Birkbeck College

School of Economics, Mathematics and Statistics

BSc Financial Economics, BSc Economics with Accounting

First Year Students’

Preparatory Mathematics Course

Monday, 17th Sept., 2012 – Friday, 28th Sept., 2012

Lecturer: Ranch Patel Email: rpatel@ems.bbk.ac.uk

Pre-Test : 1 Hour Insert your answers in the answer sheet provided. Q1. Fill in ? below:

(i) 12 - 3(?) = 3 (ii) 4(1+ ?)= 0 (iii) 8 – 2? = 0 (iv) ?3 = –8 (v) 2 x 32 =? (vi) 3(? - 6) = -15

(vii) – (3 + ?) = 5 (viii) ? 11 =-11 (ix) ? – 8 = -7

(x) √? = 9

Q2. Write expressions for

(i) the sum of a, b, c and d. (ii) the product of m and n. (iii) the reciprocal of x. (iv) e raised to the power x. (v) the cube of the product of x and y. (vi) m taken away from n. (vii) the modulus of the sum of x and y. (viii) the sum of absolute values of x and y. (ix) the arithmetic mean of p, q and r. (x) the geometric mean of p, q and r.

Q3.

(i) A plant of height 10cm increases by 5%. What is the new height?

(ii) A plant increases in height by 4% in a week and grows to a height of 108 cm. What was the height at the beginning of the week?

(iii) A investor invests £10000 for 20 years at a compunded rate of 6% per

annum. What is the value of the investment after the 20 years? Q4. The Present Value, PV of an annuity of £S per annum for n years when the interest rate is r is given by the formula:

1r1

1

1r1

1

r1

S PV

n

Find the Present value of an annuity of £10,000 per annum for 10 years when the interest rate is 7%.

Q5.(a) (i) Sketch the graph of the straight line y = 2 – 3x (ii) what is the gradient? (iii) what is the y intercept? (iv) what is the x intercept?

(b) In a linear relationship when x increases from 3 to 6, y increases from 5 to 10.

(i) Calculate x and y (ii) What is the ratio of the increase in y to the increase in x? (iii) What is the ratio of the proportional increase in y to the proportional increase in x ? Q6. Solve the equations :

(i) x 3 = 3 (ii) 2(√x – 3) =10 (iii) – 4 + 2x2 = 2 (iv) x2 – 2x – 8 = 0 (v) (x – 3)2 = 1 Q7 Simplify each of the following: (i) 3(3 – 2x) – (-2 + x) (ii) x2y + 4xy2 – xy(x + y)

(iii) 3

32

9(xyz)

zy3x (iv) 4x2 + y2 – (x – 2y)2

(v) 4x

1

2x

3

Q9 Simplify and write the following expressions with the exponents in the numerator:

i. 22x

3 ii

34

22

2x

xy8x

y iii.

5 2x

1

y iv.

23 x

x 3

2

y

y v.

1x

3

Q10 (a) Factorise the following: (i) 6x2y – 3xy2 (ii) ac + bc - ad - bd

(b) Simplify the following: (Hint: Factorise numerator and then cancel out)

1) -(x

.23x x 2

Q11 Consider the relation y: = x2 – 2x – 8 (i) Find the y intercept. (Hint: Put x=0 and evaluate) (ii) Find the x intercepts (Hint: Put y=0 and solve for x) (iii) Find the axis of symmetry (iv) Find the minimum value of y (Hint: use your answer from (iii) (v) Use the above information to sketch the graph.

Chapter 1

Number Types & Basic Operations

Natural Numbers

1.1

The set of whole numbers {1, 2, 3, 4, 5,…….} is called the set of Natural numbers. 0

and negative numbers such as -4 are not Natural numbers.

E1.1 Which of these is not a natural number?

12 yes/no; 3.4 yes/no; -4 yes/no; 0 yes/no

Integers

1.2 Extending the Natural Numbers by including 0 and negative whole numbers gives

the set of Integers.

The set of numbers { ….,-3, -2, -1, 0, 1, 2, 3, …} is denoted as .

E1.2 Which of these is not an Integer?

12 yes/no; 3.4 yes/no; -4 yes/no; 0 yes/no; 5

2 yes/no; -5.0 yes/no

A1.2 Yes; no; yes; yes; no; yes

E1.3 Are all natural numbers also Integers? Yes/No

A1.3 Yes

E1.4 Are all integers also natural numbers?

A1.4 No. The set of Natural numbers is ‘properly nested’ inside the set of integers.

Rational Numbers

1.5 Extending the Natural Numbers by including decimals and negative numbers gives

the set of Rational Numbers.

The number 2.5 lies between the integers 2 and 3. But 2.5 is not an integer. It is a

Rational number.

E1.5 Which number lies exactly between 2.5 and 3.5?

Is this number a natural number? Is it an integer?

Real Numbers,

Rational

numbers

Irrational Numbers

Integers,

Natural numbers, N

A1.5 The number 3 (or 3.0) It is a natural number and an integer.

E1.6 Are all natural numbers Rational numbers? Yes/No

Are all integers Rational numbers? Yes/no

Are all rational numbers integers? Yes/no

A1.6 Yes, all natural numbers are Rational numbers.

Yes, all integers are Rational numbers.

No: the set of integers is ‘properly nested’ in the set of rational numbers.

1.7 Every decimal can be expressed as a fraction and every fraction can be expressed as

a terminating or recurring decimal.

A recurring decimal is non terminating with repeating digits.

For example 0.123123123… is a recurring decimal. We write it more formally as a

terminating decimal with dots over the recurring digits viz:

0.123123123… = 321.0

There a calculator key that toggles the display of a rational number between a

fraction and a decimal. Find it on your calculator and ensure you know how to use it.

E1.8 Complete the table using your calculator

Decimal Fraction Type of decimal

0.5

2

1

Terminating

8

5

Terminating

0.525 Terminating

3

1

Recurring

A1.8

Decimal Fraction Type of decimal

0.5

2

1

Terminating

0.525

8

5

Terminating

0.525

8

5

Terminating

30.

3

1

Recurring

1.9 The last row shows the fraction ‘one-third’ as a non-terminating decimal, 0.3333...

Use your calculator to express the fraction 99

19as a decimal.

The calculator will show this as a ‘recurring decimal’, 0.191919…

i.e. a decimal that is not terminating but has repeating digits ‘19’ in this case.

Square Numbers

1.10 Consider the sequence of integers 0, 1, 4, 9, ....

They are called square numbers because

0x0 = 0, 1x1 = 1, 2x2 = 4, 3x3 = 9, ...

E1.10 What are the next three square numbers?

A1.10 4x4 = 16, 5x5 = 25, 6x6 = 36. So the next three square numbers are 16, 25, 36.

We use brackets to denote multiplication e.g. (4)(4) = 16

1.11 Learn this sequence of square Numbers:

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361,

400

E1.11 Complete the following:

(i) (?)(?) = 9, (ii) (?)(?) = 361 (iii) 13x13 = ?

A1.11 (i) 3 (ii) 19 (iii) 169

1.12 We have a special form of notation for expressing square numbers.

We write 32 = 9 to mean 3x3 = 9. Also 7

2 = 49.

E1.12 Replace the ? in the following:

(i) 42 = ? (ii) ?

2 = 100 (iii) 8

? = 64

A1.12 (i) 16 (ii) 10 (iii) 2

E1.13 Which is the missing square number in the sequence of square numbers?

9, 16, 25, ? , 49

A1.13 36

Square Roots

1.14 The inverse of squaring is called square rooting, for which we use the symbol √.

For example: The statment, 62 = 36 i.e. the square of 6 is 36 can be inverted to:

636 , which is read as the ‘the square root of 36 is 6’.

Here are more examples

Using Square Using Square root

42 = 16 416

92 = 81 981

152 = 225 15225

E1.14 Complete the table below

Using Square Using Square root

122 = 144

92 =

144

A1.14

Using Square Using Square root

122 = 144 144 12

92 = 81 981

122 = 144 144 12

E1.15 Fill in the ? in the following

(i) ?2 = 100 (ii) 6

2 = ? (iii) 144 ?

A1.15 (i) 10 (ii) 36 (iii) 12

E1.16 Find the value of ,81 i.e. what is the square root of the square root of 81.

A1.16 81 39

Irrational Numbers

1.17 Consider the two adjacent square numbers 16 and 25.

We know that:

the square root of 16 is 4 i.e. 416

the square root of 25 is 5 i.e. 525

So what is the square root of 20 which lies between the square numbers 16 and 25?

Clearly the answer lies between 4 and 5, which are adjacent integers.

So the answer is not an integer. Is it a rational number?

Use your calculator to find .20

Your calculator will show 4.472135955…

This is non-terminating decimal but it is not recurring. It is an IRRATIONAL

number.

E1.17 Which of these roots are irrational numbers?

i. 25 ii. 250 iii. 2500 iv. 5.2 v. 25.0

A1.17 i. No ii. Yes iii. No iv. Yes v. No

Number Line

1.18

A number line, which shows the relative sizes of the numbers is a useful graphical

aid. For example it allows to rank numbers and compare their sizes.

4

3

2

1

0

-1

-2

-3

-4

Note that when we write the number 4 we really mean the number, positive +4 but

we drop the + sign. So 4 has a silent or redundant + in front of it.

1.19

Note that we have used the symbol + to mean ‘positive’ sign and to represent the

ADD operation.

For example: +4 means positive 4.

Mathematics is a language that uses notation that is totally unambiguous and is

also as economical as possible. Dropping the (positive) + sign is economy of

notation.

You may argue that + is ambiguous because it means the sign of a positive number

and also the operation of ADD.

However the ambiguity is removed by the context.

Writing 2 + 3, clearly uses + as ADD and writing +4 clearly uses + as the positive

sign. It is DUALITY of usage not ambiguity in usage.

+3 + +4 = +7 : This means positive 3 ADD positive 4 equals positive 7.

For economy we write: 3 + 4 = 7 i.e. we remove the redundant 3 (Positive) +

symbols.

E1.19 Remove any redundant + in the following expression:

+2 + +3 + 5 + 6 = +16.

A1.19 2 + 3 + 5 + 6 = 16.

1.20

The above number line enables us to order numbers in increasing value or

decreasing value.

We use the notation:

> for “greater than” ; e.g. 6 > 4 (six is greater than 4)

< for “less than” ; e.g. -6 < -4 (minus six is less than minus 4)

≤ for “less than or equal to” and

≥ for “greater than or equal to”.

E1.20 a) Rewrite the set with its members listed in order of increasing value from

smallest to largest.

{ 7, 0, -3, -5, 4, 3 } { , , , , , }

b) State whether the following statements are true or false. Where the statements

are False rewrite them correctly.

Statement True/False Statement True/false

5 > 10 0 > -3

6 < -3 -6 > -3

-6 > -3 7 > -4 < 0

4 ≤ 3+1 5 ≥ 5 > 0

A1.20 a) { 7, 0, -3, -5, 4, 3 } { -5 ,-3, 0, 3, 4, 7}

b)

Statement True/False Statement True/false

5 > 10 F:5 < 10 0 > -3 True

6 < -3 F:6 > -3 -6 > -3 F: -6 < -3

-6 > -3 F: -6 < -3 7 > -4 < 0 True

4 ≤ 3+1 True 5 ≥ 5 > 0 True

1.21 Every integer has a sign and magnitude. The sign is either + or .

The magnitude is its size.

Integer Sign Magnitude

-11 - 11

+16 + 16

-8 - 8

7 + 7

0 - or + 0

E1.21 What is the sign of -5? What is its size?

A1.21 Sign is –ve; Size is 5

1.22 The magnitude is also called the absolute value or modulus. The notation for this is

two vertical bars enclosing the integer viz:

|-4| = 4.

This is read as: ‘the absolute value of -4 is 4’ or

‘the modulus of -4 is 4’ or ‘the magnitude of -4 is 4’.

E1.22 Complete the following table. Rewrite the false statements correctly.

Statement True/False Statement True/false

|5| = 5 |-6| > |5|

|-6| = 6 -3 > |-3|

Sign(-6) = 6 7 > |-4| < 0

A1.22

Statement True/False Statement True/false

|5| = 5 True |-6| > |5| True

|-6| = 6 True -3 > |-3| F: -3 < |-3|

Sign(-6) = 6 F:Sign of -6 is –ve 7 > |-4| < 0 F: 7 > |-4| > 0

Multiplication

1.23 Multiplication is just repeated addition.

4 + 4 + 4 is the addition of 3 copies of 4.

So we can ‘shorten’ 4 + 4 + 4 to 3 x 4 or (3)(4).

Or even more concisely as 3(4).

Note the use of the bracket to imply multiplication.

Note also that 3 + 3 + 3 + 3 = 4(3) which we can also write as 3(4).

Note also 2(5) = 2+2+2+2+2 or 2(5) = 5+5.

Is this ambiguity? Maybe but the result of both is the same.

E1.23 Write the following expressions switching ADD with MULTIPLY.

5 + 5 + 5 + 5 =

4(5) =

-3 + -3 + -3 + -3 =

3(-5) =

-3(4) =

A1.23 5 + 5 + 5 + 5 = 4(5)

4(5) = 5 + 5 + 5 + 5 or 4 + 4 + 4 + 4 + 4

-3 + -3 + -3 + -3 = 4(-3)

3(-5) = -5 + -5 + -5

-3(4) = -3 + -3 + -3 + -3

CHAPTER 2 – Directed Numbers

Addition and Subtraction

2.1 Directed numbers are positive and negative numbers. They can be put on a

number line as shown below.

+

3

+3

+2

+1

0

-1

-2

-3

So counting upwards from -3 we get -2, -1, 0, 1, 2 …

Writing this as a sequence, we get:

-2, -1, 0, 1, 2, 3

2.2 Increasing 5 by 3 takes us to 8.

We can write this mathematically as 5 + 3 = 8

Decreasing 5 by 7 takes us to -2

We can write this as 5 - 7 = -2

Increasing -2 by 2 takes us to 0.

We can write this as -2 + 2 = 0.

E2.2 Now complete the following table:

Action Result Mathematical statement

Increase -2 by 3

Decrease -2 by 5

5 + 4 = 9

Increase 0 by 2

3 -6 + … = ….

Increase -4 by … 4

Decrease … by 5 -1

A2.2

Action Result Mathematical statement

Increase -2 by 3 1 -2 + 3 = 1

Decrease -2 by 5 -7 -2 – 5 = -7

Increase 5 by 4 9 5 + 4 = 9

Increase 0 by 2 2 0 + 2 = 2

Increase -6 by 9 3 -6 + 9 = 3

Increase -4 by 8 4 -4 + 8 = 4

Decrease 4 by 5 -1 4 – 5 = -1

2.3 Now compare the following two actions:

Decrease 5 by 2 and Decrease 5 by -2.

The first result is 3 so the second result cannot be 3 as well.

The second result is 7.

Mathematically the second is written as: 5 - -2 = 7.

To decrease by a negative amount is to increase by a positive amount.

10 3 to7identical is 10 3- - 7 So

We use the symbol ≡ to mean ‘is identical to’.

So we write: 7 - -3 = 10 ≡ 7 + 3 = 10

E2.3 Complete the rows of the following Table. The first has been done for you.

Action Result Mathematical

statement

Identical

Statement

Decrease 2 by -3 5 2 – -3 = 5 2 + 3 = 5

Decrease -2 by -5

Decrease 5 by -4 5 + 4 =

Decrease -2 by -5 -2 – -5 = 3

-6 + 6 = 0

A2.3

Action Result Mathematical

statement

Identical

Statement

Decrease 2 by -3 5 2 – -3 = 5 2 + 3 = 5

Decrease -2 by -5 3 -2 – -5 = 3 -2 + 5 = 3

Decrease 5 by -4 9 5 – -4 = 9 5 + 4 = 9

Decrease -2 by -5 3 -2 – -5 = 3 -2 + 5 = 3

Decrease -6 by -6 0 -6 – -6 = 0 -6 + 6 = 0

Multiplication and Division

2.4 With multiplication and division we have the rule that ‘two negatives make a

positive’ We can summarise the rules as:

(positive)(positive) = positive e.g. (+4)(

+3) =

+12 or (4)(3) =12

(positive)(negative) = negative e.g. (+4)(-3) = -12 or (4)(-3) = -12

(negative)(positive) = negative e.g. (-4)(+3) = -12 or (-4)(3) = -12

(negative)(negative) = positive e.g. (-4)(-3) = +12 or (-4)(-3) =12

E2.4 Evaluate the following expressions:

(-5)(-7) = (-5)(-7)(-2) =

(-5)(7) = (-5)(2)(-7) =

(5)(-7) = (3)(-5)(-7) =

(5)(7) = -2(-5)(-3) =

A2.4

(-5)(-7) = 35 (-5)(-7)(-2) = -70

(-5)(7) = -35 (-5)(2)(-7) = 70

(5)(-7) = -35 (3)(-5)(-7) = 105

(5)(7) = 35 -2(-5)(-3) = -30

2.5 Similarly for division:

positivenegative

negative

negativepositive

negative

negativenegative

positive

positivepositive

positive

For example:

25

10

25

10or 2

5

10

25

10or 2

5

10

25

10or 2

5

10

E2.5 Evaluate the following quotients:

3

12

3

6

4

12

A2.5 Evaluate the following quotients:

43

12

23

6

34

12

Exercise 2

a. Evaluate the following expressions:

2 + -3 + -4 = -2 ─ -3 ─ -4 =

2 ─ -3 + -4 = -2 ─ 3 + -4 =

-2 + -3 ─ -4 = -2 + 3 ─ -4 =

2 ─ -3 ─ -4 = -2 ─ 3 ─ 4 =

b. Evaluate the following products:

(-2)(3)(-4) = (2)(3)(-4) =

(2)(3)(4) = (-2)(-3)(-4) =

(-2)(-3)(4) = (2)(-3)(4) =

(2)(3)(-4) = (2)(-3)(4) =

c. Evaluate the following quotients:

4

12

4

12

4

12

4

12

d. Four negative numbers and five positive numbers are all multiplied together.

What is the sign of the resultant number?

e. Fill in ? below:

(i) 3(?) = -12 (ii) 4(?)= 0 (iii) 8 – ? = 10

(iv) -4(?) = –8 (v) 2 + -3 ─ ? = 4 (vi) 3(?)(-4) =24

(vii) – 3 + ? = -5 (viii) ? 11 = -11 (ix) ? – 8 = -7

(x) -3(-2)(?) = -3 (xi) |-4| + 5 = (xii) |-3|(-4) =

(xiii) (|(-3)(4)|) (|-3|) = (xiv) |-3||-4 |=