Index of /Couchman and Jones 2 Unit Book 1

Writing Team Stan Jones Ken Couchman

Keith Sllnn Frank Morgan Illustrated by George Gifford

1

First published 1982 Reprinted 1983 Reprinted 1984 (twice)

Reprinted 1985 Reprinted 1986 Reprinted 1987

Reprinted 1990 (twice) Reprinted 1992 by Collins Dove

A Division of

HarperCollinsPublishers

(Australia) Pty Ltd 22-24 Joseph Street

North Blackburn Victoria 3130 For Shakespeare Head Press

© 1982 S.B. Jones & K.E. Couchman

This book is copyright. No part of it may be reproduced or transmitted without the written permission of the publisher.

ISBN 0 85558 765 2 Typeset in Hong Kong by Ascot Trade Typesetting Ltd

This text is designed to give a natural transition from the mathematics of the junior school to 2 Unit senior mathematics.

We suggest that all students should begin the course by working through Chapter 1 to familiarise themselves with the use and applications of the scientific calculator which form an integral part of this course.

It is left to the teacher's discretion whether to proceed immediately with Chapters 2-4 or to treat these sections as the need arises in conjunction with other topics.

So that competence in mathematical skills and techniques can be attained, numerous examples are given as teaching illustrations and as exercises to be undertaken independently by the student.

Additional practice and problem solving experience is given in the Practice Papers which are placed at intervals throughout the text.

When developing the calculus, the proofs which justify the rules for differentiation are given in the appendix so that students can study these when their mathematical maturity allows.

Operations - percentages - fractions -powers and roots - scientific notation - order of operations- using the memory.

Grouping symbols - binomial products -substitution - formulae - operations with surds.

Common factors - grouping - difference of two squares- quadratic trinomials- algebraic fractions.

Linear equations- equations resulting from sub-stitution- absolute value- quadratic equations -simultaneous equations- inequalities.

Points, lines and figures - angles - parallel lines - angle sum of triangles and other poly-gons - congruent triangles- special triangles -similar triangles- the theorem of Pythagoras.

Definition and properties- intercepts made by parallel lines- areas triangles and quadrilaterals.

Notation - domain and range - graphs of functions- algebraic representation of geomet-rical properties- regions in the plane.

(

The sine rule - the area of a triangle - the cosine rule.

The gradient- point gradient formula - two point formula - intercept formula - parallel and perpendicular lines - line through the intersection of two lines- the distance formula - the mid-point - perpendicular distance of a point from a line.

Gradient of a curve the gradient function -rules for differentiation -tangents and normals - maximum and minimum turning points -curve sketching - a note on continuity and limits.

Derivatives of products, quotients, functions of a function,

xn

for n rational.

Equal likely outcomes- set notation- range of probability - ways of counting

n (S) and

n

(£) - bias - probability trees - comple-mentary events - non mutually exclusive events.

A modern scientific hand calculator featuring • four basic calculations

• memory calculations • fraction calculations • trigonometric functions • logarithmic functions • powers and roots • reciprocals

CHAPTER 1

The scientific calculator has become a common and valuable aid for use in mathematics and the student of 2 unit mathematics should become adept at doing calculations both with and without the calculator.

In this first chapter much of the work can be done with the aid of a calculator and the opportunity should be taken to become completely familiar with its various functions.

Do the following worked examples, checking the read-out in each case. Should you make. an error, press the clear key and start again.

1. 9·26

+

15·8 - 12·473

2. 15·9 X 6·07

3. 560 + 13-5 4. 64 X 18·7 - 6·84

45·2 X 17·9

5

' 62·8

6. 784 8·2 X 7·6

7. Find

i

of 2154.

8. Change~~ to a decimal.

9·26

GJ

15·8 [ ] 12·4731 = 1

15·9

0

6·071 = 1

56oGJ n 5 G

64

0

18·7

G

6·84

B

45·2

0

17·9

G

62·81

=

1

784

G

8·2

G

7·6

G

2154

~

7

~

8

~

15 Q 3 2 G

Do these examples and check your answer with the given answer in each case.

1. 7·84

+

15·9- 6·275

2. 4·84 X 7·3

3. 410·76 + 12·6 4. 5·8 X 7·1 X 2·8

5. 16·8 + 360 6. 9·08 X 32 - 115·6

7. 4·8 X 13-6 + 8·5

8. 39·2 + 7·3

9 52684

0

8400

9 X 6 X 7 X 4 10. 15

11. Find ₁5

2 of 672.

12. Change /₆ to a decimal. 13. Find i~ of 81·48.

14 0 17·84 X 45

160

15. 48·6 9·3 X 6·4

16. 382 _{- 1263}

Most scientific calculators have a clear key and a clear entry key.

The clear key

c0

or

lAc

I

or

I

CA

I )

clears the entire machine except the memory. It is sound practice to press this key before beginning calculations.

The clear entry key (

~

or

0

or

~

) is used to correct mistakes. It clears the last number entered to allow for correction without the need to go back and start at the beginning.

For example:

1. 56 X 34

2. 986 + 0·6

s60

43~

34

B

986G 6

§J

· 6 G

jl)(J!j

1 Cl43.33:n

If a mistake is made by pressing the wrong operation key, it can be corrected by immediately pressing the correct operation key.

For example:

3·86 X 57·93 3·86

GJ

0

57·93 G

t opcn1

If an operation which is mathematically impossible is at-tempted (such as 6 + 0), the calculator will give an error message such as an 'E' sign and will lock the calculator. No further calculation can be done until the clear key is pressed.

An overflow error is indicated by the calculator if the answer to a calculation is outside its range. Once again the calculator is locked and no further calculations are possible until the clear key is pressed.

Do these questions on your calculator and check your answer with the given answer in each case. 1. Simplify:

(a) 541 000 - 286 597 (b) 7·2 X 2·8 X 1·06 (c) 28 x 5·37

+

17·94 (d) 939·6 + 29

(e) ~of 5562

(f) 9·08 X 7·6 - 34·74 (g) i~ of 115·02 (h) 9·6 X 7·8 + 5·2 (i) 166·05 + 4· 5 (') 15·6 X 8-4

J 0·012

2. Before doing each of these multiplications decide how many decimal places there will be in the answer. Check this on the calculator read-out. (a) 7-4 x 13 (e) 0·924 x 61 (b) 9·26 X 5·4 (f) 13 X 51·63

(c) 5·08 X 7·26 (g) 27-4 X 5·0072

(d) 38·9 X 6·004 (h) 579 X 8·27

3. Give the answer to each division correct to two decimal places.

(a) 8·74 + 12 (e) 756·9 + 52·4 (b) 56·2 + 8·1 (f) 81-46 + 0·76 (c) 9·72 + 0-43 (g) 9-4 + 2·847 (d) 476 + 3·8 (h) 24·764 + 365

This is not an operation key but changes the sign of the number displayed from plus to minus and vice versa. It can be used whenever operating with directed numbers.

For example:

1. Enter -6·8

2. ( -62·7) X 3·5

3. (-5·8) X (-7·28)

4. (-133·7) + (-5·42) 5. 7·38 - ( -4·92)

6·8

E[]

6 2 · 7 0 0 3·5

B

5·8

0 0

7·28

~

B

133·7

0

Q

5·42

0

EJ

7·38

B

4·92

0 B

6.1\

I

-719.451

Use your calculator to check the following rules for operations with directed numbers.

1. The product of two negative numbers is positive.

Try: (a) ( -4) X ( -8) (b) ( -1·6) X ( -6·2)

2. The product of a positive number and a negative number is negative.

Try: (a) 6 X (-9) (b) ( -3·8) X 8·9 3. The quotient of two negative numbers is positive.

Try: (a) (-12) + (-3) (b) (-9·6) + (-2·4)

4. The quotient of a positive number and a negative number is negative.

Try: (a) 32 + ( -4) (b) ( -15·6) + 3·2

5. To subtract a negative number is the same as adding its opposite.

Try: (a) 8- (-7)and8

+

7 (b) (-5·3)- (-2·8)and(-5·3)

+

2·8

For simple calculations the operator may prefer to calculate with positive numbers and adjust the sign of the answer, using the rules above.

Do these examples and check your answer against the given answer in each case.

1. ( -16·4) X 24·5 6. 56·2 + (-18·9)

2. 8·48 X ( -0·76) 7. 6-4 X ( -3·6) + 4·2

3. ( -10·7) X ( -4·9) 8. (- 24·7)

+ (

-18·3)

4. 4·6 X 16 X ( -2·8) 9. 84·2 + ( -2·6) + ( -8·5)

5. 18·26 - ( -7·84) 10. (-28) X (-6·4) + (-15·2)

When the reciprocal key is pressed the reciprocal of the number displayed is obtained immediately.

For example:

1

8[B 0.125

-8

1

2·64~

2·64

45

+

23 45

G

23

G

BJ

7·2 X 3·8 7·20

3·8G~

Use of the reciprocal key increases the versatility of the calculator. For example 7.8

3 : 9

5.9 can be worked in different ways. Here are two of the ways which involve the use of the reciprocal key.

. 34·9

(l) 7·8

+

5·9

(ii) 34·9 7·8

+

5·9

7·8 EJ5·9

G

BJ 0

34·9

B

7·8

G

5·9

G G

34·9

G

r:B

5

1. What is the reciprocal of:

(a) 9 (b) 6·2 (c) 58 (d) 49·7 2. Find the value of:

1

(d) 54

~

29 (g)

28

(a) 36 _15·2

+

_17·6

(b) _1 _(e) 1 (h) 45

8·5 2·8 X 3·1 (3·24)2

1 _(f) 1 . 5·26

(c) 18 x 2·3 _7·6

+

_3·7 (I) 7·1 X 3·6

Most scientific calculators do not have a percentage key, however percentage calculations can be done quite easily if you use your basic knowledge of percentages.

Examples:

1. Find 6% of $354.

We know that 6%

=

1

8

0

=

·06, thus to find 6% of a quantity we multiply by ·06.

·06

0

354

El

($21·24)

2. Find 112% of $780.

We know that 112%

= }56=

1·12, thus to find 112% of a quantity we multiply by 1·12. 1·12

0

780

I= I

STU)

I

($873·60)

3. Increase $230 by 8%.

We could find 8% of $230 and add it to $230. A quicker way is to find 108% of $230. 1·08

0

230 1 = 1 241<.4

1

($248-40)

4. Decrease $230 by 15%.

We could find 15% of $230 and subtract it from $230. A quicker way is to find 85% of $230.

·85

~

230

El

($195·50)

5. Convert 1 5

7 to a percentage.

5

B

17

0100 EJ

(29-4%)

6. What percentage is $83 of $125?

83

~

125

0

100

El

66A (66·4%)

Do these examples and check your answer with the ~iven answer in each case.

1. Find:

(a) 9% of $120 (b) 32% of $40·60

(c) 85% of $28 000 (d) 106% of $356 (e) 91% of $12·80

(f) 116% of $8-40 (g) 65% of $4850 (h) 7±% of $236

2. (a) Increase $48 by 12%. (b) Increase $130 by 24%. (c) Decrease $46·50 by 20%. (d) Decrease $250 by 16%. (e) Increase $34·60 by 8±%.

3. Convert each fraction to a percentage. Give answers correct to one decimal place.

(a) 152 (e) ~6

(b) 196 (f) g~

(c) j~ (g) l~g

(d) ~~ (h) g~

4. What percentage is

(a) 16 of 35? (d) $4·65 of $6·20? (b) 42of70? (e) $324of$950? (c) 126 of 300? (f) $20·84 of $36?

1. Mr. Thompson invests $2400 in a government savings loan offering 11·5% interest per year. How much interest does he earn in 4 years?

2. A jewellery store advertised an end-of-season sale where all goods were offered at a discount of 15%. How much would a diamond ring be sold for if it was originally marked at $1290?

3. A salesman sells an electronic organ for $2450. If he receives 6% commission for the sale, what is the net amount received by the store owner for this sale?

4. An estate agent charges 5% commission on the first $15 000 of the value of a property sold and 3% on the remaining value. What commission does he receive for selling a home unit for $72 000?

5. Sue receives a monthly salary of $740. She also receives a commission of 2t% on sales. Last month she sold goods to a value of $5860. How much did she earn altogether last month?

6. Tom Smith borrows $28 000 from his bank to help set up an importing business. He has to pay interest at the rate of 12% p.a. How much interest does he pay

(a) for the first year?

(b) for the second year, if he pays back $5600 at the end of the first year?

7. A manufacturer produces an article for a cost of $146. At what price must he sell it to make a gross profit of 45%?

8. A new motor car, bought for $8600, depreciates in value by 20% in the first year and by 15% in the second year. Find the value of the car at the end of the first year and at the end of the second year.

9. An electrical store bought transistor radios for $32 and marked them for sale at a price 35% above the cost. If he allowed a customer 10% discount, find the actual selling price.

10. An importer buys 1600 toy dolls for $5120 and sells them at a profit of 40%. What is the selling price of each doll?

11.

The value of a new production machine is

$132 000.

It depreciates in value each year by

15%

of its value at the beginning of the year. Find its value after 3 years.

12.

A book costing

$3-48

is sold for

$5·25.

Express the profit as a percentage of the cost price.

13.

A manufacturer makes an article at a cost of

$187·50.

He lists it for sale at

60%

above cost price and allows a trade discount of

30%

off the list price and a cash discount of

2±%

off the reduced price. What profit does he make per article if both discounts are given?

14.

The annual interest on a loan of

$4500

is

$382·50.

What is the interest rate charged on the loan?

15.

A commodity, costing

$475

per tonne, is sold retail at

76

cents per kilogram. If there is a wastage of

4%

find the profit as a percentage of the cost.

Some calculators have a fraction key

I

a%

I

which allows the entry and calculation of fraction examples.

Examples:

1.

Enter%.

4la%15

2.

Enter 3~.

3la%12Ja%J7

3. 2!

+

/s

2la%1 3labfcl4

G

7 Ia% 1151

=

I

4. 11

X

3i

₁

I

a% 121 a% I 3

0

3 I a% I 3 I a% I 8

I =

I

5. 5/1

X

3g

5Ja%J7Ja%111

0

3Ja%J13Ja%J151

=I

Note:

4J),

312 J/,

3,1]3 160.

5 l'i II(

(i) When an answer exceeds the fraction capacity of the calculator it is automatically converted to a decimal. See example 5 above.

(ii) Any fraction on the display can be converted to lowest terms by pressing the equals key. If the fraction key is pressed after the equals key the fraction on display is converted to decimal form.

For example:

1. Enter

4!3.

Reduce the fraction to its lowest terms and find its decimal equivalent.

2. Enter ~~6. Reduce the fraction to its lowest terms and express

it

as a decimal.

3. Change

5?

2 to decimal form.

4. Reduce 42 8

cf

4 to its lowest terms.

5. Simplify: (a)

!

+

i

+

152

(b)

?1 -

~ (c) 2~

+

4?

2

(d)

5l -

218 5

(e)

i

X ~ X 160

(f)

3t

+

I!

(g) 3i X 2~ 7

It

(h) li

+

2~ - 152

(i)

5!- I!-

I?o

(j) 4t X 2}

(k)

i

X

t

X

!o

(1) 5152 + Ii

(m) 3172 - Ii

+

2~

(n)

8g

+

I!

+

21

We have seen that a calculator can be used to convert a fraction to a decimal. In fact all fractions can be expressed as decimals which either terminate or recur.

Common fractions whose denominators involve powers of 2 and 5 only can be expressed as fractions with denominators that are powers of IO. This explains why such fractions when expressed as decimals

always terminate.

Examples:

(i)

!

=

180

=

0·8 (ii)

io

= 1~0 = 0-45 (iii)

i

=

1 6

0 2_c]i

0

=

0·625

When the denominator of a fraction includes factors other than 2 or 5 the fraction always yields a recurring decimal.

Examples:

(i)

l

= 0· 333333 ... , this is shortened to 0·

3

(ii) ₁5

1 = 0-454545 ... , this is shortened to 0·45

1. Change each of these fractions to an equivalent fraction with a denominator that is a power of ten and express as a decimal.

(a)

!

(c)

lo

(b)

!

(d)

15

(e) ~6

(f)

i

(g)

lo

(h) 470

2. Check by division that

1

= 0·6666 .... Write this recurring decimal in a shortened form.

3. The fraction~

=

0·2857I42857I4 .... We shorten this by writing~

=

0-2857I4. Write the decimal 0·586586 ... in a shorter way.

4. By using the division algorism express each of the following fractions as a decimal. (a) /1 (b) ~ (c)

i

(d) /5 (e)

t

The converse of the above result states: Any terminating or recurring decimal can be expressed as a fraction.

1. Express each of these terminating decimals as a fraction in lowest terms. (a) 0·8 (b) 0·46 (c) 6·285 (d) 0·0054 (e) 0·0875

2. To express recurring decimals as common fractions, follow these methods.

Example (i): Example (ii):

Change 0·7 to a fraction. Change 0·43 to a fraction.

Solution: Solution:

Let x = 0·7 Let x = 0·43

i.e. x

=

0·77777 .. . i.e. x

=

0·434343 .. . lOx

=

7·77777 .. . IOOx

=

43-434343 .. . By subtraction 9x = 7 By subtraction 99x

=

43

,',X=~

. 0 7' 7 .

I.e. · = ₉ I.e.

Express the following recurring decimals as common fractions. (a)

O·S

(b) o-38 (c) 0·03 (d) 5·6 (e) O·i48

3. Check the methods used for this more difficult example.

Example:

Change 0·26 to a fraction.

Solution (i):

Let i.e.

By subtraction

i.e.

X= 0·26

lOx= 2·6 IOOx

=

26·6

90x

=

24 X= ~6

= 145

0·26

=

1\

Solution (ii):

Let i.e.

By subtraction

i.e.

Express the following recurring decimals as common fractions: (a) 0·34 (b) 0·57 (c) 0·283 (d) 0·453 (e) 0·726

.·.X=~~

. . 43

0·43

=

99

X= 0·26 X= 0·26666 .. . lOx

=

2·66666 .. .

9x = 2·4 X= 2g4

- 24 - 90

_ _±_

- 15

. 4

0·26 =IS

You will remember something about index notation from your earlier mathematics. In the following exercise we revise the basic concepts.

1. An index number is used as a short way of writing certain products.

Example:

4 X 4 X 4 X 4 X 4

=

45

The index 5 means the base number 4 has been used as a factor 5 times.

Express each of the following products in index form: (a) 6 X 6 X 6 X 6 (e) 5 X 5

(b) 7 X 7 X 7 X 7 X 7 (f) 3 X 3 X 3 X 3 X 3 X 3 X 3 (c) 8 X 8 X 8 (g) 2 X 2 X 2 X 2 X 2

(d) 4 X 4 X 4 X 4 X 4 X 4 (h) 9 X 9 X 9

Q) 10 X 10 X 10 X 10

(j) 6 X 6 X 6 (k) 1 7 X 1 7 X 1 7

2. To find the value of a number expressed in index form we find the product of the factors.

Example:

53

=

5 X 5 X 5

=

125

Find the value of each of these expressions: (a) 104 (f) 43

(b) 25

(g) 27

(c) 82 _{(h) 10}6

. (d) 33 _{(i), 3}4

(e) 54 (j) (i)2

(k) (0-4)2

(1) (1!)3 (m) (0·6)2

(n) (0·1)4 (o) (~)3 .

(p) (1·2)2 (q) (0·3)4

(r) (11)3

(s) (O·St (t) (2!)4

3. When multiplying numbers expressed in index form with the same base, add the indices.

Example:

103 X 104

=

103+4

=

107

Write each of these products in index form: (a) 43 X 44 (f) 3 X 35

(b) 24 X 2 3 (g) 26 X 22

(c) 52 X 55 (h) 105 X 104

(d) 64 X 6 (i) 44 X 4

(e) 53 X 53 (j) 5 X 55

(k) a3 x a2

(1) (4 X (3

(m) y2 _{x y}

(n) m5 x m3

(o) x x x5

(p) y X y

(q)

a

4

x

_a5

(r) n2 _{x n}3

(s) x3 _{x x}

(t) a"'

x

a"

4. When dividing numbers expressed in index form with the same base subtract the indices.

Example:

56 7 54

=

56-4

=

52

Write the answer to each of these divisions in index form: (a) 36 7 32 (e) 412 7 46 (i) a4 7 a3 (b) 54 7 53 (f) 28 7 2 (j) y8 7 y2

(c) 106 7 104 (g) 57 _{7 5}3 _{(k) m}5 _{7 m}

(d) 83 7 82 (h) 107 _{7 10 (1) x}9 _{7 x}3

5. Any number raised to the zero index has the value of 1. Example:

53 50+3 53 5°

=

5° X -

= - - = - =

1

53 53 53

Write the value of: (a) 4°

(b) 10°

(e) 8

+

2°

(f) 12 - 6° 6. For any number x: X -1

=-

1

X

-2 1

X

=-z

X

-3 1

X = 3

X

-n 1

X =----;;

Example:

rs

=

_!__

=

1

25 32

Write the numerical value of the following: (a) 4-1 _{(c) 6-}2 _(e)

s-

3

(b)

r

3 _{(d) 8-}1 _(f)

r

4

(g)

r4

(h) 10-3

(i) To find the index which gives the square root of a number, study this:

at x at= at+t = a1 _{= a}

Thus (at)2 _{= a}

.'.at=

Ja

X

(m) t6 7 t2

(n) a14 _{7 a}7

(o) x11 7 x6

(p) a"' 7 a"

(g) 6 X 6°

(h) 8 - a0

(i) 12-1

(j) 4-3

Hence if a number is raised to the index! we take the square root of the number.

Example: 251 =

J25

=5

(ii) To find the index which gives the cube root of a number, study this: at x at x at

=

at+t+t

=

al

=

a

Thus (at)3 _{= a}

.".at=~

Hence if a number is raised to the index~ we take the cube root of the number. Example:

64t =

{.164

=4

at means the square root of

a,

i.e.

JG.

at means the cube root of a, i.e.~· In general, a* means the nth root of a.

Write down the value of:

1. 491 4. 27' 1 7. 125t

2. gt 5. 36± 8.

lOot

3. 161 6. 641 9. 1000' 1

Squares and powers can be found in a variety of ways on most scientific calculators.

1. Using the factors in the normal way.

272

21GJ27G 729.

273

2 7 0 2 7 0 2 7 1 3 J%83.

2. A quicker way.

272 27

0 0 G 729.

273 ₂₇

0 0 G B 196H3.

274 27

0 0 B G B 53144·1.

Note: In this method the number of times the equals key is pressed is one less than the index.

3. Some calculators have a squaring key

[2J

which gives the square directly.

272

27

[2]

4. Some calculators also have a power raising key such as

G·

This can be used to find any power as follows. 272 27

G

21 = 1 720.

273 27

G

3 ·1

=

1 19683.

274 27

G

41 = 1 531441.

275 27

G

51

=

1 14348907.

10. 1441

11. 16*

Practise the above methods on the following examples and check your read-out with the answer in each case.

1. 392 6. 174 11. 882 16. (3·72)3 2. 564 7. (18·6)2 _{12. (24·6)}2 _{17. 43}2

3. (9·2)3 8. (5·74)3 _{13. 15}3 _{18. 213}2

4. 7362 9. 622 14. 644 19. (4·8t

5. (4·86)5 10. 1383 _{15. (8·94)}5 _{20. (92·3)}3

To find the square root of a number the square root key is used. It gives directly the square root of the number on the display.

J44I

441

[£]

2], I

-J3784 3784

[£]

61.514225 1

-J259·8 259·81Zl 16.1183121

Find these square roots and check your read-out with the given answer.

1. -J31.36 5. -J338·56 9. )192 X 6·8

2.

y'500

6. -J56 X 14 10. -J7·9 X 15-4

3.

J356

7 .

.jW

11.

fW

79

4.

Jf924

8. -J216 X 45

12~

_{. 16}

If your calculator has a root key lx1_{/vl then the roots of numbers can be found as follows.}

J758T

= 7581* 7581 lx,/vl 31 = I r---1 -l-9.-64-4-55-.4--.1

~

= 7581* 7581 lx,/yl41 = 1 1 9.3310738 1

;j758f

= 7581! 7581 lxvvl 51= I

Note: If your calculator has no root key but has a power raising keyG, then roots can also be found quickly.

(i) Using a fraction key

J758T

= 7581* (ii) Using reciprocal key

{/7581 = 7581*

7581

G

I la%131 =I 19.6445541

7581

G

3

[g

1 = 1 19.6445541

Use a calculator to find the value of the following:

1. ~ 6.

\(56

11. -V06·8)2

2.~ 7.

J292

12.

~

6

3.

JI750

8. -V84 X 72 13. (92 X 170)!

4. 655* 9 . .V912 X 66 14. {/742 X 13-9

5. {)7000

wfJf

0

48 15 j84

X 69

0 72

Example:

A cube has a volume of 17 576 em 3_{• Find the side length of the cube and its surface area.}

Solution:

The volume of a cube is given by V

=

x3

So side length x

=

-V17 576 em

=

26 em (from the calculator) Surface area of a cube is given by A = 6x2

.'.A = 6 x 262

=

4056 (from the calculator) The surface area of the cube is 4056 cm2

.

1. Find the side length of:

(a) a square with an area of 1521 m2 •

(b) a cube with a volume of 5832 cm3 .

2. The area of a square field is 17 ·64 hectares. Find the length of the field in metres. 3. A metal cube has a side length of 46 mm. Find the surface area and volume of the cube.

4. The dimensions of a rectangular field are 744 metres by 217 metres. Find the diagonal distance across the field.

5. Calculate the area of a circle with radius 24 em. Use the

0

key on your calculator and give answer to the nearest square centimetre.

6. The formula r

=

R

can be used to calculate the radius of a circle given its area. Find, to the nearest metre, the radius of a circle having an area of 1000 square metres.

7. (a) If 7·2x3 = 50, find x correct to I decimal place. (b) If x5

= 300, find the value of x(I

+

x) correct to two places of decimals. 8. The volume of any spherical object is calculated by using the formula V

=

1nr

3

. What is the

9. Using the formula of question eight find the radius of a sphere having a volume of 200 m 3 . Give

answer correct to one decimal place.

10. If n

=

j?f

find n when a

=

4·62, b

=

7·38 and c

=

5·66. Give answer correct to two decimal

places.

Powers of 10 are often used by scientists and mathematicians to write large or small numbers in a shorter form.

1. Write the value of

(a) 103 _{(b) 10}5 _{(c) 10}2 _{(d) 10}6 _{(e) 10}8 _{(h) 10}9

2. Write each of these numbers as a power of 10. (a) 10000 (c) 1000

(b) 100000 (d) 1000000

(e) 10 000 000 000

(f) 100 000 000

(g) 10 000 000

(h) 1 000 000 000 3. Write the following in expanded form.

Example:

6 X 106 = 6000000

(a) 5 x 104

(b) 6 X 103

(c) 8 x 107

(d) 4 X 102

(e) 3 x 105

(f) 9 X 106

(g) 7 X 108

(h) 2 X 104 4. Complete the following statements using powers of 10.

Example:

300000 = 3 X 105 (a) 70 000 = 7 x ...

(b) 4000000 = 4 X .•.

5. Copy and complete this table:

10-1 =

_!_

10

10-2 =_I_

100

(C) 50 000 000 = 5 X

(d) 9000

=

9 X

10-3 = _1_

10-4

=

_1_

6. Write the following in fraction form:

Example:

3 X 10-4

=

3 X

10~00

=

10~00

(a) 7 X 10-3 _{(b) 6 x 10-}6 _{(c) 5 X 10-}4

7. Complete the following using negative powers of 10.

Example:

8

= 8 X 1 = 8 X 10-5 100 000 100 000

6 (a) 10000

7

(c) 1000

15

(e) 200000000 = 2 x ...

(f) 600000

=

6 X

10-5 = _1_

10-6

=

_1_

(d) 9 x 10-5

8 (d) 1 000 000

(e) 4 X 10-7

SCIENTIFIC NOTATION FOR LARGE NUMBERS These large numbers are written in scientific notation.

(i) 400 000 000 = 4 X 108

(ii) 93 000 000 = 9·3 X 107 (iii) 356 000 = 3 ·56 X 105

In scientific notation, a number is written as a number between 1 and 10 and multiplied by a power of 10.

Example:

Express 748 000 in scientific notation.

(i) Mark the spot between the first and second significant figures (see arrow). This will be the position of the decimal point in scientific notation.

(ii) Count the number of figures between the arrow and the existing decimal point in the given number. Counting to the right is positive.

1. Write these numbers in scientific notation by filling in the power of 10.

748000

r

The power of 10 is 5 ... 748 000 = 7·48 X 105

(a) 170000

=

1·7 X • . • (d) 820000

=

8·2 X • . . (g) 14600000

=

1·46 X (b) 58000

=

5·8 X . . • (e) 67000000

=

6·7 X (h) 8960000

=

8·96 X ••• (c) 6 340 000

=

6·34 X . . . (f) 8700

=

8·7 X (i) 583 000

=

5·83 X 2. Complete these:

(a) 620 000 = . . . x 105

(b) 79 000

= . . .

X 104

(c) 4200000

= ...

x 106

(d) 765 000

= ...

X 105 3. Write these numbers in scientific notation:

(a) 16000 (c) 8400 (b) 720 000 (d) 9 260 000

(e) 74000000 (f) 658 000

(e) 62400000

= ...

x 107

(f) 9450000

= ...

X 106

(g) 478 000 000 (h) 19 620 000

4. The star nearest the earth is approximately 41600 000 000 000 kilometres away. Express this distance in scientific notation.

5. The planet Pluto is approximately 7 360 000 000 kilometres from the sun. Give the distance in scientific notation.

6. When rounding off a number to a given number of significant figures one additional figure must be looked at. If this additional figure is 5 or more the final figure is rounded off upwards. If the additional figure is less than 5 the final figure remains unaltered.

For example: (i) 5·826 x 105

= 5·83 x 105

when rounded off correct to 3 significant figures. (ii) 4·081 x 108 _{= 4·1 x 10}8 _{when rounded off correct to 2 significant figures.}

(iii) 8 463 000 = 8·46 x 106 _{when rounded off correct to 3 significant figures.}

Round off (a) 4·327 x 105 correct to 3 significant figures. (b) 2·7043 x 108 _{correct to 4 significant figures.}

(c) 5·0038 x 106 _{correct to 4 significant figures.}

(d) 24 720 000 correct to 2 significant figures. (e) 1·70049 x 107 _{correct to 5 significant figures.}

SCIENTIFIC NOTATION FOR SMALL NUMBERS

Scientific notation may also be used to represent very small numbers.

Example:

Write ·000057 in scientific notation.

·000057

=

5'7

=

5·7 X 10-5 100000

Using the same quick method as before we could proceed as follows:

(i) Mark the spot between the first and second significant figures (see arrow).

(ii) Count the number of figures between the arrow and the decimal point in the given number. Counting to the left is negative.

1. Write these numbers in scientific notation by filling in the power of 10. ·000057

The power of 10 is -5

:. ·000057 = 5·7 x 10-5

(a) ·00096 = 9·6 X . • . (d) ·0075 = 7·5 X , , , (g) ·000081 = 8·1 X ...

(b) ·00375 = 3·75 X , , , (e) ·000022 = 2·2 X , , , (h) ·0468 = 4·68 X , , ,

(c) ·0000082 = 8·2 X . . . (f) ·0094 = 9·4 X , , , (i) ·000000926 = 9·26 X

2. Write these numbers in scientific notation: (a) ·0078 (f) ·0029

(b) ·000037 (g) ·000065 (c) ·068 (h) ·00729 (d) ·00074 (i) ·071 (e) ·0643 (j) ·0006

(k) ·0000064

(1) ·00363

(m) ·0918

(n) ·00077 (o) ·00463

(p) ·584 (q) ·0000078

(r) ·0000188 (s) ·00914 (t) ·000000075

3. A micron is one-millionth of a metre. Write 3 microns as a decimal of a metre, using scientific notation.

4. Write 73 millimicrons in metres, using scientific notation.

5. Round off:

(a) 2·706 X 10-3 (b) 4·0032 X 10-6 (c) 1·8326 x 10-9

(d) ·00007293 (e) 6·287 x 10-5

Example (i):

correct to 3 significant figures. correct to 4 significant figures. correct to 2 significant figures. correct to 2 significant figures. correct to 3 significant figures.

(2·8 X 106) X (4 X 10-2) = (2·8 X 4) X (106 X 10-2) = 11·2 X 104

= (1·12 X 10) X 104

=

1·12 X 105

17

Example (ii):

1·2 X 104 1·2 104

--:-4·-::-8-x---,--1 0---;;2 = -4.-8 X -~ 0---2

= ·25 X 106

= (2·5 X 10-1) X 106

=

2·5 X 105

(regrouping)

Most calculators have an exponent key IExPI which enters the power of ten and allows direct cal-culation with numbers expressed in scientific notation.

Example (iii):

6·84 IExPI5

0

3·78 IExPI 31 = I

Example (iv):

(7·3 x 103) 7 (5·94 x 10-2) 7·31ExPI3

Q

5·941ExPI21+f_ll =I

Notes:

(i) Notice particularly how the negative exponent is entered.

I

This means 2·58552 x 109

(ii) Since the answer is within the normal display capacity of the calculator it is given is 122 895·62.

If this is required in scientific notation it must be rewritten, e.g. 1· 23 x 105 correct to 3 significant figures.

Evaluate each example and give answers in scientific notation, correct to 3 significant figures.

1. (3·8 X 106) X (5·75 X 103)

2. (7-4 X 104) X (6·83 X 102)

3. (9-4 X 105) 7 (2·5 X 102)

4. (7·83 X 107) X (4·7 X 10-4)

5. (3·166 X 103) 7 (8·49 X 10-2)

7·938 X 106

6. 2

8-47 X 10

7. (30 X 8·92)2 X (4·63 X 103)

8. 7·62 X 10

3

5·169 X 10 4

9. (8·92 X 106)2

10. J8·6 x 1016

11. (80 X 5·92)2 X (4·7 X 104)

12. 79 000 X 9400

13. (7·16 X 103) 7 (4·94 X 108)

5·634 X 108 14

• 30·64

15 (80 X 40)

2

. 2·79 X 10 3

16. 4·91 X 10

5

7-46 X 109

1. Write down the value of each of these simple expressions. (a) (7 x 3)

+

6 (f) 40 + (17 - 9)

(b) (6

+

2) X 5 (g) (7 X 5) - (5 X 4)

(c) 8

+

(4 X 6) (h) [(5

+

4) X 6] + 2

(d) 30- (5 X 5) (i) 5 X [15 - (4 X 3)]

(e) (30 - 5) x 5 (j) 30 - [12 + (4 + 4)]

(k) (24 + 2) + (18 + 3) (1) [ (30 - 12) + 6]

+

7

(m) 6 X [(5 X 9)- (3 X 10)] (n) 100- [17

+

(6 x 3)]

( 0) 56 + [ ( 4 X 6) - (2 X 8)]

2. A vinculum (or fraction line) acts as a grouping symbol.

Thus\~~~

=

3 9

6

=

4.

Simplify these expressions:

(a)~

13- 7

(b) 4 X 9 7

+

5

(c) 16

+

8

16 - 8

(d) 33

(3 X 7) - 10

60

(e) 3

+

(2 x 6)

(f) 32 - 8 6+2

48

(g) (9 X 3) - 11

h 18

+ 12

( ) 24 - (7 X 2)

(1 ') 15

+

(9 X 3)

15 - 8

3. The radical sign(;-) can also act as a grouping symbol. Thus

.)9

+

16

=

J23

=

5.

Simplify these: (a) .j(5 x 7)

+

1

(b) )102 _{- 8}2

(c) .)(109 - 11) + 2

(d) .)57 - 21

(e) {/62 - 32

(f) {/96 + (7

+

5)

4. The symbol for absolute value, which consists of a pair of vertical lines on either side of a number or expression, also acts as a grouping symbol.

You will remember that the absolute value of a number is its numerical value without regard to sign.

Thus 171

=

7 and l-71

=

7.

Evaluate these: (a) 115 - 71

(b) 19 - 141

(c) 13 x ( -6)1

(d) 16 - 15 - 121

(e)

I (-

5) x 4

+

91 (f) 17 - 131

+

6

(g) 13 X ( - 7) I

+

113 - 41

(h) IC5 X 8) - (7 X 8)1 - 10

(i) 19 - 151 X 13 X (5 - 2)1

When grouping symbols are inserted in an expression they indicate the order in which the operations are to be done and thus make the expression unambiguous. Inserting a pair of grouping symbols for every operation however, would often make the expressions very unwieldy. For this reason the following convention has been universally agreed. The convention allows grouping symbols to be omitted in many expressions and still leave the meaning clear.

An expression without grouping symbols, powers or roots is simplified by first working any multiplications or divisions from left to right and then working any additions or subtractions from left to right.

Examples:

(i) 9

+

4 X 8

=

9

+

32

=

41

(ii) 5 X 9 - 36 + 12

=

45 - 3

=

42

1. Simplify each of the following expressions: (a) 5

+

4 x 7 (e) 8 x 10 + 5 (b) 20 - 12 + 4 (f) 9 X 6 - 6 X 5 (c) 6

+

4 x 8 - 5 (g) 50 + 5 - 40 + 8 (d) 3 X 6

+

3 X 4 (h) 30

+

70 X 2

+

25

(i) 36 + 9

+

6 X 7

(j) 35 - 120 + 5 (k) 72 + 4 X 2

(1) 54 - 4 X 6 2. If an expression involves powers or roots, these have priority and are evaluated first.

Examples:

(i) 8

+

32 = 8

+

9

= 17

Simplify these: (a) 62

- 32

(b) 72 - 5 X 4 (c) 5

+

6 x

Jf6

Example:

(ii) 4

+

5 X 42

=

4

+

5 X 16

= 4

+

80

=

84

(d)

J8T

+

7 X 9

(e) 15

+

5 x 23

(f) 33

+

3 X

J64

Evaluate 96·4

+

37·2 x 23·9.

(iii) 16

+

J64

X 2

=

16

+

4 X 2

=

16

+

8

=

24

(g)

J8T

+ 9 (h) 15- ~X 4

(i) 25 X 3 + 8

Using the convention for order of operations we do the multiplication first and then the addition.

37·2

0

23·9

0

96-41 =

I

t)85A8

Do these examples and check your read-out with the given answer in each case.

Answers

1. 63·9

+

7·35 X 54·7 7. 56 X 242 1. 465·945 7. 32256 2. 49·74481 8. 23·596606 3. 175·8 9. 73·2736

2. 360 + 15·8

+

26·96 8. (18-4)3 + 264

4. 325-48 1 0. 63·821766 5. 881·57142 11. 9·7467943

3. 58·6 X 9·6 + 3·2 9. (17·2 - 8·64)2

4. 75·84

+

(15·8)2

10. 9·6

+

14 X

Jf5

6. 67-449961 12. 75·646235

5. 863

+

72 + 28

+

16 11. )17

+

13 X 6 6. 35

+

9 X

Jf3

12. (32·6)3 + 458 ORDER OF OPERATION FACILITY

Some scientific calculators have a facility which automatically follows the conventions for order of operations when the operator presses the keys in the order stated in the examples.

Do this calculation 3

+

4 x 5 on your calculator by pressing the keys in the order given from left to right.

The correct answer is of course 23. If the read-out on your calculator is 23 then it has the order of operations facility. Such calculators may also have bracket facilities for handling grouping symbols when they occur.

If the read-out displayed is 35 then your calculator does not possess the order of operation facility and you must be careful to always follow the conventions discussed earlier. In this case you must do the multiplication first and then the addition.

The memory key on your calculator allows a number or part answer to be remembered while other calculations are being carried out. This facility often allows a quicker method for doing a calculation and contributes greatly to the flexibility of the operator in using the calculator.

Check your calculator to see what type of memory keys it has. The following types are common.

This key transfers the number on the display into the memory. Any previous number stored in memory is automatically cleared. If zero is entered into memory by using one of these keys, the memory is cleared.

This key transfers the number on the display to the memory positively. It is added to whatever was in the memory.

This key transfers the number on the display to the memory negatively. It is subtracted from whatever was in the memory.

This key recalls the number in the memory to the display so that it can be used in further calculations.

It does not clear the memory.

This key clears the contents of the memory.

0 37·3

(l) 5·6

+

8·7

(ii) 5·3

+

7·9 6-4- 2·5

(iii) 15

2

+

₃₆2

148

5·6

B

8·7

B

~

37·3

LJ B G

6-4 EJ2·5

G

~

5·3

G

7·9

B

LJBB

15

0

~

36

0

EJ EJ

LJ

148

EJ

0

. 6·8 X 7·5

+

5·7 X 9·2

(IV) 9·8

+

13·6

6·8

0

7·5

B

~

5·7

0

9·2

G

EJ

9·8

G

13-6

B G

EJ

B

[B

2.601\391 ()

3.20571\02

Note: The method shown for calculating each example is not necessarily the only one. Once the

operator is familiar with the memory facility he may devise different ways of doing these calculations.

Do these calculations and check your answer with the given answer in each case.

1. (16·8)2 1- (12·9)2

17·8 1- 26·7 2

' 15·9 - 6·8

646 3. (2·9)2

4. 5·87 X 17 1- (6·4)2

5. 584 21·6 1- 15·8

6. 282 1- 2 X 16 X 154

7. 15·64 4·91 1- 3·85

8 . ._)34·75 1- 6·7 X 2·85

9. 272 1- 322 1- 192

10. 245 (6·8 1- 5·9)2

11. 293 - 38 X 56

34·9 - 15·35 8·36 1- 1·99 12.

28-46 - 16·92

13. (6·2 X 5·3)2

1

14. 1 -

8-472

15. n x (5·423)

2

18·927

1

17

• (7·365)2

18 72·094 X 116·8 . (79·64)2 _{- (16·78)}2

19.

i

X 17-42 -

t

X 13-81 (4·609)5

20

' 27·8 - 19·66

Part of the micro electronics of the modern computer.

I

CHAPTER 2

Example:

Express Sx

+

3y - z - 2x

+

4y

+

2z in its simplest form.

Solution:

~+~-z-h+~+b=~-h+~+~-z+b

= 3x

+

7y

+

z

(collecting like terms)

Express the following algebraic expressions in their simplest form.

1. 6a - 3b

+

7 a - 8b 16. 4m2

+

9 - 10m - 6m2

+

3

2. 4x - 3y - Sz

+

y

+

x

+

z 17. 6 - 2x- 7x2 - 12

+

5x2

+

2x

3. 7a

+

4b

+

c - 9a - 6b

+

c 18. lOxy - y2

+

_5x2

- 2xy

+

y2 - 8xy

4. 3r - 4s

+

6t - 8r - Ss 19. 3a - 7

+

4a2 - 6a

+

3a2 - 15

5. Sm - 2n - 3n

+

8m - 4n 20. 6x2

- xy- 3x- 2x2

+

5xy

6. -2x

+

7y- z - Sy- 3x

+

6z 21. 2a2_b

+

_3ab2

+

4a2b - 7ab2

7. 3a

+

Sc - 9a

+

4b - 2a

+

2b 22. 7x3

- 3x2y

+

4xy2 - 3x3

+

4y3 - 2x2y

8. x - 7y

+

6x - z

+

Sy - 4z 23. a4

+

_3a2

- 2a - 7a2

+

a3

+

6a2

9. lOd- 9e

+

7d- 3f- Se 24. yz - zx

+

xy - yz

+

xy

+

zx

10. 4b - 3a - 9b - 7c - Sa 25. 2x3 - x2 - 3x

+

2 - x3

+

x - 1

11. 3x2

+

2x - 4

+

5x2 - 7x 26. 15xy2 - 8x2y

+

lOxyz - 4x2y

+

Sxyz

12. 4ab

+

6a - 2b - 3ab

+

7a 27. 7a2b

+

8ab2 - b3 - 5a2b

+

2ab2

13. 9a2

+

6a - 4 - 5a2

- 9b

+

6 28. 4b3

+

c3 - 2abc - a3 - b3 - 3abc

14. xy - 3x2

- 4y2

+

6xy - 2y2 29. a3 - 3ab

+

b2 - 2a3 - a2

+

3b2

+

Sab

15. 6ab

+

a2

+

2b2

- a

+

3a2 - 2ab 30. 4x3

+

3x2

+

x - 4 - 5x2 - 7x - 11

Example (i):

Remove parentheses and simplify (3x

+

7) - (2x - 4).

Solution:

(3x

+

7) - (2x - 4)

=

3x

+

7 - 2x

+

4

Remove parenthesesJ~:nd simplify:

/ .-~-<, ~

1. (Sa - 6)

+

qa

-::;4)

/

2. (4x - S)

--(x+

6)

3. (7m

+

6) - (3m::, S) 4. (8 - 3y) :..,.. (6

+

6y)

S. (9a

+

2b) - (3a - 8b) 6. (Sx- 7y)

+

(4x- lly) 7. 3x- (4x

+

6y- 4)

8. 4a - 2b - (7

+

3b - a)

9. Sxy - 2x2

+

_(3x2

- Sxy)

1. Add Sx- y

+

4 to 2x- 3y - S. 2. Subtract 4x

+

y - S from x - 6y

+

2. 3. Subtract 3x2 _{- Sx}

+

_{6 from 2x}2

- 2x - 3.

4. Add 3xy

+

2x- 4y to 7 - 3x- 8xy.

S. Subtract Sx - 6 from 9x2

- 2x - 8.

6. Find the sum of 3x

+

6, S - 2x and 8x- 9.

10. Sx2

+

_{9 - (2x}2

- 3x - 7)

11. 2a - (a

+

6 - a)

12. -x2

+

(7x2 - 3x

+

6)

13. 6ab - (2ab

+

3a - 2b - 7ab)

14. 4x2 _{- ( -Sx}2

) - (3x2

+

6)

IS. 4a - (a

+

b) - (a - b)

16. 7 - (x

+

2)

+

(3 - 2x)

17. 2a- c - (7c

+

2la)- (20a

+

8c)

18. b

+

a - c - (c - a - b)

7. Find the sum of 3x- 4y

+

3z and x - 3y- 8z. 8. Subtract 3a3

+

_2a2

- 6 from 4a2 - 2a - 8.

9. Add 3x2

+

_{7xy- Sy}2 _{to 8x}2

- 9xy- 3y2•

10. From 6a2 _{- 7ab - 12 take -4ab}

+

_2. Example (ii):

Show that 3(x - 8) - 2(Sx - 3) = -7x - 18.

Solution:

L.H.S.

=

3(x - 8) - 2(Sx - 3)

= 3x - 24 - lOx

+

6

=

-7x- 18

= R.H.S.

Show that each of the following is true.

1. 4(x - 4)

+

3(2x - 6)

=

lOx - 34

2. 3(a

+

6) - 2(a - S)

=

a

+

28 3. 2(m

+

4) - 3(m- 2) = 14-m 4. 6(3x

+

4) - S(3x

+

2)

=

3x

+

14

s.

S(2y - 4) - 3(6 - 2y)

=

16y - 38

6. 3(a- 2) - 2(4a- 3)

=

-Sa

7. Sa - 6

+

3(6 - 3a - 2a2

) = 12 - 4a - 6a2

8. 4x2

- 3x - 2(x2 - 4x

+

7) = 2x2

+

_{5x - 14}

9. 2(1 - 2x) - 3(6

+

x)

=

-7x - 16

10. 7(3x - 2y) - 3(2x - 8y)

=

15x

+

lOy

11. 5(a

+

b +c)+ 6(2a- 3b- c)= 17a- l3b- c

12. 3(4x - y

+

2z) - 5(2x - 2y

+

3z) = 2x

+

7y - 9z

13. 8(x2

+

_{2x - 5) - 5(x}2

- x

+

2) = 3x2

+

21x - 50 14. 2(a2

- 2a) - 3(2a2

+

3a - 4)

=

-4a2 - 13a

+

12

15. (3x - 2) - (4x - 5)

+

(x

+

6) = 9

16. 2(a - 1)

+

3(1 - a) - 2(2 - 3a)

=

Sa - 3 17. 3(x - y

+

z) - 4(y

+

x - z) = - x - 7y

+

7z

18. 6(a2

+

_2ab

+

_b2

) - 3(2a2 - 3ab

+

2b2)

=

21ab

19. 3(6 - 2x

+

7x2₎

+

_5(x2

- 4x - 2)

=

26x2 - 26x

+

8

20. 5(2a2

- 3a - 4) - 4(a2 - 4a - 5)

=

6a2

+

a

Example (i): 3xy2 _{X -2x}3

y

=

3 X -2 X X X x3 X y2 X y

=

-6x4y3

1. 3a x 7 6. 3m x 2n 11. -2x x -5x

2. 6 x 4x2 7. 5x2 _{x 2y 12. (3a)}2

3. 9mn x 8 8. 4a2 _{x 3a 13. 4a}2_{b x 6ab}3

4. 12 x 3xy2 _{9. 2x}2 _{x 6x3 14. a}2_{b x -3b}2_c

5. 5x x 4x 10. 6m2 x -2m 15. (-x3)2

Example (ii):

-3x(x2 _{- 2x}

+

_{5) = -3x3}

+

_6x2

- 15x

1. 2a(a2

- 3a

+

5) 9. 2mn(m2

+

4n - 3)

2. 4x(~ + 2y - 6) 10. -4z(x - y - 2z)

16. xyz x x2_yz3

17. 3a2_b3 _{x - 2a}4

b

18. (-2x2_y)3

19. 5ab3_c2 _{x -2ac}3

20. -4x2_{y x 6y3}

3. 5y(3y3

- 4y2

+

y) 11. -2xy(5x2y - 3xy2

+

4xy)

4. 2m(m2

+

_n3

- mn) 12. ab(3a3b - 2b2

+

4a - ab2)

5. 3a(6a - 2b2

+

_{c) 13. 3m}2_{n(2mn - 4mn}2

+

_6n)

6. -3a(a - b

+

c) 14. - 5xy2_{(7xy - 3x}2_{y - 2xy}2 )

7. 5x(3x3

- 4y2

+

2z) 15. -abc(abc- a2bc- ab2c)

8. -2q(3p - 2q- 4c)

Example (iii):

Remove parentheses and simplify by collecting like terms. 8a2

- 7a - 2a(a - 5)

Solution:

8a2

- 7a - 2a(a - 5)

=

8a2 - 7a - 2a2

+

lOa ·

= 6a2

+

3a

1. 3x2

+

4x

+

2x(x - 4) 11. 3a(a - 4)

+

2a(3a

+

6) 2. 6a2 - 3a

+

6

+

5a(2a

+

1) 12. 4x(x

+

6) - 3x(x - 4)

3. 3b2 _{- 4b - 2b(b}

+

_{4) 13. m(2m - 7) - 5m(4m}

+

₁₎

\

4. 4m2

+

6m - 5m(2m - 8) 14. 2y(6

+

4y) - 6y(2 - 3y)

5. 3y2 _{- 5y - 4 - 3y(2y - 6) 15. 6c(3c}

+

_{8) - 2c(c - 1)}

6. x2 - 3x

+

8 - x(5x

+

3) 16. 2b(b - 4) - b(2b

+

5)

7. 16 - 5n - 3n2

- 8n(n - 3) 17. 3a(a2 - 2a

+

1)

+

a(a2 - 3a)

8. 6a3

- 4a2 - 18a

+

3a(a2 - 2a

+

6) 18. 4x(x2

+

3x - 5) - 2x(3x2 - 5x

+

6)

9. 4x3

- 3x2

+

6x - 2x(5 - 3x

+

2x2) 19. n2(3n

+

6) - 5n(2n2 - 4n

+

1)

10. m2_{n - 2mn}2

+

mn - 3mn(m - 4n

+

5) 20. 2m2(3m2

+

2m

+

6) - 5m(3m2 - 4m)

Example:

Expand (x

+

5) (2x - 4)

Solution:

(x

+

5)(2x - 4) = x(2x - 4)

+

5(2x - 4)

= 2x2 - 4x

+

lOx- 20

=

2x2

+

6x- 20

Expand the following products: ';(

1. (x

+

2)(x

+

3) 11. (x- 4)(2x

+

3) 2. (a

+

1)(a

+

5) 12. (3a - 2)(2a

+

5) 3. (x

+

3)(x - 1) 13. (m - 7)(m

+

1)

4. (m

+

1)(m - 6) 14. (3c

+

4)(5c

+

2) 5. (y

+

5)(y - 8) 15. (4n - 3)(n

+

3) 6. (n

+

2)(2n

+

6) 16. (x- 1)(x- 3) 7. (a

+

3)(3a - 4) 17. (a- 3)(2a- 5) 8. (2b

+

4)(b - 2) 18. (3m - 1)(2m

+

1)

9. (3x

+

2)(4x - 5) 19. (2x

+

3y)(x

+

2y)

10. (2z

+

7)(z - 1) 20. (a

+

b)(3a - 4b)

27

:\;'

21. (2m - n)(3m

+

n)

22. (x - 2y)(3x - y)

23. (x

+

2)(y - 3) 24. (a

+

3)(2a - 3b)

25. (2x2

+

x)(x

+

4) 26. (a- 7)(3a2

- a)

27. (2x

+

4)(2x

+

4) 28. (a

+

b)(a - b)

STANDARD RESULT

Example:

Find the product of (2x - 3) (2x

+

3).

Solution:

(2x - 3)(2x

+

3)

=

(2x)2 - (3)2

=

4x2 - 9

Find the following products:

1. (a

+

1)(a - 1) 9. (5x

+

3)(5x - 3) 2. (x

+

4) (x - 4) 10. (9 - 7a)(9

+

7a) 3. (m - n)(m

+

n) 11. (2m - n)(2m

+

n) 4. (b - 3)(b

+

3) 12. (4x - 5y)(4x

+

5y) 5. (2y

+

1)(2y - 1) 13. (ab

+

1)(ab - 1)

6. (5 - x)(5

+

x) 14. (3p

+

q)(3p - q) 7. (1

+

m)(l - m) 15. (a - 7b)(a

+

7b) 8. (3a - 7)(3a

+

7) 16. (x2

+

y2)(x2 - y2)

STANDARD RESULTS

Example:

(3x - 2y)2 = (3x) 2 - 2(3x) (2y)

+

(2y) 2

=

9x2 - 12xy

+

4y2

1. (x

+

3)2 9. (2x - 4)2 2. (y- 1)2 _{10. (1 - a) 2}

3. (a+ 6)2 11. (1 - 5y)2 4. (m- 4)2 12. (x

+

y) 2 5. (2x

+

1)2 _{13. (3 - m) 2}

6. (3b - 1)2 _{14. (2 - 5a)2}

7. (5a

+

2)2 15. (p - q) 2 8. (6y

+

5)2 16. (1 - 2x)2

17. (2x

+

3y)2 18. (4a - 3b)2 19. (1

+

9m)2 20. (6n - 5m)2 21. (7y

+

2z)2 22. (mn

+

2) 2 23. (ab- 3)2 24. (xy- 1)2

17. (3 - a 2)(3

+

a 2) 18. (x2

+

4y)(x2 - 4y) 19. (11a

+

9b)(lla- 9b) 20. (ab

+

8c)(ab - 8c) 21. (3xy - 1)(3xy

+

1)

22. (3a2 - 5b 2)(3a2

+

5b2) 23. (x3 - 1)(x3

+

1)

24. (x2y

+

3)(x2y - 3)

25. (2ab

+

c)2 26. (3xy - z) 2 27. (a - 4bc)2 28. (x2

+

5)2 29. (a2 - 1)2

Example:

(3x - 4)(x2 - 2x

+

5)

=

3x(x2 - 2x

+

5) - 4(x2 - 2x

+

5)

= 3x3

- 6x2

+

15x- 4x2

+

8x- 20

=

3x3

- I Ox2

+

23x - 20

Find these products:

I. (a

+

1)(a2

+

3a

+

3) 2. (x - l)(x2 - 2x

+

1)

3. (y

+

3)(y2

- 4y - 3)

4. (m - 6)(2m2

+

3m

+

1) 5. (2b

+

3)(b2 - Sb - 7)

6. (Sx - 1)(x2

- 6x

+ 1)

7. (2n

+

3)(3n2 - 2n

+

4)

8. (4x - 5)(2x2

+

3x - 7)

9. (2a - 1)(4a2

+

2a - 1)

10. (4 - 3x)(x2

- 4x - 3)

Example:

Ifm

=

6, n

=

-3, find the value of .J5m- 2n

Solution:

11. (5 - a)(3 - 6a- a2 )

12. (3m

+

4)(2m3 - m2 +2m - 5)

13. (a

+

b)(a2

+

b2)

14. (x - y)(x2 - xy

+

y2)

15. (z - 3)(z2 - 6z

+

9)

16. (x2

+

2x)(3x2

+

2x- 1)

17. (3a2

+

a)(a3

- 3a2

+

6a- 3)

18. (m2 - 2m)(2m3 - 5m2

+

4m - 1)

19. (3x - 1)(5x3 - 9x2 - 2x - 2)

20. (2x2 - 3x)(x3 - x2

+

3x - 4)

.Jsm- 2n = .j(S x 6)- (2 x -3) =

J36

= 6

I. If x

=

4, find the value of each expression. (a) Sx

+

2 (f) 40 - Sx (b) S(x

+

2) (g) 3x - 2x2

(c) 3x2 _{(h) (x}

+

₂₎2

(d) (3x)2 _{(i) (x - 3)}2

(e) 4x2

+

_{2 (j) (x}

+

_{7)(x - 7)}

(k) (4x - 3)2

(1) 3x(1 - x2 )

(m) (x- 1)3

(n) .J7x- 3

( o) x2

- Sx

+ ·

6

2. If a = -2, b = 6, c = - I, evaluate each of the following expressions.

(p) (x- 1)(3x

+

5)

(q) ~x2

+

11 (r) 5x2 _{x 2x}3

(s) 4x4 + 8x (t) (3x - 10)4

(a) a

+

b

+

c (f) a2 - b (k) .Ja

+

2b

+

c (p) (a

+

b)(a - b)

(b) 3abc (g) ab

+

be (1) (a

+

b - c)2 _{(q) b(3a}2

+

_2a)

(c) a(b- c) (h) (a- b)2 (m) c3 - b3 - a3 (r) ~c

+

3b - Sa

(d) a+ 2b - 3c (i) 2a(a- 2b

+

3c) (n) (b- a)(c- a) (s) (4a2 - c2)(3b2

+

c2)

(e) ~ (') a+ c (o) a2 +a+ 2b (t) (a+ c)2

b

+

2c J a - c b

+

c2 _b2

3. If x

=

!,

y

=

t,

z

=

t,

find the value of: (a) x

+

y (e) x2

+

z

(b) y - z (f) 4x

+

6y - 8z

(c) x

+

y

+

z (g) 4x2

+

2x - 3z

(d)

~

(h)

III

z ~ y

Example:

(i) (x

+

z)(x - z)

(j) (x + y)z (k) (x - y)2

(1) X

+

y

y+z

(m) (x

+

3y

+

2z)3 (n) 24x3_yz

(o) (2x

+

3y)(6y - 8z)

(p)

~

~ yz

If l = 67·3 and g = 9·8, find the value of

if

correct to one decimal place.

Solution:

j{=ftl

~ 2·6 (from the calculator)

1. If x

=

7·8, y

=

5·2, z

=

6·7 find, correct to one decimal place, the value of: (a) xyz (c) (y

+

z)3 _{(e) {jx}

+

_y

+

_{z (g) 40 - z}2

(b) yz- x (d)

~

(f) 12x- 7y (h)

~

x

+

y 3z

y

y

2. Given m

=

3·7 x 103 _{and n}

=

_{9·3 x 10}5 _{find the value of 3 mn.}

3. Given that x

=

8·73 x 106 andy

=

4·1 x 104 find the value of

JXY.

4. Find correct to one decimal place the value of

G~

J

when a

=

19·8 and b

= -

6·9. 5. Find to the nearest integer the value o( P3 _{if P}

= )

av2

- c and a

=

11·85, v

= -

5·13 and

c = 14·74.

6. If t

=

5·3 X 10-3 and v

=

7·8 X 10-2, find to 3 significant figures the value of )20 - tv.

7. What is the value of {jl2a - 7b when a = 12·5 and b = -10·3. Give your answer correct to 2 decimal places.

8. Calculate correct to one decimal place the value of

ft

when x

=

4·37, y

=

6·74 and z

=

4·86.

9. If D

=

Jm xn

106, find D when

m

=

1·78 x 10-3 and

n

=

9·89 x 102. Give your answer

correct to 3 significant figures.

10. Find the value of

-JW

if W

=

1·86x3 _{- 4·73y}2_{, given that x}

=

_{4·97 andy}

=

_{3·98. Answer}

correct to one decimal place.

Example:

If s

=

ut

+!

at2, finds when u

=

30, t

=

4 and a

=

3. Solution:

s = ut

+

!at2

1. The distance in metres a stone falls from rest after t seconds is given by the formula d

=

!

gt2

where g

=

9·8. Find d when t

=

2.

2. The formula for converting a temperature expressed in the Celsius scale (C) to one expressed in

the Fahrenheit scale (F) is F = 9

~

+

32.

Find the Fahrenheit readings for the following Celsius readings.

(a) C

=

10 (b) C

=

25 (c) C

=

35 (d)

c

= 40

3. Water boils at 100°C. What temperature would this be on a Fahrenheit thermometer?

4. The reverse formula C

=

~ (F - 32) can be used to convert a Fahrenheit temperature reading to a Celsius reading. What is the equivalent Celsius reading for a temperature of 100°F?

5. The angle sum of a polygon with n sides is given in degrees by the formula S

=

180(n - 2)

Use the formula to find the angle sum of a polygon with seven sides.

6. (a) Find the angle sum of a dodecagon (12 sided figure). (b) Find the size of each angle if the dodecagon is regular.

7. The formula A = 180 - 360 can also be used to find the size, A degrees of each angle in a regular

n

polygon of n sides. Find the size of each angle of a regular:

(a) pentagon (b) hexagon (c) octagon (d) decagon

8. A formula to measure the volume of timber in a log using the Hoppus or quarter girth system is:

V= Q2

L

where Vis the volume in cubic metres L is the length of the log in metres

Q is the quarter girth. Find the volume of timber in a log:

(a) 7-4 metres long, having a quarter girth of 1·6 metres. (b) 10·8 metres long, having a quarter girth of 0·9 metres.

9. The approximate distance, d kilometres, of the visible horizon from a height h metres above sea level is given by the formula

d=5A

Use this formula to find the approximate distance of the horizon from a height of: (a) 4 metres (b) 16 metres (c) 100 metres.

10. The sum of the first n counting numbers 1

+

2

+

3

+

4

+

5

+ · · · +

n is given by S = ~(1

+

n)

(a) Find the sum of the numbers from 1 to 100.

(b) Find the sum of the numbers from 14 to 40 inclusive.

11. Tofindthesumof15termsoftheseries1

+

2

+

4

+

8

+

16

+

32

+ · · ·

wecanusetheformula

a(r" - 1)

S=----'----~

r- 1

where the first term, a, is equal to 1 the multiplying ratio r is equal to 2 and the number of terms n is equal to 15. Find the sum of 15 terms using the formula.

12. If a vehicle is travelling at K kilometres per hour, then its speed Min metres per second is given by the formula

5K

M=l8.

Convert the following to speeds expressed in metres per second.

(a) 60 km/h (b) 80 km/h (c) 90 km/h (d) 140 km/h

13. The sum of the squares of the first n counting numbers: 12

+

22

+

32

+

42

+

52

+

62

+ ... +

nz is given by the formula

S = n(n

+

1)(2n ₆

+

1) _.

(a) Find the sum of the squares of the first 20 counting numbers.

(b) Find the sum of the squares of all the counting numbers from 9 to 25 inclusive. (c) Find the sum ofthe squares of all the counting numbers between 15 and 35. 14. The sum of the cubes of the first n counting numbers

13

+

23

+

33

+

43

+

53

+

63

+ ... +

n3

n2(n

+

1)2 is given by S

=

4 . Find the sum of the cubes:

(a) of the first 15 counting numbers

(b) of all the numbers from 9 to 15 inclusive.

A rational number is one that can be expressed in the form ~ where a and b are integers and b i= 0. Thus all integers and fractions are rational numbers.

We have seen earlier that all rational numbers can be expressed as decimals which either terminate or recur.

For example

-i

= 0·625

i

=

0·6

On the other hand there are numbers which when expressed as decimals neither terminate nor

recur. These are called irrational numbers. ·

Let us consider the hypotenuse of this right triangle. It is exactly

J5

units in length.

If we attempt to express this length in decimal form we can only obtain an approximate value.

Thus

J5

co