematics for Management and Economics

BM.0 Introduction 2

BM.1 Sets 3

BM.2 Assertions 4

BM.3 Propositional Formulas 5

BM.4.1 Algebraic Laws 6

BM.4.2 Equivalence of Equations 8

BM.4.3 Transforming Equations 9

BM.4.4 Specific Equations with one variable 10

BM.4.5 Equation Systems 11

BM.5 Functions 12

Solutions 14

Mathematics for Management and Economics 2 BM.0 Introduction

MODULE MATHEMATICS

BM.0 INTRODUCTION The course Mathematics for Management and Economics (first semester) assumes that students are at swiss professional maturity level (Berufsmaturitätssniveau). Mathematical terms, concepts and theories associated with that level of competence will not be repeated.

This entry test helps you evaluate your mathematical competences. In case you should not be able to solve most of the problems, please try to work on this before beginning your studies. One of the many books that might be helpful in case you need some brush up is Marvin Bittinger & David Ellenbogen: Calclulus and its Applications. Pearson International Edition.

Mathematics for Management and Economics 3 BM.1 Sets

MODULE MATHEMATICS

BM.1 SETS Exercise BM.1 - 1

Let the sets A , B ,C , and D be defined as follows: A people with bloodgroup A, B women, C people aged at least 100, D people in retirement. • Describe the following sets using the symbols A through D :

a) Set of all retired men.b) Set of retired women having bloodgroup A. c) Set of retired women having a bloodgroup other than A.

• Describe the following sets by naming their peculiarities:d) CB ∩ e) DB ∩ \ A

Mathematics for Management and Economics 4 BM.2 Assertions

MODULE MATHEMATICS

BM.2 ASSERTIONS Exercise BM.2 - 1 Which of the following assertions are true: a) 10 is a natural number. b) 10 is a rational number.

c) 31 is a natural number.

d) 31 is a real number.

e) π is a rational number. f) π is a real number.

Exercise BM.2 - 2

Which of the following assertions are true: a) 10 is a natural number and smaller than 20. b) 10 is a natural number and smaller than 9.

c) 31 is a natural number and smaller than 20.

Mathematics for Management and Economics 5 BM.3 Propositional Formulas

MODULE MATHEMATICS

BM.3 PROPOSITIONAL FORMULAS Exercise BM.3 - 1

x is smaller than 10 is a propositional formula. For which of the following numbers x is the resulting assertion true:

a) 10=x

b) 9=x

c) π=x . Exercise BM.3 - 1

x is a natural number and larger than 100 is a propositional formula. For which of the following numbers x is the resulting assertion true:

a) 100=x

b) 101=x

c) π+= 300x .

Mathematics for Management and Economics 6 BM.4.1 Algebraic Laws

MODULE MATHEMATICS

BM.4.1 ALGEBRAIC LAWS Exercise BM.4.1 - 1

Complete the following equations without using parantheses:

a) =− )( avwzx

b) =+ yyx )1734(

Exercise BM.4.1 - 2 Complete the following equations by factoring out:

a) =+ avxabx

b) =− xzxy 105 Exercise BM.4.1 - 3

Complete the following equations using just one fraction bar:

a) =−a

cdc

ab

b) =−

++

aba

cba

2)(3

3)(2

Exercise BM.4.1 - 4 Complete the following equations using just one fraction bar:

a) =

2

3

4

cd

abc

b) =−

+

aba

cba

2)(3

3)(2

Exercise BM.4.1 - 5 Complete the following equations by expanding the left side term:

a) =− 2)1(x

b) =+ 3)1(z

Mathematics for Management and Economics 7 BM.4.1 Algebraic Laws

MODULE MATHEMATICS

Exercise BM.4.1 - 6

Complete the following equations using just one exponent:

a) =+− 22 )1()1( xx

b) =−+

3

3

)1()1(

zz

Exercise BM.4.1 - 6 Name the following quantities using the units in brackets:

a) 1300 2mm [in 2m ]

b) 23 secm [in

hkm ]

c) 45 3m [in 3cm ]

Exercise BM.4.1 - 7 Complete the following equations using the one log-sign just once:

a) =+ yx aa loglog2

b) =− xx aa log31log3

Mathematics for Management and Economics 8 BM.4.2 Equivalence of Equations

MODULE MATHEMATICS

BM.4.2 EQUIVALENCE OF EQUATIONS Exercise BM.4.2 - 1

Which of the following equations are equivalent:

a) 110)1)(1(?

−==⇔=+− xoderxxx

b) 102?

2 =⇔=−+ xxx

Mathematics for Management and Economics 9 BM.4.3 Transforming Equations

MODULE MATHEMATICS

BM.4.3 TRANSFORMING EQUATIONS Exercise BM.4.3 - 1

Solve the following equation for y (given 0>y and }1,1{−∉x ):

75)1(

log2

10 =−−x

y

Exercise BM.4.3 - 2 Determine the solutions of the following equation, i.e. name all real numbers satisfying thisequation:

0)1)(93( 22 =−+− xxx Exercise BM.4.3 - 3 Determine the solutions of the following equation:

0)17)(9(

)9)(9)(1(=

−−−+−

xxxxx

Exercise BM.4.3 - 4 Determine the solutions of the following equation:

12 2 64y y+= − Exercise BM.4.3 - 5 Determine the solutions of the following equation:

16)3( 4 =−x Exercise BM.4.3 - 6

Determine the solutions of the following equation: 3 5 52 4x x− =

Mathematics for Management and Economics 10 BM.4.4 Specific Equations with one variable

MODULE MATHEMATICS

BM.4.4 SPECIFIC EQUATIONS WITH ONE VARIABLE Exercise BM.4.4 - 1

How many solutions does each of the following equations have:

a) 0122 =+− xx

b) 1122 =+− xx

c) 1122 −=+− xx Exercise BM.4.4 - 2

What are the parameter values a and b so that the straight line defined by baxy += meets the following requirements: 8)2(,2)0( == yy .

Mathematics for Management and Economics 11 BM.4.5 Equation Systems

MODULE MATHEMATICS

BM.4.5 EQUATION SYSTEMS Exercise BM.4.5 - 1

Determine all solutions of the following equation system:

2412

=+=+−

yxyx

Exercise BM.4.5 - 2

Determine all solutions of the following equation system:

2212

2 −=

=+−

yyx

Exercise BM.4.5 - 3

Determine all solutions of the following equation system:

123132

=+=+=+

zxzyyx

Mathematics for Management and Economics 12 BM.5 Functions

MODULE MATHEMATICS

BM.5 FUNCTIONS Exercise BM.5 - 1

Let the function f be defined by the assignment 3)( −= xxf . What is the domain of that function, i.e. for which numbers is the assignment defined? Exercise BM.5 - 2

The public transport system of the city of Aakebrö has a unit ticket price: one ticket (price: 3 kroners) entitles you to travel as far and as often as you wish by trolleybus for one day. Let C be the function showing transportation cost [in kroners] as a function of travelling distance [in km] per day for one person. a) What is the domain of that function? b) What is the assignment of that function? c) Sketch the graph of that function in the diagram below.

Exercise BM.5 - 3 The new city council of Aakebrö decrees a different price system for the trolleybuses. That system is shown by the graph below. Explanations:

- Bold dots belong to the graph, empty dots do not. - The domain of the function extends to values beyond x=10.

Give the assignment of the new price system.

Mathematics for Management and Economics 13 BM.5 Functions

MODULE MATHEMATICS

Mathematics for Management and Economics 14 Solutions

MODULE MATHEMATICS

SOLUTIONS Exercise BM.1 - 1 a) D \ B = BD∩ b) ABD ∩∩ c) DB∩ \ A ADB ∩∩= d) CB ∩ is the set of all women aged at least 100 years. e) DB ∩ \ A is the set of all retired women having a bloodgroup other than A Exercise BM.2 - 1 a) true b) true c) false d) true e) false f) true Exercise BM.2 – 2 a) true b) false c) false Exercise BM.3 - 1 For the numbers 9 or π the assertion is true, for 10 it is false. Exercise BM.3 - 1 For 101 the assertion is true, for 100 or π+300 it is false. Exercise BM.4.1 - 1 a) xavxwzavwzx −=− )( b) 21734)1734( yxyyyx +=+ Exercise BM.4.1 - 2 a) )( vbaxavxabx +=+ b) )2(5105 zyxxzxy −=− Exercise BM.4.1 - 3

a) ac

dcbaac

cdcabaa

cdc

ab 22 −=

−=−

b) ac

bacbaaac

cbaabaa

bac

ba6

)(9)(46

3)(32)(22

)(33

)(2 −++=

−++=

−+

+

Exercise BM.4.1 - 4

a) abdc

dabcc

cd

abc

34

34

3

432

2

==

Mathematics for Management and Economics 15 Solutions

MODULE MATHEMATICS

b) )(9)(4

)(332)(2

2)(3

3)(2

bacbaa

bacaba

aba

cba

−+

=−

+=

−

+

Exercise BM.4.1 – 5 a) 12)1( 22 +−=− xxx b) 133)1( 233 +++=+ zzzz Exercise BM.4.1 - 6 a) 22222 )1()]1)(1[()1()1( −=+−=+− xxxxx

b) 3

3

3

11

)1()1(

⎥⎦⎤

⎢⎣⎡

−+

=−+

zz

zz

Exercise BM.4.1 - 6 a) 22 0013,01300 mmm =

b) h

kmh

kmm 8,82236,3sec

23 =⋅=

c) 33 000'000'4545 cmm = Exercise BM.4.1 - 7 a) )(loglog)(logloglog2 22 yxyxyx aaaaa =+=+

b) )(log)(log)(log)(log)(loglog31log3 3 83

8

31

33

13 xxx

xxxxx aaaaaaa ===−=−

Exercise BM.4.2 - 1 a) 110)1)(1( −==⇔=+− xorxxx . There is equivalence. b) 1022 =⇐=−+ xxx . The converse implication would be false, since -2 is a solution,

too. There is no equivalence. Exercise BM.4.3 - 1

)1(122102

102

10 2

10)1(12log1257)1(

log75

)1(log −=⇔−=⇔=+=

−⇔=−

−xyxy

xy

xy

Exercise BM.4.3 - 2

}{ 1,10)1)(1(010930)1)(93( 2222

−∈⇔=+−

⇔=−=+−⇔=−+−

xxxxorxxxxx

The quadratic equation 0932 =+− xx has no real solution. Exercise BM.4.3 - 3

} } }{{{ 9,117,99,9,1

0)17)(9(0)9)(9)(1(0)17)(9(

)9)(9)(1(

−∈⇔∉−∈⇔

≠−−=−+−⇔=−−

−+−

xxandx

xxandxxxxx

xxx

Mathematics for Management and Economics 16 Solutions

MODULE MATHEMATICS

Exercise BM.4.3 - 4 80735,27log2)12(2227722 2

11 ==⇔=−=−=⇔−= ++ yyyyyyy Exercise BM.4.3 - 5

{ }1,523234)3(16)3( 24 ∈⇔−=−=−⇔=−⇔=− xxorxxx Exercise BM.4.3 - 6

7575105322)2(42 105252553 −=⇔=−⇔=−⇔==== ⋅− xxxxxxxxx

Exercise BM.4.4 - 1 a) Exactly one solution (which is 1). b) Exactly two solutions (0 and 2). c) No real solution (negative discriminant). Exercise BM.4.4 - 2 362228)2(,2)0( =⇔=⇔+==== aaayby . Verification: 82232203 =+⋅=+⋅ and . Exercise BM.4.5 - 1 Multiply the second equation with 2 and add it to the first equation to obtain 59 =y , i.e.

95=y . 24 =+ yx now implies

92

92018

95492

954242 −=

−=

⋅−⋅=⋅−=−= yx .

Verification: 29

18)95(49

2

199

95)9

2(2

==+−

==+−⋅−

Exercise BM.4.5 - 2 Since 222 −=y has no real solution, the entire equation system has no solution either. Exercise BM.4.5 - 3

1=+ zx implies xz −= 1 . Inserting this in the second equation yields 233)1(3 =−+=−+ xyxy , i.e. 13 −=− xy . The resulting system is

13,132 −=+−=+ yxyx , so yxyx −=+ 332 , which implies xy =4 .

Inserting this in the first equation 111342 ==+⋅ yyy , so 111=y . This implies 11

4=x and

117

1141 =−=z .

Verification:

11111

117

114

21122

117311

1

11111

111311

42

==+

==⋅+

==⋅+⋅

Exercise BM.5 - 1

Mathematics for Management and Economics 17 Solutions

MODULE MATHEMATICS

The term below the root sign must be nonnegative, which means 3≥x . Exercise BM.5 - 2 a) ),0[ ∞=KD

b) ⎩⎨⎧

>=

=0300

)(xfürxfür

xC

c) Hint: bold points belong to the graph, empty points do not.

Exercise BM.5 - 3

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>≤<≤<≤<≤<

=

=

44332

543

2110

0

210

)(

xforxforxforxforxfor

xfor

xK