Cambridge IGCSE Mathematics Extended Practice Book

14 Further solving of equations and inequalities

3 'PS FBDI PG UIF GPMMPXJOH EJBHSBNT
ĕOE UIF JOFRVBMJUZ UIBU JT SFQSFTFOUFE CZ UIF VOTIBEFE
SFHJPO

5y 5y

44

33

22

1 1 x
–5 –4 –3 –2 –1 0 x 1 234 5
1 2 3 4 5 –5 –4 –3 –2 –1 0
–1
–1

–2 –2

–3 –3

–4 –4

–5 –5

5y 5y
44

33

2 2 x
1 1 234 5
1
–5 –4 –3 –2 –1 0 x
–1 1 2 3 4 5 –5 –4 –3 –2 –1 0

–1

–2 –2

–3 –3

–4 –4
–5 –5

4 #Z TIBEJOH UIF VOXBOUFE SFHJPOT
TIPX UIF SFHJPO EFĕOFE CZ UIF TFU PG JOFRVBMJUJFT

x y
x ¦ y BOE y ȴ o

5 (a) #Z TIBEJOH UIF VOXBOUFE SFHJPOT
TIPX UIF SFHJPO UIBU TBUJTĕFT BMM UIF JOFRVBMJUJFT
x y ȳ
x o y ȳ o BOE x ȴ o

(b) 8SJUF EPXO UIF JOUFHFS DPPSEJOBUFT x
y
XIJDI TBUJTGZ BMM UIF JOFRVBMJUJFT JO UIJT DBTF

14.4 Linear programming
t " NBUIFNBUJDBM XBZ UP FYQSFTT DPOTUSBJOUT JO CVTJOFTT BOE JOEVTUSZ UP PCUBJO HSFBUFTU QSPĕU
MFBTU DPTU
FUD
t $POTUSBJOUT UBLF UIF GPSN PG MJOFBS JOFRVBMJUJFT

5y

4

3 Exercise 14.4

2 1 *O UIF EJBHSBN
UIF TIBEFE SFHJPO SFQSFTFOUT UIF TFU PG JOFRVBMJUJFT y ȳ
x ȴ o
y ȴ x BOE
x y ȳ 'JOE UIF HSFBUFTU BOE MFBTU QPTTJCMF WBMVFT PG x y TVCKFDU UP UIFTF JOFRVBMJUJFT
1 x
–5 –4 –3 –2 ––110 12345 2 (a) 0O B HSJE
TIBEF UP JOEJDBUF UIF SFHJPO TBUJTGZJOH BMM UIF JOFRVBMJUJFT y ȴ
ȳ x ȳ

y ȳ x BOE y ȳ ox
–2
(b) 8 IBU JT UIF HSFBUFTU QPTTJCMF WBMVF PG y x JG x BOE y TBUJTGZ BMM UIF JOFRVBMJUJFT JO a

–3

–4

–5

Unit 4: Algebra 93

14 Further solving of equations and inequalities

Tip 3 "U B TDIPPM CBLF TBMF
DIPDPMBUF GVEHF JT TPME GPS B QSPĕU PG BOE WBOJMMB GVEHF GPS B QSPĕU
PG 4BMMZ IBT JOHSFEJFOUT UP NBLF BU NPTU CBHT PG DIPDPMBUF GVEHF BOE CBHT PG WBOJMMB
*EFOUJGZ ZPVS VOLOPXOT GVEHF 4IF IBT FOPVHI UJNF UP NBLF B NBYJNVN PG CBHT BMUPHFUIFS )PX NBOZ CBHT PG
BOE HJWF UIFN FBDI B FBDI UZQF TIPVME TIF NBLF UP NBYJNJTF IFS QSPĕU BOE XIBU JT UIJT NBYJNVN QSPĕU
WBSJBCMF
4 " GBDUPSZ NBLFT EJČFSFOU DPODFOUSBUFT GPS DPME ESJOLT ćF QSPDFTT SFRVJSFT UIF QSPEVDUJPO
1BZ BUUFOUJPO UP XPSET MJLF PG BU MFBTU UISFF MJUSFT PG PSBOHF GPS FBDI MJUSF PG MFNPO DPODFOUSBUF 'PS UIF TVNNFS
BU MFBTU
ABU MFBTU
ANJOJNVN
FUD UP MJUSFT CVU OP NPSF UIBO MJUSFT PG PSBOHF DPODFOUSBUF OFFET UP CF QSPEVDFE JO B
FYQSFTT UIF DPOTUSBJOUT BT NPOUI ćF EFNBOE GPS MFNPO
PO UIF PUIFS IBOE
JT OPU NPSF UIBO MJUSFT B NPOUI
JOFRVBMJUJFT -FNPO DPODFOUSBUF TFMMT GPS QFS MJUSF BOE PSBOHF DPODFOUSBUF TFMMT GPS QFS MJUSF )PX
NBOZ MJUSFT PG FBDI TIPVME CF QSPEVDFE JO PSEFS UP NBYJNJTF JODPNF

14.5 Completing the square

t $PNQMFUJOH UIF TRVBSF JT B NFUIPE VTFE GPS TPMWJOH RVBESBUJD FRVBUJPOT UIBU DBOOPU CF TPMWFE CZ GBDUPST

t &YQSFTTJPOT PG UIF GPSN x2 + ⎛ a⎞ 2 ⎛ a⎞ 2
⎝ 2⎠ ⎝ 2⎠
x DBO CF XSJUUFO JO UIF GPSN
x + −

Exercise 14.5

1 8SJUF UIF GPMMPXJOH FYQSFTTJPOT JO UIF GPSN (x + )2 + b

Tip (a) x2 + 6x 4 (b) x2 4x 7 (c) x2 + 14x + 44
(d) x2 − 12x + 30 (e) x2 + 10x + 17 (f) x2 + 22x + 141
*G UIF DPFďDJFOU PG x (g) x2 + 24x + 121 (h) x2 − 16x + 57 (i) x2 + 18x + 93
JT OPU
NBLF JU CZ (j) x2 2x 10 (k) x2 8x 5 (l) x2 + 20x + 83
EJWJEJOH UIF FRVBUJPO CZ
UIF DPFďDJFOU PG x 2 4PMWF UIF GPMMPXJOH RVBESBUJD FRVBUJPOT CZ UIF NFUIPE PG DPNQMFUJOH UIF TRVBSF
HJWJOH ZPVS

Tip ĕOBM BOTXFS UP UXP EFDJNBM QMBDFT JG OFDFTTBSZ

*G UIF FRVBUJPO JT OPU JO (a) x2 5x 6 = 0 (b) x2 x − 6 0 (c) x2 4x 3 = 0 (d) x2 6x 7 = 0
UIF GPSN
ax bx c
(e) x2 − 16x + 3 0 (f) x2 + 7x 1 = 0 (g) x2 + 9x 1 = 0 (h) x2 + 11x + 27 = 0
UIFO DIBOHF JU JOUP UIBU (i) x2 2x − 100 = 0
GPSN CFGPSF CFHJOOJOH UP
TPMWF JU 3 4PMWF UIF GPMMPXJOH FRVBUJPOT CZ DPNQMFUJOH UIF TRVBSF BOTXFST UP UXP EFDJNBM QMBDFT

JG OFDFTTBSZ


(a) 2x2 3x − 2 0 (b) x (x − 4) = −3 (c) 3x2 2(3x 2)

(d) 2 5= 3 (e) (x + 1)(x − 7) = 4 (f) x + 2x2 = 8
x

94 Unit 4: Algebra

14 Further solving of equations and inequalities

14.6 Quadratic formula

t ćF HFOFSBM GPSN PG UIF RVBESBUJD FRVBUJPO JT ax bx c

t ćF RVBESBUJD GPSNVMB JT x = −b ± b2 − 4ac
2a
t ćF RVBESBUJD GPSNVMB JT VTFE QSJNBSJMZ XIFO UIF RVBESBUJD FYQSFTTJPO DBOOPU CF GBDUPSJTFE

Tip Exercise 14.6

ćF œ JO UIF GPSNVMB 1 &BDI PG UIF GPMMPXJOH RVBESBUJDT XJMM GBDUPSJTF 4PMWF FBDI PG UIFN CZ GBDUPSJTBUJPO BOE UIFO
UFMMT ZPV UP DBMDVMBUF UXP
WBMVFT VTF UIF RVBESBUJD GPSNVMB UP TIPX UIBU ZPV HFU UIF TBNF BOTXFST JO CPUI DBTFT

Tip (a) x2 − 14x + 40 = 0 (b) x2 + 14x −120 = 0 (c) x2 + 5x 6 = 0
(d) x2 8x 15 0 (e) x2 3x 4 = 0 (f) x2 4x 4 = 0
5BLF DBSF XIFO UIF (g) x2 + 4x 12 0 (h) x2 + 10x + 25 = 0 (i) x2 + 2x 8 = 0
DPFďDJFOU PG x JT OPU (j) x2 4x 12 0 (k) x2 9x 20 0 (l) x2 + 3x 40 0
FRVBM UP
2 4PMWF FBDI PG UIF GPMMPXJOH FRVBUJPOT CZ VTJOH UIF RVBESBUJD GPSNVMB (JWF ZPVS BOTXFST
Tip
DPSSFDU UP UXP EFDJNBM QMBDFT XIFSF OFDFTTBSZ ćFTF RVBESBUJD FYQSFTTJPOT EP OPU GBDUPSJTF
:PV NVTU NBLF TVSF
UIBU ZPVS FRVBUJPO UBLFT (a) 2 + 6x 4 = 0 (b) x2 + x − 4 0 (c) x2 + 14x + 44 = 0
UIF GPSN PG B RVBESBUJD (d) x2 6x 8 = 0 (e) x2 + 10x + 17 = 0 (f) x2 + 7x 1 = 0
FYQSFTTJPO FRVBM UP [FSP (g) x2 + 24x + 121 = 0 (h) x2 + 11x + 27 = 0 (i) x2 + 18x − 93 = 0
*G JU EPFT OPU UIFO ZPV XJMM (j) x2 2x 10 0 (k) x2 8x 5 = 0 (l) x2 2x − 100 = 0
OFFE UP DPMMFDU BMM UFSNT PO
UP POF TJEF TP UIBU B [FSP 3 4PMWF FBDI PG UIF GPMMPXJOH FRVBUJPOT (JWF ZPVS BOTXFST DPSSFDU UP UXP EFDJNBM QMBDFT
BQQFBST PO UIF PUIFS TJEF
XIFSF OFDFTTBSZ (b) 6x2 11x 35 0 (c) 2x2 3x + 1 = 0
Consecutive numbers are one unit (a) 2 4x 8 = 0 2
apart.
(d) 2x2 5x = 25 (e) 4x2 13 9 = 0 (f) 8x2 4x = 18

(g) 9 2 1 6x (h) x (x −16) + 57 = 0 (i) 9( 1) x2

(j) 2x2 5x = 3 (k) 6x2 13x 6 = 0 (l) 4 2 3=0

4 5XP DPOTFDVUJWF OVNCFST IBWF B QSPEVDU PG 'PSN B RVBESBUJD FRVBUJPO UP ĕOE XIBU
UIF UXP OVNCFST BSF

5 ćF XJEUI PG B QPTUBHF TUBNQ JT UXP UIJSET JUT IFJHIU ćF QPTUBHF TUBNQ JT UP CF FOMBSHFE
UP DSFBUF B QPTUFS *G UIF BSFB PG UIF QPTUFS JT UP CF DN
XIBU XJMM UIF EJNFOTJPOT PG UIF
QPTUFS CF

Unit 4: Algebra 95

14 Further solving of equations and inequalities

14.7 Factorising quadratics where the coefficient of x2 is not 1
t ćF DPFďDJFOU PG x JT OPU BMXBZT &YUSB DBSF NVTU CF UBLFO XIFO FYQSFTTJOH TVDI B RVBESBUJD BT B QSPEVDU PG

JUT GBDUPST

Exercise 14.7

Tip 1 'BDUPSJTF FBDI PG UIF GPMMPXJOH FYQSFTTJPOT

"MXBZT MPPL GPS DPNNPO (a) 2 2 3 (b) 9 2 6x + 1 (c) 4x2 12 9 (d) 6x2 7x − 5
GBDUPST BT UIF ĕSTU (f) 14x2 − 51x + 7 (g) 3x2 11x 20 (h) 6x2 11x 7
TUFQ XIFO GBDUPSJTJOH (e) 4 2 3 (j) 3x2 7x − 66 (k) 15x2 16x − 15 (l) 8x2 25x 3
FYQSFTTJPOT
(i) 3x2 10x 25
/PUF − x
o x −

2 'BDUPSJTF DPNQMFUFMZ :PV NBZ OFFE UP SFNPWF B DPNNPO GBDUPS CFGPSF GBDUPSJTJOH UIF

USJOPNJBMT (b) 2x2 − 1 (c) 50x2 + 40x + 8
(a) 4x2 12 9 (f) 12x2 2x 2
2
(d) 6x2 7xy 5y2
(e) 4 4 2 3

(g) 2(x 1) 4x2 4x (h) (x + 1)2 + 3(x + 1) + 2 (i) 3(x 1)2 10(x + 1) 25

(j) 6x2 14x − 132 (k) (3 x2 1) + x (x + 1) (l) 8x2 25xy 3y2

14.8 Algebraic fractions
t :PV DBO VTF UIF UFDIOJRVFT GPS XPSLJOH XJUI OVNCFS GSBDUJPOT BOE GPS TJNQMJGZJOH JOEJDFT
t /VNFSBUPS BOE EFOPNJOBUPS DBO CF EJWJEFE DBODFMMFE
CZ UIF )$'
t 'BDUPSJTF
JG QPTTJCMF
ĕSTU

Exercise 14.8

Tip 1 4JNQMJGZ UIF GPMMPXJOH GSBDUJPOT

"MXBZT MPPL GPS DPNNPO (a) 2x (b) 3x (c) x (d) 12x
GBDUPST ĕSTU 14 x 7x 30

(e) 16z (f ) 34x (g) 12x2 (h) 5a
6 17xy 18xy 25a2b

(i) 24ab2
36a2b

2 4JNQMJGZ UIF GPMMPXJOH GSBDUJPOT

(a) x2 + x −6 (b) a2 b2 (c) 2x2 xy y2 (d) 3x2 10x 3
x 2x + 2ab x2 xy x2 2x 15
2 a2 b2

(e) a2 5a 14 (f ) a2 b2 b2 (g) 2x2 7x −15 (h) x4 2x2 3
a2 7a a2 + ab 22 6 x2 +1

(i) 35x2 + 49x
15x2 + 21x

96 Unit 4: Algebra

14 Further solving of equations and inequalities

Tip 3 8SJUF FBDI PG UIF GPMMPXJOH BT B TJOHMF GSBDUJPO JO JUT MPXFTU UFSNT

%JWJEJOH CZ B GSBDUJPO JT UIF (a) 8x × 5x (b) 3y × 2y (c) 5 × 7
TBNF BT NVMUJQMZJOH CZ UIF 15 16 4 9 a a
SFDJQSPDBM PG UIF GSBDUJPO
(d) 7x2 ÷ 14x (e) a ÷ ⎛ a × c⎞ (f ) 4a2 × b4 b2 ÷ 2a
5y 25 y 2 b ⎝ b a⎠ 3b2 2a 3

(g) 4a2 × b2 ÷ 3ab (h) x2 6x 8 × x + 3 (i) 4 9x2
7b 8a 1 3x 9 2 − x 62 2

Tip 4 4JNQMJGZ UIF GPMMPXJOH

" OFHBUJWF TJHO JO GSPOU (a) 3 + 2 (b) 3 − 4− p + 3 (c) 3 + 7 (d) p 5 1 + 2 4 2
PG B GSBDUJPO BČFDUT BMM UIF x y 2 8p 4p 2p 5p + p
UFSNT JO UIF OVNFSBUPS
(e) 3 + 2 (f ) 2m − 3( 2) (g) 5 + 4
+ +1 3 x +x
x 2 x 2 x2 − 6 x2 − 2

(h) 2 − 3 (i) 1 − 2
+ 22 x2 −
x2 3x x2 4x 3 1

Mixed exercise 1 4PMWF GPS x BOE y JG x y BOE x 2y = 12

p.a. stands for ‘per annum’, which 2 4PMWF GPS x BOE y JG y 4x = 7 BOE 2y 3x − 4 0
means ‘per year’.
3 .S )BCJC IBT UP JOWFTU )JT QPSUGPMJP IBT UXP QBSUT
POF XIJDI ZJFMET Q B BOE UIF
PUIFS Q B ćF UPUBM JOUFSFTU PO UIF JOWFTUNFOU XBT BU UIF FOE PG UIF ĕSTU ZFBS )PX
NVDI EJE IF JOWFTU BU FBDI SBUF

4 4JNQMJGZ UIF GPMMPXJOH JOFRVBMJUJFT

(a) −3 < + 2 (b) 5 − 2x ≥ 7

5 3FQSFTFOU −2 < ≤ 3 HSBQIJDBMMZ

6 ćF XIPMF OVNCFST x BOE y TBUJTGZ UIF GPMMPXJOH JOFRVBMJUJFT

y > 1 5
y ≤ 4
x ≥ −2
y > x + 1 BOE y ≥ x + 1

(SBQIJDBMMZ ĕOE UIF HSFBUFTU BOE MFBTU QPTTJCMF JOUFHFS WBMVFT PG x BOE y GPS UIF FYQSFTTJPO

x + y
2

7 'BDUPSJTF

(a) x2 2xy (b) a4 b2 (c) x2 + 6x 55
(d) 2 y2 13y 7
(e) −4x2 + 2x + 6 (f) (x + 1)2 5(x + 1) −14

Unit 4: Algebra 97

14 Further solving of equations and inequalities

8 4PMWF UIF GPMMPXJOH FRVBUJPOT

(a) 2a2 2a − 6 0 (b) 3 2 4=0 (c) x2 + 2x 15 0
(d) 3x2 5x + 2 0 (f) 3x2 6x = 6 2 3
(e) 5x2 3x = −3 2 5

9 6TF UIF RVBESBUJD GPSNVMB UP ĕOE UIF WBMVF T
PG x

(a) 5x2 8x − 4 0 (b) p 2 − qx + r = 0

10 4JNQMJGZ UIF GPMMPXJOH

(a) x2 y2 (b) 16 − 4x2 (c) 1 + 1
48 2 p2 5p
(x + y )2

(d) 7x2 × 15 y 2 (e) x ÷ ⎛ 2x × z ⎞ (f ) 3a − 5 a + a
5y 14x 2y ⎝ yz 2 xy 3 ⎠⎟ 2 3 5a

(g) 4 8 ÷ x2 4 − 2 − x2 2 4x
x2 + 2 x +

(h) 3x 3x (i) 1 5 − 2x2 1 1
6x2 2x2 11 x

98 Unit 4: Algebra

15 Scale drawings, bearings
and trigonometry

15.1 Scale drawings

t ćF TDBMF PG B EJBHSBN
PS B NBQ
DBO CF HJWFO JO UIF GPSN PG B GSBDUJPO PS B SBUJP
t " TDBMF PG NFBOT UIBU FWFSZ MJOF JO UIF EJBHSBN IBT B MFOHUI XIJDI JT 1
50000 PG UIF MFOHUI PG UIF MJOF

UIBU JU SFQSFTFOUT JO SFBM MJGF 'PS FYBNQMF DN JO UIF EJBHSBN SFQSFTFOUT DN PS LN
JO SFBM MJGF

REWIND Exercise 15.1
Revise your metric conversions
from chapter 13. 1 (a) ć F CBTJD QJUDI TJ[F PG B SVHCZ ĕFME JT N MPOH BOE N XJEF " TDBMF ESBXJOH PG B
ĕFME JT NBEF XJUI B TDBMF PG DN UP N 8IBU JT UIF MFOHUI BOE XJEUI PG UIF ĕFME JO UIF
A5 is half A4 and has dimensions ESBXJOH
14.8 cm × 21 cm.
(b) ć F QJUDI TJ[F
JODMVEJOH UIF BSFB JOTJEF UIF HPBM
JT N MPOH BOE N XJEF 8IBU BSF
UIFTF EJNFOTJPOT JO UIF ESBXJOH PG UIJT QJUDI

2 (a) ć F QJUDI TJ[F PG B TUBOEBSE IPDLFZ ĕFME JT N MPOH BOE N XJEF " TDBMF ESBXJOH
PG B IPDLFZ ĕFME JT NBEF XJUI B TDBMF PG 8IBU BSF UIF EJNFOTJPOT PG UIF IPDLFZ
ĕFME JO UIF ESBXJOH

(b) " TDIPPM UIBU XBOUT UP IPME B 4FWFO " 4JEF IPDLFZ UPVSOBNFOU IBT UISFF TUBOEBSE IPDLFZ
ĕFMET BU UIFJS 4QPSUT $FOUSF 8PVME JU CF QPTTJCMF UP IBWF ĕWF NBUDIFT UBLJOH QMBDF BU UIF
TBNF UJNF
JG UIF TJ[F PG UIF QJUDI VTFE GPS 4FWFO " 4JEF IPDLFZ JT N × N

3 (a) ć F TJ[F PG B UFOOJT DPVSU JT N × N 8IBU XPVME CF B HPPE TDBMF GPS B ESBXJOH
PG B UFOOJT DPVSU JG ZPV DBO POMZ VTF IBMG PG BO " QBHF &YQSFTT UIJT TDBMF BT B GSBDUJPO

(b) (i) .BLF BO BDDVSBUF TDBMF ESBXJOH
VTJOH ZPVS TDBMF *ODMVEF BMM UIF NBSLJOHT BT
TIPXO JO UIF EJBHSBN CFMPX

(ii) ć F OFU QPTUT BSF QMBDFE N PVUTJEF UIF EPVCMFT TJEF MJOFT .BSL FBDI OFU QPTU XJUI
BO × PO ZPVS TDBMF ESBXJOH

5.5 m Centre service line Base line
Net Service line 1.4 m
Singles side line
Doubles side line

Unit 4: Shape, space and measures 99

15 Scale drawings, bearings and trigonometry

4 (a) ć F LBSBUF DPNCBU BSFB NFBTVSFT N × N 6TJOH B TDBMF ESBXJOH BOE B TDBMF PG ZPVS
DIPJDF
DBMDVMBUF UIF MFOHUI PG UIF EJBHPOBM

(b) 8 IBU XPVME CF B NPSF BDDVSBUF XBZ UP EFUFSNJOF UIF MFOHUI PG UIF EJBHPOBM

15.2 Bearings
t " CFBSJOH JT B XBZ PG EFTDSJCJOH EJSFDUJPO
t #FBSJOHT BSF NFBTVSFE DMPDLXJTF GSPN UIF OPSUI EJSFDUJPO
t #FBSJOHT BSF BMXBZT FYQSFTTFE VTJOH UISFF ĕHVSFT

Exercise 15.2

1 (JWF UIF UISFF ĕHVSF CFBSJOH DPSSFTQPOEJOH UP

(a) FBTU (b) TPVUI XFTU (c) OPSUI XFTU

2 8SJUF EPXO UIF UISFF ĕHVSF CFBSJOHT PG X GSPN Y

(a) N (b) N (c) N

70° Y Y
X 45°
130° X
X

Y

3 7JMMBHF " JT LN FBTU BOE LN OPSUI PG WJMMBHF # 7JMMBHF $ JT LN GSPN WJMMBHF # PO B
CFBSJOH PG ° 6TJOH B TDBMF ESBXJOH XJUI B TDBMF PG ĕOE

(a) UIF CFBSJOH PG WJMMBHF # GSPN WJMMBHF "
(b) UIF CFBSJOH PG WJMMBHF " GSPN WJMMBHF $
(c) UIF EJSFDU EJTUBODF CFUXFFO WJMMBHF # BOE WJMMBHF "
(d) UIF EJSFDU EJTUBODF CFUXFFO WJMMBHF $ BOE WJMMBHF "

100 Unit 4: Shape, space and measures

15 Scale drawings, bearings and trigonometry

15.3 Understanding the tangent, cosine and sine ratios

t ćF IZQPUFOVTF JT UIF MPOHFTU TJEF PG B SJHIU BOHMFE USJBOHMF
t ćF PQQPTJUF TJEF JT UIF TJEF PQQPTJUF B TQFDJĕFE BOHMF
t ćF BEKBDFOU TJEF JT UIF TJEF UIBU GPSNT B TQFDJĕFE BOHMF XJUI UIF IZQPUFOVTF
t ćF UBOHFOU SBUJP JT the opposite side PG B TQFDJĕFE BOHMF
the adjacent side
opp( )
opp( )
tanθ = adj( ) opp( ) adj( ) × tanθ
adj( ) = tanθ

t ćF TJOF SBUJP JT the opposite side PG B TQFDJĕFE BOHMF hypotenuse
the hypotenuse
opp( )
opp( ) θ
sinθ = hyp opp hyp sinθ
hyp = sinθ adjacent side opposite side

t the adjacent side
ćF DPTJOF SBUJP JT the hypotenuse PG B TQFDJĕFE BOHMF

adj( ) adj( )
cosθ = hyp
adj hyp cosθ
hyp = cosθ

Exercise 15.3

1 $PQZ BOE DPNQMFUF UIF GPMMPXJOH UBCMF

(a) (b) (c) (d)
g qA
B e
ca A f r
p
b CA
A

hypotenuse
opp(A)
adj(A)

Remember, when working with 2 $PQZ BOE DPNQMFUF UIF TUBUFNFOU T
BMPOHTJEF FBDI USJBOHMF
right-angled triangles you may
still need to use Pythagoras. (a) PQQ °
=
60° BEK °
=
= y DN
x cm

30° °
= q DN
y cm °
= q DN
= p DN
(b) p cm
50°

q cm
40°

Unit 4: Shape, space and measures 101

15 Scale drawings, bearings and trigonometry

The memory aid, 3 $BMDVMBUF UIF WBMVF PG UIF GPMMPXJOH UBOHFOU SBUJPT
VTJOH ZPVS DBMDVMBUPS (JWF ZPVS BOTXFST
SOHCAHTOA, or the triangle UP UXP EFDJNBM QMBDFT XIFSF OFDFTTBSZ
diagrams
(a) UBO ° (b) UBO ° (c) UBO °
O AO (d) UBO ° (e) UBO °

S H C HT A 4 $PQZ BOE DPNQMFUF UIF TUBUFNFOUT GPS FBDI PG UIF GPMMPXJOH USJBOHMFT
HJWJOH ZPVS BOTXFS BT B
GSBDUJPO JO JUT MPXFTU UFSNT XIFSF OFDFTTBSZ
may help you remember the
trigonometric relationships.

(a) (b) (c)
B A
y
1.2 cm 55° B
x
3 cm x d cm 1 cm
1.8 cm
C
C 4 cm A
UBO A =
UBO x = UBO ° =
UBO y = x =
UBO B =
(e)
(d) A
Z 5m X 1.5 cm

12 m C 2.5 cm B
y

Y AC =
UBO B =
UBO y = UBO C =
∠X =
UBO X =

5 $BMDVMBUF UIF VOLOPXO MFOHUI UP UXP EFDJNBM QMBDFT
JO FBDI DBTF QSFTFOUFE CFMPX

(a) (b) (c)

B ym A
X

3 cm 72 m 15° 37° c cm
Y
65° B
C x cm A 2.5 cm
C

(d) (e)

B am C A
55°
B

13 m 25 cm x cm
45° C

A

102 Unit 4: Shape, space and measures

15 Scale drawings, bearings and trigonometry

6 'JOE
DPSSFDU UP POF EFDJNBM QMBDF
UIF BDVUF BOHMFT UIBU IBWF UIF GPMMPXJOH UBOHFOU SBUJPT

(a) (b) (c) (d)
(f) 1153 (g) 5 12
(e) 1 (h) 61
4 63

7 'JOE
DPSSFDU UP UIF OFBSFTU EFHSFF
UIF WBMVF PG UIF MFUUFSFE BOHMFT JO UIF GPMMPXJOH EJBHSBNT

(a) (b) (c) B
A 13.5 m A 17 cm
a 2.4 cm
13 m c
B b
0.7 cm 15 cm
C
(d) C
Z 5m X
(e)
12 m A
d
Y 55 cm

e B
C 70 cm

8 'PS FBDI USJBOHMF ĕOE

(a) (i) IZQ = (ii) BEK θ
= (iii) DPT θ = (e)
(b) (c) (d) Z yX
y x Ba
θ ac θ CP xz
θ rq θ
z c
b bθ Y
Q pR

A

9 'PS FBDI PG UIF GPMMPXJOH USJBOHMFT XSJUF EPXO UIF WBMVF GPS

(i) TJOF (ii) DPTJOF (iii) UBOHFOU PG UIF NBSLFE BOHMF

(a) (b) (c) (d) (e)

13 22 B 16 13 E
7 5x 4x
A 63
12 C D 85
10

Unit 4: Shape, space and measures 103

15 Scale drawings, bearings and trigonometry

10 6TF ZPVS DBMDVMBUPS UP ĕOE UP UIF OFBSFTU EFHSFF

(a) BO BDVUF BOHMF XIPTF TJOF JT

(b) BO BDVUF BOHMF XIPTF DPTJOF JT

(c) BO BDVUF BOHMF XIPTF DPTJOF JT

(d) BO BDVUF BOHMF XIPTF TJOF JT 3
2

(e) BO BDVUF BOHMF XIPTF TJOF JT 12

(f) BO BDVUF BOHMF XIPTF UBOHFOU JT

11 ćF EJBHSBN TIPXT UXP MBEEFST QMBDFE JO BO BMMFZ
CPUI SFBDIJOH VQ UP XJOEPX MFEHFT PO
PQQPTJUF TJEFT POF JT UXJDF BT MPOH BT UIF PUIFS BOE SFBDIFT IFJHIU H
XIJMF UIF TIPSUFS POF
SFBDIFT B IFJHIU h ćF MFOHUI PG UIF MPOHFS MBEEFS JT N *G UIF BOHMFT PG JODMJOBUJPO PG UIF
MBEEFST BSF BT TIPXO
IPX NVDI IJHIFS JT UIF XJOEPX MFEHF PG UIF POF XJOEPX UIBO UIBU PG
UIF PUIFS

7.5 m
H

h

Tip 27º 61º

3FNFNCFS
POMZ SPVOE 12 'PS FBDI PG UIF GPMMPXJOH USJBOHMFT ĕOE UIF MFOHUI PG UIF VOLOPXO MFUUFSFE TJEF DPSSFDU UP
ZPVS XPSLJOH BU UIF ĕOBM UXP EFDJNBM QMBDFT
PS UIF TJ[F PG UIF MFUUFSFE BOHMF DPSSFDU UP POF EFDJNBM QMBDF

TUFQ

(a) (b) (c) (d)
A A
Q C D
3x m 75°
x°
6 cm 12 cm B R O B
xm 5 cm cm
C 8.6 cm
15 m 6 cm
52° x° E
B y cm C D E P S x°
3 cm A 45° E

y° 5.6 cm
9m D

104 Unit 4: Shape, space and measures

15 Scale drawings, bearings and trigonometry

15.4 Solving problems using trigonometry
t *G OP EJBHSBN JT HJWFO
ESBX POF ZPVSTFMG
t .BSL UIF SJHIU BOHMFT JO UIF EJBHSBN
t 4IPX BMM UIF HJWFO NFBTVSFNFOUT

Tip Exercise 15.4

.BLF TVSF ZPV LOPX UIBU 1 "U B DFSUBJO UJNF PG UIF EBZ 4BNVFM T TIBEPX JT N MPOH ćF BOHMF PG FMFWBUJPO GSPN UIF
FOE PG IJT TIBEPX UP UIF UPQ PG IJT IFBE JT ¡ )PX UBMM JT 4BNVFM
horizontal
angle of depression 2 4BSBI T MJOF PG TJHIU JT N BCPWF UIF HSPVOE 4IF JT MPPLJOH BU UIF UPQ PG B USFF UIBU JT N
BXBZ
line of sight (a) *G UIF USFF JT N UBMM
XIBU JT UIF BOHMF PG FMFWBUJPO UISPVHI XIJDI TIF JT MPPLJOH
DPSSFDU UP UIF OFBSFTU EFHSFF

line of sight of (b) " CJSE TJUT PO UPQ PG UIF USFF $BMDVMBUF DPSSFDU UP UIF OFBSFTU EFHSFF
UIF BOHMF PG
angle EFQSFTTJPO GSPN UIF CJSE UP 4BSBIhT GFFU

elevation horizontal

15.5 Angles between 0° and 180°
t *G UXP BOHMFT TVN BEE VQ
UP ¡ UIFZ BSF TBJE UP CF TVQQMFNFOUBSZ
t "O BOHMF BOE JUT TVQQMFNFOU IBWF UIF TBNF TJOF WBMVF TJO θ = TJO ¡ − θ


t ćF DPTJOF PG BO BOHMF BOE JUT TVQQMFNFOU IBWF UIF TBNF WBMVF CVU EJČFSFOU TJHOT DPT θ oDPT ¡ o θ


t ćF UBOHFOU PG BO BOHMF BOE JUT TVQQMFNFOU IBWF UIF TBNF WBMVF CVU EJČFSFOU TJHOT UBO θ oUBO ¡ o θ



Exercise 15.5

Tip 1 &YQSFTT FBDI PG UIF GPMMPXJOH JO UFSNT PG UIF TVQQMFNFOUBSZ BOHMF CFUXFFO ¡ BOE ¡

TJO θ TJO ¡ o θ
(a) DPT ¡ (b) TJO ¡ (c) −DPT ¡ (d) TJO ¡
DPT θ –DPT ¡ o θ
(e) −DPT ¡ (f) TJO ¡ (g) DPT ¡ (h) TJO ¡
(i) DPT ¡ (j) DPT ¡

2 (JWFO UIBU θ JT BO BOHMF PG B USJBOHMF
ĕOE BMM QPTTJCMF WBMVFT PG θ CFUXFFO ¡ BOE ¡ UP UIF
OFBSFTU EFHSFF
JG

(a) TJO θ (b) DPT θ − (c) TJO θ (d) DPT θ
(e) TJO θ (f) DPT θ (g) DPT θ (h) DPT θ −
(i) TJO θ (j) TJO θ

15.6 The sine and cosine rules

t ćF TJOF BOE DPTJOF SVMFT DBO CF VTFE JO USJBOHMFT UIBU EP OPU IBWF B SJHIU BOHMF

t ćF TJOF SVMF sin A = sin B = sinC PS a = b = c
a b c sin A sin B sinC

t ćF DPTJOF SVMF a2 b2 + c2 bc cos A PS b2 a2 + c2 2ac cos B PS c2 = a2 + b2 ab cos C

Unit 4: Shape, space and measures 105

15 Scale drawings, bearings and trigonometry

Tip Exercise 15.6

3FNFNCFS
UIF TUBOEBSE 1 5P POF EFDJNBM QMBDF ĕOE UIF WBMVF PG x JO FBDI PG UIF GPMMPXJOH FRVBUJPOT
XBZ PG MBCFMMJOH B USJBOHMF
JT UP MBCFM UIF WFSUJDFT XJUI (a) sin = sin 45° (b) x = 23
VQQFS DBTF DBQJUBM
MFUUFST 11 12 sin 50° sin 72°
BOE UIF TJEFT PQQPTJUF XJUI
UIF TBNF MPXFS DBTF MFUUFST (c) sin x = sin 38° (d) x = 30
7.4 5.2 sin 35° sin 71°
Tip
(e) x = 8 (f ) sin = sin 45°
ćF TJOF SVMF JT VTFE XIFO sin 55° sin 68° 4 6
EFBMJOH XJUI QBJST PG
PQQPTJUF TJEFT BOE BOHMFT (g) sin = sin 35° (h) x = 8.5
24 36 sin 59° sin 62°

(i) sin = sin105°
4 16

2 'JOE UIF MFOHUI PG UIF NBSLFE TJEF JO FBDI PG UIF GPMMPXJOH USJBOHMFT

(a) (b) (c)
7.7 cm
7.7 cm 57.2° 5.5 cm
42° 114.5°
x cm 37°
x cm x cm
54.3° 72°

(d) (e) (f )
25.5° 4.9 cm
x cm 45°
23.4 cm x cm 9.7 cm 81.7° 71.5°

24° x cm

60°

(a) 3 'JOE UIF TJ[F PG UIF BDVUF
NBSLFE BOHMF JO UIF GPMMPXJOH USJBOHMFT
(b) (c)

θ 9.5 cm θ
9.9 cm
5.4 cm 4.8 cm
104°
129° θ 5.6 cm
42°
(f )
8 cm 18.2 cm

(d) (e) 40°

5.5 cm 124° 7.5 cm 78°
θ 10 cm θ

5.2 cm 11.8 cm
θ

106 Unit 4: Shape, space and measures

15 Scale drawings, bearings and trigonometry

4 (JWFO b2 a2 + c2 2ac cos B
DPQZ BOE DPNQMFUF UIF GPMMPXJOH FRVBUJPO
DPT B yyyyyyy

5 'JOE UIF TJ[F PG UIF BOHMF NBSLFE θ JO FBDI PG UIF USJBOHMFT CFMPX DPSSFDU UP POF
EFDJNBM QMBDF


(a) (b)

3 cm 3 cm 2.4 cm 3.5 cm

θ θ
2.4 cm 2.4 cm

(c) (d)

θ 4.7 cm 5 cm
2.4 cm
2.8 cm θ
3.2 cm
4.1 cm

Tip 6 'JOE UIF UIJSE TJEF PG UIF GPMMPXJOH USJBOHMFT DPSSFDU UP POF EFDJNBM QMBDF

(a) (b)
ćF DPTJOF SVMF JT VTFE
XIFO BMM UISFF TJEFT BSF AX
LOPXO PS XIFO ZPV LOPX
UXP TJEFT BOE UIF JODMVEFE 25° 6.8 cm
BOHMF z

5.6 m 6.7 m 31°
Y 7 cm Z

B aC P
(c) (d)

O

m 4.7 cm r
7 cm
131°
M Q
43.5°
5.4 cm
N 3.8 cm

R

Unit 4: Shape, space and measures 107

15 Scale drawings, bearings and trigonometry

15.7 Area of a triangle

t :PV DBO ĕOE UIF BSFB PG B USJBOHMF FWFO JG ZPV EP OPU IBWF B QFSQFOEJDVMBS IFJHIU
BT MPOH BT ZPV IBWF UXP TJEFT BOE
UIF JODMVEFE BOHMF
1 1 1
t "SFB PG USJBOHMF ABC area = 2 ab sin C PS area = 2 ac sin B PS area = 2 bc sin A

Exercise 15.7

Tip 1 Draw a rough sketch of each of these figures before calculating their area.

5P ĕOE UIF BSFB PG B /PUF UIF DPOWFOUJPO PG VTJOH B MPXFS DBTF MFUUFS UP OBNF UIF TJEF PQQPTJUF UIF WFSUFY PG UIF
USJBOHMF ZPV OFFE UXP TBNF MFUUFS IBT CFFO VTFE GPS TPNF PG UIFTF USJBOHMFT

TJEFT BOE UIF BOHMF
CFUXFFO UIF UXP (a) ΔABC XJUI BC DN
AC DN BOE ∠C ¡
HJWFO TJEFT (b) ΔABC XJUI a DN
b DN BOE ∠C ¡
*G ZPV BSF HJWFO UXP BOHMFT (c) ΔXYZ XJUI XZ DN
XY DN BOE ∠X ¡
BOE B TJEF
VTF UIF TJOF (d) ȾPQR XJUI q DN
∠Q ¡ BOE ∠R ¡
SVMF UP ĕOE BOPUIFS TJEF (e) ȾXYZ XJUI XY DN
∠X ¡ BOE ∠Y ¡
ĕSTU (f) ΔPQR XJUI p DN
q DN BOE ∠Q ¡

Tip 2 'JOE UIF BSFB PG B USJBOHMF XJUI TJEFT DN
DN BOE DN

5P ĕOE UIF BSFB PG PUIFS 3 'PS FBDI PG UIF QPMZHPOT CFMPX
ĕOE x BOE UIF BSFB PG UIF QPMZHPO
QPMZHPOT
EJWJEF UIF
TIBQF JOUP USJBOHMFT
ĕOE (a) (b) (c)
UIFJS BSFBT BOE BEE UIFN
UPHFUIFS x x 2.5 cm
7 cm 5 cm x

4 cm 4.5 cm

65° 74°
5 cm 6 cm

15.8 Trigonometry in three dimensions
t 1ZUIBHPSBT
USJHPOPNFUSJD SBUJPT
UIF TJOF SVMF
UIF DPTJOF SVMF BOE UIF BSFB SVMF DBO BMM CF VTFE UP TPMWF %

QSPCMFNT

t :PV DBO ĕOE BOHMFT CFUXFFO MJOFT BOE CFUXFFO QMBOFT

Exercise 15.8

1 ćF EJBHSBN TIPXT B SFDUBOHVMBS QSJTN 6TF 1ZUIBHPSBT BOE USJHPOPNFUSZ UP ĕOE UIF

Tip GPMMPXJOH EJTUBODFT DPSSFDU UP UISFF TJHOJĕHBOU ĕHVSFT
BOE BOHMFT DPSSFDU UP POF EFDJNBM

%SBXJOH TFQBSBUF SJHIU QMBDF
D
BOHMFE USJBOHMFT GPVOE JO A
UIF % ESBXJOH DBO IFMQ
B C 3 cm (a) DE˘F (b) FD
H E (c) DFE (d) GH
(e) HF (f) GHF
G 9 cm 4 cm (g) CH (h) CHF

F

108 Unit 4: Shape, space and measures

15 Scale drawings, bearings and trigonometry

2 ćF EJBHSBN TIPXT B TRVBSF CBTFE QZSBNJE
P

60°
Q

15 m

6TF 1ZUIBHPSBT BOE USJHPOPNFUSZ UP ĕOE UIF GPMMPXJOH EJTUBODFT DPSSFDU UP UISFF TJHOJĕDBOU
ĕHVSFT


(a) QS (b) QO (c) PQ (d) PO

Tip 3 ćF EJBHSBN TIPXT B USJBOHVMBS QSJTN
A
"MXBZT DIFDL UIBU UIF
TPMVUJPO ZPV IBWF GPVOE JT 8 cm
SFBTPOBCMF

BE D

8 cm

F 4 cm C

(a) $BMDVMBUF UIF MFOHUI PG BD DPSSFDU UP UXP EFDJNBM QMBDFT

(b) $ BMDVMBUF BDF
UIF BOHMF BD NBLFT XJUI UIF CBTF CDEF DPSSFDU UP POF EFDJNBM QMBDF


Mixed exercise 1 'JOE UIF WBMVF PG x

(a) sin x = sin121°
5.2 7.3

(b)

3.5 cm 5 cm
x
(c)
7.7 cm 4 cm

x 11.5 cm
39.4°

Unit 4: Shape, space and measures 109

15 Scale drawings, bearings and trigonometry

Tip 2 " USJBOHMF IBT TJEFT PG MFOHUI DN
DN BOE DN 6TF UIF DPTJOF SVMF UP ĕOE UIF TJ[F PG
UIF TNBMMFTU BOHMF DPSSFDU UP UISFF TJHOJĕDBOU ĕHVSFT

ćF TNBMMFTU BOHMF JO B
USJBOHMF JT PQQPTJUF UIF 3 *O UIF ĕHVSF
CD N 'JOE AB
TIPSUFTU TJEF A

50° 21° D
BC 530 m

4 BD DN 'JOE AC
A

70° C 50°
B 50 cm D

5 " NBO JT TUBOEJOH BU B MPPLPVU QPJOU N BCPWF UIF TFB )F TQPUT B TIBSL JO UIF XBUFS BU BO
BOHMF PG EFQSFTTJPO PG ¡ " TXJNNFS JO UIF XBUFS JT N GSPN UIF GPPU PG UIF MPPLPVU QPJOU
)PX GBS JT UIF TIBSL GSPN UIF TXJNNFS

6 ćF EJBHSBN OPU ESBXO UP TDBMF
TIPXT UXP BFSPQMBOFT
X BOE Y
ĘZJOH PWFS BO BJSĕFME ćF
BFSPQMBOFT BSF ĘZJOH EJSFDUMZ CFIJOE FBDI PUIFS BOE BSF NFUSFT BCPWF UIF HSPVOE R
S
BOE T BSF QPJOUT PO UIF HSPVOE BOE UIF BOHMF PG FMFWBUJPO PG QMBOF X GSPN T JT ¡ BOE UIF
BOHMF PG FMFWBUJPO PG Y JT ¡ 'JOE UIF EJTUBODF CFUXFFO UIF UXP BFSPQMBOFT RS
TIPXJOH BMM
XPSLJOH BOE HJWF ZPVS ĕOBM BOTXFS DPSSFDU UP UIF OFBSFTU NFUSF

XY

1500 m 77°

R 58°
ST

110 Unit 4: Shape, space and measures

16 Scatter diagrams
and correlation

16.1 Introduction to bivariate data
t 8IFO ZPV DPMMFDU UXP TFUT PG EBUB JO QBJST
JU JT DBMMFE CJWBSJBUF EBUB 'PS FYBNQMF ZPV DPVME DPMMFDU

IFJHIU BOE NBTT EBUB GPS WBSJPVT TUVEFOUT

t #JWBSJBUF EBUB DBO CF QMPUUFE PO B TDBUUFS EJBHSBN JO PSEFS UP MPPL GPS DPSSFMBUJPO o B SFMBUJPOTIJQ CFUXFFO
UIF EBUB 'PS FYBNQMF
JG ZPV XBOUFE UP LOPX XIFUIFS UBMMFS TUVEFOUT XFJHIFE NPSF UIBO TNBMMFS TUVEFOUT
ZPV
DPVME QMPU UIF UXP TFUT PG EBUB IFJHIU BOE NBTT
PO B TDBUUFS EJBHSBN

t $PSSFMBUJPO JT EFTDSJCFE BT QPTJUJWF PS OFHBUJWF
BOE TUSPOH PS XFBL 8IFO UIF QPJOUT GPMMPX OP SFBM QBUUFSO

UIFSF JT OP DPSSFMBUJPO

t " MJOF PG CFTU ĕU DBO CF ESBXO PO B TDBUUFS EJBHSBN UP EFTDSJCF UIF DPSSFMBUJPO ćJT MJOF TIPVME GPMMPX UIF
EJSFDUJPO PG UIF QPJOUT PO UIF HSBQI BOE UIFSF TIPVME CF NPSF PS MFTT UIF TBNF OVNCFS PG QPJOUT PO FBDI TJEF PG
UIF MJOF :PV DBO VTF B MJOF PG CFTU ĕU UP NBLF QSFEJDUJPOT XJUIJO UIF SBOHF PG UIF EBUB TIPXO *U JT OPU TUBUJTUJDBMMZ
BDDVSBUF UP QSFEJDU CFZPOE UIF WBMVFT QMPUUFE

Exercise16.1

1 .BUDI FBDI HSBQI CFMPX UP B EFTDSJQUJPO PG UIF DPSSFMBUJPO TIPXO

ABC

DE

(a) XFBL OFHBUJWF (b) OP DPSSFMBUJPO (c) TUSPOH QPTJUJWF
(d) TUSPOH OFHBUJWF (e) XFBL QPTJUJWF

Unit 4: Data handling 111

16 Scatter diagrams and correlation

Distance jumped (m)2 4PPLJF DPMMFDUFE EBUB GSPN TUVEFOUT JO IFS TDIPPM BUIMFUJDT UFBN 4IF XBOUFE UP TFF JG UIFSF
XBT B DPSSFMBUJPO CFUXFFO UIF IFJHIU PG UIF TUVEFOUT BOE UIF EJTUBODF UIFZ DPVME KVNQ JO UIF
MPOH KVNQ FWFOU 4IF ESFX B TDBUUFS EJBHSBN UP TIPX UIF EBUB
Student heights compared to distance jumped
6.0

5.5

5.0

4.5

4.0

150 155 160 165 170 175 180
Height (cm)

(a) $PQZ UIF EJBHSBN BOE ESBX UIF MJOF PG CFTU ĕU PO UP JU
(b) 6TF ZPVS MJOF PG CFTU ĕU UP FTUJNBUF IPX GBS B TUVEFOU DN UBMM DPVME KVNQ
(c) ' PS UIF BHF HSPVQ PG 4PPLJF T TDIPPM UFBN
UIF HJSMT SFDPSE GPS MPOH KVNQ JT N

)PX UBMM XPVME ZPV FYQFDU B HJSM UP CF XIP DPVME FRVBM UIF SFDPSE KVNQ
(d) %FTDSJCF UIF DPSSFMBUJPO TIPXO PO UIF HSBQI
(e) 8 IBU EPFT UIF DPSSFMBUJPO JOEJDBUF BCPVU UIF SFMBUJPOTIJQ CFUXFFO IFJHIU BOE IPX GBS

ZPV DBO KVNQ JO UIF MPOH KVNQ FWFOU

3 ćF UBCMF CFMPX TIPXT UIF BHFT PG UFO TUVEFOUT BOE UIF EJTUBODF UIFZ DBO TXJN JO IBMG BO IPVS

Tip Student Age (years) Distance (m)
"NZ
8IFO UJNF JT POF PG UIF #FUI 8
EBUB QBJST
JU JT OPSNBMMZ $IFSJF
UIF JOEFQFOEFOU WBSJBCMF
%BOJ
TP ZPV XJMM QMPU JU PO UIF &NNB 9
IPSJ[POUBM BYJT 'SBO
(JUB
)BOOBI
*OHF 9
Jen

(a) 8IBU JT UIF EFQFOEFOU WBSJBCMF
(b) 1MPU B TDBUUFSHSBN
(c) %FTDSJCF UIF DPSSFMBUJPO
(d) %SBX B MJOF PG CFTU ĕU
(e) ) PX PME XPVME ZPV FTUJNBUF ,BUF UP CF JG TIF JT BCMF UP TXJN N JO IBMG BO IPVS
(f) )PX SFMJBCMF JT ZPVS BOTXFS UP e

(g) ) PX GBS XPVME ZPV FTUJNBUF -ZOOF XIP JT
DBO TXJN JO IBMG BO IPVS

112 Unit 4: Data handling

16 Scatter diagrams and correlation

Mixed exercise 1 4UVEZ UIF TDBUUFS EJBHSBN BOE BOTXFS UIF RVFTUJPOT
(a) 8IBU EPFT UIJT EJBHSBN TIPX
Accidents at a road junction (b) 8IBU JT UIF JOEFQFOEFOU WBSJBCMF
50 (c) $PQZ UIF EJBHSBN BOE ESBX B MJOF PG CFTU 6TF ZPVS CFTU ĕU MJOF UP QSFEJDU
(i) U IF OVNCFS BDDJEFOUT BU UIF KVODUJPO XIFO UIF BWFSBHF TQFFE PG WFIJDMFT JT LN I
40 (ii) XIBU UIF BWFSBHF TQFFE PG WFIJDMFT JT XIFO UIFSF BSF GFXFS UIBO BDDJEFOUT
(d) %FTDSJCF UIF DPSSFMBUJPO
30 (e) 8 IBU EPFT ZPVS BOTXFS UP d
UFMM ZPV BCPVU UIF SFMBUJPOTIJQ CFUXFFO TQFFE BOE UIF
OVNCFS PG BDDJEFOUT BU B KVODUJPO
20
2 " CSBOE OFX DBS .PEFM 9
DPTUT .S 4NJU XBOUT UP ĕOE PVU XIBU QSJDF TFDPOE IBOE
10 .PEFM 9 DBST BSF TPME GPS )F ESFX UIJT TDBUUFS HSBQI UP TIPX UIF SFMBUJPOTIJQ CFUXFFO UIF
QSJDF BOE UIF BHF PG DBST JO B TFDPOE IBOE DBS EFBMFSTIJQ
0
0 35 70 105 140 Comparison of car age with re-sale price
Average speed (km/h)
Number of accidents 15
Price ($ 000)
12

9

6

3

0
12345
Age (years)

(a) %FTDSJCF UIF USFOE TIPXO PO UIJT HSBQI
(b) #FUXFFO XIJDI UXP ZFBST EPFT UIF QSJDF PG UIF DBS GBMM JO WBMVF CZ UIF MBSHFTU BNPVOU
(c) %FTDSJCF XIBU IBQQFOT UP UIF QSJDF XIFO UIF DBS JT UP ZFBST PME
(d) )PX PME XPVME ZPV FYQFDU B TFDPOE IBOE .PEFM 9 DBS UP CF JG JU XBT BEWFSUJTFE GPS

TBMF BU
(e) 8IBU QSJDF SBOHF XPVME ZPV FYQFDU B ZFBS PME .PEFM 9 UP GBMM JOUP

Unit 4: Data handling 113

17 Managing money

17.1 Earning money
t 1FPQMF XIP BSF GPSNBMMZ FNQMPZFE NBZ CF QBJE B TBMBSZ
XBHF PS FBSO DPNNJTTJPO

o " TBMBSZ JT B ĕYFE BNPVOU GPS B ZFBS PG XPSL
VTVBMMZ QBJE JO NPOUIMZ JOTUBMNFOUT

o " XBHF JT BO BHSFFE IPVSMZ SBUF GPS BO BHSFFE OVNCFS PG IPVST
OPSNBMMZ QBJE XFFLMZ

o 8PSLFST XIP TFMM UIJOHT GPS B MJWJOH BSF PęFO QBJE B DPNNJTTJPO ćJT JT B QFSDFOUBHF PG UIF WBMVF PG
UIF HPPET‫ڀ‬TPME

t "EEJUJPOBM BNPVOUT NBZ CF QBJE UP FNQMPZFFT JO UIF GPSN PG PWFSUJNF PS CPOVTFT
t &NQMPZFST NBZ EFEVDU BNPVOUT GSPN FNQMPZFFT FBSOJOHT TVDI BT JOTVSBODF
VOJPO EVFT
NFEJDBM

BJE BOE UBYFT

t ćF BNPVOU B QFSTPO FBSOT CFGPSF EFEVDUJPOT JT UIFJS HSPTT FBSOJOHT ćF BNPVOU UIFZ BDUVBMMZ BSF
QBJE BęFS EFEVDUJPOT JT UIFJS OFU FBSOJOHT

Exercise 17.1

Casual workers are normally paid 1 " XPNBO XPSLT B IPVS XFFL 4IF FBSOT 8IBU JT IFS IPVSMZ SBUF PG QBZ
an hourly rate for the hours they
work. 2 8IBU JT UIF BOOVBM TBMBSZ PG B QFSTPO XIP JT QBJE QFS NPOUI

Tip 3 "O FMFDUSJDJBO T BTTJTUBOU FBSOT QFS IPVS GPS B IPVS XFFL )F JT QBJE UJNFT IJT
IPVSMZ SBUF GPS FBDI IPVS IF XPSLT BCPWF IPVST )PX NVDI XPVME IF FBSO JO B XFFL JG
3FNFNCFS
XPSL PVU BMM IF‫ڀ‬XPSLFE
QFSDFOUBHF EFEVDUJPOT PO
UIF HSPTT JODPNF BOE UIFO (a) IPVST (b) IPVST (c) IPVST (d) 42 1 IPVST
TVCUSBDU UIFN BMM GSPN UIF 2
HSPTT
4 4BOEJMF FBSOT B HSPTT TBMBSZ PG QFS NPOUI )JT FNQMPZFS EFEVDUT JODPNF UBY

Some workers are paid for each JOTVSBODF BOE GPS VOJPO EVFT 8IBU JT 4BOEJMF T OFU TBMBSZ
piece of work they complete. This
is called piece work. 5 " DMPUIJOH GBDUPSZ XPSLFS JO *OEPOFTJB JT QBJE UIF FRVJWBMFOU PG QFS DPNQMFUFE
HBSNFOU )PX NVDI XPVME IF FBSO JG IF DPNQMFUFE HBSNFOUT JO B NPOUI

6 /BBEJSB SFDFJWFT BO BOOVBM TBMBSZ PG 4IF QBZT PG IFS XFFLMZ HSPTT FBSOJOHT JOUP
IFS QFOTJPO GVOE "O BEEJUJPOBM JT EFEVDUFE FBDI XFFL GSPN IFS TBMBSZ $BMDVMBUF

(a) IFS XFFLMZ HSPTT FBSOJOHT
(b) IFS XFFLMZ QFOTJPO GVOE QBZNFOU
(c) IFS OFU JODPNF QFS XFFL

114 Unit 5: Number

17 Managing money

17.2 Borrowing and investing money

t 8IFO ZPV CPSSPX NPOFZ ZPV NBZ QBZ JOUFSFTU PO UIF BNPVOU CPSSPXFE
t 8IFO ZPV JOWFTU PS TBWF NPOFZ ZPV NBZ FBSO JOUFSFTU PO UIF BNPVOU JOWFTUFE
t *G UIF BNPVOU PG JOUFSFTU QBJE PS DIBSHFE
JT UIF TBNF GPS FBDI ZFBS
UIFO JU JT DBMMFE TJNQMF JOUFSFTU
t 8IFO UIF JOUFSFTU GPS POF ZFBS JT BEEFE UP UIF JOWFTUNFOU PS EFCU
BOE UIF JOUFSFTU GPS UIF OFYU ZFBS
JT DBMDVMBUFE PO UIF JODSFBTFE JOWFTUNFOU PS EFCU

JU JT DBMMFE DPNQPVOE JOUFSFTU

t ćF PSJHJOBM BNPVOU CPSSPXFE PS JOWFTUFE JT DBMMFE UIF QSJODJQBM
t 'PS TJNQMF JOUFSFTU
UIF JOUFSFTU QFS BOOVN = JOUFSFTU SBUF × QSJODJQBM PSJHJOBM TVN JOWFTUFE

PRT
t ćF GPSNVMB VTFE UP DBMDVMBUF TJNQMF JOUFSFTU JT I = 100 XIFSF

o 1 = UIF QSJODJQBM

o 3 = UIF JOUFSFTU SBUF

o 5 = UIF UJNF JO ZFBST


t 'PS DPNQPVOE JOUFSFTU
LOPXJOH IPX UP VTF B NVMUJQMJFS DBO IFMQ ZPV EP UIF DBMDVMBUJPOT GBTUFS

o 'PS FYBNQMF
JG UIF DPNQPVOE JOUFSFTU JT
UIFO UIF NVMUJQMJFS JT 110005 = 1 05

o .VMUJQMZ UIF QSJODJQBM CZ B QPXFS PG UIF NVMUJQMJFS ćF OVNCFS PG ZFBST PG UIF JOWFTUNFOU UFMMT ZPV

XIBU UIF QPXFS JT 4P
JG JU ZPV JOWFTU B TVN GPS UISFF ZFBST BU
ZPV XPVME NVMUJQMZ CZ
ćJT HJWFT

ZPV UIF ĕOBM‫ڀ‬BNPVOU

t )JSF QVSDIBTF )1
JT B NFUIPE PG CVZJOH UIJOHT PO DSFEJU BOE QBZJOH UIFN PČ PWFS BO BHSFFE QFSJPE PG UJNF
/PSNBMMZ ZPV QBZ B EFQPTJU BOE FRVBM NPOUIMZ JOTUBMNFOUT

‘per annum’ (p.a.) means Exercise 17.2
‘per year’
1 $BMDVMBUF UIF TJNQMF JOUFSFTU PO
Tip
(a) JOWFTUFE GPS B ZFBS BU UIF SBUF PG QFS BOOVN
:PV DBO DIBOHF UIF TVCKFDU (b) JOWFTUFE GPS ĕWF ZFBST BU UIF SBUF PG QFS BOOVN
PG UIF TJNQMF JOUFSFTU (c) JOWFTUFE GPS UXP ZFBST BU UIF SBUF PG QFS BOOVN
GPSNVMB (d) JOWFTUFE GPS FJHIU ZFBST BU UIF SBUF PG QFS BOOVN
(e) JOWFTUFE GPS NPOUIT BU UIF SBUF PG QFS BOOVN
I = PRT
100 2 JT JOWFTUFE BU QFS BOOVN TJNQMF JOUFSFTU )PX MPOH XJMM JU UBLF GPS UIF BNPVOU UP
SFBDI
P = 100I
RT 3 ćF UPUBM TJNQMF JOUFSFTU PO JOWFTUFE GPS ĕWF ZFBST JT 8IBU JT UIF QFSDFOUBHF SBUF
QFS BOOVN
R = 100I
PT 4 ćF DBTI QSJDF PG B DBS XBT ćF IJSF QVSDIBTF QSJDF XBT B EFQPTJU BOE
JOTUBMNFOUT PG QFS NPOUI GPS UXP ZFBST )PX NVDI NPSF UIBO UIF DBTI QSJDF XBT UIF
T = 100I IJSF QVSDIBTF QSJDF
PR
5 -FCP DBO QBZ DBTI GPS B OFX DBS PS IF DBO CVZ JU PO )1 CZ QBZJOH B EFQPTJU BOE
NPOUIMZ QBZNFOUT PG )PX NVDI FYUSB XJMM IF QBZ CZ CVZJOH PO )1

Unit 5: Number 115

17 Managing money

Using the multiplying factor for 6 $BMDVMBUF UIF DPNQPVOE JOUFSFTU PO
compound interest gives the new
amount. Subtract the principal to (a) JOWFTUFE GPS B ZFBS BU UIF SBUF PG QFS BOOVN
find the actual interest. (b) JOWFTUFE GPS ĕWF ZFBST BU UIF SBUF PG QFS BOOVN
(c) JOWFTUFE GPS UXP ZFBST BU UIF SBUF PG QFS BOOVN
(d) CPSSPXFE GPS FJHIU ZFBST BU UIF SBUF PG QFS BOOVN
(e) CPSSPXFE GPS NPOUIT BU UIF SBUF PG QFS BOOVN

7 )PX NVDI XJMM ZPV IBWF JO UIF CBOL JG ZPV JOWFTU GPS GPVS ZFBST BU JOUFSFTU

DPNQPVOEFE BOOVBMMZ

8 .ST (FOBSP PXOT B TNBMM CVTJOFTT 4IF CPSSPXT EPMMBST GSPN UIF CBOL UP ĕOBODF
TPNF OFX FRVJQNFOU 4IF SFQBZT UIF MPBO JO GVMM BęFS UXP ZFBST *G UIF CBOL DIBSHFE
IFS‫ڀ‬DPNQPVOE JOUFSFTU BU UIF SBUF PG QFS BOOVN
IPX NVDI EJE TIF PXF UIFN BęFS
UXP‫ڀ‬ZFBST

17.3 Buying and selling

t ćF BNPVOU B CVTJOFTT QBZT GPS BO JUFN JT DBMMFE UIF DPTU QSJDF ćF QSJDF UIFZ TFMM JU UP UIF QVCMJD GPS JT
DBMMFE UIF TFMMJOH QSJDF ćF BNPVOU UIF TFMMFS BEET POUP UIF DPTU QSJDF UP NBLF B TFMMJOH QSJDF JT DBMMFE B

NBSL VQ 'PS FYBNQMF
B TIPQLFFQFS NBZ CVZ BO JUFN GPS BOE NBSL JU VQ CZ DFOUT UP TFMM JU GPS

t ćF EJČFSFODF CFUXFFO UIF DPTU QSJDF BOE UIF TFMMJOH QSJDF JT DBMMFE UIF QSPĕU JG JU JT IJHIFS UIBO UIF DPTU
QSJDF
PS UIF MPTT JG JU JT MPXFS UIBO UIF DPTU QSJDF


o 1SPĕU = TFMMJOH QSJDF o DPTU QSJDF

o -PTT = DPTU QSJDF o TFMMJOH QSJDF

t profit (or loss)
ćF SBUF PG QSPĕU PS MPTT
JT UIF QFSDFOUBHF QSPĕU PS MPTT
Rate of profit (or loss) = cost price ×100%

t " EJTDPVOU JT BO JOUFOUJPOBM SFEVDUJPO JO UIF QSJDF PG BO JUFN
%JTDPVOU = PSJHJOBM TFMMJOH QSJDF o OFX NBSLFE QSJDF

t ćF SBUF PG EJTDPVOU JT B QFSDFOUBHF Rate of discount = discount price × 100%
original selling

Exercise 17.3

1 'JOE UIF DPTU QSJDF JO FBDI PG UIF GPMMPXJOH

(a) TFMMJOH QSJDF
QSPĕU
(b) TFMMJOH QSJDF
QSPĕU
(c) TFMMJOH QSJDF
MPTT
(d) TFMMJOH QSJDF
MPTT 33 13
2 'JOE UIF DPTU QSJDF PG BO BSUJDMF TPME BU XJUI B QSPĕU PG

3 *G B TIPQLFFQFS TFMMT BO BSUJDMF GPS BOE MPTFT PO UIF TBMF
ĕOE IJT DPTU QSJDF

4 " EFOUJTU PČFST B EJTDPVOU UP QBUJFOUT XIP QBZ UIFJS BDDPVOUT JO DBTI XJUIJO B XFFL
)PX NVDI XJMM TPNFPOF XJUI BO BDDPVOU PG QBZ JG UIFZ QBZ QSPNQUMZ JO DBTI

116 Unit 5: Number

17 Managing money

5 $BMDVMBUF UIF OFX TFMMJOH QSJDF PG FBDI JUFN XJUI UIF GPMMPXJOH EJTDPVOUT

(a) EJTDPVOU

(b) EJTDPVOU

(c) EJTDPVOU 5 1
2

Mixed exercise 1 /FSJOB FBSOT QFS IPVS )PX NBOZ IPVST EPFT TIF OFFE UP XPSL UP FBSO

(a)
(b)
(c)

2 " NFDIBOJD XPSLT B IPVS XFFL GPS B CBTJD XBHF PG QFS IPVS 0WFSUJNF JT QBJE BU UJNF
BOE B IBMG PO XFFLEBZT BOE EPVCMF UJNF PO XFFLFOET $BMDVMBUF IJT HSPTT FBSOJOHT GPS B XFFL
JG IF XPSLT IJT OPSNBM IPVST QMVT

(a) UISFF IPVST PWFSUJNF PO ćVSTEBZ

(b) POF FYUSB IPVS QFS EBZ GPS UIF XIPMF XFFL

(c) UXP IPVST PWFSUJNF PO 5VFTEBZ BOE 1 1 IPVST PWFSUJNF PO 4BUVSEBZ
2

3 +BNJSB FBSOT B NPOUIMZ TBMBSZ PG

(a) 8IBU JT IFS BOOVBM HSPTT TBMBSZ
(b) 4 IF QBZT UBY BOE IBT B GVSUIFS EFEVDUFE GSPN IFS NPOUIMZ TBMBSZ $BMDVMBUF

IFS OFU NPOUIMZ JODPNF

4 " JOWFTUNFOU FBSOT JOUFSFTU BU B SBUF PG Q B ćJT UBCMF DPNQBSFT UIF TJNQMF BOE
DPNQPVOE JOUFSFTU

Years 12345678

Simple interest

Compound interest

(a) $PQZ BOE DPNQMFUF UIF UBCMF
(b) 8IBU JT UIF EJČFSFODF CFUXFFO UIF TJNQMF JOUFSFTU BOE DPNQPVOE JOUFSFTU FBSOFE BęFS

ĕWF ZFBST
(c) % SBX B CBS DIBSU UP DPNQBSF UIF WBMVF PG UIF JOWFTUNFOU BęFS POF
ĕWF BOE ZFBST

GPS CPUI UZQFT PG JOUFSFTU $PNNFOU PO XIBU ZPVS HSBQI TIPXT BCPVU UIF EJČFSFODF
CFUXFFO TJNQMF BOE DPNQPVOE JOUFSFTU

5 'JOE UIF TFMMJOH QSJDF PG BO BSUJDMF UIBU XBT CPVHIU GPS BOE TPME BU B QSPĕU PG

6 $BMDVMBUF UIF TFMMJOH QSJDF PG BO JUFN PG NFSDIBOEJTF CPVHIU GPS BOE TPME BU B
QSPĕU‫ڀ‬PG‫ ڀ‬

Unit 5: Number 117

17 Managing money

7 " HBMMFSZ PXOFS EJTQMBZT QBJOUJOHT GPS BSUJTUT 4IF QVUT B NBSL VQ PO UIF QSJDF BTLFE
CZ UIF BSUJTU UP DPWFS IFS FYQFOTFT BOE NBLF B QSPĕU "O BSUJTU TVQQMJFT UISFF QBJOUJOHT BU
UIF QSJDFT MJTUFE CFMPX 'PS FBDI POF
DBMDVMBUF UIF NBSL VQ JO EPMMBST
BOE UIF TFMMJOH QSJDF
UIF HBMMFSZ PXOFS XPVME DIBSHF
(a) 1BJOUJOH "

(b) 1BJOUJOH #

(c) 1BJOUJOH $


8 "O BSU DPMMFDUPS XBOUT UP CVZ QBJOUJOHT " BOE # GSPN RVFTUJPO
)F BHSFFT UP QBZ
DBTI PO DPOEJUJPO UIBU UIF HBMMFSZ PXOFS HJWFT IJN B EJTDPVOU PO UIF TFMMJOH QSJDF PG
UIF‫ڀ‬QBJOUJOHT

(a) 8IBU QSJDF XJMM IF QBZ
(b) 8IBU QFSDFOUBHF QSPĕU EPFT UIF HBMMFSZ PXOFS NBLF PO UIF TBMF

9 " CPZ CPVHIU B CJDZDMF GPS "ęFS VTJOH JU GPS UXP ZFBST
IF TPME JU BU B MPTT PG
$BMDVMBUF UIF TFMMJOH QSJDF

10 *U JT GPVOE UIBU BO BSUJDMF JT CFJOH TPME BU B MPTT PG ćF DPTU PG UIF BSUJDMF XBT
$BMDVMBUF UIF TFMMJOH QSJDF

11 " XPNBO NBLFT ESFTTFT )FS UPUBM DPTUT GPS UFO ESFTTFT XFSF "U XIBU QSJDF TIPVME TIF
TFMM UIF ESFTTFT UP NBLF QSPĕU

12 4BM XBOUT UP CVZ B VTFE TDPPUFS ćF DBTI QSJDF JT 5P CVZ PO DSFEJU
TIF IBT UP QBZ
B EFQPTJU BOE UIFO NPOUIMZ JOTUBMNFOUT PG FBDI )PX NVDI XJMM TIF TBWF CZ
QBZJOH DBTI

118 Unit 5: Number

18 Curved graphs

18.1 Plotting quadratic graphs (the parabola)
t ćF IJHIFTU QPXFS PG B WBSJBCMF JO B RVBESBUJD FRVBUJPO JT UXP
t ćF HFOFSBM GPSNVMB GPS B RVBESBUJD HSBQI JT y = ax + bx + c
t ćF BYJT PG TZNNFUSZ PG UIF HSBQI EJWJEFT UIF QBSBCPMB JOUP UXP TZNNFUSJDBM IBMWFT
t ćF UVSOJOH QPJOU JT UIF QPJOU BU XIJDI UIF HSBQI DIBOHFT EJSFDUJPO ćJT QPJOU JT BMTP DBMMFE UIF WFSUFY PG UIF HSBQI

o *G UIF WBMVF PG a JO UIF HFOFSBM GPSN PG B RVBESBUJD FRVBUJPO JT QPTJUJWF
UIF QBSBCPMB XJMM CF B AWBMMFZ TIBQF BOE UIF
y WBMVF PG UIF UVSOJOH QPJOU B NJOJNVN WBMVF

o *G UIF WBMVF PG a JO UIF HFOFSBM GPSN PG B RVBESBUJD FRVBUJPO JT OFHBUJWF
UIF QBSBCPMB XJMM CF B AIJMM TIBQF BOE UIF
y WBMVF PG UIF UVSOJOH QPJOU B NBYJNVN WBMVF

Exercise 18.1

Remember, the constant term 1 $PQZ BOE DPNQMFUF UIF GPMMPXJOH UBCMFT 1MPU BMM UIF HSBQIT POUP UIF TBNF TFU PG BYFT 6TF
(c in the general formula) is the WBMVFT PG − UP PO UIF y BYJT
y-intercept.
(a) x
y = −x2 + 2 −3 −2 −1 0 1 2 3

(b) x −3 −2 −1 0 1 2 3
y = x2 − 3

(c) x −3 −2 −1 0 1 2 3
y = −x2 − 2

(d) x −3 −2 −1 0 1 2 3

y = −x2 − 3

(e) x −3 −2 −1 0 1 2 3

y = x2 + 1
2

Unit 5: Algebra 119

18 Curved graphs

2 .BUDI FBDI PG UIF ĕWF QBSBCPMBT TIPXO IFSF UP POF PG UIF FRVBUJPOT HJWFO

y (a)
13
12

11 (b)

10

9 (c)

8

7

6

5

4

3

2

1
x

–4 –3 –2 –1 –10 1234

–2

–3

–4

–5

–6 (d)

–7

–8

–9

–10

–11

–12
–13 (e)

y 4 x2 y x2 + 3 y 3+ x2 y = x2 +2 y x2

3 $PQZ BOE DPNQMFUF UIF UBCMF PG WBMVFT GPS FBDI PG UIF FRVBUJPOT HJWFO CFMPX 1MPU UIF QPJOUT
PO TFQBSBUF QBJST PG BYFT BOE KPJO UIFN
XJUI B TNPPUI DVSWF
UP ESBX UIF HSBQI PG UIF
FRVBUJPO

(a) x −2 −1 0 1 2 3 4 5

y = x2 − 3x + 2

(b) x −3 −2 −1 0 1 2 3

y = x2 − 2x − 1

(c) x −2 −1 0 1 2 3 4 5 6

y = −x2 + 4x + 1

120 Unit 5: Algebra

18 Curved graphs

Height in metres4 " UPZ SPDLFU JT UISPXO VQ JOUP UIF BJS ćF HSBQI CFMPX TIPXT JUT QBUI

10

8

6

4

2

0
012345
Time in seconds

(a) 8IBU JT UIF HSFBUFTU IFJHIU UIF SPDLFU SFBDIFT
(b) )PX MPOH EJE JU UBLF GPS UIF SPDLFU UP SFBDI UIJT IFJHIU
(c) )PX IJHI EJE UIF SPDLFU SFBDI JO UIF ĕSTU TFDPOE
(d) 'PS IPX MPOH XBT UIF SPDLFU JO UIF BJS
(e) &TUJNBUF GPS IPX MPOH UIF SPDLFU XBT IJHIFS UIBO N BCPWF HSPVOE

18.2 Plotting reciprocal graphs (the hyperbola)

t ćF HFOFSBM GPSNVMB GPS B IZQFSCPMB HSBQI JT y = a PS xy = a
x
t x ȶ BOE y ȶ
t ćF x BYJT BOE UIF y BYJT BSF BTZNQUPUFT ćJT NFBOT UIF HSBQI BQQSPBDIFT UIF x BYJT BOE UIF y BYJT CVU OFWFS
JOUFSTFDUT XJUI UIFN

t ćF HSBQI JT TZNNFUSJDBM BCPVU UIF y = x BOE y = −x MJOF

Tip Exercise 18.2

3FDJQSPDBM FRVBUJPOT IBWF 1 %SBX HSBQIT GPS UIF GPMMPXJOH SFDJQSPDBM HSBQIT 1MPU BU MFBTU UISFF QPJOUT JO FBDI PG UIF UXP
B DPOTUBOU QSPEVDU ćJT RVBESBOUT BOE KPJO UIFN VQ XJUI B TNPPUI DVSWF
NFBOT JO xy = a
x BOE
y BSF WBSJBCMFT CVU a JT B (a) xy = 5 (b) y = 16 (c) xy = 9
DPOTUBOU x

(d) y = − 8 (e) y = − 4
x x

2 ćF MFOHUI BOE XJEUI PG B DFSUBJO SFDUBOHMF DBO POMZ CF B XIPMF OVNCFS PG NFUSFT ćF BSFB

PG UIF SFDUBOHMF JT N2

(a) %SBX B UBCMF UIBU TIPXT BMM UIF QPTTJCMF DPNCJOBUJPOT PG NFBTVSFNFOUT GPS UIF MFOHUI
BOE XJEUI PG UIF SFDUBOHMF

(b) 1MPU ZPVS WBMVFT GSPN a
BT QPJOUT PO B HSBQI
(c) +PJO UIF QPJOUT XJUI B TNPPUI DVSWF 8IBU EPFT UIJT HSBQI SFQSFTFOU
(d) " TTVNJOH
OPX
UIBU UIF MFOHUI BOE XJEUI PG UIF SFDUBOHMF DBO UBLF BOZ QPTJUJWF WBMVFT

UIBU HJWF BO BSFB PG N2
VTF ZPVS HSBQI UP ĕOE UIF XJEUI JG UIF MFOHUI JT N

Unit 5: Algebra 121

18 Curved graphs

18.3 Using graphs to solve quadratic equations
t *G B RVBESBUJD FRVBUJPO IBT SFBM SPPUT UIF HSBQI PG UIF FRVBUJPO XJMM JOUFSTFDU XJUI UIF x BYJT ćJT JT

XIFSF y =

t 5P TPMWF RVBESBUJD FRVBUJPOT HSBQIJDBMMZ
SFBE PČ UIF x DPPSEJOBUFT PG UIF QPJOUT GPS B HJWFO y WBMVF

Exercise 18.3

1 6TF UIJT HSBQI PG UIF SFMBUJPOTIJQ y x2 − x − 6 UP TPMWF UIF GPMMPXJOH FRVBUJPOT

8
y = x2 − x − 6

6
4
2

Tip –4 –3 –2 –1 0 1 2 34 5
–2
'PS QBSU (c) ZPV NJHIU ĕOE
JU IFMQGVM UP SFBSSBOHF UIF –4
FRVBUJPO TP UIF MFę IBOE
TJEF NBUDIFT UIF FRVBUJPO –6
PG UIF HSBQI
J F TVCUSBDU
GSPN CPUI TJEFT –8

(a) x2 x − 6 0 (b) x2 x − 6 4 (c) x2 x = 12

2 (a) %SBX UIF HSBQI PG y x2 − x + 2 GPS WBMVFT PG x GSPN − UP

(b) 6TF ZPVS HSBQI UP ĕOE UIF BQQSPYJNBUF TPMVUJPOT UP UIF FRVBUJPOT
(i) −x2 − x + 2 = 0
(ii) −x2 − x + 2 = 1
(iii) −x2 − x + 2 = −2

3 (a) 6TF BO JOUFSWBM PG ¦ ȳ x ȳ PO UIF x BYJT UP ESBX UIF HSBQI x2 − x − 6

(b) 6TF UIF HSBQI UP TPMWF UIF GPMMPXJOH FRVBUJPOT
(i) −6 = 2 − − 6
(ii) x2 x − 6 0
(iii) x2 x = 12

18.4 Using graphs to solve simultaneous linear and non-linear equations
t ćF TPMVUJPO JT UIF QPJOU XIFSF UIF HSBQIT JOUFSTFDU

Exercise 18.4

1 %SBX UIFTF QBJST PG HSBQIT BOE ĕOE UIF QPJOUT XIFSF UIFZ JOUFSTFDU

(a) y = 4 BOE y 2x + 2
x

(b) y = x2 + 2x − 3 BOE y x +1

(c) y x2 + 4 BOE y = 3
x

122 Unit 5: Algebra

18 Curved graphs

2 6TF B HSBQIJDBM NFUIPE UP TPMWF UIF GPMMPXJOH FRVBUJPOT TJNVMUBOFPVTMZ

(a) y 2x2 + 3x − 2 BOE y = x + 2

(b) y = x2 + 2x BOE y x + 4
(c) y 2x2 + 2x + 4 BOE y 2x − 4

(d) y 0 5x2 + x +1.5 BOE y 1 x
2

18.5 Other non-linear graphs
t " DVCJD FRVBUJPO IBT UISFF BT UIF IJHIFTU QPXFS PG JUT WBSJBCMF

o *G x JT QPTJUJWF
UIFO x JT QPTJUJWF BOE ox3 JT OFHBUJWF

o *G x JT OFHBUJWF
UIFO x JT OFHBUJWF BOE ox3 JT QPTJUJWF

t $VCJD FRVBUJPOT QSPEVDF HSBQIT DBMMFE DVCJD DVSWFT
t ćF HFOFSBM GPSN PG B DVCJD FRVBUJPO JT y ax3 bx2 cx d
t &YQPOFOUJBM HSPXUI JT GPVOE JO NBOZ SFBM MJGF TJUVBUJPOT
t ćF HFOFSBM FRVBUJPO GPS BO FYQPOFOUJBM GVODUJPO JT y xa

Exercise 18.5

Tip 1 $POTUSVDU B UBCMF PG WBMVFT GSPN o ȳ x ȳ BOE QMPU UIF QPJOUT UP ESBX HSBQIT PG UIF

#FGPSF ESBXJOH ZPVS GPMMPXJOH FRVBUJPOT
HSBQIT DIFDL UIF SBOHF PG
y WBMVFT JO ZPVS UBCMF PG (a) y x3 − 4x2 (b) y = x3 + 5 (c) y 2x3 + 5x2 + 5
WBMVFT (d) y x3 + 4x2 − 5 (e) y = x3 + 2x − 10 (f) y 2x3 + 4x2 − 7
(g) y x3 − 3x2 + 6 (h) y 3x3 + 5x
Tip
2 B
$ PQZ BOE DPNQMFUF UIF UBCMF PG WBMVFT GPS UIF FRVBUJPO y x3 − 5x2 + 10
3FBSSBOHF UIF FRVBUJPO
TP UIBU UIF -)4 PG UIF x o –2 o –1 o 0 1 2 3 4 5 6
FRVBUJPO JT FRVJWBMFOU y
UP UIF HJWFO FRVBUJPO UP
IFMQ ZPV TPMWF UIF OFX (b) 0O B TFU PG BYFT
ESBX UIF HSBQI PG UIF FRVBUJPO y x3 − 5x2 + 10 GPS − ≤ x ≤
FRVBUJPO
(c) 6TF UIF HSBQI UP TPMWF UIF FRVBUJPOT
Tip
(i) x3 5x2 10 0 (ii) x3 5x2 10 10
8IFO ZPV IBWF UP QMPU
HSBQIT PG FRVBUJPOT (iii) x3 5x2 10 x 5
XJUI B DPNCJOBUJPO PG
MJOFBS
RVBESBUJD
DVCJD
3 $POTUSVDU B UBCMF PG WBMVFT GPS o ȳ x ȳ GPS FBDI PG UIF GPMMPXJOH FRVBUJPOT BOE ESBX
SFDJQSPDBM PS DPOTUBOU
UFSNT ZPV OFFE UP ESBX UIF HSBQIT
VQ B UBCMF PG WBMVFT XJUI
BU MFBTU FJHIU WBMVFT PG x UP (a) y x − 1 (b) y = x3 + 1 (c) y = x2 + 2 − 4
HFU B HPPE JOEJDBUJPO PG x x x
UIF TIBQF PG UIF HSBQI
(d) y 2x + 3 (e) y x3 − 2 (f ) y x2 − x + 1
x x x

4 0O UP UIF TBNF TFU PG BYFT ESBX UIF HSBQIT PG

(a) y = 2x BOE y = 2−x GPS o ȳ x ȳ
(b) = 10x BOE y 2x −1 GPS ȳ x ȳ

Unit 5: Algebra 123

Growth in cm18 Curved graphs

5 " TJOHMF DFMMFE BMHBF IBT CFFO EJTDPWFSFE UIBU TQMJUT JOUP UISFF TFQBSBUF DFMMT FWFSZ IPVS
(a) 3FQSFTFOU UIF ĕSTU ĕWF IPVST PG HSPXUI PO B HSBQI
(b) (i) 0O UP UIF TBNF TFU PG BYFT VTFE GPS a
TLFUDI UIF HSBQI PG 12x + 1
XIJDI
SFQSFTFOUT UIF HSPXUI PG BOPUIFS TJOHMF DFMMFE PSHBOJTN XJUI B ĕYFE SBUF PG HSPXUI
(ii) 8IBU JT UIF HSPXUI SBUF PG UIF TFDPOE PSHBOJTN
(c) 6TF ZPVS HSBQI UP BOTXFS UIF GPMMPXJOH
(i) AęFS IPX NBOZ IPVST BSF UIFSF FRVBM OVNCFST PG FBDI PSHBOJTN
(ii) HPX NBOZ DFMMT PG FBDI PSHBOJTN JT UIJT

18.6 Finding the gradient of a curve
t " DVSWF EPFT OPU IBWF B DPOTUBOU HSBEJFOU
t ćF HSBEJFOU PG B DVSWF BU B QPJOU JT FRVBM UP UIF HSBEJFOU PG UIF UBOHFOU UP UIF DVSWF BU UIBU QPJOU
t y change vertical change

(SBEJFOU = x change = horizontal change

Exercise 18.6

1 %SBX UIF HSBQI PG y x2 − 2x − 8 'JOE UIF HSBEJFOU PG UIF HSBQI
(a) BU UIF QPJOU XIFSF UIF DVSWF JOUFSTFDUT XJUI UIF y BYJT
(b) BU FBDI PG UIF QPJOUT XIFSF UIF DVSWF JOUFSTFDUT XJUI UIF x BYJT

2 (a) %SBX UIF HSBQI PG y x3 −1 GPS o ȳ x ȳ
(b) 'JOE UIF HSBEJFOU PG UIF DVSWF BU UIF QPJOU A



3 ćF HSBQI TIPXT IPX UIF SBUF PG HSPXUI PG B CFBO QMBOU JT BČFDUFE CZ TVOMJHIU 'JOE UIF SBUF
PG HSPXUI XJUI BOE XJUIPVU MJHIU PO UIF UIJSUFFOUI EBZ
Growth rate of beans
y
30
25
20
15 with light
without light
10
5
x
0 5 10 15
Days

124 Unit 5: Algebra

18 Curved graphs

Mixed exercise 1

(a) 8SJUF BO FRVBUJPO GPS FBDI PG UIF HSBQIT BCPWF
A
B BOE C
(b) (i) 4PMWF FRVBUJPOT A BOE C TJNVMUBOFPVTMZ

(ii) )PX XPVME ZPV DIFDL ZPVS TPMVUJPO HSBQIJDBMMZ
(c) 8IBU JT UIF NBYJNVN WBMVF PG B

2 (a) %SBX UIF HSBQIT PG UIF GPMMPXJOH FRVBUJPOT PO UIF TBNF HSJE y x2 BOE y x3

(b) 6TF ZPVS HSBQI UP TPMWF UIF UXP FRVBUJPOT TJNVMUBOFPVTMZ
(c) #Z ESBXJOH B TVJUBCMF TUSBJHIU MJOF
PS MJOFT
PO UP UIF TBNF HSJE
TPMWF UIF FRVBUJPOT

(i) 2 = 4
(ii) x3 + 8 0
(d) 'JOE UIF SBUF PG DIBOHF GPS FBDI HSBQI BU UIF QPJOUT XIFSF x

3 ćF EPUUFE MJOF PO UIF HSJE CFMPX JT UIF BYJT PG TZNNFUSZ GPS UIF HJWFO IZQFSCPMB

y
8
7
6
5
4
3
2
1x
–8 –7 –6–5–4–3–2––110 1 2 3 4 5 6 7 8
–2
–3
–4
–5
–6
–7
–8

(a) (JWF UIF FRVBUJPO GPS UIF IZQFSCPMB
(b) (JWF UIF FRVBUJPO GPS UIF HJWFO MJOF PG TZNNFUSZ
(c) $ PQZ UIF EJBHSBN BOE ESBX JO UIF PUIFS MJOF PG TZNNFUSZ
HJWJOH UIF FRVBUJPO GPS

UIJT MJOF

Unit 5: Algebra 125

19 Symmetry and loci

19.1 Symmetry in two dimensions
t 5XP EJNFOTJPOBM ĘBU
TIBQFT IBWF MJOF TZNNFUSZ JG ZPV BSF BCMF UP ESBX B MJOF UISPVHI UIF TIBQF TP UIBU

POF TJEF PG‫ڀ‬UIF MJOF JT UIF NJSSPS JNBHF SFĘFDUJPO
PG UIF PUIFS TJEF ćFSF NBZ CF NPSF UIBO POF QPTTJCMF MJOF

PG TZNNFUSZ JO B‫ڀ‬TIBQF

t *G ZPV SPUBUF UVSO
B TIBQF BSPVOE B ĕYFE QPJOU BOE JU ĕUT PO UP JUTFMG EVSJOH UIF SPUBUJPO

UIFO JU IBT SPUBUJPOBM TZNNFUSZ ćF OVNCFS PG UJNFT UIF TIBQF ĕUT PO UP JUT PSJHJOBM QPTJUJPO EVSJOH B

SPUBUJPO JT DBMMFE UIF PSEFS PG SPUBUJPOBM TZNNFUSZ

If a shape can only fit back into Exercise 19.1
itself after a full 360° rotation, it
has no rotational symmetry. 1 'PS FBDI PG UIF GPMMPXJOH TIBQFT

(a) DPQZ UIF TIBQF BOE ESBX JO BOZ MJOFT PG TZNNFUSZ
(b) EFUFSNJOF UIF PSEFS PG SPUBUJPOBM TZNNFUSZ

A BC

D EF

GH

2 (a) ) PX NBOZ MJOFT PG TZNNFUSZ EPFT B SIPNCVT IBWF %SBX B EJBHSBN UP TIPX
ZPVS‫ڀ‬TPMVUJPO

(b) 8IBU JT UIF PSEFS PG SPUBUJPOBM TZNNFUSZ PG B SIPNCVT
3 %SBX B RVBESJMBUFSBM UIBU IBT OP MJOFT PG TZNNFUSZ BOE OP SPUBUJPOBM TZNNFUSZ
126 Unit 5: Shape, space and measures

19 Symmetry and loci

19.2 Symmetry in three dimensions
t ćSFF EJNFOTJPOBM TIBQFT TPMJET
DBO BMTP CF TZNNFUSJDBM
t " QMBOF PG TZNNFUSZ JT B TVSGBDF JNBHJOBSZ
UIBU EJWJEFT UIF TIBQF JOUP UXP QBSUT UIBU BSF NJSSPS JNBHFT PG FBDI

PUIFS

t *G ZPV SPUBUF B TPMJE BSPVOE BO BYJT BOE JU MPPLT UIF TBNF BU EJČFSFOU QPTJUJPOT PO JUT SPUBUJPO
UIFO JU IBT SPUBUJPOBM
TZNNFUSZ ćF BYJT JT DBMMFE UIF BYJT PG TZNNFUSZ

Tip Exercise 19.2

ćJOL PG B plane of 1 'PS FBDI PG UIF GPMMPXJOH TPMJET
TUBUF UIF OVNCFS PG QMBOFT PG TZNNFUSZ
symmetry BT B TMJDF PS DVU (a) (b) (c) (d)
UISPVHI B TPMJE UP EJWJEF
JU JOUP UXP IBMWFT UIBU BSF (e) (f) (g) (h)
NJSSPS JNBHFT PG FBDI
PUIFS

Tip 2 8IBU JT UIF PSEFS PG SPUBUJPOBM TZNNFUSZ BCPVU UIF HJWFO BYJT JO FBDI PG UIFTF TPMJET
(a) (b) (c)
ćJOL PG BO axis of
symmetry BT B SPE PS BYMF
UISPVHI B TPMJE 8IFO UIF
TPMJE UVSOT PO UIJT BYJT
BOE SFBDIFT JUT PSJHJOBM
QPTJUJPO EVSJOH B UVSO
JU
IBT SPUBUJPOBM TZNNFUSZ

(d) (e) (f)

Unit 5: Shape, space and measures 127

19 Symmetry and loci

19.3 Symmetry properties of circles
t " DJSDMF IBT MJOF TZNNFUSZ BCPVU BOZ EJBNFUFS BOE JU IBT SPUBUJPOBM TZNNFUSZ BSPVOE JUT DFOUSF
t ćF GPMMPXJOH UIFPSFNT DBO CF VTFE UP TPMWF QSPCMFNT SFMBUFE UP DJSDMFT

o UIF QFSQFOEJDVMBS CJTFDUPS PG B DIPSE QBTTFT UISPVHI UIF DFOUSF

o FRVBM DIPSET BSF FRVJEJTUBOU GSPN UIF DFOUSF BOE DIPSET FRVJEJTUBOU GSPN UIF DFOUSF BSF FRVBM JO MFOHUI

o UXP UBOHFOUT ESBXO UP B DJSDMF GSPN UIF TBNF QPJOU PVUTJEF UIF DJSDMF BSF FRVBM JO MFOHUI

Tip Exercise 19.3 (b)

:PV OFFE UP MFBSO UIF 1 'JOE UIF TJ[F PG UIF BOHMFT NBSLFE x BOE y
DJSDMF UIFPSFNT 4UBUF UIF (a)
UIFPSFN ZPV BSF VTJOH A
XIFO TPMWJOH B QSPCMFN JO
BO FYBN

230° O
x x
X 80° Z
B
y
Y

2 *O UIF EJBHSBN
MN BOE PQ BSF FRVBM DIPSET S JT UIF NJEQPJOU PG MN BOE R JT UIF
NJEQPJOU PG PQ MN DN 'JOE UIF MFOHUI PG SO DPSSFDU UP UXP TJHOJĕDBOU ĕHVSFT

MP

S OR

9 cm
NQ

3 *O EJBHSBN (a)
MN BOE PQ BSF FRVBM DIPSET BOE MN DN S BOE T BSF UIF NJEQPJOUT PG

MN BOE PQ SFTQFDUJWFMZ SO DN

*O EJBHSBN (b)
DIPSE AB NN

'JOE UIF MFOHUI PG UIF EJBNFUFS PG FBDI DJSDMF
BOE IFODF DBMDVMBUF JUT DJSDVNGFSFODF
DPSSFDU

UP UXP EFDJNBM QMBDFT

(a) P (b)
M C

ST AO
O
E D
NQ B

128 Unit 5: Shape, space and measures

19 Symmetry and loci

19.4 Angle relationships in circles
t 8IFO B USJBOHMF JT ESBXO JO B TFNJ DJSDMF
TP UIBU POF TJEF JT UIF EJBNFUFS BOE UIF WFSUFY PQQPTJUF UIF

EJBNFUFS UPVDIFT UIF DJSDVNGFSFODF
UIF BOHMF PG UIF WFSUFY PQQPTJUF UIF EJBNFUFS JT B SJHIU BOHMF ¡


t 8IFSF B UBOHFOU UPVDIFT B DJSDMF
UIF SBEJVT ESBXO UP UIF TBNF QPJOU NFFUT UIF UBOHFOU BU ¡
t ćF BOHMF GPSNFE GSPN UIF FOET PG B DIPSE BOE UIF DFOUSF PG B DJSDMF JT UXJDF UIF BOHMF GPSNFE CZ UIF FOET PG UIF

DIPSE BOE B QPJOU BU UIF DJSDVNGFSFODF JO UIF TBNF TFHNFOU


t "OHMFT JO UIF TBNF TFHNFOU BSF FRVBM
t ćF PQQPTJUF BOHMFT PG B DZDMJD RVBESJMBUFSBM BEE VQ UP ¡
t &BDI FYUFSJPS BOHMF PG B DZDMJD RVBESJMBUFSBM JT FRVBM UP UIF JOUFSJPS BOHMF PQQPTJUF UP JU

Tip

*G ZPV MFBSO UIF DJSDMF UIFPSFNT XFMM ZPV TIPVME CF BCMF UP TPMWF NPTU
DJSDMF QSPCMFNT

The angle relationships for Exercise 19.4
triangles, quadrilaterals and
parallel lines (chapter 3), as well 1 *O UIF EJBHSBN
O JT UIF DFOUSF PG UIF DJSDMF
∠NPO ¡ BOE ∠MOP ¡
as Pythagoras’ theorem (chapter N
11), may be needed to solve circle
problems. MQ
O

70°
P 15°

'JOE UIF TJ[F PG UIF GPMMPXJOH BOHMFT
HJWJOH SFBTPOT

(a) ∠PNO (b) ∠PON (c) ∠MPN (d) ∠PMN

2 O JT UIF DFOUSF PG UIF DJSDMF
BC

55°

25°
O

AD

$BMDVMBUF (b) ∠AOD (c) ∠BDC
(a) ∠ACD

Unit 5: Shape, space and measures 129

19 Symmetry and loci

3 $BMDVMBUF UIF BOHMFT PG UIF DZDMJD RVBESJMBUFSBM
E

D
115°
A

C

B
4 AB JT UIF EJBNFUFS PG UIF DJSDMF $BMDVMBUF UIF TJ[F PG BOHMF ABD

B
125° C

D
A

5 SPT JT B UBOHFOU UP B DJSDMF XJUI DFOUSF O SR JT B TUSBJHIU MJOF XIJDI HPFT UISPVHI UIF DFOUSF
PG UIF DJSDMF BOE ∠PSO ¡ 'JOE UIF TJ[F PG ∠TPR
S

29°

P OQ

R
T

6 (JWFO UIBU O JT UIF DFOUSF PG UIF DJSDMF CFMPX BOE ∠BAC ¡
DBMDVMBUF UIF TJ[F PG ∠BOC
A

72° C
O

BD

130 Unit 5: Shape, space and measures

19 Symmetry and loci

7 *O UIF EJBHSBN
AB ]] DC
∠ADC ¡ BOE ∠DCA ¡

D
A

64°

22°

B
C

$BMDVMBUF UIF TJ[F PG

(a) ∠BAC (b) ∠ABC (c) ∠ACB

8 SNT JT B UBOHFOU UP B DJSDMF XJUI DFOUSF O ∠QMO ¡ BOE ∠MPN ¡

S N
M T

18°
O

Q
56°

P

'JOE UIF TJ[F PG (b) ∠MOQ (c) ∠PNT
(a) ∠MNS

19.5 Locus
t " MPDVT JT B TFU PG QPJOUT UIBU NBZ PS NBZ OPU CF DPOOFDUFE
UIBU TBUJTGZ B HJWFO SVMF " MPDVT DBO CF B TUSBJHIU MJOF
B

DVSWF PS B DPNCJOBUJPO PG TUSBJHIU BOE DVSWFE MJOFT

t ćF MPDVT PG QPJOUT FRVJEJTUBOU GSPN B HJWFO QPJOU JT B DJSDMF
t ćF MPDVT PG QPJOUT FRVJEJTUBOU GSPN B ĕYFE MJOF JT UXP MJOFT QBSBMMFM UP UIF HJWFO MJOF
t ćF MPDVT PG QPJOUT FRVJEJTUBOU GSPN B HJWFO MJOF TFHNFOU JT B ASVOOJOH USBDL TIBQF BSPVOE UIF MJOF TFHNFOU
t ćF MPDVT PG QPJOUT FRVJEJTUBOU GSPN UIF BSNT PG BO BOHMF JT UIF CJTFDUPS PG UIF BOHMF
t 5P ĕOE UIF MPDVT PG QPJOUT UIBU BSF UIF TBNF EJTUBODF GSPN UXP PS NPSF HJWFO QPJOUT ZPV IBWF UP VTF B

DPNCJOBUJPO PG UIF MPDJ BCPWF UP ĕOE UIF JOUFSTFDUJPOT PG UIF MPDJ

REWIND Exercise 19.5

Make sure you know to bisect an 1 $PQZ UIF EJBHSBNT BOE ESBX UIF MPDVT PG UIF QPJOUT UIBU BSF DN GSPN UIF QPJOU PS MJOFT JO
angle as this is often required in FBDI DBTF
loci problems (see chapter 3).

(a) A (b) A B (c) A

BC

Unit 5: Shape, space and measures 131

19 Symmetry and loci

2 $POTUSVDU B DJSDMF XJUI DFOUSF O BOE B SBEJVT PG DN
(a) %SBX UIF MPDVT PG QPJOUT UIBU BSF DN GSPN UIF DJSDVNGFSFODF PG UIF DJSDMF
(b) 4IBEF UIF MPDVT PG QPJOUT UIBU BSF MFTT UIBO DN GSPN DJSDVNGFSFODF PG UIF DJSDMF

3 *O UIJT TDBMF EJBHSBN
UIF TIBEFE BSFB SFQSFTFOUT B ĕTIQPOE ćF ĕTIQPOE JT TVSSPVOEFE CZ
B DPODSFUF XBMLXBZ N XJEF $PQZ UIF EJBHSBN BOE ESBX UIF MPDVT PG QPJOUT UIBU BSF N
GSPN UIF FEHFT PG UIF ĕTIQPOE 4IBEF UIF BSFB DPWFSFE CZ UIF DPODSFUF XBMLXBZ

1m

4 /JDL MJWFT LN GSPN UIF SPBE BOE LN GSPN UIF TDIPPM ćF DMPTFTU EJTUBODF PG UIF TDIPPM
GSPN UIF SPBE JT LN 6TJOH B TDBMF PG DN UP LN
DPOTUSVDU UIF MPDJ UP TIPX UXP QPTTJCMF
QPTJUJPOT GPS /JDL T IPNF

Road 4 km

SCHOOL

School

5 "OOB MJWFT BU A BOE #FUUZ MJWFT BU B ćFZ XBOU UP NFFU BU B QPJOU FRVJEJTUBOU GSPN CPUI
IPNFT #Z DPOTUSVDUJPO
TIPX UIF MPDVT PG QPTTJCMF NFFUJOH QPJOUT

132 Unit 5: Shape, space and measures

19 Symmetry and loci

6 'PMMPX UIF JOTUSVDUJPOT DBSFGVMMZ UP ĕOE UIF MPDBUJPO PG B XBUFS NBJO PO B DPQZ PG UIF
EJBHSBN CFMPX

Cell tower Radio mast
A C

30 m 15 m

B Flagpole Scale 1 cm = 5 m

(a) ' JOE UIF MPDVT PG QPJOUT UIBU BSF FRVJEJTUBOU GSPN UIF DFMM UPXFS BOE UIF ĘBHQPMF
UIFO
UIF TJOHMF MPDVT UIBU JT FRVJEJTUBOU GSPN AB BOE BC -BCFM UIJT QPJOU D BOE ESBX B MJOF
UP KPJO JU UP UIF CBTF PG‫ڀ‬UIF ĘBHQPMF

(b) ćF XBUFS NBJO JT UFO NFUSFT GSPN QPJOU D BOE PO UIF MJOF TFHNFOU KPJOJOH QPJOU D UP
UIF CBTF PG UIF ĘBHQPMF .BSL UIJT BT QPJOU X PO ZPVS EJBHSBN

Mixed exercise 1 )FSF BSF ĕWF TIBQFT (c) (d) (e)

(a) (b)

'PS FBDI POF

(i) JOEJDBUF UIF BYFT PG TZNNFUSZ JG BOZ

(ii) TUBUF UIF PSEFS PG SPUBUJPOBM TZNNFUSZ
2 4UVEZ UIF EJBHSBN

(a) 8IBU UZQF PG TPMJE JT UIJT
(b) 8IBU JT UIF DPSSFDU NBUIFNBUJDBM OBNF GPS UIF SPE UISPVHI UIF TPMJE
(c) 8IBU JT UIF PSEFS PG SPUBUJPOBM TZNNFUSZ PG UIJT TPMJE
(d) )PX NBOZ QMBOFT PG TZNNFUSZ EPFT UIJT TPMJE IBWF

Unit 5: Shape, space and measures 133

19 Symmetry and loci

3 *O FBDI PG UIF GPMMPXJOH O JT UIF DFOUSF PG UIF DJSDMF 'JOE UIF WBMVF PG UIF NBSLFE BOHMFT
(JWF SFBTPOT GPS ZPVS TUBUFNFOUT

(a) (b) (c) (d)
A
A B A B
x 28° O
yz O O x D A 60°
160° 29° Ox
D w B
C x B C y
C C

4 'JOE UIF MFOHUI PG x BOE y JO FBDI PG UIFTF EJBHSBNT O JT UIF DFOUSF PG UIF DJSDMF JO FBDI DBTF

(a) (b)

18 cm y 250 mm

O yx
x 15 cm 120 mm

O

5 +FTTJDB UFMMT B GSJFOE UIBU IFS IPVTF JT LN GSPN UIF QFUSPM TUBUJPO PO 3PVUF BOE LN GSPN
3PVUF JUTFMG 6TF B TDBMF PG DN UP LN UP TIPX BMM QPTTJCMF MPDBUJPOT GPS +FTTJDB T IPVTF

Route 66

Petrol station

6 $POTUSVDU MJOF TFHNFOU AB DN MPOH 4IPX UIF MPDVT PG QPJOUT UIBU BSF FRVJEJTUBOU GSPN
CPUI A BOE B BOE BMTP DN GSPN A

134 Unit 5: Shape, space and measures

20 Histograms and frequency
distribution diagrams

20.1 Histograms

t " IJTUPHSBN JT B TQFDJBMJTFE HSBQI *U JT VTFE UP TIPX HSPVQFE OVNFSJDBM EBUB VTJOH B DPOUJOVPVT TDBMF
PO UIF IPSJ[POUBM BYJT ćJT NFBOT UIFSF BSF OP HBQT JO CFUXFFO UIF EBUB DBUFHPSJFT o XIFSF POF FOET
UIF

PUIFS CFHJOT


t #FDBVTF UIF TDBMF JT DPOUJOVPVT
FBDI DPMVNO JT ESBXO BCPWF B QBSUJDVMBS DMBTT JOUFSWBM
t 8IFO UIF DMBTT JOUFSWBMT BSF FRVBM
UIF CBST BSF BMM UIF TBNF XJEUI BOE JU JT DPNNPO QSBDUJDF UP MBCFM UIF WFSUJDBM TDBMF

BT GSFRVFODZ

t 8IFO UIF DMBTT JOUFSWBMT BSF OPU FRVBM
UIF WFSUJDBM BYJT TIPXT UIF GSFRVFODZ EFOTJUZ

Frequency density = frequency ćF BSFB PG FBDI ACBS SFQSFTFOUT UIF GSFRVFODZ
class interval

t " HBQ XJMM POMZ BQQFBS CFUXFFO CBST JG BO JOUFSWBM IBT B GSFRVFODZ GSFRVFODZ EFOTJUZ PG [FSP

Exercise 20.1

1 ćF UBCMF TIPXT UIF NBSLT PCUBJOFE CZ B OVNCFS PG TUVEFOUT GPS &OHMJTI BOE .BUIFNBUJDT JO
B NPDL FYBN

Remember there can be no gaps Marks class interval English frequency Mathematics frequency
between the bars: use the upper o
and lower bounds of each class o 3
interval to prevent gaps. o 7 5
o 6
REWIND o
You met the mode in chapter 12. o 51
o
o 6
o 3
o 1

(a) % SBX UXP TFQBSBUF IJTUPHSBNT UP TIPX UIF EJTUSJCVUJPO PG NBSLT GPS &OHMJTI BOE
.BUIFNBUJDT

(b) 8IBU JT UIF NPEBM DMBTT GPS &OHMJTI
(c) 8IBU JT UIF NPEBM DMBTT GPS .BUIFNBUJDT
(d) 8 SJUF B GFX TFOUFODFT DPNQBSJOH UIF TUVEFOUT QFSGPSNBODF JO &OHMJTI BOE

.BUIFNBUJDT

Unit 5: Data handling 135

20 Histograms and frequency distribution diagrams

2 4UVEZ UIJT HSBQI BOE BOTXFS UIF RVFTUJPOT BCPVU JU Frequency

Ages of women in a clothing factory
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0

19 21 23 25 27 29 31
Age (years)

(a) )PX EP ZPV LOPX UIJT JT B IJTUPHSBN BOE OPU B CBS HSBQI
(b) )PX NBOZ XPNFO BHFE o XPSL JO UIF DMPUIJOH GBDUPSZ
(c) )PX NBOZ XPNFO XPSL JO UIF GBDUPSZ BMUPHFUIFS
(d) 8IBU JT UIF NPEBM DMBTT PG UIJT EBUB
(e) &YQMBJO XIZ UIFSF JT B CSPLFO MJOF PO UIF IPSJ[POUBM BYJT

Tip 3 4BMMZ EJE B TVSWFZ UP ĕOE UIF BHFT PG QFPQMF VTJOH BO JOUFSOFU DBGÏ ćFTF BSF IFS SFTVMUT

*U IFMQT UP SFNFNCFS UIBU Age (a) ȳ a ȳ a ȳ a ȳ a ȳ a
UIF GSFRVFODZ EFOTJUZ
UFMMT ZPV XIBU UIF DPMVNO No. of people
IFJHIU TIPVME CF *G POF
DPMVNO JT UXJDF BT XJEF %SBX BO BDDVSBUF IJTUPHSBN UP TIPX UIFTF EBUB 6TF B TDBMF PG DN UP ĕWF ZFBST PO UIF
BT BOPUIFS
JU XJMM POMZ CF IPSJ[POUBM BYJT BOE BO BSFB TDBMF PG POF TRVBSF DFOUJNFUSF UP SFQSFTFOU POF QFSTPO
IBMG BT IJHI GPS UIF TBNF
GSFRVFODZ 4 ćJT IJTUPHSBN TIPXT UIF OVNCFS PG IPVTFT JO EJČFSFOU QSJDF SBOHFT UIBU BSF BEWFSUJTFE JO B
QSPQFSUZ NBHB[JOF

House prices in property magazine
30

25

20
Frequency density 15

10

5

0 200
20 40 60 80 100 120 140 160 180
Price ($thousands)

(a) )PX NBOZ IPVTFT XFSF JO UIF o QSJDF SBOHF
(b) )PX NBOZ IPVTFT XFSF JO UIF o QSJDF SBOHF
(c) )PX NBOZ IPVTFT BSF SFQSFTFOUFE CZ POF TRVBSF DFOUJNFUSF PO UIJT HSBQI

136 Unit 5: Data handling

20 Histograms and frequency distribution diagrams

20.2 Cumulative frequency
t $VNVMBUJWF GSFRVFODZ JT B ASVOOJOH UPUBM PG UIF DMBTT GSFRVFODJFT VQ UP FBDI VQQFS DMBTT CPVOEBSZ
t 8IFO DVNVMBUJWF GSFRVFODJFT BSF QMPUUFE UIFZ HJWF B DVNVMBUJWF GSFRVFODZ DVSWF PS HSBQI
t :PV DBO VTF UIF DVSWF UP FTUJNBUF UIF NFEJBO WBMVF PG UIF EBUB
t :PV DBO EJWJEF UIF EBUB JOUP GPVS FRVBM HSPVQT DBMMFE RVBSUJMFT ćF JOUFSRVBSUJMF SBOHF *23
JT UIF EJČFSFODF

CFUXFFO UIF VQQFS BOE MPXFS RVBSUJMFT 23 ¦ 21


t %BUB DBO BMTP CF EJWJEFE JOUP FRVBM HSPVQT DBMMFE QFSDFOUJMFT ćF UI QFSDFOUJMF JT FRVJWBMFOU UP UIF NFEJBO

Exercise 20.2

1 ćF UBCMF TIPXT UIF QFSDFOUBHF TDPSFE CZ B OVNCFS PG TUVEFOUT JO BO FYBNJOBUJPO

Percentage Number of students
ȳ m
ȳ m 3
ȳ m 7
ȳ m
ȳ m
ȳ m 51
ȳ m
ȳ m
ȳ m 3
ȳ m 1

Tip (a) %SBX B DVNVMBUJWF GSFRVFODZ DVSWF UP TIPX UIJT EBUB 6TF B TDBMF PG DN QFS UFO QFSDFOU

:PV XJMM OPSNBMMZ CF HJWFO PO UIF IPSJ[POUBM BYJT BOE B TDBMF PG DN QFS UFO TUVEFOUT PO UIF WFSUJDBM BYJT
B TDBMF UP VTF XIFO ZPV
IBWF UP ESBX B DVNVMBUJWF (b) 6 TF ZPVS DVSWF UP FTUJNBUF UIF NFEJBO
21 BOE 23
GSFRVFODZ DVSWF JO BO (c) &TUJNBUF UIF *23
FYBNJOBUJPO
(d) ćF QBTT SBUF GPS UIJT UFTU JT 8IBU QFSDFOUBHF PG UIF TUVEFOUT QBTTFE UIF UFTU
Tip
(e) *OEJDBUF UIF UI BOE UI QFSDFOUJMFT PO ZPVS HSBQI 8IBU EP UIFTF QFSDFOUJMFT
5P ĕOE UIF QPTJUJPO
PG UIF RVBSUJMFT GSPN B JOEJDBUF Height of professional
DVNVMBUJWF GSFRVFODZ basketball players
Dn2V SBWOFE
V34TnF U/IPF UGFP USINBVU MUBIFJ Tn4
2 ćJT DVNVMBUJWF GSFRVFODZ DVSWF TIPXT UIF IFJHIU JO
JT EJČFSFOU UP UIF GPSNVMBF DFOUJNFUSFT PG QSPGFTTJPOBM CBTLFUCBMM QMBZFST Cumulative frequency 200
VTFE GPS EJTDSFUF EBUB
(a) &TUJNBUF UIF NFEJBO IFJHIU PG QMBZFST JO 150
UIJT TBNQMF
100
(b) &TUJNBUF 21 BOE 23
(c) &TUJNBUF UIF *23 50
(d) 8IBU QFSDFOUBHF PG CBTLFUCBMM QMBZFST BSF
0
PWFS N UBMM 125 140 155 170 185 200

Height (cm)

Unit 5: Data handling 137

20 Histograms and frequency distribution diagrams

Mixed exercise 1 ćJT QBSUJBMMZ DPNQMFUFE IJTUPHSBN TIPXT UIF IFJHIUT PG USFFT JO B TFDUJPO PG USPQJDBM GPSFTU
Heights of trees
Frequency
10

8

6

4

2

0
0 2 4 6 8 10 12
Height in metres

(a) " TDJFOUJTU NFBTVSFE ĕWF NPSF USFFT BOE UIFJS IFJHIUT XFSF N
N
N

N BOE N 3FESBX UIF HSBQI UP JODMVEF UIJT EBUB

(b) )PX NBOZ USFFT JO UIJT TFDUPS PG GPSFTU BSF ȴ N UBMM
(c) 8IBU JT UIF NPEBM DMBTT PG USFF IFJHIUT

2 " OVSTF NFBTVSFE UIF NBTTFT PG B TBNQMF PG TUVEFOUT JO B IJHI TDIPPM BOE ESFX UIF GPMMPXJOH UBCMF

Mass (kg) Frequency
≤ m <
≤ m < 7
≤ m < 13
≤ m <
≤ m < 11

(a) %SBX B IJTUPHSBN UP TIPX UIF EJTUSJCVUJPO PG NBTTFT
(b) 8IBU JT UIF NPEBM NBTT
(c) 8IBU QFSDFOUBHF PG TUVEFOUT XFJHIFE MFTT UIBO LH
(d) 8IBU JT UIF NBYJNVN QPTTJCMF SBOHF PG UIF NBTTFT

3 ćF UBCMF TIPXT UIF BWFSBHF NJOVUFT PG BJSUJNF UIBU UFFOBHFST CPVHIU GSPN B QSF QBJE LJPTL
JO POF XFFL

Minutes ≤ m < ȳ m < ȳ m < ȳ m < ȳ m < ȳ m <

No. of 15

teenagers

%SBX BO BDDVSBUF IJTUPHSBN UP EJTQMBZ UIJT EBUB 6TF B TDBMF PG DN UP SFQSFTFOU UFO NJOVUFT
PO UIF IPSJ[POUBM BYJT BOE BO BSFB TDBMF PG DN QFS ĕWF QFSTPOT

4 ćJSUZ TFFEMJOHT XFSF QMBOUFE GPS B CJPMPHZ FYQFSJNFOU ćF IFJHIUT PG UIF QMBOUT XFSF
NFBTVSFE BęFS UISFF XFFLT BOE SFDPSEFE BT CFMPX

Heights (h cm) ≤ h ≤ h < ≤ h < ≤ h <
Frequency 3 15

(a) 'JOE BO FTUJNBUF GPS UIF NFBO IFJHIU
(b) %SBX B DVNVMBUJWF GSFRVFODZ DVSWF BOE VTF JU UP ĕOE UIF NFEJBO IFJHIU
(c) &TUJNBUF 21 BOE 23 BOE UIF *23

138 Unit 5: Data handling

21 Ratio, rate and proportion

21.1 Working with ratio
t " SBUJP JT B DPNQBSJTPO PG UXP PS NPSF RVBOUJUJFT NFBTVSFE JO UIF TBNF VOJUT *O HFOFSBM
B SBUJP JT

XSJUUFO JO UIF GPSN a b

t 3BUJPT TIPVME BMXBZT CF HJWFO JO UIFJS TJNQMFTU GPSN 5P FYQSFTT B SBUJP JO TJNQMFTU GPSN
EJWJEF PS
NVMUJQMZ CZ UIF TBNF GBDUPS

t 2VBOUJUJFT DBO CF TIBSFE JO B HJWFO SBUJP 5P EP UIJT ZPV OFFE UP XPSL PVU UIF OVNCFS PG FRVBM QBSUT

JO UIF SBUJP BOE UIFO XPSL PVU UIF WBMVF PG FBDI TIBSF 'PS FYBNQMF
B SBUJP PG NFBOT UIBU UIFSF BSF

FRVBM QBSUT 0OF TIBSF JT 35 PG UIF UPUBM BOE UIF PUIFS JT 25 PG UIF UPUBM

Remember, simplest form is also Exercise 21.1
called ‘lowest terms’.
1 &YQSFTT UIF GPMMPXJOH BT SBUJPT JO UIFJS TJNQMFTU GPSN
Tip
(a) 2 3 3 2 (b) 1 1 IPVST NJOVUFT
:PV DBO DSPTT NVMUJQMZ 4 2
UP NBLF BO FRVBUJPO BOE 3
TPMWF GPS x (c) DN UP N (d) H UP UISFF LJMPHSBNT

(e) H UP H

2 'JOE UIF WBMVF PG x JO FBDI PG UIF GPMMPXJOH

(a) x (b) x (c) x (d) x

(e) x (f ) 2 = x (g) 5 = 16 (h) x = 10
7 4 x 6 4 15

(i) x = 1 (j) 5 = 3
21 3 x 8

3 " MFOHUI PG SPQF DN MPOH NVTU CF DVU JOUP UXP QBSUT TP UIBU UIF MFOHUIT BSF JO UIF SBUJP

8IBU BSF UIF MFOHUIT PG UIF QBSUT

4 5P NBLF TBMBE ESFTTJOH
ZPV NJY PJM BOE WJOFHBS JO UIF SBUJP $BMDVMBUF IPX NVDI PJM BOE
IPX NVDI WJOFHBS ZPV XJMM OFFE UP NBLF UIF GPMMPXJOH BNPVOUT PG TBMBE ESFTTJOH

(a) NM (b) NM (c) NM

5 ćF TJ[FT PG UISFF BOHMFT PG B USJBOHMF BSF JO UIF SBUJP A B C 8IBU JT UIF TJ[F PG
FBDI BOHMF

6 " NFUBM EJTD DPOTJTUT PG UISFF QBSUT TJMWFS BOE UXP QBSUT DPQQFS CZ NBTT
*G UIF EJTD IBT B
NBTT PG NH
IPX NVDI TJMWFS EPFT JU DPOUBJO

Unit 6: Number 139

21 Ratio, rate and proportion

21.2 Ratio and scale
t 4DBMF JT B SBUJP *U DBO CF FYQSFTTFE BT

MFOHUI PO UIF ESBXJOH SFBM MFOHUI

t "MM SBUJP TDBMFT NVTU CF FYQSFTTFE JO UIF GPSN PG n PS n
t 5P DIBOHF B SBUJP TP UIBU POF QBSU
ZPV OFFE UP EJWJEF CPUI QBSUT CZ UIF OVNCFS UIBU ZPV XBOU UP CF

FYQSFTTFE BT 'PS FYBNQMF XJUI
JG ZPV XBOU UIF UP CF FYQSFTTFE BT
ZPV EJWJEF CPUI QBSUT CZ

ćF SFTVMU JT

Tip Exercise 21.2 A

8JUI SFEVDUJPOT TVDI BT 1 8SJUF UIFTF SBUJPT JO UIF GPSN PG n
NBQT
UIF TDBMF XJMM CF JO
UIF GPSN n
XIFSF n (a) (b) N LN (c) NJOVUFT 1 1 IPVST
8JUI FOMBSHFNFOUT UIF TDBMF 2
XJMM CF JO UIF GPSN n

XIFSF n n NBZ OPU CF 2 8SJUF UIFTF SBUJPT JO UIF GPSN PG n
B XIPMF OVNCFS
(a) (b) N DN (c) H UP NH

Exercise 21.2 B

1 ćF EJTUBODF CFUXFFO UXP QPJOUT PO B NBQ XJUI B TDBMF PG JT NN 8IBU JT UIF
EJTUBODF CFUXFFO UIF UXP QPJOUT JO SFBMJUZ (JWF ZPVS BOTXFS JO LJMPNFUSFT

2 " QMBO JT ESBXO VTJOH B TDBMF PG *G UIF MFOHUI PG B XBMM PO UIF QMBO JT DN
IPX MPOH JT
UIF XBMM JO SFBMJUZ

3 .JHVFM NBLFT B TDBMF ESBXJOH UP TPMWF B USJHPOPNFUSZ QSPCMFN DN PO IJT ESBXJOH
SFQSFTFOUT N JO SFBM MJGF )F XBOUT UP TIPX B N MPOH MBEEFS QMBDFE N GSPN UIF GPPU
PG B XBMM

(a) 8IBU MFOHUI XJMM UIF MBEEFS CF JO UIF EJBHSBN
(b) )PX GBS XJMM JU CF GSPN UIF GPPU PG UIF XBMM JO UIF EJBHSBN

4 " NBQ IBT B TDBMF PG

(a) 8IBU EPFT B TDBMF PG NFBO
(b) $PQZ BOE DPNQMFUF UIJT UBCMF VTJOH UIF NBQ TDBMF

Map distance (mm) 50

Actual distance (km) 50

5 .BSZ IBT B SFDUBOHVMBS QJDUVSF NN XJEF BOE NN IJHI 4IF FOMBSHFT JU PO UIF
QIPUPDPQJFS TP UIBU UIF FOMBSHFNFOU JT DN XJEF

(a) 8IBU JT UIF TDBMF GBDUPS PG UIF FOMBSHFNFOU
(b) 8IBU JT UIF IFJHIU PG IFS FOMBSHFE QJDUVSF
(c) *O UIF PSJHJOBM QJDUVSF
B GFODF XBT NN MPOH )PX MPOH XJMM UIJT GFODF CF PO UIF

FOMBSHFE QJDUVSF

140 Unit 6: Number

21 Ratio, rate and proportion

21.3 Rates
t " SBUF DPNQBSFT UXP RVBOUJUJFT NFBTVSFE JO EJČFSFOU VOJUT 'PS FYBNQMF TQFFE JT B SBUF UIBU DPNQBSFT

LJMPNFUSFT USBWFMMFE QFS IPVS

t 3BUFT DBO CF TJNQMJĕFE KVTU MJLF SBUJPT ćFZ DBO BMTP CF FYQSFTTFE JO UIF GPSN PG n
t :PV TPMWF SBUF QSPCMFNT JO UIF TBNF XBZ UIBU ZPV TPMWFE SBUJP BOE QSPQPSUJPO QSPCMFNT 6TF UIF VOJUBSZ

PS SBUJP NFUIPET

Tip Exercise 21.3

ćF XPSE AQFS JT PęFO 1 "U B NBSLFU
NJML DPTUT QFS MJUSF )PX NVDI NJML DBO ZPV CVZ GPS
VTFE JO B SBUFT 1FS DBO
NFBO AGPS FWFSZ
AJO FBDI
2 4BN USBWFMT B EJTUBODF PG LN BOE VTFT MJUSFT PG QFUSPM &YQSFTT IJT QFUSPM DPOTVNQUJPO
APVU PG FWFSZ
PS APVU PG BT B SBUF JO LN M
EFQFOEJOH PO UIF DPOUFYU
3 $BMDVMBUF UIF BWFSBHF TQFFE PG UIF GPMMPXJOH WFIJDMFT
Remember, speed is a very
important rate. (a) " DBS UIBU USBWFMT LN JO IPVST
speed = distance (b) " QMBOF UIBU USBWFMT LN JO POF IPVS NJOVUFT
(c) " USBJO UIBU USBWFMT LN JO NJOVUFT
time
4 )PX MPOH XPVME JU UBLF UP USBWFM UIFTF EJTUBODFT
distance = speed × time
(a) LN BU LN I (b) LN BU LN I
time = distance (c) LN BU LN I (d) N BU LN I
speed

5 )PX GBS XPVME ZPV USBWFM JO 2 1 IPVST BU UIFTF TQFFET
2

(a) LN I (b) LN I

(c) NFUSFT QFS NJOVUF (d) UXP NFUSFT QFS TFDPOE

21.4 Kinematic graphs
t %JTUBODFoUJNF HSBQIT TIPX UIF DPOOFDUJPO CFUXFFO UIF EJTUBODF BO PCKFDU IBT USBWFMMFE BOE UIF UJNF

UBLFO UP USBWFM UIBU EJTUBODF ćFZ BSF BMTP DBMMFE USBWFM HSBQIT

t 5JNF JT OPSNBMMZ TIPXO BMPOH UIF IPSJ[POUBM BYJT CFDBVTF JU JT UIF JOEFQFOEFOU WBSJBCMF %JTUBODF JT
TIPXO PO UIF WFSUJDBM BYJT CFDBVTF JU JT UIF EFQFOEFOU WBSJBCMF

t :PV DBO EFUFSNJOF TQFFE PO B EJTUBODFoUJNF HSBQI CZ MPPLJOH BU UIF TMPQF TUFFQOFTT
PG UIF MJOF ćF TUFFQFS UIF
MJOF
UIF HSFBUFS UIF TQFFE B TUSBJHIU MJOF JOEJDBUFT DPOTUBOU TQFFE VQXBSE BOE EPXOXBSE TMPQFT SFQSFTFOU NPWFNFOU

JO PQQPTJUF EJSFDUJPOT BOE B IPSJ[POUBM MJOF SFQSFTFOUT OP NPWFNFOU

o

t 4QFFEoUJNF HSBQIT TIPX TQFFE PO UIF WFSUJDBM BYJT BOE UJNF PO UIF IPSJ[POUBM BYJT
t 'PS B TQFFEoUJNF HSBQI
HSBEJFOU BDDFMFSBUJPO 1PTJUJWF HSBEJFOU BDDFMFSBUJPO
JT BO JODSFBTF JO TQFFE /FHBUJWF

HSBEJFOU EFDFMFSBUJPO
JT B EFDSFBTF JO TQFFE

t %JTUBODF TQFFE ¨ UJNF :PV DBO ĕOE UIF EJTUBODF DPWFSFE JO B DFSUBJO UJNF CZ DBMDVMBUJOH UIF BSFB CFMPX FBDI
TFDUJPO PG UIF HSBQI "QQMZ UIF BSFB GPSNVMBF GPS RVBESJMBUFSBMT BOE USJBOHMFT UP EP UIJT

Unit 6: Number 141

21 Ratio, rate and proportion

Exercise 21.4

1 ćF HSBQI CFMPX TIPXT UIF EJTUBODF DPWFSFE CZ B WFIJDMF JO B TJY IPVS QFSJPE

Distance (km) 600
500
400
300
200
100

0 123456
Time (hours)

(a) 6TF UIF HSBQI UP ĕOE UIF EJTUBODF DPWFSFE BęFS

(i) POF IPVS (ii) UXP IPVST (iii) UISFF IPVST

(b) $BMDVMBUF UIF BWFSBHF TQFFE PG UIF WFIJDMF EVSJOH UIF ĕSTU UISFF IPVST

(c) %FTDSJCF XIBU UIF HSBQI TIPXT CFUXFFO IPVS UISFF BOE GPVS

(d) 8IBU EJTUBODF EJE UIF WFIJDMF DPWFS EVSJOH UIF MBTU UXP IPVST PG UIF KPVSOFZ

(e) 8IBU XBT JUT BWFSBHF TQFFE EVSJOH UIF MBTU UXP IPVST PG UIF KPVSOFZ

2 %BCJMP BOE 1BN MJWF LN BQBSU GSPN FBDI PUIFS ćFZ EFDJEF UP NFFU VQ BU B TIPQQJOH
DFOUSF JO CFUXFFO UIFJS IPNFT PO B 4BUVSEBZ 1BN USBWFMT CZ CVT BOE %BCJMP DBUDIFT B USBJO
ćF HSBQI TIPXT CPUI KPVSOFZT

200 Pam
Pam

100
Distance (km)
Dabilo

0 Dabilo
60 120 180 240

Time (min)

(a) )PX NVDI UJNF EJE %BCJMP TQFOE PO UIF USBJO

(b) )PX NVDI UJNF EJE 1BN TQFOE PO UIF CVT

(c) "U XIBU TQFFE EJE UIF USBJO USBWFM GPS UIF ĕSTU IPVS

(d) )PX GBS XBT UIF TIPQQJOH DFOUSF GSPN

(i) %BCJMP T IPNF (ii) 1BN T IPNF

(e) 8IBU XBT UIF BWFSBHF TQFFE PG UIF CVT GSPN 1BN T IPNF UP UIF TIPQQJOH DFOUSF

(f) )PX MPOH EJE %BCJMP IBWF UP XBJU CFGPSF 1BN BSSJWFE

(g) )PX MPOH EJE UIF UXP HJSMT TQFOE UPHFUIFS

(h) )PX NVDI GBTUFS XBT 1BN T KPVSOFZ PO UIF XBZ IPNF

(i) *G UIFZ MFę IPNF BU B N
XIBU UJNF EJE FBDI HJSM SFUVSO IPNF BęFS UIF EBZ T PVUJOH

142 Unit 6: Number