CGP Maths for GCSE and IGCSE Textbook, Higher (for the Grade 9-1) 1847626874, 9781847626875

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Mathematics for GCSE & IGCSE” _

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CGP has the best practice for

9-1 GCSE Maths — by miles! ——— "GCSEMathematicsEdexcel— | GCSE OCR |

Mather?atics vse

tere

GCSE

eiteiened CCSE AQA athematics

CoP

For the Grade 9-1 Cour

Practice

Workbooks, Exam Practice Workbooks, Practice Exam Papers and Textbooks... we have all the GCSE Maths questions you could ever wish for!

All out now at cgpbooks.co.uk

Mathematics for GCSE

& IGCSE

Thousands of practice questions and worked examples covering the new Grade 9-1 GCSE and IGCSE® Maths courses.

Higher Level

Editors: Rob Harrison, Shaun Harrogate, Paul Jordin, Sarah Oxley, Andy Park, David Ryan, Jonathan Wray

Contributors: Katharine Brown, Eva Cowlishaw, Alastair Duncombe, Stephen Green, Philip Hale, Phil Harvey, Judy Hornigold, Claire Jackson, Mark Moody, Charlotte O’Brien, Rosemary Rogers, Manpreet Sambhi, Neil Saunders, Jan Walker, Kieran Wardell, Jeanette Whiteman

Published by CGP

This book is not endorsed by Cambridge International Examinations. ® IGCSE 1s a registered trademark of Cambridge International Examinations. ISBN: 978 1 78294 437 9 Clipart from Corel® Printed by Elanders Ltd, Newcastle upon Tyne. Text, design, layout and original illustrations All rights reserved.

© Coordination Group Publications Ltd. (CGP) 2016

Photocopying more than 5% of this book is not permitted, even if you have a CLA licence. Extra copies are available from CGP with next day delivery * 08001712712 * www.cgpbooks.co.uk

Contents Number, Ratio and Algebra

Section 7 — Powers and Roots

Section 1 — Arithmetic, Multiples and Factors Dlg

Calcitonin... eerie) arp Nae.

2

ie 1.3

Multiples alc actors: 125 Pua). ots) 5 Prime Numbers and Prime Factors................ 7 IME EC reece c i MRR eres Reh nts ve 9

Section 2 — Approximations cee NAIA ctr ei macs ESR RRSARTNS 12 2) ppekand Lowen Bounds. 2.250403. a 14

Section 3 — Fractions One i aoe 3.4 3.5 MOE cuenme

Pquvmlent MEaCHOns Wo Anis soe cesesst Mime Nima bers tere steep ecoanscsanttooneeea Ordering Broctions i 2 tec cciettictess Adding and Subtracting Fractions ............... Multiplying and Dividing Fractions ............ TACHIONS ANG DCCIIMAlS ee eee foawesetecet ens FeePCTS EPODICMIS oo ee ot on oan ee

17 18 19 21 Ze 24 |

Section 4 — Ratio and Proportion ar Ae

APN De Pe eee acai sap caste eset oe 29 Wavadins ia aiotveit RAO | 2.2.20. 2ieacsceereses 32 ANS memVION ATOR EROD CIS i ee wcrc sssee-coceetrccee es 34 Zee TENOR NORTE ee os oro ne sparing ceoearetenniiee 35 4.5 Ratio and Proportion Problems .................-. 38

Section 5 — Percentages DON BE CECE CCS cok orem areas ace ccyome coda roveboatnespins 40 5.2 Percentages, Fractions and Decimals .......... 42 5.3 Percentage Increase and Decrease............... 44

5.4 Die

Compound Percentage Change..............044 46 PC RC CIITA SCSik COONG eee or ot cacevanrierezenn

k) 155*— 21a? + 18ab

I) 22pq? — \1p3q°

m)xyt+xeytxytxy

mn) l6x*y—8xy?+ 2x77?

0) 36x"? + 8x2y”

p) 5x* + 3x34 — 25x3y

3!) Factorise the following expressions.

ay 13x27 + 223° + 20x?

b) 16a°b° — a*b> — ab?

c) 21p°q? — 14pq? + Tp*q

d) 14xy4 + 1323 — Sx°4 g) 16a°b* + 8a*b?

e) Ixy? + 4y° + 17ey? h) 187k + 217°k® — 157?k?

f) 16c%d° — 14c? + 8c3d° i) 36x24 — 72x°y’ + 18xy3

j) 20a*bS + 4a2b* — 5a5b'S

k) 11x2y3 + 11x8y? + 66xy"

I) 2jk-97k-12P

6.4

Factorising — Quadratics

A quadratic expression is one of the form ax” + bx + c where a, b and c are constants (a # 0). You can factorise some quadratics.

Example 1 Factorise: 1.

aye Flore Be)

Dee

x 15

Find two numbers that add up to 6 and multiply to 8:

a)

4+2=6and4x2=8

Then x? + 6x+ 8 = (x +4)(x + 2) 2.

You can check this by expanding the brackets.

1.

You need two numbers that multiply to —-15, so one must be positive and the other must be negative.

2.

They add up to +2, so the positive one is bigger.

@+4)@+ 2) =2?+2x+4x+8 b)

5+-3=2and5

x -3=-15

Then x? + 2x — 15 =(«+5)(x - 3)

Exercise 1 1

Factorise each of the following expressions.

a) ety 6

dja

624 12

Dita sya12

C)Exarte Say,

eye ox 4

f) v+6v+9 Section 6 — Expressions

2

Factorise each of the following expressions. b) x°—6x+8 a) x°+4x4+3 e) 7 +3y—10 d) x —5x+4

c) x°-7x+ 10 1) x +2x-8 i) P—4t-12

15 h) x?-—2x-

g) s°+3s5—18

Example2 Factorise 2x? — 5x — 3.

3K 2k ee

2x = Sx

1. The 2x? must come from

C= | |

multiplying 2x by x, so put 2x in one bracket and x in the other.

2.

(2x+ 3)

For the other numbers in the

1) = 2x? = 2x + 3x33 = 2

+x = 3

7K

brackets, try combinations of

numbers that multiply to give —3.

(2x+ I@— 3) = 2x7 — 6x +x—3 = 2x°— 5x3

ae

so 2x* —5x—3 = 2x + Dx —3) Exercise 2 1

Factorise the following expressions. a) 2374 3x7 1 b) 3x°- 16x +5 e) 2x?- 13x+6 f) 3b°>-7b-—6

c) 5x*-17x+6 g) 5x*+12x-9

d) 2P —5t-12 h) 2x?-3x+1

i) 7a’+19a—6

k) 727+ 38z + 15

l) 3y?—26y + 16

j) 1x’? - 62x - 24

Example 3 Factorise 6x — 19x + 10 1.

The 6x2 could come from 6x x x or from 3x x 2x.

2.

The second term in the quadratic is negative (—) but the third term is positive (+). This implies that both factors are of the form (ax — b).

Py (Oe).

(be = 1

The — 61x tells you that you need

to avoid products like 6x = 10

10

iol ene

(3x — 2)(2x — 5) = 6x2 — 15x —4x + 10 = 6x2 — 19x + 10

which give too large answers.

Try (3x — 2)(2x — 5) instead. 2

OV br 0

So 6x° — 19x + 10 = 2)2x Bx = — 5)

Factorise the following expressions.

a) 6° 5x>F 1 e) 25a’ + 65a + 36

b) 6x*— 13x +6 f) 12x?-19x+5

c) 15x°-x-2 g) 12u+6u-—6

d) 10x*— 19x +6 h) 14w?+25w+9

Difference of Two Squares Some quadratic expressions have no middle term, e.g. x — 49. These quadratics have factors that are the same, except that one has a positive term and one has a negative term. For example, x? — 49 factorises to (x — 7)(x + 7).

More generally:

a*?—b?=(a+b)(a—b)

This is known as the difference of two squares.

Example 4 Factorise x* — 16

16=4 Difference of two squares says: x* — a? = (x + a)(x — a).

x —16=x°-4 Section 6 — Expressions

So,x-16=(+4)~-4)

Exercise 3 1

2

Factorise each of the following expressions. a) x°—25 b) x°-9

c) x?- 36

d) x°-81

Factorise each of the following expressions. a) 4x°-—49 b) x°- 64

c) 36x*-4

e) b?- 121

g) 25x’- 16

d) 9x*— 100 h) 27 — 12

A quadratic expression is of the form 16x — 9.

a) Find /l6x b) Write the expression 16x? — 9 in the form a? — b?. c) Hence factorise 16x? — 9, 3

f) 16z*-1

Exercise 4 — Mixed Exercise 1

Factorise each of the following expressions.

a) x°-4x—-5

b) P—8t+ 16

c) x°—36

d) yw -—y-20

e) x°-x-12

1) y+ Sy + 12

g) x

h) x*-4x—45

i) s?—10s+ 16

2

ee

132

5x —6

k) b°?+b-6

Il) f-9t+ 14

c) 100x* — 64

d) y+ 19y+ 84

Factorise each of the following expressions. a) l6a*—25

b) b?>—7b-18

e) 2-1

f) 36x?— 169

g) 5x°-3x-2

h) 81/— 121

i) 4c?— 196

j) @+9a-36

k) 2 —15z+ 56

I) 144)? — 225

m) 22? + 21z+ 40

n) 3x?—17x—28

0) 2-2 +1

p) 7+ 6f— 16

q) 4y°-2y-2

r) 6z?—27z+ 12

S$) Sx + 19x +6

te2x7— 13x

6.5

Algebraic Fractions

Simplifving Algebraic Fractions Algebraic fractions can be simplified by factorising the numerator and/or the denominator, and then cancelling out any common factors. For algebraic fractions with quadratic expressions, factorise the quadratics first, then look for any factors that can be cancelled out.

Example 1 xy Simplify 2 21xy q

| aye Boer cy a BOT Sy

1.

3isacommon factor, so it can be cancelled.

—=

2.

xis acommon factor, so it can be cancelled.

= Bee

3.

jis acommon factor, so it can be cancelled.

es -S v

ze

4

a

dake ay + 8a'b’

een

3cd ¢) 8c + 6c?

48t — 6¢ 8s°t

o

Example 2 cd x’ — ae 16 Simplify eke

3

1.

Factorise the numerator and the denominator.

2.

Cancel any common factors.

yelp

ee (+4) (x — 4) _x-4

x +8x+16

@+r4)~+4)

x+4

Simplify:

Ast + 85° 4) St + 16st 4

.

6y + 8 2

15xz + 15z

15y + 20

S

25xyz — 25yz

Oxy — 6x

3a b+ Sab?

3y —3

Tab

w

Simplify each of the following fractions as far as possible.

a

2x — 8 x°—5x+4

6a — 3 DF aoa

x — Ix + 12

x? + 7x + 10 9) x’ + 2x — 15

x + 4x

poEayeate

f

a

)

2¢ +t—45

P81

Adding and Subtracting Algebraic Fractions Algebraic fractions should be treated in just the same way as numerical fractions. This means that to add or subtract them, they need to have a common denominator.

Example 3 Express * z : or ay 2 as a single fraction. 1.

The fractions are being added, so first find a common denominator.

2.

Here, the common denominator is 4 x 3 = 12.

3.

3

4

4(x—2)

Sa

3(2x+3)

a =

Then convert both fractions into fractions

with denominator 12, so they can be added.

=

ee ae tet Soe 12

ewe Oe Sun 102

Exercise 2 1

Express each ofthese as a single fraction, simplified as far as possible.

Xx

x_x

Pee 2

Uae

cae

2DeeD

57

6

AZ

9)

Express each of these as a single fraction, simplified as far as possible. HD a 0 |

Vie

23

Section 6 — Expressions

Lael

nebo

ee

Sle

al

heat

e+2

cel

Omnia:

Example 4 3

— 1 Express 2x et a

2¥-— 1-3

single fraction.

eel ek voc 2

1. The fractions are being subtracted,

re DG 72)

Bor) |

(sk IG —=2)) © (Gee 1G

so first find a common denominator.

2)

2. To dothis, multiply both the numerator and denominator of = __ (2x — I)(w— 2) — 3(@@ + 1) the left-hand side by (x — 2), and the right-hand side by (x + 1). (+ 1)@—-2) 3.

Subtract the fractions and expand the brackets.

Oe

Simplify by collecting like terms.

—

SS

See

&+1)@- 2)

pane

oh el

heel |

(x + 1I)@ —- 2)

3

Express each of these as a single fraction, simplified as far as possible. 2

l

he

Xt2.

a

3

*—3

yg

Veo ye pet

2G

Meee

7aes sama

2x+1

~

2x+3

a

i)

a

Lv

oa a

hei

sc)

x—3

-ys3

3

Ora Twg I

Sosa I

ee? a =| 8)

eta

Dire | ped

x-—4 a

Example 5 x eae aes

Express

x— 1 . : aD as a single fraction.

: : Factorise the terms in each denominator.

x x—1 eee) Pile

2.

Here the denominators have a common factor, (x + 2).

3.

The fractions need to be added, so their denominators need to be the same.

= Geri &a) a

4.

; ie)

Multiply both the numerator and the denominator

ape teats Ava @=De+)

of the right-hand fraction by the same term, (x + 1). Add the fractions and simplify the

(+ DO + 2) +241) _ x + @aba axe )) xe

numerator by collecting like terms.

~~

(«+ )1I)@~+2)

~—

&@+)D@4+2)

Exercise 3 1

Express each of the following expressions as a single fraction, simplified as far as possible.

2

ae 5

Ge iyeGo) OS ea 2

ae el

soe.

[eae

ene)

OT Gea) PeGAYE)

Express each of the following expressions as a single fraction, simplified as far as possible. 2X x= 3 10 Zz

ee I

2) yar x—3

©) aA

3

3x

eee)

y ee a+]

3

eae A

a+1

“oH 4a43 as far as possible. simplified fraction, single a as expressions following Express each ofthe 2 coe Lan b ta Cea 6 55x ao ee 2 fee pa ttre 3x

ee eee

QO Plaa3 | ee—Sxe4

o

C28

d i

as ee

F-Ht2

ee

Sy +6 Section 6 — Expressions

ei

Multiplying and Dividing Algebraic Fractions Example6

eT

Express—

eaa

Pores

1 eo 8

as a single fraction.

e

2.

Factorise each term as far as possible. Cancel any factor which appears on

3.

the top of either fraction and on the bottom of either fraction. Multiply the terms.

x= 2

Tie Gkoikee

v

2x +4

ee

eee 20-42) 5 ~ GB + 4) fe (x + 4) aa 2 ae ee

=

44

GO

by

Exercise 4 1

Express each of the following as a single fraction, simplified as far as possible.

te3

5b

) Xe 2

645

ae arose

l

3%

Oa

2

ba

eT haat

l

3z

x’ — 16 x+3 Sx tO x4 x : —3x-4.

Vase!

2

xe+4x+3, Deer aie

x +4044

OR (yee ts

Express = = = = 1.

x+4 oeere

z+ 3z510 x2+ Oz 4D d)) ———_. z=) pe Aes

weed

Example 7

ue ae as a single fraction.

:

aa + ia

This is a division, so turn the second fraction

2 34+ 3x + 6.

over and change the sign to a multiplication.

+ 4x +3

TO]

Factorise each term as far as possible.

ead

3(x-+2y

& =)

Cancel any factor which appears on both the

7 G7 \Ge8))

top and the bottom of either fraction.

chef LA Cote ineee(paral ~ (x —3)

l

peere) —

x3

Express each of the following as a single fraction, simplified as far as possible. a)

5

(eas

Ne aeo el

Express each of the following as a single fraction, simplified as far as possible.

2

4

3x7

oy * T6xy

Express each of the following as a single fraction, simplified as far as possible.

ea 3

t . 24st

») ap a?

Ax

2k

Sage

Sy et2i7

)

6b? ° 20b

ye 4x°y = 38

d)

ab ——

+

ea =

Express each of the following as a single fraction, simplified as far as possible.

y= oy

2.y-2 ¥

Ay*

Section 6 — Expressions

b) 2c+1. cd 30

LS

5

Savy 5

d) an)

ee

rset

Gir

AE

6

Express each of the following as a single fraction, simplified as far as possible. a 38 x — KX— 6 a) xRoe b x°—4x+4. 2x-—4 +9x+20° x +7x+12 laseageeasS axeaNs y —5y+6.

y-2

d) x—4x+4 . x +3x-10 Nae aied00) exe =

c) Vey a0s sy 12 e)

6.6

K+ 2x lox + 4x 3 Ke Die Ag xx= 8

f)

feOats fa OCG. O+3t+2° £4+8t4+7

Expressions Problems

Exercise 1 1

Simplify each of the following expressions by collecting like terms. a) 3x—2x+5x-x

b) 8y+ 2y—Sy + 3y

c) Sa—3a+2a10a

d) 3x+2y—4x + 5y

e) 6x + 3y—2x-y h) 3x?+ 5x—7+ 2x*?-3x+4

f) i)

g) 9p9-5+3p+4q+3q-7

8s+4+ 3t—-2s—S5t+2

AXXO eee

Expand and then collect like terms in each of the following expressions. a) 5@—2)+-3+) b) 6x +3) + 3(x—4)

4(a + 3)-2(@+ 2) 5(2x — 1) + 4(3x — 2) 2x(2x + 3) + 3(x — 4)

e) 2(2x + 3)—3(2x+1) h) x(x + 3) + 2(x—1)

d) S5(p + 2)—3(p—4) g) 3Gb-— 1) —S(b-3)

9

A triangle has sides of lengths (x + 2) cm, (3x — 1) cm and (2x + 4) cm. Find a simplified expression for the perimeter of the triangle. A rectangle is (x + 5) cm long and (x — 2) cm wide.

Find expanded expressions for a) the perimeter of the rectangle, b) the area of the rectangle.

A sports ground is a rectangle (2x + 3) m long and (2x —5)m wide. a) Write an expression for the length of a white line painted around the edge of the ground. b) Write an expression for the number of square metres of turf that would be needed to re-lay the whole ground.

10 Rufina makes 36 scones to sell. The ingredients cost £x and she sells the scones for y pence each. Write an expression to describe the profit, in pence, that she makes.

A square has sides (3x — 2) cm long. Find expanded expressions for a) the perimeter of the square,

b) the area of the square.

11 A swimming pool is (2x — 3) m long, (x + 2) m wide and 2 m deep. Write an expression for (x + 2) m pee) the volume of the pool

Zeke has 3 gems, each valued at £(2x + 3), and Sharon has 4 gems, each valued at £(3x — 4). Write

an expression for the total value of the 7 gems.

a) in cubic metres. b) in litres.

I travel at 2x miles per hour for 3 hours, and then at (x + 20) miles per hour for 2 hours. Write an expression to describe the total distance I travel.

of

12 A music stand is made of 3 strips of metal, each (x + 20) cm long, and 5 strips of metal, each (2x —5) cm long. Write an expression for the total length of metal required to make the music stand.

Write an expression for the total cost of3 widgets, each costing (x — 2) pence, and 4 widgets, each costing (2x + 3) pence.

13 Expand and simplify each ofthe following expressions. a) Gee DG

3)

d) (v+2)(y-6)

DyE(f22 (si

c) @—3)at 4)

5)

f) (t—6)(¢+ 1)

e) (x-—4)(x—5)

14 Expand and simplify each of the following expressions. c) (3t—2)2t+ 1) b) Gree Dex 5) a) 2% +3)@=3) f) y—3)2y+ I) 2) (oe Oy) e) (4x + 1+ 3) k) (x—-2) i) (Cx ty j) (+5Y

m) (x + 1)@— 2) + 3)

n) (y—5)y—3)y + 2)

d) (2a —5)(3a —2)

h) (2z —3)(2z + 3)

0) —(¢—8)(¢— (t+ 1)

Il) Gx+4y p) 3(2 + 4)(z—3)(z + 2)

Section 6 — Expressions

Exercise 2 Factorise each of the following expressions as far as possible. b) 6a+3 a) 4x-8 e) 6x—15y+ 12 d) 3x + 6xy

1

c) 5t-—10

Factorise each of the following expressions as far as possible. b) 8xy— 12x? a) 15x — 10x?

2

c) y+ 4xy f) a@b—2ab+ ab’ i) jR-27R

e) pq’ —p'q h) 16x? + 12x*y — 8xy’

d) 6xy + 9x’? g) 6c?d—9cd* + 12c

125=>16

f) 8a

Factorise each of the following expressions as far as possible.

3.

c) 6ab>—4ab* + 8ab? f) 8Pk—O6pPk

b) 4p’q° — 2p’°q° e) 14x3 + Tx’y — Txy’

a) ox yt 2x d) 9c?d—-9c3d* + 15c? Factorise each quadratic.

4

5

a) a’+6a+8 e) x°—8x+ 15

b) x°+4x+3 f) x2 +9x+ 20

c) 2-5z+6 g) x°+2x-24

d) x°+ 3x-18 h) x*—3x- 10

Factorise each quadratic. a) P+ 5t-14 e) x°+ 4-21

b) x°+x—-20 f) x?— 100

ce) a —4a—5 g) r—3r-18

d) m?+7m-+ 10 h) x? —12x+27

b) 3m?—8m+4 f) 16%’°-9

c) 3x7=5x=2 g) 5x?+2x-3

d) 49°+ 4g+ 1 h) 10z*+ llz—6

Factorise each quadratic.

6

A)ex Ox 2 e) 6n?+ 13n +6

Exercise 3 1.

Express each of the following as a single, simplified, algebraic fraction.

2

a) 449

ea

c) BS

Dea ts

Med ea

Dig

oeae

Simplify the following expressions.

Sega 2

Pi

Wrnanco= Pa

ieee a

ae

‘

3

aac eae ae

»

a

D

eat ie 12 x Ane eet)

Simplify the following expressions. sy)

eee aan

cee

a—Ta+10,. o

sae

vee

°)A |

a’+2a—3 ©

oon

ok

ane ge

g) b+ DRO E 2D 6 b 6b. 5" 36-423 Section 6 — Expressions

210

S00! d) ve aX

Spas 3s

1) Gy aa Oemake HA h) x se 8x Se 15 x PAS IE

|

ie 4x 8 ee = Sree?

Section 7 — Powers and Roots 7.1

Squares, Cubes and Roots

Squares and cubes are written using powers, e.g. 4 x 4=4 and4x 4x 4=4, The square of any number is positive. The cube of a positive number is always positive, but the cube of a negative number is always negative.

Example 1

Find:

a) #

b) (-4y

4=4x4=16

CA)

(4)? =—4 x -4 = 16

(—4)° =-4 x -4 x -4=_64

Exercise 1 1

Find the square of each of the following numbers, without using a calculator.

at f) —5 2

3

b) 7 g) —ll

c) 100 h) -30

d) 20 i) 0.1

ej j) 0.5

Find the cube of each of the following numbers, without using a calculator. a) 2

b) 5

c) —10

d) 6

e) —8

f) -l

g) 9

h) 20

i) -300

j) —0.

Qo h) —(10°)

d) (3) il) @Y

e) (8) ) GY

Without using a calculator, find:

a) 3° f) 12?

b) 3° gyiesy’

Every positive number has two square roots, one positive and one negative. The symbol -/_ is used for the positive square root. Negative numbers don’t have square roots. Every number has exactly one cube root. The symbol 3/ is used for cube roots.

Example 2 a)

Find both square roots of 16. 1. 4°=16, so the positive square root is 4. 2. There’s also the negative square root.

16 =4 —/16 =-4

b) Find the cube root of 64.

/64 =4

43 = 64, so the cube root is 4.

Exercise 2 1.

Without using a calculator, find:

d) v81

a) 736

b) v10000

oc) —16

e) v121 i) The square roots of 49

Loy Oo

h) 400 g) —v144 000 000 9 of roots j) The square

Section 7 — Powers and Roots

65

2

Without using a calculator, find:

a) VB

b) V27

c) -¥i000

d) ¥=1

e) ¥—125

f) 7/512

g) The cube root of 216

hh) The cube root of —8

Example 3 Calculate

10° — 6°.

The square root and cube root symbols act like brackets. Evaluate the expression inside before you find the root.

710° — 6’ = 7100 — 36 = /64

=o Exercise 3 1

2

Without using a calculator, find:

a) 232

b) 5°— 10?

¢) Fay

d) (11 —6)

€) eT 3

‘ay (oeee

8) vy looe

h) 4/5.

jess

j) /4 (10°) x

k) V3 +45)

l) ¥3?x 2412?

Use your calculator to find the following, giving your answers to 2 decimal places where appropriate.

a) 12?

b) 3.4

C) 2020.

d) 4.58?

e) 24.2?

f) /344

g) /894

h) ("2 7)

i) V5 7

j) 3.4

k) 8.6

Il) 144

m) 38.6

n) 738 + 43

0) (3.8 + 4.6)

7.2.

Indices and Index Laws

Index notation (or ‘powers’) can be used to show repeated multiplication of a number or letter. For example, 2 x 2 x 2 = 23. This is read as “2 to the power 3”. Powers have an index and a base:

23

index

base =>

Example 1 a) Rewrite the following using index notation.

(1) 3 X3P Bi 31x 3 3x3x3x3x3=35 Gbxbxbxbxcxe

bxbxbxbxcxc=h xe?

= bie

b) Rewrite 100 000 using powers of 10.

100 000 = 10x 10x 10x 10x 10

= 10°

Exercise1 1

Write the following using index notation.

A) 3s 8 res d) 9x9x9x9x9x9x9xY Section 7 — Powers and Roots

b) 2x2x2x2x2

CC)

e) 22a

tex 17

Dee 2

No) a ee aI eee

2

3

Using index notation, simplify the following. a) hxhxhxh b) tx¢txtxtxt

C) SXSXSXSXSX5SX5

d) axaxbxbxb

f) mxXmxnxmxnxm

e) KXKXkxXkxkxfxfxf

Rewrite the following as powers of 10. a) 10x 10x 10 b) 10 million

c) 10000

d) 100 thousand

e) 100 000 000

d) i) n) s)

e) i) 0) t)

Exercise 2 1

Evaluate the following using a calculator.

a) 3°

b)i2'

c) 6°

f) 139

g) 3

h) 95

k) 8 +2°

I) 150-34

m) 2’+ 10°

py. 8?

q) 10* x 105

pro

4%

2?

8 3x28 3x 23 (93 +4

4 8

f)

(2°.

g)

Ooeee ne

;

D tor

8° x 8* aAe

4’ x

Tesla

h)) =ape x

4

ea) sei

10"

1)

K) Poe

;

ae

(8S) 2 Seee

) “Bx en

a) Write:

(i) 4 as a power of2

(ii) 4 as a power of2

(iii) 2° x 4° as a power of2

b) Write:

(i) 9 x 3° as a power of 3

(ii) 5 x 25 x 125 asapowerof5

(iii) 16 x 2° as a power of4

(ii) 1007 x 10°

(iii) 3° x 2?

c) Evaluate: (i) 2? x 4

Negative Indices Where a and m are any numbers or variables: an"=

1

a

Exercise 5 1

2

Write the following as fractions. a) 4! lt) ee

d) 5°

e) 2 pase

Write the following in the form a”. a)

3

fs

15

b)

a8Mm

c)

di By

|

d) 37

e)

Sa

I

Simplify the following. ay

a) (5)

|

Ze =I

b) (3)

Section 7 — Powers and Roots

i =2

% (3)

5

=3

4) (5)

7

) (75)

2

Example 4 Simplify the following. abil

a) Yr i

1. Rewrite > using negative indices.

aes ale=yr+y? y

2. Subtract the indices.

Say

by ze xi(zty 1. Multiply the indices to simplify (z*)°.

4

as x

a

2. Now add the indices.

= 78+C8) = 70

3. Anything to the power 0 is 1.

=]

78

Without using a calculator, simplify the following. Leave your answers in index form.

aor

>

b) 81° x 8’

5) e) (ry

el f) 2+ J,

: (=)

Yee

VoL eeee

pe

i

D 794 5

XA)

6

Oe ”

¢) ge = 2

d) (3°)

eel

h) 7x7?.

k) 45+4%x 44

6"

(=)

Cn)

m’ +m

) 3°34

nS

eee

LOR aon

Gey

s) (10° = 10°

VR

22h

ol

3-3

a) Write the number 0.01 as: (i) a fraction of the form 7

(ii) a fraction of the form via

b) Rewrite the following as powers of 10. (i) 0A (ii) 0.00000001

(iii) a power of 10

(iii) 0.0001

(iv) 1

Example 5 Evaluate 2* x 5°. Give the answer as a fraction. 4

1. Turn the negative index into a fraction. 2.

6

24 x 5% =2' x a= 33

Evaluate the powers.

= it

Evaluate the following without using a calculator.

Write the answers as fractions.

a) 3

b) 32

A).

esoearal aire)

c) 3x5? Pr Rad eo (56.

hs )

©) Gi 10s) P


.

12 in

48‘ Sac

In Questions 2-7, give all answers to 3 significant figures. 2

5 A field in the shape of an equilateral triangle has sides of length 32 m. Find the area of the field.

Find the area of the

triangle shown. 6.5 cm

6

13m 9m

& 5 cm 3

15m

Find the area of the triangle shown.

a) Use the cosine rule to calculate x. b) Find the area of the triangle. 4in

iS A\ 4

A triangular piece of paper has a side of length 4.5 cm and a side of length 7.1 cm. The angle between these two sides is 67°. Find the area of the piece of paper.

:

10 cm

a) Use the sine rule to calculate y. b) Find the area of the triangle. Section 27 — Pythagoras and Trigonometry

265

27.5

Sin, Cos and Tan of Larger Angles

Sin, cos and tan are really useful for working out acute angles (angles between 0° and 90°). They can also be-used to find obtuse angles (angles between 90° and 180°) and reflex angles (angles between 180° and 360°).

Sin of Obtuse and Reflex angles The graph shows y = sin x for —180° < x S$ 360°. For each value of y (except y = 1, y=—1l and y = 0) there are two positive values of x between 0° and 360°.

x

oy”

For 0 131m

7m

a

| 59m

4| mm

34 mm

f)

e)

Pee 16.5 mm

7.96 m

1.12 m 18.01 m

Ge Find the exact value of the following, without using your calculator. e)iacos SID. b) cos 300° a) cos 330°

Section 27 — Pythagoras and Trigonometry

Tan of Obtuse and Reflex Angles. The graph shows y = tan x for —90° < x < 360°. The values of tan x for obtuse angles are negative, but putting tan 'y into a calculator gives a negative x-value. To find the obtuse angle, add on 180°. To find the reflex angle from an acute or obtuse angle, add on 180°.

Example 3 Find the obtuse angle x for which tan x = —10.

tan '(—10) = —84.29° (to 2 d.p.) For the obtuse angle, x =—84.29° + 180° = 95.7° (to | d.p.)

Exercise 3 1

Find the acute and reflex angles that satisfy each of the following equations. Give your answers to | d.p. where appropriate. a) tanx=6.4

2

b) tanx=3.6

c) tanx=1

d) tanx=2.1

Find the three values in the range —90° < x < 360° that satisfy each of the following equations. Give your answers to | d.p. where appropriate. a) tanx=-2.7

b) tan x =-10.5

c) tanx=—l

d) tanx=-7.1

Exercise 4 — Mixed Exercise Unless told otherwise, round your answers to these questions to | d.p. where appropriate.

In each of the following, find the size of the obtuse angle x. a) 6 cm

7cm

11 cm

2

= mm 10

=

28 cm S

18 cm 16 cm

a) Find the obtuse angle x for which tan x =—6.4 b) Find angles x and y in the range 0° < x adj. You could also have found the length of

tany = v85 x =

tan

4

(zs)

‘

DF first, then used the formula for sin x.

p

ZBDF = 23.5° (to 1 d.p.)

Exercise 1 1

For the cube shown with sides of length 3 m, find:

4

a) the exact length AF b) the exact length FC c) the angle AHC, to | dp.

C s

ae

o

For the cuboid shown, find:

a) the exact length AH

2

For the square-based

b) the angle EDG, to | dp.

pyramid shown, find: a) the exact vertical height b) the angle BCE, to | d.p. c) the angle AEB, to 1 dp.

3

5

For the triangular prism shown, find:

a) the exact length BC b) the angle EDF, to 1 d.p.

c) the exact length DC

6cm

For the triangular prism shown, where M is the midpoint of AC, find: a) the perpendicular height, BM, to3 s.f. b) the length EM, to 3 sf.

Section 27 — Pythagoras and Trigonometry

269

Exercise 2 1.

The diagram shows the triangular prism J/KLMN.

M

a) Use the cosine rule to find the size of angle J/K,

y

correct to 1 d.p. b) Hence find the area of triangle //K, correct to | d.p.

5m

c) Hence find the volume of the triangular prism IJKLMN, to the nearest m?.

2

L

The diagram shows the triangle CEH drawn inside the cuboid ABCDEFGH.

K

11m

E

a)

Find the exact length CE. : : b) Find the exact length CH.

F

a. alan

A

c) Find the exact length EH.

ae.

ee

m

d) Hence use the cosine rule to find the size of angle ECH, correct to | d.p.

&

9m

4m

a

The diagram shows the cuboid PORSTUV W. a) Use Pythagoras and the cosine rule to find the size of angle PSU, correct to 1 d.p.

b) Hence find the area of triangle PSU, correct to | d.p.

ee a

an

Ky

Exercise 1 1

Find the value of the letter in each of the following. Give your answers to 3 s-f.

a)

b)

6.7 cm

Scm

3cm

a

c) 2.7m

3.9 cm

a

e)

f) 17m

e

g) f

h) 8 ft

§

4m

3.5m

es

CT]

30 m

i)

28cm

e..

j) 72m

6cstern

A

\

9.5. ft

k) 12 ft

15 fi

Section 27 — Pythagoras and Trigonometry

4.5m

4

l a

17cm

y

1)

iz

is —

CI

Sn

Find the size of the missing obtuse angle g, to 1 dp.

125m

5

a) Find the length of

DA 14m

The square PORS is drawn inside the square

a

yoo!

corrugated iron needed to make the roof of my bike shed, to 3 s-f.

Ee i.

b) Find the angle of elevation of the roof of my bike shed, to 1 d.p.

B |

ABCD. Find the side

Q

length of ABCD, to 3 s-f.

+19m—> 6

S

Using the sine and/or cosine

rules, find the angles x,

[_

D

wee

a w9o'c

Al

eer

©

and z in the triangle shown, correct to 1 d.p.

5S

oO

S

Find the side lengths of the rhombus shown, to 3 s-f.

a)

7

b)

The shape shown is made up of two right-angled triangles. Find the value of x.

/s\

lcm

c)

12 cm

36°

a

2 x v

i “—(6> 0.3m

10 A bird is flying in the sky. Point P is on flat, horizontal ground. The horizontal distance between the bird and point P is 37.5 m. The angle of elevation between point P and the bird is 61°. Find the vertical height of the bird above the ground, to the nearest metre.

O

11 Logan cycles 6 miles due north from P to Q, then 8.5 miles from Q to R. Find the bearing ofR from Q (to the nearest degree), given that the direct distance from P to R is 11 miles.

6 ‘a N

8.5 miles

iirniles

©

fp

Section 27 — Pythagoras and Trigonometry

R

12 A pilot flies 950 km from Rambleside to Alverston, on a bearing of 030°. He then flies a further 950 km to Marrow on a bearing of 120°. Find the direct return distance from Marrow to Rambleside to the nearest kilometre. 13 Leo has a triangular piece of fabric with dimensions as shown.

on

a) Find the size of the missing angle p, to 3 s.f. b) Hence find the area of the fabric, to3 s.f. Lb

12 cm

14 Find the vertical heights of the following symmetrical houses, to 3 s.f.

a)

AXe

ST

b)

+—6 m—>

ee

+wop>

15 Find the exact areas of the following triangles, without using a calculator.

a)

272) oe LD

=

b)

8 cm

>

A cm

367 5/2 cm

16 Gabrielle has a wooden crate in the shape of a cuboid, as shown. a) Find: (i) the exact length ED

(ii) the angle FDH, to 1 d.p. (iii) the angle CHD, to | d.p. b) Gabrielle wants to pack a 10 ft metal pole in the crate. Will it fit?

&® Points P, O and R are plotted on a grid of 1 cm squares. P has coordinates (1, 3), O has coordinates (5, 4) and R has coordinates (7, 1).

a) (i) Find the exact distance PO. (ii) Find the exact distance PR.

b) (i) Find the bearing of O from P, to 2 d.p. (ii) Find the bearing ofR from P, to 2 d.p. c) Find the area of the triangle POR.

@ The diagram on the right shows a shape made up of the square-based pyramid EF'GHI and the cuboid ABCDEFGH. P is the centre of the square EFGH and O is the centre of ABCD. P lies on the vertical line O1. a) (i) Find the length A/, to 2 dp. (ii) Find the angle OA/, to | d.p.

b) Find the angle OAP, to 1 d.p.

ee

Section 27 — Pythagoras and Trigonometry

Section 28 — Vectors 28.1

Vectors and Scalars

A vector has magnitude (size) and direction. A vector can be written as a column vector or drawn as an arrow.

Column vector: as ee, 3 \ X-component: 3 units right ES ee () 2) « ; 8 2 6 5 4

3 | 0

Sm oF

h)

Te

|

r

okae

i)

7 él

|

6

5

5

4

:

4

5

oo

1 2 2

He

SASS 6 7 Sa

Section 31 — Transformations

Ee eet

listesihaal

ta

6 5-423)—1) Oe eoeamansnG ee

D5 25 AS

Grae meat,

Example 3 : Shape A, given by the matrix A Draw shapes A and B.

Fool nae , 1s transformed by M beeezet |

1. Put the matrices in the right order. The first transformation goes to the left of A, and the second to the left of that. So EMA = B. 2.

3.

oe then by E 0 -l

ube 012

into shape B.

B—(

Multiply the matrices to find B.

as (

(You could find MA first, and multiply the result by E, but here it’s easier to find EM first, then multiply by A.)

2 2 = iE ie

Use matrices A and B to draw the shapes.

Exercise 3 1

For each of the following, (i) find the matrix B, and (ii) draw shapes A and B.

ae a) Shape A ( 3 |is transformed by the matrix L e 5)then by the matrix M f ‘into shape B.

a;

0 2 4\. b) Shape A 5 5 ais transformed by the matrix L i 3)then by the matrix M

5iinto shape B.

Shape A, shown here, is transformed by the matrix P ‘ :to give shape B. B is then transformed by Q ( :5)to give shape C.

a) Copy the diagram shown and draw shape B. b) Draw shape C. c) (i) Find the matrix R that maps A directly onto C. (ii) Describe the two consecutive transformations that R represents. Sw

Shape A, shown, is transformed by the matrix X ( 1)to give shape B. 0 1

=

7; 0 B is then transformed by Y ie |to give shape C.

a) Copy the diagram shown and draw shape B. b) Draw shape C. c) (i) Find the matrix Z that maps A directly onto C. (ii) Describe the two consecutive transformations that Zrepresents. = §=s-y7-3—

Shape A

, ee / 124 4)\. is transformed by the matrixM oe , then by matrixN Sa

a) Draw shapes A and C. b) Find the matrix, L, that maps shape A directly onto shape C. c) Describe the single transformation that matrix L represents.

a) Find the matrix, T, that maps shape A onto shape B, shown. b) Find the matrix, U, that maps shape B onto shape C. c) Hence find the matrix, V, that maps shape A directly onto shapeC. d) Describe the two consecutive transformations that V represents.

—1

t+wb Baa ©bo

1

0\. es into shape C.

ey

|

| ae

eases

OT

Section 31 — Transformations

8x

Section 32 — Congruence and Similarity 32.1

Congruence and Similarity

Congruent shapes are exactly the same shape and size. The images of rotated and reflected shapes are congruent to the original shapes. Similar shapes are exactly the same shape but different sizes. Similar shapes can be in different orientations. The image of a shape after it has been enlarged is similar to the original shape.

Congruent Triangles Two triangles are congruent if they satisfy any of the following ‘congruence conditions’.

b a

G

a

y

a

a

a

b

b

Side, Angle, Side:

Side, Side, Side:

Two sides and the angle between on one triangle are the same as two sides and the angle between on the other triangle.

The three sides on one triangle are the same as the three sides on the other triangle.

MY

a yy

y Z

:

a

h

oS

h

Angle, Angle, Side: Two angles and a side on one triangle are the same as two angles and the corresponding side on the other triangle.

Right angle, Hypotenuse, Side: Both triangles have a right angle, both triangles have the same hypotenuse and one other side is the same.

Example 1

4m

Are these two triangles congruent? Give a reason for your answer.

2 on =

Ss a er 4cm

Two of the sides and the angle between them are the same on both triangles. So the triangles are congruent.

Exercise 1 1

The following pairs of triangles are congruent.

Find the value of the angles and sides marked with letters.

b)

4cm

4cm

s

ra c)

§

rs

a

£

ee

§/ 80°

LZ

9>

92°

Xe

cs

28°

9om e) ot

2

a:

~

=

oe

4cm

f)

O

'S

xe

520°,

:

4cm

g)

92°

a 28°

is

or

6

h)

10 em

29 cm

ZLiem

oo 26 cm

10 cm

49°

21cm

ANOS,

29 cm

Exercise 2 Show that the two triangles below are congruent.

[ke 2

3.

ABCDE isa regular pentagon. Prove that triangles ABC and CDE are congruent.

pao

B

ABCD isa parallelogram. Prove that triangles ABC and ADC are congruent. B

C

D q

4

ABCD isakite and O is theA point where the diagonals of the kite intersect. Prove that BOC and DOC are congruent triangles.

Similar Triangles Two triangles are similar if they satisfy any one of the following conditions. If two triangles are similar, then all of the conditions are true.

A «&

All the angles on one triangle are the same as the

All the sides on one triangle are in the same ratio as the corresponding

angles on the other triangle.

sides on the other triangle.

ab Two sides on one triangle are in the same ratio as the corresponding sides

on the other triangle, and the angle between is the same on both triangles.

Section 32 — Congruence and Similarity

Example 2 Are these two triangles similar? Give a reason for your answer.

All the angles in one triangle are the same as the angles in the other triangle. So the triangles are similar. z

Exercise 3 1

For each of the following, decide whether the two triangles are similar. In each case explain either how you know they are similar, or why they must not be. a)

b)

as MS

OF

2

2

8 cm

oo

a

ea

2)

yaa 9cm

.SB

Sem

d) 9

3S

os 3

25cm

i

-=

Ronn

WA) ian

9 cm

30°

Rear

ae

(\ W

ss

1.

Use properties of similar triangles to find the corresponding angle in POR.

x = angle ORP, as corresponding angles in similar triangles are equal.

2.

Calculate the size of this angle.

Angle ORP = 180° — 110° — 30° = 40° Sox = 40°

Exercise 5 1

y.

Triangles POR and XYZ are similar.

O

a) Find the ratio of the length YY

ies

to the length PQ. b) Find the length YZ.

2

4m

3m

po

R

y=

7

The diagram shows two similar triangles, ABC and ADE. a) Find the ratio of the length AB to the length AD.

b) Find the length AE. c) Find the length BC.

1 3

The diagram below shows two similar triangles, ORU and OST. Find the following lengths.

a) ST

b) OT

5

c) UT

12m

a

The diagram below shows two similar triangles, STW and SUV. Find the following lengths.

a) TW

b) SU

c) TU

S

SES

ae

vig

O

e

6 4

The triangles shown below are similar. Find y.

The diagram below shows two similar triangles,

JKN and JLM. Find the following lengths. a) JL b) JN c) NM

100° =S

7

ee

Oa

The triangles shown below are similar. Find z.

Ls

wi

Section 32 — Congruence and Similarity

32.2

Areas of Similar Shapes

For two similar shapes, where one has sides that are twice as long as the sides of the other, the area of the larger shape will be four times (i.e. 2? times) the area of the smaller shape. In general, for an enlargement of scale factor n, the area of the larger shape will be n? times the area of the smaller shape.

Area = np < ng

q|

Area=pq

= npg

If the ratio of sides is a: b, the ratio of the areas is a”: b’.

Pp

Example 1 Triangles A and B are similar. The area of triangle A is 10 cm? and the area of triangle B is 250 cm?. What is the scale factor of enlargement? 250 = 10n?2 SOW 250

B

OS

n=5

250 cm?

The scale factor is 5.

Not to scale

Exercise 1 1

Work out the area of the square formed when square A is enlarged by a scale factor of 4.

4

A

The rectangles below are similar. Calculate the area of shape A. Not to scale

10 cm

2

Triangles A and Bare similar. Calculate the area ofB

"

Not to scale

3cm

3.

ie

5

A triangle has perimeter 12 cm and area 6 cm’. It is enlarged by a scale factor of 3 to produce a similar triangle. What is the perimeter and area of the new triangle?

6

Find the dimensions of rectangle B, given that the rectangles are similar.

2 crm

The shapes below are similar. Work out the base length of shape Q.

Not to scale

Not to scale

Q 400 cm? Lt cm

4cm

Exercise 2 1

Squares A and B have side lengths given by the ratio 2:3. Square A has sides of length 8 cm.

@

The ratio of the sides of two similar shapes is 4:5. The area of the smaller shape is 20 cm’. Find the area of the larger shape.

&

Stuart is drawing a scale model of his workshop. He uses a scale of 1 cm:50 cm. The area ofthe bench on his drawing is 3 cm’. What is the area of his bench in real life?

a) Find the side length of B. b) Find the area of B. c) Find the ratio of the area of A to the area of B.

2

The ratio of the areas of two similar shapes is 2.25:6.25. Find the ratios of their side lengths. Section 32 — Congruence and Similarity

The rules about enlargements and areas of 2D shapes also apply to surface areas of 3D shapes.

Example 2 Shapes A and B on the right are similar. Find the surface area of shape B. 10 cm

1.

Find the surface area of A.

2.

Compare corresponding side lengths

3.

Surface area of A = 2[(2 x 4) + (4 x 10) + (2 x 10)

= 2(8 + 40 + 20) = 136 cm?

to find the scale factor of enlargement.

10 cm + 4 cm = 2.5, so the scale factor is 2.5

Use these to find the surface area of B.

Surface area ofB = 136 x 2.5* = 136 x 6.25 = 850 cm?

Exercise 3 1

A and B are similar prisms. B has a surface area of 1440 in’. Calculate the surface area ofA.

Pa

B

4

——

Cylinder A has surface area 63x cm?. ee the surface area of os similar cylinder B.

ae

5

as 6 in 2

3

12 in

5

A sphere has surface area 4007 mm’. What is the surface area of a similar sphere with a diameter three times as large? The 3D solid P has a surface area of 60 cm’. Q, a similar 3D solid, has a surface area of 1500 cm?. If one side of shape P measures 3 cm, how long is the corresponding side of shape Q?

The vertical height of pyramid A is 5 cm. Find the vertical height of similar pyramid B. nS

.

Surface area=32cm2 6

Surface area = 72 cm?

Similar shapes P, Q and R have surface areas in the ratio 4:9: 12.25. Find the ratio of their sides.

32.3

Volumes of Similar Shapes

For two similar 3D shapes, when the side lengths are doubled, the volume is multiplied by a factor of 8 (= 2°). In general, if the sides increase by a scale factor of n, the volume increases by a scale factor of n’.

For two similar shapes with sides in the ratio a: b, their volumes will be in the ratio a? : b’.

Example 1

Volume = xyz

A

Volume = nx X ny X nz = nWxyz

Not to scale

Triangular prisms A andB are similar. Find the volume of B. 15 +5 =3, so B is 3 times bigger than A.

Volume of B = 3° x volume of A =27 x 80 = 2160 cm’

cai

Valine=.SULCRA.

Section 32 — Congruence and Similarity

(ote

Exercise 1 1.

6

Rectangular prisms A and Bare similar. Find the dimensions of B. ae A

a 2 CU |he :

S5cm

a

x= 1920 cm? Vol a Boks ; 2 A 3D solid of volume 18 m? is enlarged by scale vol lid? f th factor 5. What is the yee tev es nae Sie f werent Ses Beene Ee ROD IS Se AUR ae B A

Vol

= =3 ee

:

7

8

ere

O

4

LI

Sa ; A cube has sides of length 3 cm.

a

ae

3

height vertical heigh and vertical A pyramid has volume 32 cm° ° and 8 cm. A similar pyramid has volume 16 384 cm’. What is its vertical height? Two similar solids have volumes of 20 m?* and 1280 m*. James says that the sides of the larger solid are 4 times as long as the sides of the smaller solid. Claire says that the sides are 8 times longer.

Who is correct? 9

.

.

:

The radius of the planet Uranus is approximately 4 times the size of the radius of Earth. What does this tell you about the volumes of these planets?

10 The rectangular prisms below are similar. Calculate the volume of shape A.

PDS endo lveeyot simular cube Wize volume is 3.375 times as big. 5

Two similar solids have side lengths in the Tato 2): a) What is the ratio of their volumes? b) The smaller shape has a volume of 100 mm’. What is the volume of the larger shape?

Cone A below has a volume of 607 cm’. Find the volume of cone B, given that the shapes are similar. A

B

aa

32.4

3

6cm

Volume = 3600 cm?

15 cm Xx

eames, Ce

Congruence and Similarity Problems

Exercise 1 1

For each of the following, decide whether the two triangles are congruent, similar or neither. In each case, explain your answer.

a)

b) 2cm

2cm

ea

(05°

“=