Cambridge IGCSE Mathematics Study Guide Answer Key

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CambridgeInternationalExaminations

Answer key and markscheme

CONTENTSAnswer Key

Chapter 1 Numbers 1

Chapter 2 Algebra 9

Chapter 3 Geometry 21

Chapter 4 Mensuration 27

Chapter 5 Coordinate Geometry 28

Chapter 6 Trigonometry and Pythagorasʼ Theorem 30

Chapter 7 Vectors and Matrices 33

Chapter 8 Probability 38

Chapter 9 Statistics 41

Practice test markscheme

Practice Test (Paper 2) 47

Practice Test (Paper 4) 51

1

Answer Key

© 2017 NTK Publishing Limited

Chapter 1 NumbersExercise 1.1

1 (a) 7, −2, 111, 25, 0

(b) 7, −2, 34

, 111, 2.4, 25, 0, 512

(c) π , 3

(d) 25, 0

2 (a) 71, 81

(b) − 4, −3

(c) 6.2, − 4, 89

, −3, 71, 81, 3.1

(d) 81

3 (a) 3, 7, 13 (b) 2, 17

(c) 23, 29, 31 (d) 5, 37, 43

(e) 41, 43, 47 (f) 53, 61, 79

4 (a) 1, 3, 25 (b) 5

(c) −2, 2.1, 1, 3, 5, 4.62, 43

, 25

(d) 3, 25

5 7, 14, 56, 63, 70

6 (a) 1, 2, 3, 6 (b) 1, 2, 3, 4, 6, 12

(c) 1, 5, 25 (d) 1, 13

(e) 1, 2, 3, 5, 6, 10, 15, 30

(f) 1, 2, 4, 8, 16, 32, 64

7 (a) 4 (b) 8

8 (a) 24 (b) 60

9 (a) 32 × 5 (b) 32 × 23

(c) 7 × 3 × 22 (d) 23 × 3 × 5

(e) 2 × 7 × 9 (f) 5 × 33

(g) 2 × 3 × 5 × 7 (h) 32 × 5 × 7

10 (a) H.C.F = 6 L.C.M = 1560

(b) H.C.F = 45 L.C.M = 1350

(c) H.C.F = 15 L.C.M = 4725

(d) H.C.F = 12 L.C.M = 11880

(e) H.C.F = 36 L.C.M = 2592

(f) H.C.F = 90 L.C.M = 1080

Exercise 1.2A

1 A 11,13,17,19{ }=

2 (a) P

Q

4, 8,12,16

3, 6, 9,12,15,18

{ }{ }

=

= (b)

P

Q

4, 8,12,16

3, 6, 9,12,15,18

{ }{ }

=

=

3 B 1, 3, 9,13{ }=

4 (a) All elements in A are in B

(b) Yes, it is because 26 is not in B

5 (a) 3, 7, 9{ } (b) 2, 3, 5, 7, 8, 9,11{ }

6 (a) 4 (b) 4

(c) 0 (d) 2 , 2, 4 , 42 2{ } { }{ }

7 (a) 9 is a multiple of 3

(b) 12{ } (c) 11

8 (a) 2, 3, 5, 7{ } (b) 4

(c) True, since 6.5 is not a member of set A.

9 (a) A 7, 6, 5, 4, 2, 1, 2, 3, 4, 6{ }′ = − − − − − −

(b) 3, 5{ }−

(c) 8, 5, 3, 0,1, 2, 5, 7, 8{ }− − −

2

Cambridge IGCSE Mathematics Study Guide


10 (a) 8, 7, 6, 5, 2, 1, 0{ }− − − − − −

(b) 4, 2{ }−

(c) 8, 7, 6, 5, 3, 2, 1, 0,1, 3{ }− − − − − − −

Exercise 1.2B

1 (a) 5,10,15{ } (b) φ

(c) 5, 7, 8, 9,10,15{ }

2 (a) 1, 2,13,15,17{ } (b) 5, 8,10,13,15,17, 30, 33{ } (c) 30, 33{ }

3 (a) 5, 9{ } (b) 22, 24{ } (c) φ

4 (a) 4, 5, 6{ } (b) 4, 5{ } (c) 3, 8, 20, 22{ }

5 (a) ℰ

A

ab

ef

d

c

B

(b) c{ }

6 (a) ℰ

A

B C

3

248

9

10 5

1 7

(b) A B CA B C 10

φ

( ) { }∩ ∩ =∪ ∩ =

7 ℰ

A

B C

2

4.2

933

15

5

107

8 (a) ℰ

A B

(b) A Bℰ

(c) A Bℰ

(d) A

B C

ℰ

9 (a) 7 (b) 8

10 (a) 0 (b) 2

(c) 5 (d) 7

(e) 18 (f) 1

11 (a) 1 (b) 0

(c) 13

3

Answer Key


12 A

1 2 3

5

B

ℰ

Exercise 1.2C

1 14

2 (a) ℰ

C

50 50 100

T

(b) 50

3 (a) 28 (b) 5

(c) 5

4 (a) ℰ

M

5 5

5

10

S

(b) 15 (c) 5

(d) 5

5 (a) ℰ

G

5 5

5

15

S

(b) 5 (c) 15

6 (a) ℰ

B

7 8

20

15

F

(b) 8 (c) 20

(d) 7

7 (a)

x

Sandwich

Chocolate Coffee

20

15

8 − x

7 − x

9

10

ℰ

(b) 4

8 (a) 15 (b) 10

Exercise 1.3

1 (a) 121 (b) 6.25

(c) 49 (d) 3

10

(e) 2710

(f) 16

2 (a) 125 (b) −64

(c) −1 (d) 0.2

(e) 1.5 (f) 25

3 13

4 −4.913

5 −62.9

6 4 9 2 3 5

4 9 13 5

+ = + =

+ = ≠

7 41 or −41

Exercise 1.4

1 (a) 17°C (b) 30°C

(c) Moscow

2 8084

3 (a) −4°C (b) 0.5°C

4



Exercise 1.5

1 (a) 2330

(b) 2328

(c) 528

(d) 59

(e) 2924

(f) 8135

(g) 192

(h) 12152

2 (a) 113

(b) 34

(c) 12

(d) 3

3 (a) 2.4 (b) 2400 mL

4 (a) 0.34, 1.5, 2.4, 3.7, 6.45

(b) 1.4, 7.85, 7.94, 9.3, 10.7

5 (a) Terminating decimals: 0.35, 0.6, 4.7

Recurring decimals: 2.6, 2.87

(b) Terminating decimals: 1.3, 3.56, 2.43, 4.7

Recurring decimals: 3.8

6 (a) 45

(b) 2750

(c) 3325

(d) 7625

(e) 23

(f) 4599

(g) 2 753999

(h) 2 118

7 (a) 1

250 (b)

1425

(c) 2310

(d) 131

5000

8 Fractions Decimals Percentages

(a)25

0.4 40%

(b)123100

1.23 123%

(c)625

0.24 24%

(d)29

0.2 22.2%

(e)51750

10.34 1034%

(f)115

2.2 220%

Exercise 1.6

1 (a) 7 > 2 (b) −1 < 0

(c) 0.23 < 0.32 (d) −0.1 > −1

(e) >312

3 (f) <9 16

2 (a) −1, 0

(b) −8, −7, −6, −5, −4, −3

(c) 3, 2, 1

3 (a) < < < <0.2 0.2 0.2 0.2 10.2

3 2 3

(b) < <0.2% 0.2 0.2

(c) < < < <0.001 1101

1100

0.011 1.2%

(d) < < <14

% 0.025 14

23

Exercise 1.7

1 (a) 27 (b) 32

(c) 360 (d) 51597

2 (a) 24 × 34 × 7 (b) 52 × 112

(c) 23 × 33 × 132 (d) 23 × 7 × 11 × 13

5

Answer Key


3 (a) 254

(b) 19

(c) 13 (d) 4

(e) 1

32 (f) −8

(g) 1

16 (h)

58

4 (a) 4.98 × 10−9 (b) 9.999 × 1010

5 (a) 124 000 000 (b) 0.000 000 62

6 (a) 1.512 (b) 1.6 × 104

(c) 4.2 × 108 (d) 9.72 × 102

(e) 3.33 × 1010 (f) 2.30 × 106

(g) 2.32 × 1010 (h) 3.06 × 10−1

7 7−23, by 40 353 607 times.

8 500 s

9 ×

× ×− − −122 10

, 4.2 10 , 2 10 , 234 2

23

Exercise 1.8

1 (a) −1 (b) −6

(c) 0.6 (d) −0.2

(e) −16 (f) 21

(g) 64 (h) −43.47

(i) −8 (j) 12

(k) 4 (l) 16

2 (a) −1 (b) 25−

(c) 20 (d) 33

(e) −6 (f) 3

3 (a) 32

(b) 56

(c)

3258

125

= (d)

10332

209

=

−

−

Exercise 1.9

1 (a) 232.12 (b) 0.01

(c) 132.13 (d) 0.54

2 (a) 3 720 000 (b) 0

(c) 6000 (d) 93 000

3 (a) 70 000 (b) 0.2

(c) 10 (d) 0.7

4 (a) 1030 (b) 1 500 000

(c) 737 000 000 (d) 93.8

5 (a) 8009.21 (b) 8010

(c) 8000 (d) 8009.2

(e) 8009.206

6 (a) 7 8

4×

(b) 14

7 14.02

8 (a) 4 26

43

×= (b) 2 2

28

2( )+=

(c) 48

13

=

Exercise 1.10

1 upper bound: 9.8 0.12

9.85+ = ; lower bound

9.8 0.12

9.75− =

6



2 (a) lower bound: 5 12

8 12

= 33.75

(b) upper bound: 8+ 1

2

5 12

= 179

3 number of small tanks: 900+ 10

2

3.5 0.12

= 262 2269

4 (a) upper bound: 4 6.81+ 0.012

= 27.26

(b) lower bound: 4 6.81 0.012

= 27.22

5 (a) lowest possible: 26 12

25.5 kg− =

(b) upper bound: 5 26+ 12

= 132.5 kg

6 (a) upper bound: 65+ 12

58 12

= 8 kg

(b) lower bound: 65 12

58+ 12

= 6 kg

7 (a) lower bound:

2 3.14 0.012

3.56 0.012

9.81+ 0.012

= 3.77

(b) upper bound:

2 3.14+ 0.012

3.56+ 0.012

9.81 0.012

= 3.79

8 upper bound: 6 7.4+ 0.12

2

= 333.015

lower bound: 6 7.4 0.12

2

= 324.135

Warm up 1.11

1 (a) 1:3 (b) 3:5

(c) 8:11 (d) 1:2:8

(e) 15:48:20

2 (a) a:b:c = 1:2:6

(b) a:b:c = 5:4:8

(c) a:b:c = 40:15:50 = 8:3:10

(d) a:b:c = 9:12:8

Exercise 1.11

1 (a) =P w14

3 (b) 7294

(c) 36 m/s

2 (a) = × −d L3 10 5 (b) 125 000 mm

3 35 km

4 6

5 78 kg

6 (a) 743.04 (b) 639.6

7 (a) =tn

80 (b) 16

(c) decrease by 1 hour

8 (a) =×xf

3.6 105

(b) f = 1440

(c) 240

9 40 minutes

10 1 day

11 7 days

12 4

7

Answer Key


13 (a) (i) C = 4500N (ii) =tN

192

(b) (i) C = 4500 192t

C = 864000t

(ii) Inverse proportion

(c) (i) Decrease, it is because the salary budget is inversely proportional to the time required to finish the project.

(ii) =

=t

t

108000 864000

8 4 months

14 (a) 8:12 (b) 36:9

(c) 115:92 (d) 144:192

(e) 4104:912

15 (a) 8:12 (b) 36:9

(c) 115:92 (d) 144:192

(e) 4104:912

16 (a) 280 (b) $2240

17 (a) 5.5 kg (b) £22

(c) £12.8

18 (a) 3 km (b) 3.6 km/h

Warm up 1.12

1 (a) $2 (b) £13.176

(c) 4.2 kg (d) 6.4 g

(e) 9 (f) 464.4

2 (a) 10.2 (b) 105.176

(c) 64.5 (d) 4185.44

(e) 12.84 (f) 773.85

3 (a) 11.76 (b) 173.4

(c) 595.92 (d) 56.55

(e) 62.328 (f) 8.75

4 (a) 10% increase (b) 10% decrease

(c) 1.18% increase (d) 1.83% decrease

(e) 7.24% decrease (f) 13% decrease

Exercise 1.12

1 15.9%

2 (a) 85% (b) 3:17

3 60

4 750

5 4.62%

6 £96

7 11%

8 5.6%

9 £403.2

10 1% decrease

11 150,000

12 20

13 £1416

14 (a) £2940 (b) £1230

15 1 783 000 m3

8



Exercise 1.13

1 (a) −5.12 (b) 3.50

(c) 102400 (d) 0.26

(e) 0.80 (f) −11.57

2 (a) 0.75 (b) 1

(c) 0.231 (d) −5.20

(e) 0.125 (f) 1.20

3 ( ) ( )° ° ° °cos45 , cos45 , cos45 , cos453 2 3

4 (a) > (b) >

(c) < (d) <

Exercise 1.15

1 (a) €6.12 (b) €10.51

(c) €765.31 (d) €0.13

2 (a) $9.8 (b) $98

(c) 3.49 (d) $58 800

3 4.7% increase

4 £2415

5 (a) 17 800 (b) £600

(c) £588

6 (a) €40 (b) £34.4

(c) £3405.6

Exercise 1.16

1 6%

2 3200

3 £13 665.27

4 (a) £1416 (b) £1429.2

5 P = £4754.8

6 £29166.8

7 6.37%

Exercise 1.17

1 (a) 20 (b) 30, 418.5

2 (a) 1000

(b) 0 1 2 31000 874.1 764 667.8

3 (a) 13 (b) 3.25 grams

(c) 2 years

4 (a) ( )=B 100 2t (b) 6000

5 (a) 3 114 918 (b) 13 788 382

(c) Population of species A keeps decreasing.

(d) 4:1

Chapter 2 AlgebraExercise 2.1

1 (a) 2p + 2q + 2r (b) −2x + 10y

(c) ab (d) −

aq9

2 (a) y = c + mx (b) =+

r pa b

2

(c) = +R x y1 2 1

2

3 (a) ( )= +d u v t12

(b) d = xt

4 (a) S = 28 (b) V = 810

9

Answer Key


(c) =r 14

(d) =r 83

5 (a) =−y T x23

(b) + =

=−

fu fv uv

u fvv f

(c) =−+

y x RxR

2 53 1

(d) =−+

x yy

2 21

2

2

(e) =+s A t25

6 (a) 1. =−−

x q pp q

2 2. x = −3

(b) 1. =−

s bkrb a

2. s = 2

(c) 1. =−

b R aR1

2 2

2 2. b = 9

(d) 1. =−

−x y b

y a34

2

2 2. = −x 314

(e) 1. =+−

x y PyP

5 42 8

2. =−

x 174

Exercise 2.2A

1 (a) 6xy − 6y (b) 3x − 2xy + 5

2 (a) 9p2 + 2q (b) 3p2q + p2

3 (a) 0 (b) 2abc

4 (a) −5s2t − 4st (b) 2s2 − 6s2t

5 (a) 3xyz2 (b) −5xyz + 7xy2z

Exercise 2.2B

1 (a) a2 + 2a (b) 4b − ab

(b) 2b + 3ac (c) 2a2 − ab + 4b

(e) x2 − x − 6 (f) x2 − 3x − 10

(g) 6x2 − 19x − 11 (h) −5x2 − 18x + 8

(i) 20y2 + 3y − 2 (j) −y − 3y + 18

(k) x3 + x2 − 3x + 9 (l) −3x3 + 5x2 + 4

2 ( )− −x x12

12 7 102

Exercise 2.2C

1 (a) x2 − 4y2 (b) 9 − 25x2y2

(c) 4x4 − 14

(d) x2 − x9

2

(e) 5x2 + 10xy + y2 (f) 49 − 14xy + x2y2

(g) 4x2 + 4x + x1

2 (h) − +

y xy x1 2 1

2 2

(i) x4 + 2x3 + 3x2 + 2x + 1

(j) x6 − 2x4 + 6x3 + x2 − 6x + 9

(k) x6 − 4x4 + 6x2 + x1

2 − 4

(l) x2 + y2 + z2 + 2xy + 2xz + 2yz

2 (a) 5x2 + 20x − 146 (b) 4xy

(c) 2y3 − 3y2 + 1 (d) 6z4 + z3 − 16z2 − 7z + 4

Exercise 2.2D

1 (a) 2by2 (b) ax

(c) ( )−a 2 (d) ay

2 (a) ( )+ax x a2 (b) ( )+by y b4 22

(c) ( )+x x7 7 3 (d) ( )+ +x ax a x2 42 2

3 (a) ( )( )− −p q r2 1 (b) ( )( )− +a x y5

(c) ( )( )− +b x a c3 1 (d) ( )( )− − −x x a2 2

(e) ( ) ( )− − −x y x y5 22

(f) ( )−x b c2 22

4 (a) ( )( )+ +x y1 2 (b) ( )( )+ +r t s5

(c) ( )( )+ +p q m n (d) ( )( )− −a y z7

(e) ( )( )− −a b a2 (f) ( )( )− +y y1 5

(g) ( )( )− −y x y1 (f) ( )( )( )+ − +x x x2 1 1 1

10



(i) ( )( )+ − −y y h k12 (j) ( ) ( )− −x a b12

Exercise 2.2E

1 (a) ( )( )− +x x2 2 (b) ( )( )− +y y4 3 4 3

(c) ( )( )− +z z5 1 2 1 2 (d) ( )( )− +x y y7 5 5

(e) ( )( )− − − +a b a b2 6 2 6

(f) ( )( )− +p p33 3 1

2 (a) ( )+x 32 (b) ( )−x 7

2

(c) ( )−x2 52 (d) ( )−− y 11

2

(e) ( )−a b3 22 (f) ( )+− a b5 3 2

2

3 (a) ( )( )+ +− p q p q4 3 7 11 7

(b) ( )−x yz42

(c) ( )( )− − −p p q2 1 2 2

(d) ( )( )+ − + +a b a b2 6 2 6

(e) ( )−s t s20 5 3

(f) ( )( )− − − +a b c a b c2 7 6 2 7 6

(g) ( )( )+ − + + + +m n x m n x3 1 3 1

(h) ( )( )( )− + +x x x3 1 3 1 9 12

Exercise 2.2F

1 (a) ( )( )+ +x x6 1 (b) ( )( )+ −x x6 1

(c) ( )( )+ +x x2 3 (d) ( )( )− +x x3 2

(e) ( )( )− +x x8 7 (f) ( )( )− +x x5 2

(g) ( )+− x 92 (h) ( )( )+ −x x7 4

2 (a) ( )( )− −− p p2 5 (b) ( )( )+ −− a a6 3

(c) ( )( )− −m m2 3 1 (d) ( )( )− −y y4 4 7

(e) ( )( )+ −n n6 5 3 (f) ( )( )+ −4 6 8 6 1

(g) ( )−− q2 72 (h) ( )( )+ −− s s5 7 3

(i) ( )( )− +x x7 3 2 (j) ( )( )− −y y11 11 1

(k) ( )( )− +− z z3 10 9 (l) ( )( )− −− t t2 11 3

Exercise 2.2G

1 (a) ( )( )+ +x x2 1 3 (b) ( )( )+ −x x2 1 3

(c) ( )( )− +y y3 2 3 (d) ( )( )+ −y y3 1 3 4

(e) ( )( )− −a a5 2 1 1 (f) ( )( )− +a a3 4 1 5

(g) ( )( )+ +s s2 6 5 1 (h) ( )( )+ −s s3 4 1 3 2

(i) ( )( )− +− m m4 2 1 5 2 (j) ( )( )− −− m m3 2 1 1

(k) ( )( )− −− p p2 1 2 3 (l) ( )( )+ −− p p2 3 7 2

2 (a) ( )( )+ −x y x y11 12 (b) ( )( )+ −m n m n4 2 3

(c) ( )( )+ −p q p q2 5 5 (d) ( )( )− −s t s t8 3 3 8

3 (a) ( )( )+ −x x12 7 2 3 (b) ( )( )− −x x12 29 2 9

4 (a) ( )( )− −x y x y14

4

(b) ( )( )( )( )+ − − +x y x y x y x y14

2 2

Exercise 2.2H

1 ( )+a11 6 2

2 ( )−xy y x

3 ( )+a a b6

4 ( )−xy y x

5 ( )( )+ −n m1 1

6 ( )( )− −m x m 1

7 ( )( )+ +s s2 6 5 1

8 ( )( )− −p qx p q

9 ( )( )− +s t t1 1

10 ( )( )+ −− x y x y6 2 2

11 ( )( )− −y y12 13

11

Answer Key


12 ( )+n2 32

13 ( )−a b5 112

14 ( )( )− +c c3 5 7

15 ( )+x x2 22

16 ( )( )+ +a x x5 6 4

17 ( )( )− + − +m n m n6 6

18 ( )( )( )− + −q p p2 2 1 2 1

19 ( )( )+ −x x2 8

20 ( )+x3 22

21 ( )( )−p q q4

22 ( )( )( )( )− − + +p p p p1 4 1 1

23 ( )−x116

2 14

24 ( )( )( )− + +a a a116

2 1 2 1 4 12

Exercise 2.3A

1 (a) 6 (b) 67

(c) 125

(d) 1

2 (a) −103

(b) 32

(c) 0 (d) 32

3 (a) a3

(b) −a

14

(c) c13

(d) bc5

(e) −xy4

(f) −

y5

4

(g) zy8

(h) − zx3 2

2

4 (a) 2a + 1 (b) 3a + b

(c) zy8

(d) − zx3 2

2

(e) −xy

2 (f) xz

(g) −yy

2 55

(h) − xx

23

(i) −+

xx

22 1

(j) +−

xx

2 35

(k) −ac

(l) − x1

2

(m) +x1

1 (n)

−x

x 3

(o) +x

x 2 (p) +x 1

2

Exercise 2.3B

1 (a) a4

5 (b) −

a2

2 (a) bcad6

(b) −14

3 (a) −xx1

3 (b) − x1

32

4 (a) +xx

1 (b)

yx7

5 (a) ( )+−

xb y2 1

(b) −bx2

Exercise 2.3C

1 (a) LCM: 12x2 HCF: 2x

(b) LCM: 36x2y2 HCF: 3xy

(c) LCM: ( )( )+ −x x2 2 2 HCF: 1

(d) LCM: ( )−x x2 12 HCF: ( )−x 1

(e) LCM: ( )( )+ −a a b a b60 2

HCF: 5

(f) LCM: ( )− m n18 13

HCF: ( )− m3 1

12



(g) LCM: ( )( )( )+ − −x x x1 1 2 32

HCF: ( )−x2 3

2 (a) a

2110

(b) + aab

4 63

(c) −x

x2

1 (d)

( )−−

y xx y

22

(e) ( )

++

mm m7 4

2 (f)

( )−n9

2 3

(g) +−

xx

4 24 12

(h) ( )( )−

−

y y xx y x

2

Exercise 2.3D

1 (a) ( )−a

b b a9 4

(b) +

− aba b4

(c) −−

xx1

3 (d) −

+xx

2 32 3

(e) −x

x3

3 (f) 1 − x

(g) +−

xx

12 1

(h) −+

xx

33

2 (a) ( )( )( )− − +x x x3 2 2 1

(b) ( )( )( )−

− − +x

x x x5 4

3 2 2 1

3 3x + 5

4 (a) a3 − b3 (b) ( )( )− + +x y x xy y252 2 2

Warm up 2.4

1 (a) 19

(b) 4

(c) 13

(d) 3

(e) 1 (f) 1

(g) 0 (h) 1

(i) 125 (j) 4

(k) 18

(l) 19

(m) 1625

(n) 12

Exercise 2.4

1 (a) 59 (b) 78

(c) 37a5 (d) a9b7

(e) a (f) a b

13 6

(g) a14 (h) 8a2b11

(j) −25a3b4 (j) 1

2 (a) 133 (b)

162

(c) 2 (d) xy

2

3

(e) x4 (f) x y

1

213

(g) x92 (h) x

y2

3 (a) 3 (b) −2

(c) −2 (d) 7

(e) 3

4 (a) 116

(b) 5

(c) 2 (d) 34

(e) 521

(f) 32

(g) −4

Exercise 2.5A

1 (a) 6 (b) 4

13

Answer Key


(c) 4

2 (a) 15 (b) 1

(c) 1

3 (a) 2 (b) 7

(c) 2 (d) 2

4 (a) 6 (b) 74

(c) −76

5 (a) 1

18 (b)

6988

(c) 29

6 (a) −65

(b) 623

(c) 39

7 (a) 0 (b) 8

(c) 367

Exercise 2.5B

1 5

2 19

3 14

4 13

5 (a) 7 (b) 12, 16, 20

(b) 96

6 7

7 7

8 90

9 (a) 36 (b) 204

10 (a) 12 (b) 98

Exercise 2.5C

1 (a) x = 0 or x = 3 (b) x = 4 or x = −52

(c) x = 53 or x = −

94

(d) x = 3 or x = 4

2 (a) x = 0 or x = 4 (b) x = 0 or x = 5

(c) x = 0 or x = 75

(d) x = 0 or x = −4

3 (a) x = 0 or x = 3 (b) x = −52

or x = 5

(c) x = 72

or x = −5 (d) x = 32

or x = 6

4 (a) x = −2 or x = −1 (b) x = 3 or x = −4

(c) x = 8 or x = −3 (d) x = 12 or x = −3

5 (a) x = 13 or x =

32 (b) x = −5 or x =

74

(c) x = −65

or x = 23 (d) x =

23 or x =

37

6 (a) x = 3 or x = 5 (b) x = −1 or x = 52

(c) x = −56

or x = 32 (d) x = −

45

or x = 74

Exercise 2.5D

1 (a) x = 0 or x = 3 (b) x = 13 or x = −13

(c) x = 2 2 or x = −2 2

2 (a) x = 7 or x = −7 (b) x = 0 or x = 329

(c) x = −197

or x = 197

3 (a) x = −4 or x = −12

(b) x = +−2 3 74

or x = +2 3 74

14



(c) x = −5

(d) x = −−87

3 27

or x = −3 2

787

4 (a) ( )− +x 1 32

(b) x 52

214

(c) 2 x + 32

2

+ 9.5 (d) 2 x 32

2135

4

5 (a) x = 1 or x = 7

(b) x = −−52

132

or x = −132

52

(c) ( )= +x 2 1 3 or ( )= −x 2 3 1

(d) = −x 3 19 or = +x 3 19

(e) x = −−2 332

or x = −332

2

(f) x = −32

or x = 45

Exercise 2.5E

1 (a) x = 12

or x = 3

(b) x = −1 13

or x = +1 13

(c) x = +9 216

or x = −9 216

(d) = −x 43

(e) = −x 52

(f) x = 6

(g) = −x 4 433

or = +x 4 433

(h) = −x 512

14512

or = +x 512

14512

2 (a) = −−x 32

132

or = −x 132

32

(b) = −x 3 7 or = +x 3 7

(c) = −−x 112

79312

or = −x 79312

112

(d) = +x 2 23 or = −x 2 23

Warm up 2.5F

1 13, 15

2 6, 12

Exercise 2.5F

1 26

2 18

3 =x 23

or x = 4

4 length = 15 width = 12

5 6

6 12

7 30

8 12 km/h

9 (a) 60 km/h (b) 9:30 p.m.

Exercise 2.5G

1 (a) x = 3, y = 3 (b) x = 2, y = 3

(c) x = −3, y = 4 (d) x = 53, y = −

52

2 (a) x = 2, y = 3 (b) x = 5, y = −2

(c) x = 3, y = −4 (d) x = 2, y = −5

15

Answer Key


3 (a) x = 23

, y = −32

(b) x = 15, y = 12

(c) x = 6, y = −5 (d) x = 34

, y = 27

4 Adrian: £12 Billy: £20

5 Chocolate: £0.5 Crisps: £1.6

6 Solution A: 12.5 litres Solution B: 1.6 litres

Exercise 2.6A

1 (a) x < −2 (b) 4x < −2

(c) x ≤ 34

(d) 4x < −52

(e) x < 32

(f) 4x ≥ 2

2 (a) x < 26 (b) x < 143

(c) x ≥ 97

(d) x ≤ 8

(e) x < 2569

(f) x ≥ 326

3 (a) x < −3

28 (b) x > 101

46

Exercise 2.6B

1 (a)

x

y

4

4

(b)

x

y

4

8

(c)

x

y

3

4

(d)

x

y

3

5

2 (a)

x

y

3

3

6

16



(b)

x

y

7

4

(c)

x

y

0

1

3

42

2

1

(d)

x

y

2

4 93

6

2

43

6

3 (a)

x

y

4

5

4

(b)

x

y

4

5

4 5

55

4 54

(c)

x

y

12

2 6

3

(d)

5

8

14

x

y

4 (a) (i) 30x + 60y ≥ 1800 x + y ≤ 50

(ii) x ≥ 0 y ≥ 0

(b)

x

y

50

30

50 60

(c) 40

17

Answer Key


5 5

6 90

7 (a) 10

(b) 6p stamp: 28 8p stamp: 4

Exercise 2.7

1 (a) T1 = 1, T2 = 3, T3 = −7

(b) T1 = 1, T2 = 10, T3 = 37

(c) T1 = 1, T2 = 7, T3 = 79

2 (a) 5, 17, 29 (b) 2, 1, 23

(c) −3, 0, 5

3 (a) 4n − 3 (b) −4 + 34

(c) −2n−2x

4 (a) Tn = 6n − 17 (b) T20 = 103

5 ( )( )+ − +

= −

n n n

n

1 2

6 (a) 1 5 10 10 5 1

(b) n + 1

(c) 2n

7 (a) Pattern 4: 20; Pattern 5: 29; Pattern 6: 40

(b) n2 + 4

8 (a) 4n−1 (b) No

Exercise 2.8

1 932

2 (a) ( )= +y z14

13 (b) ( ) −y4 1

13

3 =p 8 5

4 = −n 12

5 a 1 2 8

b12

14

116

Exercise 2.9A

1 (a) (i) 2.5 m/s (ii) 3.33 m/s

(iii) 5 m/s

(b) (i) 50 m (ii) 200 m

(c) 2.5 m/s

2 (a) 70 km (b) 100 m/h

(c) 33 minutes (d) No

(e) 75 km/h

3 (a) 2.25 ms−2 (b) −1.2 ms−2

(c) 423 m (d) 12.1 ms−1

4 (a) (i) 0.6 ms−2 (ii) 0.4 ms−2

(b) 512 m

(c) 992 m

(d) 12.4 ms−1

(c) 792 m

5 (a) (i) 0.48 ms−2

(ii) Yes, the gradients of the two lines are equal.

(b) (i) 91.5 seconds

(ii) 588 m

(c) (i) 83.75 seconds

(ii) 495 m

18



Exercise 2.9B

1 (a)

5 10 15 20 250

2

4

6

8

10

L (c

m)

W (g)

(b) 1.6 cm

(c) 0.24, it means the spring will extend 0.24 cm when an extra 1 g of load is added.

2 (a) (i) 420 HKD (ii) 20 GBP

(b) (i) 12, it is the exchange rate of GBP to HKD.

(ii) 200 m

(c) 0

3 (a)

201020 30 40 500

20

20

40

60

80

100

10

°F

°C

(b) (i) 1.8 (ii) 32 m

(c) (i) m = 1.8, c = 32

(ii) 70

4 (a)

10 20 30 40 500

10

20

30

40

50

60

Temperature (°C)

Time (minutes)

(b) (i) 54 °C (ii) −0.6 °C / minute

(c) (i) T = −0.6t + 54 (ii) 45 minutes

5 (a) F = 4

M = 0.0004

K = 0.0006

t0 = 500

(b) (i) CA = 2 + 0.004t

(ii) CB = 0.006t

(c) t > 1000

6 (a) (i) 84 USD (ii) 60 NZD

(iii) 70 NZD

(b) The exchange rate of EUR or NZD to USD.

(c) 1.52

7 (a) Time t 5 10 15 20 25t2 25 100 225 400 625

Height H above the

ground (m)4375 4000 3375 2500 1375

(b) H = −5t2 + 4500

(c) (i) 4500 m (ii) t = 30 seconds

Exercise 2.10

1 (a) x 0.1 0.5 1 1.5 2 0.5 3y −29.9 −5.5 −2 −0.5 0.5 1.3 2

(b)

x

y

10

5

00.5 1 1.5 2 2.5 3

5

10

15

20

25

30

19

Answer Key


(c) 1.7 (accept 1.8)

2 (a) x −3 −2 −1 0 1 2 3y 1 0 −1 −1 0.5 3.75 8.875

(b) y

x1

2

4

8

6

10

2 312

4

6

23

3 (a) (i) y = x + 1

(ii) 1.5, 7.5 (accept 1.4 – 1.6, 7.4 – 7.6)

(b) (i) y = 3

(ii) 1.4, 6.6 (accept 1.3 – 1.5, 6.5 – 6.7)

4 (a) y = −2x + 2

(b) −2, 3

5 (a) x −3 −2 −1 0 1 2 3

y18

14

12

1 2 4 8

(b) y

x1

2

4

8

6

10

2 312

4

6

23

(c) x = 1

6 (a) (i) y = 2x + 4

(ii) 1, −0.8, 4.8 (accept 0.9 – 1.1, −0.9 – −0.7, 4.7 – 4.9)

(b) (i) y = 3x − 10

(ii) −1.7, 2.5, 4.2 (accept −1.8 – −1.6, 2.4 – 2.6, 4.1 – 4.3)

7 (a) x −3 −2 −1 0 1 2 3 4 5y −35 0 15 16 9 0 −5 0 21

(b) y

x1

02 3 54123

10

20

30

40

10

20

30

(c) y = 0

(d) −2, 2, 4

8 (a) 0.5, 3 (accept 0.4 – 0.6, 2.9 – 3.1)

(b)

Exercise 2.11

1 (a) −1.5 (accept −1.7 – −1.3)

(b) 1.5 (accept 1.3 – 1.7)

(c) −2 (accept −2.2 – −1.8)

(d) −9 (accept −9.5 – −8.5)

(e) 5.6 (accept 5 – 6)

(f) 1.3 (accept 1.2 – 1.4)

2 (a) x −2 −1 0 1 2 3y −11 −1 1 1 5 19

20



(b)

x

y

1 2 315

10

2

5

10

15

20

15

20

(c) (i) 5 (ii) 1

Warm up 2.12A

1 (a) +f x x: 5 7 (b) −f x x: 4 92

(c) +f x x: 2 1 (d)

−f x

x: 1

4 3

2 (a) 0 (b) 2

(c) 27 (d) 2

3 ( )( )

=

=

fg x x

gf x x

250

10

3

3

4 ( )

( )

=−

= −

fg xx

gf xx

32 46 4

Exercise 2.12A

1 (a) (i) 1.4 (ii) 163

(iii) 163

(iv) −1.4

(b) denominator = 0

2 (a) x = 4 (b) x = 0

3 (a)

( )( )

= +

= − +

−f x x

g x x x

10 17

5 14 122

(b) − +25

295

, 25

295

4 ( )( )( )

=

=

= −

f

f

f

2 4

4 0

6 3

5 ( ) = −fg x x2 4 1

6 f 2x

= 12x2

7 (a) ( ) =+

fg xx

14 152 (b)

116

(c) 9

151

8 (a) 8.04 (b) −4.5

(c) 2633

(d) 52

9 (a) a = 3 (b) 34

(c) ( )++

− xx3 1

3

10 (a) +xx

2 (b) denominator = 0

11 (a) ff 2( ) = 2, ff 32

= 32

, gg 2( ) = 4, gg 32

= 5

(b) gf s( ) = 4, fg 3( ) = 0, gh 3( ) = 3, gf 92

= 0

(c) hgf 1( ) = 0, hgf 0( ) = 6, hgf 32

= 32

(d) 2, 5

12 (a) a = 7 (b) b = −1 c = −1

21

Answer Key


Exercise 2.12B

1 (a) ( ) = −−f x x 31 (b) ( ) = +−f x x 72

1

(c) ( ) = +−f x x 35

1

2 (a) ( ) = +−f x x4 73

1 (b) ( ) = −−f x x15 910

1

(c) ( ) = −−f x x5 68

1

3 (a) ( ) = −−f xx1 21 (b) ( ) = −−f x

x2 3

21

(c) ( ) = +−f xx53

31

4 (a) ( ) = +−

−f x xx

5 31

1 (b) ( ) = +−

−f x xx

2 54 3

1

(c) ( ) = −+

−f x xx

7 52 3

1

5 (a) (i) ( ) =ff x x

(ii) f is a self-inverse function

(c) =

=

m

n

256

Chapter 3 GeometryExercise 3.1

1 (a) trapezium (b) isosceles triangle

(c) parallelogram (d) hexagon

(e) pentagon (f) equilateral trangle

2 (a)

(b)

(c)

(d)

3 (a) similar (b) neither

(c) congruent (d) neither

(e) congruent (f) neither

Exercise 3.2

1 (a) A

B

O

22



(b)

A

B

O

2 (a) A

M

B

O

112°28°

(b) A

N

B

O

112°70°

3 (a) A

B

C

(b) They meet at the in centre (Centre of the circle inscribed in ΔABC)

4 (a) A

M BC

60°

∠AMC = 120°

(b) A

M BC

60°

N

∠ANM = 90°

5 (a) A

B

(b)

A

B

23

Answer Key


6 (a) A

BC

(b) The lines (medians) meet at the centroid.

7 (a)

A B

C

(b) They meet at the circum-centre. (Centre of the circle circumscribing ΔABC)

(c)

A B

C

The circle circumscribes ΔABC.

8

A

B

C

10 (a)

A30° 30°

B

(b) 4.6 cm

Exercise 3.3

1 (a) B

A C

3 cm 4 cm

4.5 cm

(b) PQ = 1.5 m, QR = 2 m, PR = 2 m

2 (a) 1 : 6000000

(b) 565 km

3

A

1 cm represents 5 km

4.4 cm45°

N

Exercise 3.4A

1 (a) True (b) True

(c) False

2 (a) x = 9; y = 12 (b) x = 45; y = 4; θ = 65°

3 (a) y = 5; α = 53; β = 65°

(b) x = 8.75; y = 2; α = 20°; β = 70°

24



4 (a) x = 24; y = 16; g = 3; k = 2

(b) x : g = 8 : 1; y : k = 8 : 1

(c) s1 : s2 : g = n : 1 : n 1

2−

5 (a)

(b)

4 cm 4 cm 8 cm

2 cm

8 cm

1 cm

1 cm

2 cm

6 (a) ΔAMN ∼ ΔABC (b) ΔAOB ∼ ΔPOQ

7 (a) ΔAXC ∼ ΔDXB (b) ΔBED ∼ ΔBCA

(c) ΔDAE ∼ ΔBAC (d) ΔABD ∼ ΔCAB

8 (a) α = 50°; x = 9 (b) x = 6; y = 12

(c) x = 8; α = 60° (d) x = 12; α = 40°

9 (a) x = 3.98 (b) x = 1.61

(c) r = 8; t = 7.5; α = 80°

(d) x = 203

10 (a) x y36 77

; 27 77

= =

(b) x = 15

(c) x = 5.25; y = 8.75;

(d) x = 16; y = 15; z = 9; t = 25

11 (b) (i) 14.4 cm (ii) 360 cm2

Exercise 3.4B

1 (a) x = 13.5 (b) 5

2 (a) x = 1215 (b) x = 3

3 (a) x = 200 (b) x = 18

4 (a) 24π cm2 (b) 12π cm3

(c) (i) 216π cm2 (ii) 324π cm3

5 (a) 6 cm (b) 3888π cm3

6 (a) (ii) 33.3%

(b) (i) 5.77 cm (ii) 11.6 cm2

7 (a) 4 3 cm

(b) (i) 5 3 cm (ii) 15 5 3 cm( )−

8 (a) (i) 14.3 cm; 3.71 cm

(ii) 121 cm

(b) (i) XY = 18.3 cm; YZ = 13.7 cm

(ii) 2.60 cm

9 (a) 144π cm2 (b) 827

(c) (i) 16.0 cm (ii) 2.65 cm

10 (a) 4.17% (b) 8.17%

Exercise 3.5

1 (a) yes; 4 (b) yes; 2

(c) no (d) no

(e) no (d) yes; 6

25

Answer Key


2 (a) (b)

(c) (d)

(e) (f)

3 (a) (b)

(c) (d)

4 (a) 2 (b) 3

(c) 2 (d) Infinite

5 4

Exercise 3.6A

1 a = 102°; b = 78°; c = 102°

2 a = 150°; b = 60°

3 x = 20°

4 y = 18°

5 x = 26°; y = 32°

6 x = 22°; y = 46°

7 x = 55°

8 a = 111°; b = 69°

9 x = 78°; y = 102°

10 x = 65°; y = 55°; z = 65°

11 x = 20°; y = 45°

12 x = 9°

Exercise 3.6B

1 (a) x = 60° (b) x = 111°

(c) x = 50°; y = 10° (d) x = 67.5°

(e) x = 80° (f) x = 25°; y = 50°; z = 80°

2 36°

3 25°

4 60°

5 90°

6 (a) x = 108°; y = 72° (b) x = 18°

(c) x = 27°; y = 30° (d) x = 40°; y = 55°

(e) x = 25°

7 x = 112.5°

8 n = 6

9 180°

10 x = 60°

26



Exercise 3.6C

1 (a) x = 30° (b) y = 45°

(c) x = 30° (d) y = 50°

(e) x = 35°; y = 65° (f) x = 30°; y = 60°

2 (a) x = 9°; y = 18° (b) x = 112°; y = 112°

(c) x = 100°; y = 30° (d) no solution

(e) x = 118°; y = 59° (f) x = 70°

3 (a) C

B

A

O

(b) 8.5 cm

BC is the diameter of the circle.

4 (b) a = 120°

5 Base angles in an isosceles triangle are equal.

Exercise 3.6D

1 (a) x = 60° (b) x = 90°; y = 40°

(c) x = 90°; y = 60° (d) x = 35°

2 (a) x = 67°

(b) x = 80°; y = 80°

(c) x = 40°; y = 70°; z = 30°

(d) x = 30°

(e) x = 25°; y = 90°

(f) x = 118°

3 x = 100°; y = 40°

Exercise 3.6E

1 (a) x = 82° (b) x = 60°; y = 102°

(c) x = 36°; y = 72° (d) x = 20°

(e) x = 100° (f) x = 86°; y = 22°

2 (a) x = 100°; y = 80° (b) x = 90°; y = 60°

(c) x = 60° (d) x = 49°; y = 30°

(e) x = 12°; y = 15° (f) x = 44°; y = 68°

3 (a) No.

(b) Yes. Sum of any two opposite angles is 180°.

(c) No.

(d) Yes. ∠A + ∠B = 180°, ∠B = ∠C, ∠A + ∠C = 180°, same as ∠B + ∠D

(e) Yes. ∠A + ∠C = 180° and ∠B + ∠D = 180°

(f) Yes. Converse of angles in the same segment.

4 ∠CAB = 70°

Exercise 3.7

1 A

D

C

B

2 (a) A

B

C

27

Answer Key


(b) A

B

C

(c) A

B

C

3 A B

CD

4 (a)

O 1.5 cm

5 C B

A

D

Chapter 4 MensurationExercise 4.1

1 (a) 0.23 (b) 120 000

(c) 0.045 (d) 0.15

(e) 250 (f) 6 520 000

2 (a) 3750 (b) 0.012

(c) 0.00112 (d) 0.04

3 (a) 3.5 × 106 (b) 600

(c) 6.5 (d) 5 × 10−5

(e) 23,000 (f) 5.1 × 106

4 (a) 30 (b) 1 × 107

(c) 300 (d) 100 000

(e) 0.0031 (d) 6500

Exercise 4.2

1 110 cm

2 100 cm

3 a = 5

4 32.8

5 23

6 179

7 (b) x = −4 or x = 1 (c) x = 1

8 40 cm2

9 +3 23 92

28



Exercise 4.3

1 (a) perimeter = 25.1 m, area = 50.3 m2

(b) perimeter = 18.8 m, area = 28.3 m2

2 (a) arc length = 27.9 m, area = 69.8 m2

(b) arc length = 15.7 cm, area = 78.5 cm2

3 r = 7.14 cm

4 (a) 4.71 m (b) 16.7 m

(c) 14.1 m

5 (a) 44.6 m2 (b) 18.8 m2

(c) 12.6 m2 (d) 2.46 cm2

6 (a) 37.7 cm (b) 22.8 m

(c) 25.1 m (d) 46.8 cm

7 perimeter = 15.7 cm, area = 4.03 cm2

8 perimeter = 30.9 cm, area = 61.4 cm2

Exercise 4.4

1 (a) 1550 cm2 (b) 294 cm2

(c) 193 cm2 (d) 84.3 cm2

2 (a) 225 cm2 (b) 424 cm2

(c) 384 cm2 (d) 85 cm2

3 (a) surface area = 22 cm2, volume = 157 m3

(b) surface area = 438 m2, volume = 335 m3

(c) surface area = 462 cm2, volume = 718 cm3

(d) surface area = 113 cm2, volume = 113 cm3

4 ππ

=−h A rr

22

2

5 (a) 1256.64 cm2 (b) 4720 cm3

6 surface area = 63.2 cm2, volume = 56.5 m3

Exercise 4.5

1 (a) 113 m3 (b) 500 m3

(c) 48 m3 (d) 136 cubic units

2 (a) 615.0 cm3 (b) 41.9 m3

(c) 1282.9 cm3 (d) 173.7 cm3

3 471 cm3

4 1367.1 cm3

5 (a) 864 cm2 (b) 1832.6 cm3

6 (a) 3π x3 (b) x = 3

(c) 210 cm2

7 (a) 12.15, 80 (b) 530.93

Chapter 5 Coordinate Geometry

Exercise 5.1

1 (a)

( ) ( ) ( )( ) ( ) ( )

− − −

− −

A B C

D E F

4,2 ; 3, 1 ; 0, 4 ;

3,5 ; 1,5 ; 7, 1

(b) E, A, F

2 (b) A straight line

Exercise 5.2

1 (a) 3 (b) −12

(c) 0 (d) undefined

2 (a) −3 (b) −1

(c) undefined (d) undefined

29

Answer Key


3 (a) no (b) no

(c) yes (d) yes

4 (a) c = 1 (b) q = −15

5 (a) = = = =− −m m m m2; 13

; 2; 13AB AD DC BC

(b) same gradient

6 (a) (i) 0 (ii) undefined

(iii) 53

(b) 152

Exercise 5.3

1 (a) undefined (b) −38

(c) −34 (d) −

52

2 (a) = = =AB AC BC5; 5; 2 5

(b) +5 3 5

3 (c) Area = ( )( ) =12

20 20 10 sq.units

4 (a) ( )6,2 (b) ( )2,1

(c) 12

, 12

(d) ( )−2,3

(e) 5a4

, b2

2 (f) ( )+a b0, 2

5 c = −4; d = −3

6 (a) height = 9 units (b) ( )3,6

Exercise 5.4

1 (a) y = −x + 3 (b) =y x45

(c) y = −2 (d) = −−y x23

43

2 (a) y = x − 3 (b) = +y x518

89

(c) x = 5 (d) y = −bx + ab

3 (a) = +−

y x311

11711

(b) No

(c) = =a b6011

; 892

4 (a) L1: 5x − 7y + 25 = 0; L2: x + 5 = 0.

(b) L1: 2x − 3y − 8 = 0; L2: y − 4 = 0.

5 (a) gradient = 14

, y-intercept = 58

(b) gradient = undefined

y-intercept = does not exist

(c) gradient = −118

, y-intercept = −114

(d) gradient = 34

, y-intercept = −1

6 (a) = −y x12

35

(b) = −y x27

2

(c) = −−y x 53

(d) = +−y x911

3

Exercise 5.5

1 (a) L2 and L6

2 (a) −4 (b) −72

3 = +y x23

113

4 = +y x12

3

30



5 (a) = −k 43

(b) x-intercept = 92 y-intercept =

32

(c) 6y + 4x − 9 = 0

(d) 3y + x − 9 = 0

Exercise 5.6

1 L3 and L5 L1 and L4

2 (a) 6 (b) 197

(c) −6 or 1 (d) 1 or −1

3 (a) 2 (b) 3

(c) 25

4 (a) −53 (b) b = 1

(c) y-intercept = 85

, x-intercept = −83

5 (a) L1: y = −x + 7; L2: y = x + 1

(b) L1: x-intercept = 7; L2: x-intercept = −1

(c) L1

6 (a) L1: = +−y x34

3

L2: = −−y x34

154

(b) L1: = +−y x52

15

L2: = −y x25

185

7 (b) k = 3 or k = −1

(c) For k = 3, L1: 3x + 3y − 4 = 0L2: −x + y − 3 = 0

For k = −1, L1: −x + 3y − 4 = 0L2: 3x + y − 3 = 0

Chapter 6 Trigonometry and Pythagorasʼ Theorem

Exercise 6.1

1 (a) N

070°

(b) N

230°A

(c) N

300°A

2 (a) 050° (b) 280°

(c) 270°

3

N

N

R

P

Q

080°4 km

5 km

310°

31

Answer Key


4 200°

5 090°

Warm up 6.2A

1 (a) x = 12.8 (b) x = 13.7

(c) x = 7.21 (d) x = 5.83

2 (a) θ θ θ= = =pr

qr

pq

sin ; cos ; tan

(b) θ θ θ= = =ac

bc

ab

sin ; cos ; tan

(c) θ θ θ= = =sin 45

; cos 35

; tan 43

(d) θ θ θ= = =sin 15

; cos 2 165

; tan 612

3 (a) x = 4.11 (b) x = 13.7

(c) x = 6.53 (d) x = 5.64

(e) x = 40 (f) x = 50

4 (a) θ = 53.1° (b) θ = 31°

(c) θ = 63.9° (d) θ = 49.5°

Exercise 6.2A

1 OD = 5

2 EC = 9.90 cm

3 AE = 17.3

4 (a) 3x2 − 24x = 40 (b) x = 8

5 =r 83

6 26 km

7 1.3 km

8 39.1 km

9 8.66 cm

10 (a) x = 8.94; φ = 63.4°; θ = 26.6°

(b) x = 11.5; φ = 27.5°; θ = 62.5°

(c) x = 6; φ = 53.1°; θ = 36.7°

(d) x = 7.62; φ = 66.8°; θ = 23.2°

11 θ = 52.0°

12 (a) 8.94 m (b) 35.3°

13 2.25 m

14 20.5 cm

15 270 m

16 53.1°

17 12.0 m

18 (a) 10 (b) 21.8°

19 (a) 41.5° (b) 51.3°

20 248°

21 112°

22 (a) 175 km (b) 323°

23 1.73 km

Exercise 6.2B

1 0.866 m

2 40 m

3 109 m

4 12.7 m

32



5 83.9 m

6 4.1 m

7 55.6 m

8 10.8 m

9 710 m

Exercise 6.2C

1 (a) −cos 60° (b) −cos 150°

(c) −cos 90° (d) −cos 30°

(e) cos 157° (f) −cos 13°

2 (a) sin 150° (b) sin 135°

(c) sin 60° (d) −sin 0°

(e) sin 155° (f) sin 4°

3 (a) (i) 1 (ii) 1

(iii) 1

(b) sin2 θ + cos2 θ = 1, for any angle θ .

Exercise 6.3A

1 (a) 25.8 cm2 (b) 50.3 cm2

(c) 36.8 m2

2 (a) x = 12.0 (b) x = 14.1

(c) x = 9.67

3 (a) θ = 71.8° (b) α = 48.59°

(c) β = 48°

4 (a) 70.9 cm2 (b) 26.8 cm2

(c) 65.0 cm2

5 (a) 12.0 cm2 (b) 2.93 cm2

(c) 61.4 cm2

6 78.6 cm2

Exercise 6.3B

1 (a) a = 3.54 (b) x = 17.91

2 (a) x = 5.87; y = 13.50

(b) a = 12.62; b = 4.68

3 (a) θ = 42.55° (b) θ = 36.23°

4 (a) α = 56.76°; x = 17.72

(b) θ = 30.95°; β = 113.05°

5 (a) 12.42 cm

(b) AD = 6.62 cm; DC = 9.38 cm

6 (a) PQ = 8.91 cm; PS = 3.03 cm

(b) 28.96 cm2

Exercise 6.3C

1 (a) x = 7.93 (b) x = 12.26

2 (a) x = 16 (b) x = 18.60

3 (a) α = 60.61°; β = 40.80°

(b) α = 95.74°; β = 33.56°

4 (a) x = 5.18; α = 34.40°; β = 98.60°

(b) x = 20.84; α = 22.58°; β = 15.42°

5 28.59 km2

6 (a) PR = 11.34 cm; QS = 13.63 cm

(b) ∠ SQR = 14.11°; ∠ SPR = 30.50°

(c) RS = 9.76 cm

33

Answer Key


Exercise 6.3D

1 BD = 10.58 cm; AC = 17.44 cm

2 (a) QS = 7.13 cm

(b) ∠ QRP = 23.33°; ∠ QRS = 46.67°

(c) 35.93 cm2

3 (a) BM = 11.28 cm (b) ∠ AMB = 117.1°

(c) MC = 14.91 cm

5 (a)

1200 m860 m

068°

130°

N

N

CA

B

(b) 1774.43 m

(c) 273.34°

6 (a) (i) 1857.54 m (ii) 1717.26 m

(b) VA:VB = 1.08:1

7 (a) 996.11 km (b) 7.88°

8 (a) (i) 12.59 cm (ii) 39.16 cm

(b) (i) AP = 12.63 cm; PB = 26.53 cm

(ii) ∠ABP = 24.87°; ∠APB = 93.13°

Exercise 6.4

1 (a) 59.04° (b) 51.34°

(c) 45°

2 (a) 45° (b) 45°

(c) 35.26°

3 (a) 45° (b) 51.34°

4 67.01°

5 (a) 33.69° (b) 53.13°

(c) 30.81°

6 8.94 cm

7 (a) 54.74° (b) 35.26°

8 (a) 500 m (b) 23.75°

9 (a) 14.42 cm (b) 29.02°

(c) 25.09°

10 (a) 4.29 cm (b) 16.08°

11 (a) 14 cm (b) 39.87°

12 37.76°

13 (a) 61.93° (b) 51.24°

14 55.47°

15 (a) 207.85 km/h (b) 32.21°

Chapter 7 Vectors and MatricesExercise 7.1

1 (a) � ��OA (b)

� ��CD

(c) � ��BA (d)

� ��HK

2 (a) 31

(b) 13

(c) 04

(d) 13

(e) 51

34



3

x

y

0 2

2

4

4

1

1

3

3

A

C

E

D

B

11

2

3

4

234

4 (a) 74

(b) 84

(c) 04

(d) 71

(e) 54

(f) 38

5 (a)

a a + b

b (b) a

b

a + b

(c)

a b

a + b

(d) a

b

a + b

6 (a) a

a − b

b

(b) a

a − b

b

(c)

a

a − b

b

(d) a

a − b

b

7 (a) (i) a + b (ii) −c + a + b

(iii) −a + c + b (iv) a + b + c

(b) (i) −a + b (ii) c

(iii) a = b − c

8 (a) b + g (b) g + f

(c) −f + b (d) −f + b

(e) g + f (f) g + b

9 (a) a12

(b) b12

(c) +a b12

(d) +a b12

10 (a) 2519

(b) 3928

(c) 1436

(d)

143236

Exercise 7.2

1 (a) ( )9,2 (b) ( )−17,15

(c) ( )2,4 (d) ( )+ −a a4,4 2

2 (a)

0 2 31

1

2

3

4

5

2 11

2

3

45 3x

y

A

B

C

D

(b)

0 2 31

1

2

3

4

5

2 11

2

3

45 3x

y

A1

B1

C1

D1

35

Answer Key


3 (a) ( )− −5, 2 (b) ( )−17,15

(c) ( )2,4 (d) ( )−5,4

4 (a) ( )3,4 (b) ( )− −3, 4

(c) ( )3,2 (d) ( )− −7, 4

5

x

A

BC

y

0 2

2

1

3

4

4 51 3 6234

2

11

3

4

6 (a) ( )− −2, 4 (b) ( )2,4

(c) ( )−4, 2

7 (a) ( )−2,3 (b) ( )−2, 3

(c) ( )− −3, 2

8

x

P

P2

P1

y

0 21

2

4

5

1

3

43 524 135

2

4

5

1

3

9 (a)

xk = 3

C

y

(b)

x

C

y

k = 2

(c)

x

C

y

k = 12

36



(d)

x

C

yk = 1

3

10

x

y

0 21

2

4

5

6

1

3

43 524 135

2

4

1

3

C

BA

11 (a) reflection about x = 0

(b) enlargement with centre at ( )−6,1 , scale factor 2

(c) rotation with centre ( )0,0 clockwise 90°

(d) reflection about y = −1

Exercise 7.3

1 (a) c − b (b) ( )−c b12

(c) ( )−b c12

(d) ( )+b c12

2 (a) +q r23

(b) ( )+q r34

(c) −r q34

14

(d) ( )+q r14

3 (a) c + a (b) a − c

(c) −− a c27

57

(d) +c a27

57

4 (a) (i) g − d (ii) ( ) +−d g23

23

(iii) ( )−g d23

(c) (i) ( )−k g d23

(d) 2:1

5 (a) (i) b − a (ii) −b − a

(iii) −b (iv) 2b

(v) 2b + 2a

(b) 4

(c) � �� BE EF CF FC AD DA, , , , , .

6 (a) (i) s + 2t (ii) s + t

(b) (i) = +� ��QX s t (ii) =

� ��XS t

7 (a) (i) 3p (ii) −p + u

(iii) 2u (iv) −4p + 2u

(v) −2p + 4u (vi) 2u + p

(c) (i) 6 (ii) 6

(iii) 6 (iv) 12

(v) 18

8 (a) +− a b12

12

(b) +− a b12

12

(c) +a b12

12

(d) −−a b12

9 (a) (i) b + v (ii) −b + u

(iii) ( )−u v12

(b) (i) 34

(ii) 30

37

Answer Key


Exercise 7.4A

1 (a) 2 × 1 (b) 1 × 3

(c) 2 × 4 (d) 3 × 3

2 (a) a = 0, b = 4

3 (a) 313

(b) ( )− −12 2

(c) 1 1511 9

(d) does not exist

(e) 3 9 06 12 9

(f) 8 6 42 10 6

10 2 6

(g) 38

(h) 16 814 7

(i) does not exist (j) 14 422 1610 10

4 (a) 8 27 9

(b) 1 41 12

(c) 18 217 7

(d) 12 165 1

5 a = 5, b = 0

Exercise 7.4B

1 (a) 1 (b) 2

(c) −21 (d) −1

(e) 2 (f) −1

12

(g) −5

12 (h) 0

2 (a) 12

1 32 4

(b) does not exist

(c) 1

566 78 0

(d) does not exist

3 (a) 2 91 4

(b) 3 55 8

(c) 12

6 44 3

(d) 1

197 44 5

(e) 13

7 95 6

(f) 3

29

103

43

72

32

4 (a) A3 = 9A + 8I

A4 = 17A + 72I

A5 = 89A + 136I

(b) (i) 11 12 6

(ii) 403 89178 42

5 k = −4 or k = −1

6 (a) 4 28 1

(c) (i) −3 (ii) −3

(e) (i) 1 (ii) 1

7 a 13

= or 13

−

8 (b) (ii) 21

(c) x = 2, y = 1

Exercise 7.5

1 (a) Reflection about the x-axis.

(b) Enlargement with centre 0,0( ) and scale factor 2.

(c) Reflection about the x-axis, enlargement with centre 0,0( ) and scale factor 2.

(d) Reflection about the line y = −x.

(e) Rotation about the origin through 90° clockwise.

(f) Reflection about the line = x, enlargement with centre 0,0( ) and scale factor 3.

38



(g) Rotation about the origin through 90°.anti-clockwise, enlargement with centre 0,0( ) and scale factor 2.

(h) Rotation about the origin through 90° anti-clockwise.

2

x

y

0 2 31

2

1

3

4

5

6

4 5 623 141

2

3

4

3 (a) 11

(b) 01

4 (a) ( )− −1, 3 (b) 3, 1( )−

(c) 3,1( ) (d) 3, 1( )− −

5 (a) 0 22 0

(b)

x

y

0 2 31

2

1

3

4

5

6

4 5 62

2

3

5

1

4

6

4 3 16 5

A1

B1

C1

(c) 14

0 22 0

(d)

x

y

0 2 31

2

1

3

4

5

6

4 5 62

2

3

5

1

4

6

4 3 16 5A2

B2

C2

Chapter 8 ProbabilityExercise 8.1

1 (a) 14

(b) 512

(c) 23

2 (a) (i) 211

(ii) 0

(b) I

3 (a) 12

(b) 12

(c) 1

4 (a) 10

(b) (i) 12

(ii) 14

(iii) 310

5 (a) 730

(b) 430

(c) 13

39

Answer Key


Exercise 8.2

1 (a) certain (b) certain

(c) impossible (d) equally likely

2 (a) P(rainy day) = 0.64

(b) P(red ball) = 49

(c) P(boy) = 50%

3 2, 4, 1, 3

Exercise 8.3

1 (a) Not getting a 3 when a fair die is rolled.

(b) There is at least one boy in the class.

(c) There are no girls in the room.

(d) Get at least one tail when a fair coin is flipped twice.

2 (a) 56

(b) Yes

3 45

4 0.52

Exercise 8.4

1 (a) 23

(b) 80

2 (a) 411

(b) 104

3 (a) 15

(b) 10

4 920

5 (a) 16 (b)

9100

(c) It is normal to have variation between theoretical and experimental probability.

(d) Experimental probability will be close to theoretical probability.

Exercise 8.5

1 (a)

H

T

HT

H

T

121

2

12

12

12

H

T

12

12

12

HT

12

12

HT

12

12

HT

12

12

(b) (i) 3 (ii) 38

2 (a)

Red

Green

Red

Green

Red

Green

712

611

512

511

711

411

(b) (i) 2833

(ii) 1522

(iii) 722

(iv) 3566

40



3 (a) 13

(b) (i)

Hit

Miss

Hit

Miss

Hit

Miss

23

23

13

13

23

13

(ii) 49

(iii) 89

(c) (i) 227

(ii) 2 13

n

4 (a) ℰ

M

1 7

14

8

S

(b) (i) 130

(ii) 310

(iii) 715

(c) 89

5 (a) ℰ

F

12 15

4

9

B

(b) (i) 36 (ii) 12

(iii) 9

(c) (i) 940

(ii) 310

(iii) 38

6 (a) p = 16, q = 15, r = 7, s = 10

(b) (i) 56 (ii) 56

(iii) 10

(c) (i) 855

(ii) 43110

(iii) 83110

(iv) 411

7 (a) 34

(b) 14

(c) (i) 49400

(ii) 780

(iii) 1225

8 (a) (i) 58

(ii) 18

(iii) 38

(b) (i) 964

(ii) 116

(iii) 1116

(iv) 18

9 (a)

Rain0.4

0.35

0.65

0.54

0.46

0.6 norain

Rain

norain

Rain

norain

(b) (i) 0.14 (ii) 0.464

(iii) 0.584 (iv) 0.724

41

Answer Key


10 Dice 2

Dice 1

1 2 3 4 5 6

1 1,1( ) 1,2( ) 1,3( ) 1,4( ) 1,5( ) 1,6( )

2 2,1( ) 2,2( ) 2,3( ) 2,4( ) 2,5( ) 2,6( )

3 3,1( ) 3,2( ) 3,3( ) 3,4( ) 3,5( ) 3,6( )

4 4,1( ) 4,2( ) 4,3( ) 4,4( ) 4,5( ) 4,6( )

5 5,1( ) 5,2( ) 5,3( ) 5,4( ) 5,5( ) 5,6( )

6 6,1( ) 6,2( ) 6,3( ) 6,4( ) 6,5( ) 6,6( )

(b) 1

12 (c) 1

9

11 (a) 13

(b) 1936

(c) 536

Chapter 9 StatisticsExercise 9.1

1 (a) qualitative (b) quantitative

(c) qualitative (d) qualitative

2 (a) discrete (b) continuous

(c) continuous (d) discrete

3 (a) discrete

(b) Number of movies Tally Frequency

0 1

1 3

2 3

3 3

4 3

5 2

(c) 53.3%

4 (a) continuous

(b) Height Tally Frequency

140 – 150 1

150 – 160 3

160 – 170 5

170 – 180 3

180 – 190 3

Exercise 9.2

1

50 10 2015 3025

Freq

uenc

y de

nsity

Length of telephone call (minutes)

2 (a) Height (h mm) Frequency

100 < h ≤ 105 5

105 < h ≤ 110 7

110 < h ≤ 120 10

120 < h ≤ 135 42

135 < h ≤ 140 20

(b) 84

3 (a) 14 (b) 84

42



4 (a) 12 (b) 16

5 (a)

1 2 43 65

Freq

uenc

y de

nsity

Distance (km)

0

15

30

45

60

75

90

7

(b) Distance (d km) Frequency

0 < d ≤ 1 15

1 < d ≤ 2 60

2 < d ≤ 3 90

3 < d ≤ 4 75

4 < d ≤ 6 60

6 (a)

120 130 150140 170160

Freq

uenc

y de

nsity

Height (cm)

0 180

(b) Height (h cm) Frequency

120 < h ≤ 130 5

130 < h ≤ 140 10

140 < h ≤ 150 20

150 < h ≤ 160 30

160 < h ≤ 170 5

(c) 147.9 cm

7 33

Exercise 9.3

1 (a) mean = 5.7, median = 5.5, mode = 5.5

(b) mean = 42.4, median = 45, mode = 45

(c) mean = 18.375, median = 19.5, mode = 20

2 mean = 27.75, median = 25, mode = 25

3 (a) 42 (b) 40.3

4 90

5 13

6 17.2

7 3, 5, 12, 15, 15

Exercise 9.4

1 mean = 6, mode = 7

2 mean = 23, mode = 24

3 (a) 4 < x ≤ 6 (b) 5.39

4 (a) 3030

(b) We do not know the exact income of each staff member.

(c) 3000 < x ≤ 4000

5 (a) 52 mm (b) 0.071

43

Answer Key


Exercise 9.5

1 (a) 1.5 < x ≤ 2

(b) Length (x cm) Cumulative frequency

x ≤ 0.5 3

x ≤ 1 8

x ≤ 1.5 18

x ≤ 2 33

x ≤ 2.5 45

x ≤ 3 50

(c)

0.5 1 1.5 2 2.5 30

5

10

15

20

25

30

35

40

45

50

Length (cm)

Cum

ulat

ive

freq

uenc

y

(d) 1.8

2 (a) 30 < x ≤ 35

(b) Age Cumulative frequency

x ≤ 20 1

x ≤ 25 4

x ≤ 30 12

x ≤ 35 24

x ≤ 40 30

(c)

15 20 25 30 35 400

5

10

15

20

25

30

Age

Cum

ulat

ive

freq

uenc

y

(d) 32

3 (a) 6.8

(b) Time (hour) Cumulative frequency

t ≤ 2 1

t ≤ 4 3

t ≤ 6 11

t ≤ 8 21

t ≤ 10 27

t ≤ 12 30

(c)

2 4 6 8 10 120

5

10

15

20

25

30

Time (hour)

Cum

ulat

ive

freq

uenc

y

(d) 6.8

4 (a) 43.1

44



(b) Speed (v miles/hour)

Cumulative Frequency

v ≤ 30 3

v ≤ 40 11

v ≤ 50 21

v ≤ 60 28

v ≤ 70 30

(c)

20 30 40 50 60 700

5

10

15

20

25

Speed (miles/hour)

Cu

mu

lati

ve fr

equ

ency

(d) 48 miles/hour

5 (a) 40 (b) 20

(c) 5 (d) 1.7 kg

6 (a) 15% (b) 10%

(c) 3 hours

7 (a) 95 cm (b) 98 cm

(c) 92 cm (d) 6 cm

8 (a) 48 cm (b) 5 mins

(c) 10

9 (a) 100 (b) 96 minutes

(c) 18 minutes (d) 88

Exercise 9.6

1 (a) y

x

Moderate positive correlation

(b) y

x

Strong positive correlation

(c) y

x

Zero correlation

(d) y

x

Strong negative correlation

(e) y

x

Weak negative correlation

45

Answer Key


(f) y

x

Strong negative correlation

2 (a) positive (b) positive

(c) positive (d) negative

(e) negative

3 (a)

30 35 40 45 50 55 60 65 700

20

22

24

26

28

30

32

Number of cold drinks sold

Tem

pera

ture

(C°)

(b) Positive correlation

(c)

30 35 40 45 50 55 60 65 700

20

22

24

26

28

30

32

Number of cold drinks sold

Tem

pera

ture

(C°)

(d) 43

4 (a)

50 55 60 65 700

30

35

40

45

50

Age (x)

Num

ber o

f day

s (y)

(b) Moderate positive correlation

(c)

50 55 60 65 700

30

35

40

45

50

Age (x)

Num

ber o

f day

s (y)

(d) 41

5 (a)

50 60 70 80 90 100 110 120 130 140 1500

10

20

30

40

50

60

70

80

90

100

Weight (kg)

Scor

e

(b) No correlation

MARKSCHEME

CAMBRIDGE INTERNATIONAL MATHEMATICS

Practice Test

Paper 2 (Extended)

48



1 ℰA B

2 9 (a) = +−y x clet

( )put 3,1 into the equation,

= +== +

−

−

ccy x

1 34

4

1

1

2 (a) 2 1 (b) yx

x

let 0,4 0

44,0( )

=+ =

=−

1

10

( )

−−

=

− − =− + =

=

x x

x xx x

x

21

31

3 2 1 63 2 2 6

4

111

(b) 2 1 11 xxx

xx

2 39 45

52.2 (correct to 2 sig. fig)

2 2 2

2

2

+ == −=

==

1

1

1

3 8.44 1

4 S − πr2 = 2πrh

ππ

−=

S rr

h2

2

1

1

12

x

R

y

0 2 3

2

1

1

4

5

6

7

3

4 5

x = 2

6 7 2

2

11

y = 2x + 3

2y x 2 = 0

5

5( )−

=nn

180 2 150

− ===

n nnn

180 360 15030 360

12

2

1

6 3000 × 1.52 = 4560 2

7 ===

= =

xxx

x

let 0.12100 12.12

99 121299

433

1

113 x y

x y

x y

x y

y

y

3 6 2 .......... 12 5 13 ....... 21 3:

6 12 4 ...... 32 3:6 15 39 ...... 44 3 :

3 35353

( )( )

( )( )

( )( )

( ) ( )

− =+ =

×− =×+ =×

=

=

−

−

−

1

1

8 (a) ( )( )( )

− += − −

x xx x

2 3 22 2 1

2 1

1

(b)

( )( )( )

−

= −= − +

x

xx x

4 1

2 12 1 2 1

2

2 2 1

1

49

Practice Test Paper 2 (Extended) Markscheme


∴

3x 6 353

= 2

3x = 2+ 70

x = 683

1

18 (a) Reflection about the line y = x. 2

(b) 0 11 0 2

19 (a) Total distance = area under the graph

=×

+ × +×

= + +=

10 22

15 2 5 22

10 30 545 km

1

1

14 (a) 34

3

= V120

V = 4058

1

1

(b) V r h

h

h

13

4058

13

3

3 4058 3

5.37 cm

2

2

2

π

π

π ( )

( )

( )

=

=

= =

1

1

(b) The train is travelling at constant speed 2 km/min for 15 minutes.

15 × 60 = 900 seconds

1

1

(c) From the graph, the speed of the train at 8 minutes is 1.6 km/min.

= ×=

1.6 60 km/h96 km/h

1

115 (a) = +

= ° + °= °

BAD OAD OABangle angle angle 20 3050

1

1 20 (a) OP a34

= 2

(b) angle BCD + angle BAD = 180°(Supplementary angle of cyclic quad)angle BCD = 130°

If BC = CD,angle OCB = angle OCD = angle ODC

=°

= °130

265

1(b) position vector of Q OQ c1

3

= = 2

(c) = +

= −−

BQ BC CQ

a c23

1

1

21

B

A

C 5

16 ( )( )= + − °=

=

ACAC

AC

2 3 2 2 3 cos607

7

2 2 2

2

111

17 =

=

=

p ks

k

k

34

6

∴ =ps

6

when s = 9,

= =

=

p 69

632

1

1

1

MARKSCHEME

CAMBRIDGE INTERNATIONAL MATHEMATICS

Practice Test

Paper 4 (Extended)

52 © 2017 NTK Publishing Limited

1 (a) (i) sand 375 kg

stones 125 kg

1

1

(d)

0.7

0.8

0.4

0.6

0.2

0.3

3

(ii) 50 Litres 2

(iii) $100 2

(b) 10.7 cm 3 (e) 0.14 1

2 (a) 38.5 m 1 (f) 0.68 2

(b) 26.3 m 3 (g) 16 2

(c) 18.6 m 3 5 (a) (i)14

1

(d) 18 packets 4 (ii) 5 2

3 (a)−a

b a2 (b) 4 + 3x2 2

(b) (i) > −n 5 2 (c)94

3

(ii) − − − −4, 3, 2, 1 3 (d)− x43

2

(c) x = 9z2 y

or x = 3z

2

y

3 (e)43 3

6 (a) 12, 17, 35 3

(b) (i) 56 1

(d) (i) 2 (ii) 36 1

(ii) −0.732, 2.73 4 (iii) 48 2

4 (a) ℰP

22

0

0

1 1

3

3

T

S

4 (c) 56.8 3

7 (a) −2.5, 2.5, 3.33 2

(b)

0 210.5 3

23

1

456

42 1

23

1

456

x

y

4

(b) (i) 6 1

(ii) 2 1

(iii) 8 1

(c) [ ]25

or 0.4 2


53© 2017 NTK Publishing Limited

(c) x = 0.5, 2 (must be exact) 4 9 (a) 14 12 4

2

(d) (i)

0 210.5 3

23

1

456

42 1

23

1

456

x

y 2(b) 1

62 12 2

or

13

16

13

13

2

(c) det C = 0 [accept matrix is singular] 1

(d) m = 4

n = 3

2

2

(ii) slope = 3 accept 4 to 2 3 (e) −17,17 4

8 (a) (i)

(ii)

x

y

0 21 3

2

3

1

4

5

6

4 52 13

2

3

1

4

5

6

45

R′

S′

2

2

10 (a) AB, AC, AD, BC, BD, CD 2

(b) BD, BE, CD, CE, DE 2

(c) 1 1

(d) (i) 1 3

(ii) 45 1

(iii) [ ]166

or 0.0152 4

(b) Enlargement, scale factor = −12

,

centre ( )0,0 3

(c) 2c − 2a − 2b 2

(d) a + b 2

(e) 3

(f) parallelogram 1

Practice Test Paper 4 (Extended) Markscheme

Cambridge IGCSE Mathematics Study Guide Answer Key

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