Answers - The Robert Napier School

Answers 


Chapter 1 Answers 

1.1 Get Ready 

1 a Neither b Factor c Factor d Multiple 

e Neither f Factor 

2 a No b Yes c No d Yes e No 

Exercise 1A 

1 Yes, for example 2 3 5 is prime. 

2 4 

3 n 2, m 3, p 7 

4 a 2, 24 b 5, 10 c 2, 20 d 6, 18 

5 a 24 2 3 3, 60 2 2 3 5 

b 12 c 120 

6 a 72 2 3 3 2 , 120 2 3 3 5 

b 24 c 360 

7 a 18, 180 b 18, 216 c 12, 480 d 36, 720 

8 a 6 b 2520 

9 a 40 b 126 000 

10 a 11, 13, 17 and 19 are prime numbers between 10 and 20. 

b 23, 29, 31 and 37 are prime numbers between 20 and 40. 

c 37, 41, 43, 47, 53, 59, 61 and 67 are prime numbers 

between 34 and 68. 

11 Every 2 minutes 

12 3 boxes of burgers and 4 packets of buns 

13 No. If one of the prime numbers is 2 you will get an even 

number. 

1.2 Get Ready 

1 a 36 b 8 c 9 

Exercise 1B 

1 a 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 

b 1, 8, 27, 64, 125 

2 a i 64, 1, 49, 9 ii 64, 1, 8 

b i 4, 16 ii 125, 27 

c i 64, 81, 144 ii 125, 64 

d i 100, 81, 169, 64 ii 125, 64 

Exercise 1C 

1 a 9 b 49 c 64 

d 1000 e 121 

2 a 6 b 4 c 9 

d 1 e 8 

3 a 36 b 8 c 81 

d 1 e 144 

4 a 2 b 3 c 1 

d 4 e 10 

5 a 17 b 50 c 250 

d 14 e 1 f 8 

g 34 h 11 i 9 

j 50 k 10 l 6 

1.3 Get Ready 

1 a 18 b 16 c 10 

Exercise 1D 

1 a 25 b 13 c 6 d 4 

e 5 f 8 g 6 h 2 

i 1 j 5 k 14 l 52 

m 32 n 15 o 8 p 70 

2 a 49 b 25 c 243 d 123 

e 72 f 69 g 7 h 7 

3 a 25 b 4 c 8 d 31 

1.4 Get Ready 

a 20 000 b 100 c 49 

Exercise 1E 

1 a 31 b 6.4 c 17 d 1.5 

e 32 

2 a 39.36 b 32.65 c 5.76 d 155.125 

3 a 219.5 b 305.7 c 22.6 d 410.9 

4 a 5.17 b 5.34 c 3.16 d 1.67 

5 a 2.77 b 7.68 c 205 d 455 000 

6 a 0.917 b 1.08 c 8.67 d 15.8 

Exercise 1F 

1 a 0.25 b 1.6 c 0.156 25 d 2 3 8 

1.5 Get Ready 

1 32 

2 125 

3 729 

Exercise 1G 

1 a 6 12 b 4 5 c 7 6 

d 5 6 e 3 10 

2 a 100 000 b 125 c 64 

d 9 e 64 

3 a 5 b 3 c 5 

d 4 e 9 

4 a 3 4 b 5 9 c 2 6 

d 6 5 e 4 2 

5 a 9 b 16 c 16 

d 10 000 e 49 

6 a 3 b 5 c 2 

d 4 e 3 

Review exercise 

1 a 96 b 33 c 23 

2 a 4 b 4 c 2 d 4 

e 2 

3 a 2 b 2 c 16 

637 

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4 a 100 cans b 14 (with 4 cans spare) 

c 200 cans d 14 (with 8 cans spare) 

5 a 5 arrangements: 1 by 36, 2 by 18, 3 by 12, 4 by 9, 6 by 6 

b 3 arrangements: 1 by 18, 2 by 9, 3 by 6 

c 3 arrangements: 1 by 12, 2 by 6, 3 by 4 

6 64 is the next number which is both a square number and 

a cube number. 

7 a 5 760 000 b 5 760 000 c 160 

d 57 600 e 2304 

8 a 3 b 30 c 0 

9 a 3 2 b 4 c 2 12 d 5 2 

10 a 2 2 3 2 7 

b 2 3 3 3 7 

11 9, 15 

12 84 2 2 3 7 

168 2 2 6 3 2 7 2 

13 120 000 miles 

14 1.258 048 316 

15 3 

16 a False, e.g. 3 5 8 

b False, e.g. 4 9 13 

c False, e.g. 5 3 2 

d False, e.g. 2 3 6 

e True 

17 a 2 7 128 b 2020 


2.1 Get Ready 

1 4a 2 8c 3 2p 2 

Exercise 2A 

1 a 7x 4y b 10w 2z c 4p 5q 

d 3a b e 6c 2d f 2m 3n 

g 4e 7f h 2x 10y 2 i 2p 3q 5 

j 13 5b 4a 

2 5x 9 

3 5x 13y 

2.2 Get Ready 

1 4x 2y 12 2 6y 2x 4 3 8x 2y 6 

Exercise 2B 

1 a 1 b 5 c 16 d 15 

2 a 4 b 19 c 5 d 11 

e 17 f 11 

2.3 Get Ready 

1 4 11 2 7 7 3 6 6 

Exercise 2C 

1 a m 5 b 6p 2 c 20q 3 

2 a a 11 b n 4 c x 6 d y 9 

3 a 12p 6 b 12a 5 c 5b 9 d 18n 3 

4 a 20t 8 u 5 b 6x 6 y 7 c 7a 5 b 6 d 8c 2 d 9 

e 24m 6 n 5 

Exercise 2D 

1 a a 3 b b 4 c c 3 d d 

2 a 2q 2 b 3p 5 c 4x d 10y 7 

3 a 5a 2 b 4 b 5pq 3 c 4c 2 d 4 d 3x 5 

e 10m 2 n 

Exercise 2E 

1 a a 14 b b 15 c c 9 d d 16 

2 a 4p 6 b 81q 8 c 25x 8 d 

3 a 16x 12 y 8 b 49e 10 f 6 c 125p 15 q 3 d 

2.4 Get Ready 

1 a 18 2 27y 15 3 ___ 4a2 

b 6 

Exercise 2F 

12 m__ 

8 

____ 8x 9 

27y 6 

1 a 

1__ b 

1__ 

a 

b 2 c __ 1 

c 2 d ___ 1 

d 3 

2 a 

1__ 

e 6 b __ 1 

f 8 c x2 d y 

3 a 1 b 1 c _____ 1 

5p 2 q 4 

d ______ 1 

27c 9 d 3 e _____ 9r 4 

4p 6 q 2 

Exercise 2G 

1 a 3a 2 b 2c 1_ 

2 

c 

2 a 

__ 1 

a 2 b __ 1 

2c 

2.5 Get Ready 

c 

3_ 

___ 3e 

3 

d 10x 

f 

____ 1 

d ____ 1 

2x 9_ 

5 

y x 1_ 

2 

y 3_ 

2 

1 12, 14, 16 2 34, 39, 44, 3 11, 13, 15 

Exercise 2H 

1 a add 3 b 14, 17 c 29 

2 a add 6 b 20, 26 c 50 

3 a subtract 7 b 9, 16 c 44 

4 a add 1 to the difference of consecutive terms 

b 15, 21 c 55 

5 a add 2 to the difference of consecutive terms 

b 20, 30 c 90 

2.6 Get Ready 

1 a add 3 b 13, 16 c 28 

2 a add 3 b 11, 14 c 23 

3 a subtract 6 b 94, 88 c 70 

Exercise 2I 

2 

y 5_ 

2 

1 a i 2 ii 2 

b i 4 

ii 11 

c i 5 ii 19 

2 a 6n 5 b i 67 ii 295 

3 a 4n 3 b i 63 ii 403 

4 a 37 5n b i 63 ii 963 

5 7n 11 103 has no integer solution 

6 a 4n 3 b 4n 3 453 has no integer solution 

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1 a 5x 5y b 6m 10n 

2 a 240B 114A, B British stamp, A Australian stamp 

b 375B 212A 

3 a 3 b 8 c 7 d 30 e 62 

4 a y 3 b 3x 2 c z 8 d p 7 e 16a 7 

5 a a 3 b b 5 c 7p 3 d 8x 3 e 8a 

6 a subtract 3 b 87, 84 c 69 

7 a 5 b 8 

8 a 216 12n b i 60 ii 972 

9 The nth term in the sequence is 3n 2. If 3n 2 140, 

n 46. So 140 is the 46th term in the sequence. 

10 a 55 cans 

b Students’ proofs 

c 19 high (with 10 cans spare) 

11 a 5 1_ hours b 3.03 hours 3 hours 2 mins 

2 

c Naismith’s formula is for fit experienced walkers. 

12 a a 20 b 9b 8 c 27e 15 f 3 

13 Three consecutive even numbers are 2n, 2n 2, 2n 4. 

Their sum 2n 2n 2 2n 4 6n 6, which is 

always a multiple of 6. 

14 ____ 6x 2 y 

4y 3 ___ 3x2 

2 

. Squared numbers cannot be negative. 

2y 

15 a 

3p 2 ___ 

2y 

b 

1 ___ 

4q 6 

16 64 cubes 8 have 0 sides painted 

24 have 1 side painted 

24 have 2 sides painted 

8 have 3 sides painted 

c 

2y ___ 

x 2 

Sides of cube 0 1 2 3 

n by n by n (n 2) 3 6(n 2) 2 12(n 2) 8 


3.1 Get Ready 

1 8_ 9 2 __ 19 

8 3 9 2_ 

5 

Exercise 3A 

8 

1 a __ 

11 b 5_ 

9 c __ 11 

15 d 2_ 

5 

7 

2 a __ 

10 b __ 10 

21 c __ 29 

35 d 8_ 

9 

13 

e __ 

20 f __ 13 

18 g __ 7 

18 h 3_ 

4 

3 a 

1_ 

4 b __ 1 

12 c __ 19 

40 d 2_ 

9 

e 

1_ 

4 f __ 5 

12 g __ 1 

10 h __ 1 

18 

4 a 1 11 

40 b 1 __ 11 

20 c 1_ 

6 d __ 19 

20 

e 1 11 

15 f 1 5_ 

8 g 1 1_ 

5 h 1 __ 5 

18 

Exercise 3B 

1 a 8 1_ 

4 b 5 __ 3 

10 c 11 __ 5 

42 d 18 __ 3 

20 

2 7 1 miles 

22 

3 1 12 

35 lb 

Exercise 3C 

1 a 1 1_ 

4 b 2 3_ 

8 c 3_ 

4 d 3 1_ 

3 

2 a 

3_ 

4 b 1 __ 7 

12 c 2 __ 24 

35 d 3 4_ 

9 

3 3 3_ 

8 kg 

4 2 7_ 

8 pints 

3.2 Get Ready 

1 32 2 45 3 __ 19 

5 4 4 2_ 

3 

Exercise 3D 

1 a 

__ 3 

10 b __ 3 

20 c __ 6 

11 d 2_ 

9 

__ 4 

__ 

__ 

__ 

e 

2 a 

2_ 

3 b 3_ 

4 c 3 3_ 

5 d 15 

3 a 21 kg b 6 2_ 

3 m c 7 1_ 

1_ 

litres d 7 

2 2 pints 

4 21 

5 £32.65 

5 

6 a __ 

12 b 4_ 

5 c 4 1_ 

8 d 7 

21 f 9 

20 g 3 

10 h 15 

32 

e 3 f 4 3_ 

8 

7 14 5_ 

8 minutes 

8 20 5_ 

8 lb 

3.3 Get Ready 

1 

__ 6 

35 2 1_ 6 3 __ 24 

7 

Exercise 3E 

g 10 h 22 

1_ 

2 

5 

1 a __ 

12 b __ 3 

16 c 2 2_ 

3 d 1 1_ 

2 

e 

3_ 

4 f 1 1_ 

5 g 1 __ 11 

14 h 5_ 

6 

2 a 

1_ 

2 b 28 c 2 __ 1 

12 d 2 __ 4 

13 

e 2 f 3 2_ 

3 g 6 h 1 7_ 

8 

3 16 

4 36 

5 11 __ 2 

11 days 

3.4 Get Ready 

1 7_ 8 2 __ 4 

45 3 3 5_ 

8 

Exercise 3F 

1 a 

5_ 

6 b 1_ 

6 

2 880 m 3 

3 £5.60 

4 £78 

5 525 

6 

3_ 

1_ 

36 13 which is not a whole number 

8 2 

7 4 __ 1 

12 hours 

8 

1_ 

2 

7_ 

8 km 

9 

10 192 

11 £8.40 


1 a 

2 a 

2_ 

3 b 4_ 

5 c 2_ 

3 d 2_ 

3 e __ 4 

11 

__ 17 

5 b 5 1_ 

6 

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3 

Name 

Hourly rate 

Hourly rate at 

time and a half 

Aaron £8.50 £12.75 £17.00 

Chi £12.00 £18.00 £24.00 

Mahmood £14.40 £21.60 £28.80 

__ 13 

Hourly rate at 

double time 

4 a 

20 b 1 __ 7 

12 c 5 __ 7 

12 

5 a £374 b 6 hours 

1 

6 a __ 

12 b 3 __ 13 

30 c 1 __ 11 

12 

7 a 4 1_ 

6 m2 b 8 1_ 

3 m c 5_ 

6 m 

8 Yes, the part is 6 2 cm long. 

16 

9 720 000 m 2 

10 3 1 

12 miles 

11 a 8 7 inches 

16 

1_ 

b 114 12 a 8 

15 , b 1_ 

3 , c 1_ 

5 , d 4 

15 

13 64 

14 350 

15 a 

1_ 

4 b 1_ 

8 c 1_ 

4 d 3_ 

8 


4.1 Get Ready 

1 8.02, 8.09, 8.092, 8.2, 8.29, 8.9, 8.92 

2 a 

1_ 

3 b __ 7 

12 c 7_ 

8 

Exercise 4A 

13 

d 84 __ 

24 

1 0.8, 0.85, ___ 86 

100 , __ 9 

10 , 0.98 

2 a terminating b terminating c recurring 

d recurring e terminating f recurring 

3 Mitch is correct, as there is a factor of 3 in the denominator. 

4.2 Get Ready 

1 a 18.79 b 5.18 c 32.74 

Exercise 4B 

1 a 0.12 b 0.0012 c 0.04 d 0.0063 

2 a 2.536 b 1.263 c 0.043 38 d 2.52 

e 13.02 f 0.504 g 0.046 72 h 0.323 

3 15p 

4 £6.86 

5 a 60 b 14 c 640 d 65 

e 25 f 2040 g 0.05 h 0.092 

6 a 231 b 642 c 41.3 d 42.2 

7 £26.13 

8 5 

4.3 Get Ready 

1 a 0.5772 b 160.3 . c £1.41 

Exercise 4C 

1 a 6.4 b 5.7 c 16.9 

d 0.1 e 1.0 

2 a 5.67 b 8.06 c 0.13 

d 3.04 e 0.08 

3 a 6.446 b 0.079 c 5.079 

d 6.008 e 0.020 

4.4 Get Ready 

1 a 3 b 0 c 9 

2 e.g. 438, 48, 6798 

Exercise 4D 

1 a 3900 b 230 c 46 

d 6.5 e 5.1 f 0.43 

2 a 2500 b 39.0 c 4.90 

d 4.09 e 0.0110 

3 a 3000 b 40 c 3 

d 8 e 20 f 1 

4.5 Get Ready 

1 a 5000 b 20 c 7 

2 a 600 b 15 000 c 540 000 

3 a 60 b 300 c 0.025 

Exercise 4E 

1 a 4200 b 7000 c 6000 

d 200 000 e 80 000 

2 a 35 b 2 c 10 d 5 e 6 

3 a 8000, overestimate b 4, overestimate 

c 10, underestimate d 300, overestimate 

4 a 50, overestimate b 4, underestimate 

c 40, overestimate d 8, overestimate 

5 2400 

4.6 Get Ready 

1 a 0.3 b 0.05 c 0.001 

2 a 6 b 150 c 0.054 

3 a 6000 b 30 000 c 25 

Exercise 4F 

1 a 2.4 b 0.15 c 1 d 0.0018 

2 a 20 b 50 c 100 d 4 

e 0.2 

3 a 1.5, underestimate b 0.2, overestimate 

c 5, underestimate d 200, underestimate 

4 a 400, overestimate b 210, overestimate 

c 25, underestimate d 4000, underestimate 

5 0.25 

4.7 Get Ready 

1 a 60 b 600 c 6000 

2 a 30 b 3 c 0.3 

Exercise 4G 

1 a 1792 b 1792 c 17.92 d 0.017 92 

2 a 146.4 b 1.464 c 0.1464 d 0.014 64 

3 a 348 b 3480 c 34.8 d 348 

640 

CS4-Z01_EMHA_SB_GCSE_0839_ANS.indd 640 

8/18/12 12:54 PM


4 a 128.8 b 230 c 56 d 2 

5 a 0.026 b 340 c 0.034 d 100 

6 a 13 b 13 c 13 000 d 0.065 

4.8 Get Ready 

1 a terminating b recurring c recurring 

Exercise 4H 

1 

7_ 

9 2 __ 34 

2 

4 __ 

11 5 ___ 317 

323 

7 ___ 

990 8 ___ 347 

10 6 83 

99 11 2 __ 7 

66 

99 3 ___ 305 

333 

999 6 __ 1 

18 

495 9 __ 7 

30 

317 

12 7 900 

4.9 Get Ready 

1 a 6.1 b 7.0 c 6.5 d 6.5 

2 a 0.3 b 0.3 c 0.3 d 0.3 

Exercise 4I 

1 a 84.5, 83.5 b 84.05, 83.95 c 84.005, 83.995 

2 a 0.95, 0.85 b 0.905, 0.895 c 0.095, 0.085 

3 a 118.5 cm b 117.5 cm 

4 a 6450 g b 6350 g 

5 a 48.05 l b 47.95 l 

6 a 1.005 m b 0.995 m 

4.10 Get Ready 

1 10.1 

2 2.1 

3 a e and f b c and f c e and f d c and f 

Exercise 4J 

1 a 535 b 26 612.25 c 229 920.25 

2 a 6.88 b 10.823 575 c 24.35 

3 a 11 275 568.625 b 151 474.75 

4 a 11 b 1.3606… 

5 a 0.8 b 1.3137… 

6 a 31.3375 b 29.8275 c 30 (1 s.f.) 

7 2.05 10 17 (3 s.f.), 2.28 10 17 (3 s.f.) 

8 a 50 b 15.5% 


1 £153.90 

2 

2_ 

3 

3 0.47, 12 

25 , 3_ 

5 , 31 

50 

4 £42.96 

5 a 4780 b 107 c 3.23 10 15 

d 7000 e 57.0 

6 a 46 b 31 c 0.046 

d 20 e 4.1 

7 a 400 b 40 c 1 10 18 

d 0.005 e 3 

8 a 3.1 b 0.6 c 2.1 d 4.0 

77 

9 a ___ 

400 b __ 77 1_ 

19 

4 4 c _____ 770 000 

192 500 

4 

10 76 923 1p pieces 

11 a Students’ checks 

b 

A B C Best deal 

John £56.74 £62.27 £64.33 Avery Energy 

Vijay £192.37 £186.17 £280.42 Brawn Power 

12 a 3000 b 4000 c 24 

d 350 000 e 360 

13 a 5 b 5 c 80 

d 250 e 1 

14 37 minutes 

15 a 

3_ 

1 £3 b __ 30 

1.5 £20 c ___ 1500 

150 £10 

10 

d __ 

2.5 £4 e ___ 300 

50 £6 f ___ 2000 

10 £200 

16 a £21 b 32 

17 a 75, underestimate b 150, underestimate 

c 3.6, overestimate d 6000, overestimate 

e 45 000, overestimate 

18 Volume 3 8 9 216 m 3 . 

Number of people 216 

4 54 

19 £429.56 

20 

1_ 

3 

21 a 17.1 b 31.95 c 60.1425 d 24.5025 

22 a 200 b 645 c 1.8163… d 0.0638… 

23 lower bound for length is 199.5 cm, so rod may fit into slot 

of length 199.8 cm 

24 8.75 km/l 

25 LB of cylinder’s capacity 325 ml, so the cylinder always 

contains more than stated on the label. 


5.1 Get Ready 

1 a 135°, b 45°, c 135° 2 40° 3 48° 

Exercise 5A 

1 a 63° (corresponding angles) 

b 49° (corresponding angles) 

c 68° (angles in a triangle add up to 180°) 

2 p 113° (corresponding angles and angles on a straight line) 

q 67° (corresponding angles) 

r 113° (alternate angles or angles on a straight line) 

3 l 81° (vertically opposite angles) 

m 54° (alternate angles) 

n 45° (angles of a triangle add up to 180°) 

4 y 58° (alternate angles) 

z 58° (alternate angles and angles on a straight line) 

5 g 57° (isosceles triangle and alternate angles) 

h 180 2 57 66° (angles of a triangle add up to 180° 

and alternate angles) 

k 114° (angles of a triangle add up to 180° and angles on 

a straight line) 

6 a 50° ( 2 sets of alternate angles) 

7 a a and p, b and q, c and s, d and r 

b a and r, b and s, c and q, d and p 

c a and b, b and d, d and c, c and a, p and q, q and r, 

r and s, s and p, a and q, a and s, b and p, b and r, 

c and p, c and r, d and q, d and s 

The angles are on a straight line, or correspond to 

angles on a straight line. 

8 Angle BAC Angle DCE 56°, so they are corresponding 

angles 

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7/30/12 5:18:59 PM


5.2 Get Ready 

1 a 50°, b 80° 2 c 29°, d 29° 

3 e 60° 

Exercise 5B 

1 angles in a triangle, angles on a straight line 

2 same angle, same angle, angles in a triangle, angles in a 

triangle 

3 a b d (alternate angles) 

a c (corresponding angles) 

so a b c d 

b The exterior angle is equal to the sum of the opposite 

two interior angles. 

5.3 Get Ready 

j 143° 

Exercise 5C 

1 141° (angle sum of quadrilateral / equilateral triangle) 

2 126° (angles on a straight line / exterior angle of a triangle) 

3 a a 132° (symmetry), b 37° (angle sum of 

quadrilateral) 

b 66°, 114°, 114° (symmetry / angle sum of quadrilateral) 

4 113° (vertically opposite angles / angles on a straight line / 

angles at a point / angle sum of a quadrilateral) 

5.4 Get Ready 

a alternate angles b 90° 

Exercise 5D 

1 alternate angles 

2 

A 

30° 

5.5 Get Ready 

1 a East b South c North-west 

Exercise 5E 

30° 

1 a 073° b 225° c 243° 

2 a 070° b 218° c 102° 

3 249° 

4 312° 

5 a 111° b 239° 

5.6 Get Ready 

a 43° (alternate angles) 

b 72° (opposite angles) 

c 65° (angles sum of a triangle) 

d 64° (angles on a straight line) 

e 58° (angle sum of an isosceles triangle) 

f 54° (opposite angles of a parallelogram) 

g 126° (angles at the end of a parallelogram) 

Exercise 5F 

1 a 124° (isosceles triangle / angle sum of a triangle / 

vertically opposite angles) 

b 56° (angles on a straight line / corresponding angles) 

2 L 70°, M 55°, N 55° (alternate angles / angles on a 

straight line) 

3 a p 57° (exterior angle of a triangle / angle sum of a 


b q 117° (exterior angle of a triangle / corresponding 

angles) 

4 a alternate angles 

b b d (alternate angles) 

a c, part a 

so a b c d 

c Opposite angles in a parallelogram are equal. 

5 a b c d 360° (angles in a quadrilateral) 

a c 180° (given in question) 

so b d 180° 

5.7 Get Ready 

1 Equilateral triangle 

2 A square is a quadrilateral with equal sides and equal 

angles. 

Exercise 5G 

1 a 

Polygon 

Number 

of sides 

(n) 

Number of 

diagonals 

from one 

vertex 

Number 

of 

triangles 

formed 

Sum of 

interior 

angles 

Triangle 3 0 1 180° 

Quadrilateral 4 1 2 360° 

Pentagon 5 2 3 540° 

Hexagon 6 3 4 720° 

Heptagon 7 4 5 900° 

Octagon 8 5 6 1080° 

Nonagon 9 6 7 1260° 

Decagon 10 7 8 1440° 

b i n 3 ii n 2 iii (n 2) 180° 

2 Angles not equal 

Exercise 5H 

1 a 15 b 18 c 160° 

2 a 120° b 144° c 168° 

3 a 142° b 103° 

4 360 is not divisible by 25 

5 a a 32°, b 30°, c 42°, d 63°, e 44°, f 27°, 

g 59°, h 63° 

b 360° 

6 a 135° (angle sum of isosceles triangle / angles on a 

straight line) 

b i The interior angles are all the same. 

ii Not all the sides are the same length. 

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7/30/12 5:19:03 PM


Exercise 5I 

1 a i 110° ii 143° 

b i 50° ii 36° 

2 a 72° b 45° c 30° d 14.4° 

3 a 24 b 3960° 

4 144°, 98°, 129°, 128°, 107° and 114° 

5 Exterior angle 12°, interior angle 168°, angle BCA 6° 

(angle sum of isosceles triangle) 

6 Angle BCD 180 e (angles on a straight line) 

So angle BCO (180 e) 2 (by symmetry) 

And angle CBO (180 e) 2 (isosceles triangle) 

So c 180 {(180 e) 2} {(180 e) 2} (angles in 

a triangle) giving c e 


1 88 96 180, so the lines are not parallel. Ben is right. 

2 Angles on a straight line add up to 180°, 

but 120 50 180. 

3 a x 30° 

b vertically opposite angles 

c Angles around a point add up to 360°, 

but 125 135 125 360. 

4 a i x ________ 180 54 

63° 

2 

ii angle sum of an isosceles triangle 

b y 54 63 117° (exterior angle of a triangle) 

or 

y 180 63 117° (angles on a straight line) 

5 

N 

N 

50° 

60° 

Q 

P 

6 a y 58° 

b alternate angles 

7 x 130° (angles on a straight line) 

y 50° (alternate angles) 

8 x 180 (360 50 119 105) 94° (angle sum of a 

quadrilateral / angles on a straight line) 

9 a x 180 2 52 76° 

b angle sum of an isosceles triangle 

10 angle ABQ 90° (angle in a square) 

angle ABC 180 ___ 360 120° (exterior angle of a regular 

6 

polygon / angles on a straight line) 

x 360 90 120 150° (angles around a point) 

11 a angle BDA 180 127 53° (angles on a straight 

line) 

angle BAD 180 2 53 74° (angle sum of an 

isosceles triangle) 

angle DAC 90 74 16° 

b angle DCA 180 127 16 37° (angle sum of a 

triangle) 

12 angle in equilateral triangle 60° 

base angle in isosceles triangle ________ 180 57 

61.5° 

2 

(angle sum of a triangle) 

p 360 60 61.5 238.5° (angles around a point) 

13 a i w 25° (base angles of an isosceles triangle) 

ii x 180 2 25 130° (angle sum of a triangle) 

b angle SQR 180 130 50° (angles on a straight 

line) 

y ________ 180 50 

65° (angle sum of an isosceles 

2 

triangle) 

14 a 030° 

b N B 

A 

135° 

15 exterior angle ___ 360 

10 36° 

x 180 36 144° (angles on a straight line) 

16 a x _________ 180 120 

30° (angle sum of an isosceles 

2 

triangle) 

b angle ABD 180 30 150° (angles on a straight 

line) 

y 360 150 54 108 48° (angle sum of a 


17 angle ACB angle ABC x 20 (base angles of an 

isosceles triangle) 

angle BAC 2(x 20) 180° (angle sum of a triangle) 

angle BAC 140 2x 

18 angle EFD 360 2x 65 x 90 205 3x 

(angles around a point) 

angle FED angle EFD 205 3x 

(base angles of an isosceles triangle) 

19 a interior angle 180 exterior angle 

(angles on a straight line) 

interior angle 180 2_ interior angle 

3 

interior angle 108° 

b exterior angle 72° ___ 360 

n 

n 5 

20 Angle CAB 360 x 90 80 x 

(Angles at a point add up to 360°) 

190 2x angle ABC 

(Base angles in an isosceles triangle are equal) 

Angle BCD 190 2x 190 2x 

(exterior angle of a triangle sum of two interior opposite 

angles) 

380 4x 4(95° x) 

21 Angle PQR angle PRQ (180° 20°) 2 80° 

(base angles of an isosceles triangle are equal and Angles 

in a triangle add up to 180°) 

Angle PQY 80° 60° 20° 

Since angle PQY 20° and angle QPY 20°, triangle PQY 

is isosceles. 

Thus PY QY 

(sides opposite the equal angles in an isosceles triangle 

are equal) 

22 5 minutes and 27 seconds past 1 

643 


05/09/12 5:15 PM



6.1 Get Ready 

a Ask each classmate how much lunch money they get and 

calculate the mean of this data. 

b Use an internet air ticketing site or the Manchester airport 

site. 

c Search the internet for a government site that gives 

information on voting figures. 

Exercise 6A 

1 a secondary b secondary c primary 

2 a quantitative b qualitative c quantitative 

d quantitative 

3 a discrete b continuous c continuous 

d discrete 

4 a Drug A is effective at curing malaria OR Drug A is not 

effective at curing malaria. 

b Collect data on patients that have been treated with 

Drug A and those that haven’t. 

6.2 Get Ready 

If the answers are written down: 15 seconds 

If the answers are given orally (i.e. one student at a time): 

number of students in class 15 seconds. 

Exercise 6B 

1 For example, rolling a dice or using the random function on 

a calculator or from a random number table 

2 A fraction of the population is chosen at random. 

3 For example, assign the numbers 1 to 60 to the workers, 

then take the first eight different numbers under 61 that are 

generated by the calculator 

(21, 32, 54, 34, 26, 45, 35, 22) 

6.3 Get Ready 

15 

1 a __ 

25 3_ 

2 a 80 b 

5 b __ 10 

25 2_ 

5 

3_ 

8 

Exercise 6C 

1 23 boys, 27 girls 

2 Randomly select 15 employees with less than six months’ 

experience and 40 employees with more than six months’ 

experience. 

3 

Office Factory floor Managers 

workers workers 

Females 5 25 1 

Males 8 49 2 

They should be picked by simple random sampling. 

6.4 Get Ready 

1 a 4 b 6 

2 

Exercise 6D 

1 a Vehicle Frequency 

Car 

Bus 

HGV 

Bike 

Motorbike 

b Motorbike 

c Car 

2 Number of DVDs bought Frequency 

0–3 9 

4–7 11 

8–11 6 

12–15 4 

3 a Weight Frequency 

57 w 60 7 

60 w 63 3 

63 w 66 5 

66 w 69 5 

w 69 4 

b 57 w 60 c 60 w 63 

6.5 Get Ready 

Tallying 

Exercise 6E 

1 It is a biased question. 

2 A: open, B: closed, C: open, D: closed 

3 a No option for dissatisfied customers. 

New suitable question: 

What do you think of the new amusements? 

Very good Good Satisfactory Poor 

b Options overlap. 


How much money would you normally expect to pay for 

each amusement? 

£5–£7 £7.01–£8 more than £8 

c Not clear what the options mean. 


How often do you visit the park each year? 

0–2 times 3–5 times 6–8 times 

more than 8 times 

4 Do you like the new layout? Yes/No 

6.6 Get Ready 

1 a 15 girls b 6 students 

644 


05/09/12 5:15 PM


Exercise 6F 

1 a Plain Salt and Cheese and Total 

Vinegar Onion 

Males 7 7 14 28 

Females 5 6 12 23 

Total 12 13 26 51 

b 13 c 51 

2 a Orange 

Juice 

Grapefruit 

Juice 

Total 

Men 22 8 30 

Women 18 12 30 

Total 40 20 60 

b 40 

3 a Supervisors Office Shop floor Total 

staff workers 

Males 10 3 82 95 

Females 2 11 38 51 

Total 12 14 120 146 

b 51 c 146 

Exercise 6G 

1 Yes, only one area sampled 

2 A: Biased. Not everyone in the hospital’s area has a chance 

of being asked. 

B: Biased. Only people with phones have a chance of being 

asked and only in 10 towns, the sample is too small. 

C: Not biased. 

D: Biased. Only people already using the recycling facility 

are being asked. 

3 Students' own discussions 

6.8 Get Ready 

The internet, supermarkets, high street shops 

Exercise 6H 

1 a 41 000 tonnes b Cars c 2004 

d Cars 

2 a 6.3 days b May c May d May 

3 a 799 b Stays the same 

c More dairy d Numbers have decreased 


1 a It does not allow for sending no text messages. It does 

not include a time frame, e.g. per week. 

b It only includes people of one age. 

2 a The categories overlap. It does not include a time frame, 

e.g. per day. It does not allow for people who use their 

computer for more than 6 hours. 

b It only includes people of one age. 

3 a It does not allow for never visiting the cinema. It is hard 

to decide what the categories mean. It does not include 

a time frame, e.g. per month. 

b On average, how many times do you go to the 

cinema each month? 

0–1 times 2–3 times 4–5 times 


4 On average, how many emails do you send each week? 

0–5 6–10 11–15 16–20 

more than 20 

5 It only includes women. It does not include people who 

never go to the cinema. 

6 

Animal Tally Frequency 

7 

Lions 

Tigers 

Elephants 

Monkeys 

Giraffes 

Country Tally Frequency 

France 5 

Spain 7 

England 4 

Italy 4 

8 a The first question does not allow for people who 

never visit the park and it is hard to decide what each 

category means. 

The second question has overlapping categories. 

b On average, how often do you go to the County 

Park each month? 

Never 1–3 times 4–6 times 


How old are you? 

0–10 years 11–20 years Over 20 years 

9 a Biased, because it only includes those working on the 

night shift. 

b Not biased, because it uses a simple random 

sample. 

c Biased because the question starts with ‘Do you 

agree…’ 

10 6 girls 

11 12 boys 

12 a 5 students b 26 students 

13 19 students 

14 

London York Total 

Boys 23 14 37 

Girls 19 24 43 

Total 42 38 80 

645 


05/09/12 5:15 PM



7.1 Get Ready 

1 a 150 g b 3 cm c 5 m d 300 ml 

Exercise 7A 

1 a 600 cm b 21 cm c 510 cm 

d 84 cm e 5.9 cm f 48.3 cm 

g 300 000 cm h 6700 cm 

2 a 3000 kg b 8200 kg c 6 kg 

d 0.9 kg e 0.43 kg f 4.7 kg 

3 a 2 litres b 7 litres c 5.9 litres 

d 45 litres 

4 7 litres 

Exercise 7B 

1 8.8 pounds 

2 50 kg 

3 4 litres 

4 8 gallons 

5 17.5 pints 

6 360 cm 

7 60 miles 

8 96 km 

9 585p per gallon 

Exercise 7C 

1 a 80 ounces b 6 pounds 

2 180 inches 

3 a 132 pounds b 60 kg 

7.2 Get Ready 

1 60 minutes 

2 1_ 4 

3 0.6 

4 18 minutes 

5 5 hours 42 minutes 


Exercise 7D 

1 a 13 km/litre b 6 litres 

2 a 240° b 30 seconds 

3 a 60 litres/min b 18 minutes 30 seconds 

4 0.0625 litre 

7.3 Get Ready 

1 a 18 km/l b 6.5 km/l c 13.2 km/l 

Exercise 7E 

1 4.24 km/h 2 10 km/h 

3 1 hour 12 minutes 4 306 km/h 

5 The speed for the 100 m was 10.32 m/s. The speed for the 

200 m was 10.36 m/s. The 200 m race was won with the 

faster average speed. 

7.4 Get Ready 

1 a 8.3 b 13.3 c 26.0 d 12.9 

Exercise 7F 

1 2.5 g/cm 3 2 0.777 cm 3 3 1.96 g/cm 3 

4 The aluminium block has the greater mass by 155 kg. 


1 a 110 m, 70 g, 40 litres b 400 cm c 1.5 kg 

2 No, 1.5 km is 1500 m. 

3 80 km/hour 

4 432 miles/hour 

5 John’s speed: 20 km/hour. Kamala’s speed: 21 km/hour. 

Kamala had the greater average speed. 

6 320 seconds 

7 1.14 g/cm 3 

8 1930 g 


8.1 Get Ready 

1 Not necessarily, one could be an enlargement of the other. 

2 No 

3 No 

Exercise 8A 

1 QS is the hypotenuse of both triangles 

angle QPS angle SRQ 90° (given) 

PS QR (given) 

So the triangles are congruent (RHS) 

2 angle YZX angle WVX (alternate angles) 

angle ZYX angle VWX (alternate angles) 

YZ WV (given) 

So the triangles are congruent (AAS) 

And YX XW, so X is the mid-point of WY 

3 PQ PR (given) 

QS RT (given) 

angle PQS angle PRT (isosceles triangle) 

So triangle PQS is congruent to triangle PRT (SAS) 

Therefore PS PT and so triangle PST is isosceles 

4 Let the point where the line from L cuts the base at right 

angles be X. 

Now LM LN (given) 

LX is common to both triangles 

angle LXM angle LXN 90° (given) 

So triangles LMX and LNX are congruent (RHS) 

As the triangles are congruent then MX NX, so the line 

from L bisects the base. 

5 AD DB (as D is the midpoint of AB) 

angle ADE angle DBF (corresponding angles) 

angle DAE angle BDF (corresponding angles) 

So the triangles ADE and DBF are congruent (AAS) 

8.2 Get Ready 

When you fold it over, the star fits exactly on top of itself. 

When you rotate the star, it fits exactly on top of itself in four 

different positions. 

646 


8/3/12 11:00:30 AM


Exercise 8B 

1 a Yes b Yes 

b e.g. 

c No 

d No 

e Yes f No 

c e.g. 

2 a No rotational symmetry 

b Rotational symmetry of order 2 

c Rotational symmetry of order 3 

d Rotational symmetry of order 8 

e No rotational symmetry 

f Rotational symmetry of order 2 

3 a 

8.3 Get Ready 

1 a i An isosceles triangles has two sides the same length 

and two angles the same. 

ii An equilateral triangle has all three sides the same 

length and all three angles the same. 

b Yes 

2 A polygon with four sides. 

Exercise 8C 

1 a 

4 a 

Line of 

symmetry 

b Isosceles triangle 

b It is also an isosceles triangle. 

2 a No. It could be a rectangle, square, rhombus, 

parallelogram, trapezium or isosceles trapezium. 

b No. It could be a rectangle, rhombus or parallelogram. 

c No. It could be a rectangle or rhombus. 

d Yes. It is a rectangle. 

3 e.g. 

b 

4 e.g. 

Rotational symmetry of order 2 

8.4 Get Ready 

A and C 

5 a e.g. 

Exercise 8D 

1 a Similar b Not similar 

2 All corresponding angles are equal so the pentagons are 

similar 

(angle CDE angle HIJ 150°) 

647 


7/30/12 5:19:16 PM


Exercise 8E 

1 15 cm 

2 a 5.14 cm b 0.448 m 

3 15 cm 

8.5 Get Ready 

By inspection 

Exercise 8F 

1 a i AC and FE, BC and DF, AB and DE 

ii angle ABC angle EDF, angle BCA angle DFE, 

angle BAC angle DEF 

b i JK and GH, KL and HI, JL and GI 

ii angle KJL angle HGI, angle JKL angle GHI, 

angle KLJ angle HIG 

c i PN and MN, QN and ON, PQ and MO 

ii angle NPQ angle NMO, angle PQN angle NOM, 

angle PNQ angle MNO 

d i SR and WU, ST and WT, RT and UT 

ii angle STR angle UTW, angle RST angle TWU, 

angle SRT angle TUW 

2 a 5.525 cm b 7.225 cm 

3 a angle DEB angle ACB (given) 

angle DBE angle ABC (same angle) 

Therefore angle BDE angle BAC (angles in a triangle) 

So triangle ABC is similar to triangle DBE 

b 6.46 cm c 2.26 cm d 1.99 cm 

4 a angle BDE angle BAC (given) 

angle DBE angle ABC (same angle) 

Therefore angle DEB angle ACB (angles in a triangle) 

So triangle ABC is similar to triangle DBE 

b 6.36 cm c 10 cm 

5 a angle ABM angle CDM (alternate angles) 

angle BAM angle DCM (alternate angles) 

angle AMB angle CMD (vertically opposite angles) 

So triangle ABM is similar to triangle CDM 

b i 8 cm ii 12 cm 


1 a b 

2 A and C 

3 Y and Z 

4 a angle ADB angle ADC 90° (given) 

AB AC (sides of equilateral triangle) 

AD is common 

So triangle ADC is congruent to triangle ADB (RHS) 

b BD CD (corresponding sides) 

BD CD BC AB 

So BD 1_ 

2 BC 

5 AB BC (given) 

AD CD (given) 

BD is common 

So triangle ADB is congruent to triangle CDB (SSS) 

6 angle ACB angle CED (vertically opposite angles) 

angle CAB angle CED (alternate angles) 

angle ABC angle EDC (alternate angles) 

So triangle ABC is similar to triangle EDC 

a 12 cm b 9 cm 

7 12.5 cm 

8 a angle HJI angle GJF (same angle) 

angle IHJ angle FGJ (corresponding angles) 

angle HIJ angle GFJ (corresponding angles) 

So triangle HIJ is similar to triangle GFJ 

b 3.43 cm 

9 angle CAD angle BAE (same angle) 

angle ACD angle ABE (corresponding angles) 

angle ADC angle AEB (corresponding angles) 

So triangle ACD is similar to triangle ABE 

a 2 cm b 5.25 cm 


9.1 Get Ready 

1 a 10x b 12x 2 c 2x 2 

d x 4 e x 2 5x 6 

f x 2 3x 2 

Exercise 9A 

1 a 2x 6 b 3p 6 c 4m 4n 

d 15 3q e 4x 2y 6 f 10c 5 

g 4x² 8 h 3n² 6n 3 

2 a y² 2y b g 2 3g c 2x 2 10x 

d 4n n 2 e ab ac f 3s 2 4s 

g 6t² 3t h 4x 3 12x 2 

3 a 2m 6 b 6x 6 c m 2 5m 

d 8y 2 12y e 5p 10 f 3q 3q 2 

g 2s 2 6s h 12mn 3n 2 15n 

Exercise 9B 

1 a 8t 3 b 9p 6 c 11w 6 

d 7d 2 e 5a 3b f 5x 3y 5 

2 a y 20 b 9a 6 c 4x 15 

d q 2 3 e 5n f 11m² 2m 

3 a t 16 b x 19 c g² g 

d 13c 2 22c e 4s² 14s 2 f p² q 2 

4 a 3s 4 b 15m 6 c 5f 2 3f 

d n 2 4n e 2x x 2 xy f 2p 2 5p 

9.2 Get Ready 

1 a 2 b 5 c 4 d 3y 

Exercise 9C 

1 a 3(x 2) b 2(y 1) c 5(p 2q) 

d 7(2t 1) e 2(4s t) f 9(a 2b) 

g 5(3u v 2w) h t(x y) 

i c(a 1) j 3(2x 2 3x 1) 

648 


7/30/12 5:19:20 PM


k 2p(p 1) l q(q 1) m x(4x 3) 

n h(2 5h) o p(p 2 2) p s 2 (1 s) 

2 a 5x(y t) b 3a(d 2c) c 2p(3q 2h) 

d 4y(2x 1) e 2p(2q s 4t) 

f mn(1 k) g 2x(x 2) h 12s(s 2) 

i 2f 2 (3 f) j y 2 (y 2 1) k cd(3d 5c) 

l ab (a 2 b 2 ) m 2pr(4q 5s) n 7ab(2a b 3) 

o 5x 2 y(3 7y) p 3y(3y 1) 

Exercise 9D 

1 a (x 3)(x 5) b (x y)(x y) 

c p(p 1) d (2t s)(2t s 1) 

e (a 5) (a 7) f 2(2d 1)(d 1) 

2 a 2(y 2)(y 4) b 5(x 1)(3x 5) 

c 2(p 5)(4p 25) d 3(q 1)(2q 5) 

e 7(a b) (a b 2) f 2x(x 1) (2x 3) 

9.3 Get Ready 

1 24 cm 2 2 x (x 2) 

Exercise 9E 

1 a x² 7x 12 b x² 3x 2 

c x² 3x 10 d y² y 6 

e y² y 2 f x² 5x 6 

g a² 9a 20 h x² 4x 4 

i p² 8p 16 j k² 14k 49 

k a² 2ab b² l a² 2ab b² 

2 a 2x² 3x 1 b 3x² 2x 1 

c 2x² 11x 12 d 3y² 8y 3 

e 2p² 7p 3 f 6t² 7t 2 

g 6s² 19s 10 h 4x² 4x 15 

i 12y² 5y 2 j 6a² 7a 2 

k 9x² 12x 4 l 4k² 4k 1 

3 a x² 3xy 2y² b x² xy 2y² 

c x² xy 2y² d x² 3xy 2y² 

e 6p² 7pq 3q² f 6s² 7st 2t² 

g 4a² 12ab 9b² h 4a² 12ab 9b² 

9.4 Get Ready 

1 a 1 and 6, 6 and 1, 2 and 3, 3 and 2 

b 1 and 15, 1 and 15, 3 and 5, 3 and 5 

2 5 and 2 

3 3 and 5 

Exercise 9F 

1 a 3, 5 b 6, 4 c 6, 3 

d 4, 2 e 4, 2 f 3, 3 

2 a (x 3)(x 5) b (x 1)(x 7) c (x 4)(x 5) 

d (x 3) 2 e (x 1)(x 5) f (x 1)² 

g (x 3)(x 6) h (x 6)(x 3) i (x 4)(x 7) 

j (x 4)(x 3) k (x 4)(x 6) l (x 2)(x 2) 

m (x 9)(x 9) 

Exercise 9G 

1 a (x 6)(x 6) b (x 7)(x 7) 

c (y 12)(y 12) d (5 y)(5 y) 

e (w 50)(w 50) f (100 a)(100 a) 

g (x 1)(x 3) h y(18 y) 

i 4ab 

2 a 2800 b 50 c 0.75 d 20 000 

3 a (2x 7)(2x 7) b (3y 1)(3y 1) 

c (11t 20)(11t 20) d (q 1)(q 3) 

e 8t f 4(p q) 

g 4(5p q 2)(5p q 3) 

h 100st 

4 a 3(x 2)(x 2) b 5(y 5)(y 5) 

c 10(w 10)(w 10) d 4(p 4q)(p 4q) 

e 3(2a 3b)(2a 3b) f 8x 

Exercise 9H 

1 a (5x 1)(x 3) b (2x 1)(x 5) 

c (3x 1)(x 1) d (4x 1)(2x 1) 

e (3x 2)(2x 3) f (6x 1)(x 1) 

g (5x 2)(x 1) h (4x 1)(3x 2) 

i (4x 3)(2x 1) j (2x 3)(x 5) 

k (7x 2)(x 3) l (3x 2)(x 4) 

m (2y 1)(2y 5) n (6y 1)(y 2) 

o (3y 5)(2y 5) 

2 a 2(3x 4)(x 1) b 3(2y 1)(y 2) 

c 5(x 2)(x 1) 

3 a (x y)(x 2y) b (x y)(2x 5y) 

c (3x 2y)(2x 3y) 


1 5x 2 

2 a 5(m 2) b y(y 3) 

3 a (a b)(x y) b (a b)(c d) 

4 x 2 x 12 

5 a a 2 4a 4 b c 2 6c 9 

c d 2 2d 1 d x 2 2xy y² 

6 a x² 15x 50 b y² 18y 81 

c x² 2x 8 d x² x 6 

e t² 7t 6 f 2x² 11x 12 

g 6p² p 1 h 4c² d² 

i 16y² 8y 1 

7 a (t 5)(t 6) b (x 7)² c (p 5)(p 3) 

d (y 6)² e (x 4)(x 1) f (s 8)(s 8) 

8 a (x 1)(x 7) b 11 17 

9 2(Y 3), 26 red flowers 

10 a (x 20)(x 20) b (3t 2) (3t 2 

c (10 y)(10 y) d (5 2p)(5 2p) 

11 a 41 b 1.99 c 16 

12 8000 

13 Three consecutive numbers are n, n 1, n 2. 

(n 1)(n 2) n(n 1) (n 2 3n 2) (n 2 n) 

2n 2 2(n 1) 

14 6(x 2) 

15 a Each team plays 3 games at home against the other 

teams. 

So total number of games 4 3 12 

b 380 c a 2 a a(a 1) 

16 a (2x 1)(x 2) b (2w 1)(w 3) 

c (3a 2)(a 4) d (3z 2)(10z 1) 

e (8y 1)(y 3) f (3p q)(2p q) 

649 


7/30/12 5:19:25 PM


17 Let the top left number in the 2 by 2 square be n. 

n n 1 

n 6 n 7 

Difference of products from opposite corners 

(n 1)(n 6) n(n 7) (n 2 7n 6) (n 2 7n) 

6 



1 5 cm, 45 cm 2 

Exercise 10A 

1 a 40 cm 2 b 18 m 2 c 35 cm 2 

d 54 mm 2 e 30 cm 2 f 54 cm 2 

2 Area 15 cm 2 , area 25 cm 2 , base 6 cm, area 32 cm 2 , 

height 8 cm 

3 a 5 cm b 12 cm 

Exercise 10B 

1 a 56 cm 2 b 96 m 2 c 75 cm 2 d 98 cm 2 


a A lw 

b A l 2 

c A 1_ 

2 bh 

d A bh 

e A 1_ (a b)h 

2 

Exercise 10C 

1 a 28 m b 37 m 2 

2 60 

3 30 

4 6 cm 

5 a i 32 m 2 ii 14 m 2 

b 24 m c 13 d 9 

6 a 66 cm b 234 cm 2 

7 30 cm 2 

8 100 cm 2 

Exercise 10D 

1 a 22.0 cm b 40.5 mm c 17.6 cm 

d 126 cm e 68.8 m 

2 a 145 cm b 25.1 cm 

3 6.00 cm 

4 a 210 cm b 2.1 km c 2859 

5 a 7.54 cm 

b i 45.2 cm ii 2.51 cm 

6 a 408 cm b 14.6 cm 

7 71.7 cm 

Exercise 10E 

1 a 201 cm 2 b 507 cm 2 c 2550 mm 2 

d 297 cm 2 e 499 m 2 

2 a 452 cm 2 b 54.1 cm 2 c 0.709 m 2 

d 2680 mm 2 e 262 cm 2 

3 a 19.6 m 2 b 18.8 m 2 c £36.80 

4 a 10 800 m 2 b £2480 

5 70.7 cm 2 

6 13 cm 

7 193 cm 2 


1 a b c d 

Exercise 10F 

1 

2 Any six out of: 


1 

Segment 

Chord 

Diameter 

Radius 

Tangent 

Sector 

Arc 

650 


05/09/12 5:16 PM


3 a Cylinder b Cone 

c Triangular-based pyramid d Square-based pyramid 

4 a e.g. 

Exercise 10G 

1 a 

Plan 

3 cm 

Front elevation 

Side elevation 

3 cm 

b 

Plan 

5 cm 

b e.g. 

5 cm 

2 cm 

3 cm 

6 cm 



6 cm 

c 

Plan 


1 a 

4 cm 

5 cm 

5 cm 

3 cm 

d 

5 m 

5 m 


Plan 

3 m 

2 m 2 m 

3 m 


3 cm 

above 

side 

6 m 

3 cm 

2 m 

3 cm 

5 m 

5 m 

b 

4 cm 

cm 

2 cm 

cm 

front 

4 cm 

cm 

4 cm 

cm 

3 cm 

cm 

e 

2 m 


Plan 

4 cm 

6 cm 

6 m 


3 cm 

] 

3 cm 

cm 

2 cm 

cm 

above 

above 

6 cm 

cm 

front 

front 

side 

side 

5 cm 

4 cm 

6 cm 


3 cm 

3 cm 


5 cm 

651 


7/30/12 5:19:35 PM


f 

Plan 


5 cm 

1 a 28.3 cm 2 (3s.f.) b 19.6 cm 2 (3s.f.) 

c 314 cm 2 (3s.f.) 

Exercise 10J 

4 cm 

2 cm 


5 cm 


4 cm 

1 a 251 cm 3 b 13 600 000 mm 3 

c 236 cm 3 d 8930 cm 3 

2 a 360 cm 3 b 650 cm 3 

c 0.101 25 m 3 

3 1600 m 3 

4 114.4 cm 3 

5 31.92 mm 

6 3.56 m 3 3560 litres 

g 

Plan 

4 cm 


1 

5 cm 

8 cm 


5 cm 

8 cm 


2 a b 

c 


1 a 192 m 3 b 576 cm 3 

Exercise 10H 

1 a 396 cm 3 b 378 cm 3 c 204 cm 3 

2 400 cm 3 


1 a a 3 b 2a 3 c a 3 

Exercise 10I 

1 a 78 cm 3 b 2250 mm 3 c 0.498 75 m 3 

d 216 cm 3 

2 a 225 cm 3 b 10 500 cm 3 c 80.43 cm 3 

d 84 000 cm 3 

3 9 cm 

4 Volume 1_ 

2 (x 3x) 2x 2x 8x3 cm 3 

5 h 4.5y 

2 20 cm 3 

3 70 cm 2 

4 a b 

5 5 cm 

6 150 boxes 

652 


7/30/12 5:19:44 PM


7 a 5 cm b i 19 boxes ii 8 chocolates 

8 a 118.12 cm 2 b 83.52 cm 3 

9 a 

6 cm 

9 cm 

4.5 cm 

Plan 

6 cm 

9 cm 

4.5 cm 



b 189 cm 2 

10 Skirting board: 3 4 2 2 1, therefore cheapest 

3 £30.50 1 £18.75 1 £14.00 £124.25 

Coving: 20 m, therefore cheapest 

6 £27.50 1 £22.00 £187.00 

Total £311.25 

11 Area 28 m 2 , therefore ___ 550 £19.64 to spend per m2 

28 

Amy can afford Natural Twist with either underlay or 

Medium Blend with Cushion. 

12 Paving 20 £40 £800, grass 110 £15 £1650, 

total £2450 

13 2 2.55 16 m 32 roses, so cost £134.40 

14 Area 4 4 (4 1 0.5 1 1.8 1 2 ) 

7.958 m 2 

Cost 7.958 4.60 £36.61 

15 a Volume of cup ____________ 

10 4.52 

159 ml, so the cup 

4 

can hold 150 ml. 

b Volume of squash required 30 150 3 13 500 ml 

13.5 litres 

1 bottle makes 0.8 7 5.6 litres 

___ 13.5 

2.4, so 3 bottles are needed. 

5.6 

c £3.75 

16 a 700 cm 3 b 13.51 kg 

17 Volume __ 25 

2 (1 3) 10 500 m3 , 

so time 250 minutes 4 hours 10 minutes 

18 a 308.5 m 

b Add 45.8 m to the straights making each one 105.8 m or 

add 29.1 m to the diameter of the bends, making each 

diameter 89.1 m 

19 200 50 1 2 42.9 cm 2 

20 Volume of oil 60 2 180 2 036 752 cm 3 , 

so mass of oil 8754 kg 

Surface area of tank 2r 2 2rh 90 478 cm 2 , 

so mass of tank 253 kg 

Total mass 9007 kg 9 tonnes 

21 a 315 cm 3 b 0.6 g/cm 3 



1 a 1 2 3 4 5 5 7 8 8 9 10 12 b 3.5, 3.5, 4.5, 4.6, 6.2, 8.7, 12.5 

Exercise 11A 

1 2 

2 16 litres 

3 a 16 b 16 

4 227 


1 a 44 b 20.4 

Exercise 11B 

1 3 

2 a 136 b 134 

3 £1292 

4 a 10 b 20 c 24 


a Median 20 

Mode 20 

b Mean 21 to 2 s.f. 

Exercise 11C 

1 a Mean £47, mode £17, median £22 

b The median is best. The mean has been affected by one 

high value and the mode is the lowest value. 

2 One advantage from: Is the most popular measure. Can 

be used for further calculations. Uses all the data. One 

disadvantage from: Affected by extreme values. Actual 

value may not exist. 

3 a 28 b 38 

c 64 (to the nearest whole person) 

d The mean because it is the highest average. 


Number 1 2 3 4 5 6 

Frequency 1 4 4 3 3 1 

Exercise 11D 

1 a Frequency number of siblings: 0, 8, 18, 12, 12, 0, 12, 7 

Total f 30 

Total f x 69 

b 2 c 2 d 2.3 

2 a 104 b 104 c 104 

3 a 17 b 18 c 18.5 


Class interval 

Frequency 

1–4 11 

5–9 5 

10–14 7 

15–19 7 

653 


7/30/12 5:19:47 PM


Exercise 11E 

1 a 70.0 x 70.1 b 69.9 x 70.0 

2 a £281£320 b £321£360 

3 a 0.45 x 0.50 b 0.40 x 0.45 


a 60 b 0.8 c 0.000 55 

Exercise 11F 

1 42.5 2 181.0 seconds 3 54.23 seconds 


1 14 16 16 18 21 23 27 32 38 43 45 49 

2 36.0 kg 43.4 kg 43.5 kg 49.9 kg 56.2 kg 56.2 kg 

Exercise 11G 

1 Median 

2 a Q 1 50, Q 2 62, Q 3 70 b 20 c 42 

3 a Q 1 8, Q 2 13, Q 3 19 b 11 c 19 

4 a Q 1 32, Q 2 45, Q 3 52 b 20 c 47 


1 a 3.5 b 3.6 c 4 

2 a Mode £4, median £5, mean £9 

b The median, because the mode is close to the 

lowest value and the mean is affected by the 

single large amount of £38. 

3 a One advantage from: unaffected by extreme values; 

can be used with qualitative data. 

One disadvantage from: may be more than one mode; 

may not be a mode. 

b Advantage: not influenced by extreme values. 

Disadvantage: actual value may not exist. 

c One advantage from: can be used for further 

calculations; uses all the data. 

Disadvantage: affected by extreme values. 

4 a 86 b 3.6 

5 a The mode is 7, the number of rooms that occurs most 

frequently. Ali has given the maximum number of rooms. 

b 6.3 to 1 d.p. 

6 2.4 to 1 d.p. 

7 45 

8 7.7 

9 a 10.5–10.7 b 10.2–10.4 

10 a 

Class Interval Frequency 

(f) 

Class 

mid-point 

f x 

26 w 29 4 27.5 110 

29 w 32 7 30.5 213.5 

32 w 35 15 33.5 502.5 

35 w 38 12 36.5 438 

38 w 41 2 39.5 79 

Totals 40 1343 

b 33.6 kg to 3 s.f. 

11 84.8 mm to 3 s.f. 

12 19 minutes 

13 a 30 t 40 b 27.4 minutes 

14 13.0 minutes to 3 s.f. 

15 Year 11 – 26 pets (Year 9 – 76 pets, Year 10 – 72 pets) 

16 a Q 1 42 kg, Q 2 47 kg, Q 3 49 kg 

b 7 kg 

c 11 kg 

17 a 18 minutes b 15 minutes 

c Before: Q 1 14 minutes, Q 3 22 minutes 

After: Q 1 11 minutes, Q 3 19 minutes 

d Before: interquartile range 8 minutes 

After: interquartile range 8 minutes 

e Before the introduction of the traffic management 

scheme the mean time taken to travel to work was 

higher than after the scheme was introduced. The 

interquartile range stayed the same, so the spread of 

times was similar before and after the scheme started. 

18 Mean ______ 300 000 

£30 000. The owner could say that the 

10 

average salary is £30 000 and it is high enough. 

Mode/median £10 000. The workers could say that the 

average salary is £10 000 and too low. 

19 If 1 dog in 100 had three legs, then the mean number of 

legs ___ 399 

100 3.99 

The majority of dogs have four legs. 



Students' accurate drawings 

Exercise 12A 

1–5 Students' drawings 

6 The sum of the two shorter sides is less than the longest 

side. 


1 

2 

3 

A 

4 cm 

6 cm 

5 cm 5 cm 

B 

654 


7/30/12 5:19:52 PM


Exercise 12B 

1 

5 cm 4 cm 

3 a b c 

2 a 

B 

b 

A 

d 

e 

P 

P 

4 

A 

B 

4 cm 

3 

P 

5 

Q 

4 cm 


1 

3 cm 


1 

3 cm 

2,3 

60° 

any points 

on this circle 

Exercise 12C 

2 

1 a 

A 

b B 

C 

A 

B 

B 

C 

A 

3 

2 a R 

any points 

on this line 

Exercise 12D 

Q 

P 

1 3 cm 

A 

B 

b 

Q 

2 

3.5 cm 

P 

R 

P 

655 


7/30/12 5:20:14 PM


3 

4 

P 

5 cm 

1.5 cm 

Exercise 12F 

1 a 1 cm represents 2 km 

b i 6 km ii 9.5 km iii 4.2 km 

2 a 13.5 cm b 312 km 

3 b 14.4 km c 146° 

4 

Q 

R 


1 

3 cm 

any points 

inside this circle 

2 

A 

B 

any points 

this side of the 

perpendicular 

bisector of AB 

5 a 1 : 50 b 10 cm 

6 a 1 : 5 000 000 b 3.66 cm 


3 

any points 

inside this region 

1 By inspection 




5 a 250 mm b 324 m 

6 

Beeham 

Corting 

Exercise 12E 

1 

2 cm 

P 

2 

4 cm 

2.6 cm 

7 

Alston 

A 

Deetown 

3 

4 

D 

G 

3 cm 

H 

E 

F 

B 

C 

5 1.4 m 

8 Construction of angle of 30° at P 

0.8 m 

9 a 

25 cm 

C 

8 cm 

D 


1 a 50 km b 2.5 km 

2 a 400 000 cm b 30 000 cm 

b 

3 cm 

4.5 cm 

3 cm 3 cm 

3 cm 

656 


7/30/12 5:20:29 PM


10 a Students' drawings 

b 034° c 214° 

11 Construction of bisector of angle ABC 

12 A B 

C 

1.2 cm 

1 cm 

13 Construction of perpendicular bisector of a line 7 cm long 

14 Construction of perpendicular to the line ST from a point 

above the line M 

15 a Locus of centre of wheel 

16 

b 

Road 



1 a 4 2p 5q b 3 6z c 12m 2 36m 

Exercise 13A 

1 a 4 2 b 7 

3 c 2.5 4 d 1.8 

5 e 1.5 6 f 4 

7 g 6 8 h 0.5 

9 k 2.6 10 m 3.5 


1 a x 6 b b 4 c q 8 

Exercise 13B 

1 x 3 2 y 3 

3 x 2.25 4 y 2 

5 x 0.25 6 w 4 

7 z 0.25 8 x 1_ 

D 

__ 

9 

3 

9 x 1 10 y 13 


1 a x 8 b a 2 c b 2 

Exercise 13C 

1 a 2.5 2 b 2.5 

3 c 3 4 d 2 

5 e 1 6 f 9 

10 0.9 

7 x 15 8 x 2 

9 x 3 10 x 3_ 

5 0.6 


1 a x 2 b x 6_ 

7 c x 0.9 

Exercise 13D 

1 p 20 2 q 10 

3 m 30 4 x 24 

5 y 53 

8 6 x 33 

7 n 5 

42 8 t 44 

9 x 6.5 10 y 76 


1 a a 2 b b 1_ 

2 

Exercise 13E 

1 x 5 

2 31 

3 Jessica 80%, Mason 60%, Zach 70% 

4 x 8 

5 23 1_ 

4 hours 

3(x 6) 

6 a (x 4) _______ b 15 cm 

5 

7 149 units 


1 a 6 b 2 c 4 

Exercise 13F 

43 

c x 11 

2_ 

3 

1 formula 2 identity 

3 expression 4 equation 

5 formula 6 formula 

7 expression 8 equation 

9 formula 10 equation 

11 equation 12 identity 


1 t 4.5 

2 x 4.5 

3 a x 2.5 b y 2.5 

4 58° 

5 57 cm 

6 Uzma £18, Hajra £38, Mabintou £76 

7 A £8, B £12, C £4 

8 x __ 44 

3 

9 x 5.5 

10 y 10 

14 

2_ 

3 

657 


7/30/12 5:20:35 PM




1 a i 1_ 

5 

ii 0.2 b i 

1_ 

8 

c i 3_ 

5 

ii 0.6 

7 

d i 40 

ii 0.125 

ii 0.175 

Exercise 14F 

1 a 1.728 b 0.6561 c 1.0608 d 0.52 

2 £1102.50 

3 a 1.1136 b £66 816 

4 No. It is the same as an increase of 68%. 

5 8 years 

Exercise 14A 

1 a £180 b 6 kg c £1.62 d 4.96 kg 

e £6 f 45 g 2.52 km h £52.50 

2 12 

3 £10 

4 1058 

7 

5 __ 

22 


1 £75 2 £16 3 1_ 4 

Exercise 14B 

1 a 1.64 b 1.03 c 1.14 d 1.4 

e 1.134 f 1.125 g 1.15 h 1.0236 

2 a 1.4 b £21.56 

3 Helen £12 504, Tom £25 008, Sandeep £33 344 

4 £621 

5 a £144 b 70 kg c 2.784 m 

d £1370.20 e 128.52 cm 

Exercise 14C 

1 a 0.93 b 0.8 c 0.84 d 0.73 

e 0.944 f 0.975 g 0.9275 h 0.992 

2 a £255 b £34 c £1020 

3 77.9 kg 

4 £748 

5 a £5840 b £4672 


1 £12.75 2 £1300 3 

Exercise 14D 

1 a 50% b 25% c 40% d 20% 

e 25% f 25% g 60% h 12.5% 

2 90% 

3 45% 

4 a 20% b 60% c 11.25% d 8.75% 

Exercise 14E 

__ 9 

40 

1 a 50% b 60% c 12% d 8% 

2 15% 

3 40% 

4 Shop C – as percentage increases are 

A 5.19%, B 4.77%, C 5.50% 

5 20% profit 

6 2.7% 


1 2 5 2 0.3 3 1.25 


1 1.15 2 0.85 3 1.04 4 0.96 

Exercise 14G 

1 £24 000 2 £280 3 £620 

4 £180 5 421 000 6 £270 


1 a £240 b 5 kg c 10.5 kg d £10.50 

2 16.7% 

3 18 years 

4 A (A £510, B £512, C £517.50) 

5 CompuSystems (Able £23 000, Beta £23 400, 

CompuSystems £24,240, Digital £24 000) 

6 72% 

7 £7800 

8 a 62.5% b 

1_ 

4 

9 19.9% 

10 a It will be worth 32.8% of its original value. 

b 0.64 

11 £275 

12 £8400 

13 £1600 

14 £665 

15 B (A 9.2% over two years, B 9.2025% over two years) 



1 a 7 b 1 

2 a y 1 b y 1.5 

Exercise 15A 

1 a y 

5 

x 3 

4 

3 

2 

1 

5 4 3 2 1 O 1 2 3 4 5 

1 

2 

3 

4 

5 

x 

658 


7/30/12 5:20:38 PM


b 

y 

5 

e 

y 

5 

x 1.5 

4 

4 

3 

2 

y 2 

3 

2 

1 

1 

5 4 3 2 1 O 1 2 3 4 5 

1 

x 

5 4 3 2 1 O 1 2 3 4 5 

1 

x 

2 

2 

3 

3 

4 

4 

5 

5 

c 

y 

5 

4 

3 

2 

1 

x 0 

f 

y 

5 

4 

3 

2 

1 

5 4 3 2 1 O 1 2 3 4 5 

1 

2 

3 

4 

5 

x 

5 4 3 2 1 O 1 2 3 4 5 x 

1 

2 

3 

4 

y 5 

5 

d 

y 

5 

4 

3 

2 

1 

y 0 

5 4 3 2 1 O 1 2 3 4 5 

1 

x 

2 a x 4 b y 2 c y 5 d x 1_ 

2 

3 a (1, 3) b (4, 2) c ( 1_ 

2 , 3) 

4 22 units, 28 units squared 

Exercise 15B 

1 a 

b 

x 2 1 0 1 2 3 4 

y 8 6 4 2 0 2 4 

y 

8 

2 

6 

3 

4 

4 

2 

5 

2 O 2 4 

2 

x 

4 

y 4 2x 

659 


05/09/12 5:17 PM


2 a y 

8 

6 

4 

2 

y 4x 8 

Exercise 15C 

1 y 

6 

4 

O 

2 

2 4 

x 

2 

3x 4y 12 

4 

6 

8 

4 

2 

O 

2 4 

x 

10 

2 

12 

b i (2, 0) ii (0, 8) 

c i 2 ii 3.5 

3 a y 

8 

2 

6 

4 

2 

O 

2 

4 

6 

8 

y 2x 

y x 

2 

y 3x 

x 

y x 

2 a y 

10 

5 

10 5 O 

5 

10 

5 

10 

x 

x y 7 

x y 2 

x y 4 

b They are parallel to each other. 

3 y 

4 

2 

2x 3y 12 

b all pass through (0,0) 

4 a y 

40 

30 

20 

10 

4 3 2 1 O 

10 

20 

30 

40 

b (1.5, 0) 

5 y 

4 y x 

y 2x 5 

2 

y 0 

6 4 2 O 2 x 

2 

4 

6 

6.25 square units 

y 6x 9 

1 2 3 4 

x 

y 9 6x 

2 

(3, 2) 

2 

4 a y 

10 

O 2 4 

x y 1 

8 

6 

4 

2 

4 2 O 

2 

2 

2 

4 

x 

x 

5y 2x 8 

6 

2y 5x 20 

b They intersect at right angles. 

c e.g. 2x 3y 6 and 3x 2y 12 


a 5 b 5.5 c 17.25 d 0.225 

e 1 f 8.5 

660 


05/09/12 5:17 PM


Exercise 15D 

1 a (1, 2) b (3, 0.5) c (1.5, 3) 

d (2.5, 0.5) e (4, 0.5) f (0.5, 2) 

g (2.5, 0.5) h (0.5, 2.5) i (2, 3) 

j (4.5, 0.5) 

2 a (1.5, 0.5) b (4, 1) c (1.5, 1) 

d (3.5, 2.5) 

3 a (4, 4) b (2, 2.5) c (1.5, 3.5) 

d (3, 4.5) e (2, 2.5) f (2.5, 1.5) 


1 y 

y 3x 

5 

4 

3 

2 

1 

5 4 3 2 1 O 1 2 3 4 5 

1 

2 

3 

4 

5 

2 y 

5 

4 

3 

2 

1 

y 2x 

y x 

5 4 3 2 1 O 1 2 3 4 5 x 

1 

2 

1 

y 

2 

x 

3 

4 

5 

y 2x 

y x 

x 

3 

y 

y 3x 

5 

4 

3 

2 

1 

5 4 3 2 1 O 1 2 3 4 5 

1 

Exercise 15E 

2 

3 

4 

5 

y 3x 2 

y 3x 3 

__ 10 

1 a 30 b 10 c 

3 

d 2_ 

5 e 1_ 

5 f __ 1 

40 

2 a 2, 3 b 2, 4 

c 4, 6 

d 

3_ 

4 ,1 

3 a y 

b 

5 

4 

3 

2 

1 

5 4 3 2 1 O 1 2 3 4 5 

1 

2 

3 

4 

5 

y 

5 

4 

3 

2 

1 

x 

y x 2 

y 2x 3 

x 

5 4 3 2 1 O 1 2 3 4 5 

1 

2 

3 

4 

5 

x 

661 


7/30/12 5:20:48 PM


c 

d 

e 

y 

5 

4 

3 

2 

1 

5 4 3 2 1 O 1 2 3 4 5 

1 

2 

3 

4 

5 

y 

5 

4 

3 

2 

1 

1 

y 

2x 4 

5 4 3 2 1 O 1 2 3 4 5 

1 

2 

3 

4 

5 

y 

5 

4 

3 

2 

1 

y 3x 3 

1 

y 

3 

x 2 

x 

x 

b Gradient 1.8. This represents the number of degrees 

Fahrenheit for each degree Celsius. 

3 a A 15, B 10, C 4 

b The gradients represent speed. A is the car (the fastest 

vehicle), B is the lorry, C is the cycle (the slowest 

vehicle). 

4 a 

Depth (d cm) 

250 

200 

150 

100 

50 

0 

0 50 100 150 200 250 

Time (t seconds) 

b 5_ 

6 

c The depth of water is going down by 5_ cm per second. 

6 

The swimming pool is being emptied. 


1 2, 1_ 

2 , 1 

The gradient is the same as the coefficient of x. 

2 1, 2, 2 

The y-intercept is the same as the value of the number in 

the equation. 

Exercise 15G 

1 y 2x 5 

2 a i 4 ii 1 

b i 3 

ii 4 

c i 2_ 

3 ii 4 

d i 0.4 ii 4 

e i 1 1_ ii 4 

3 

f i 1_ 

2 ii 0 

3 A y 2x 8 B y 1_ 

3 x 2 C y 5 1_ 

D y 4 x E y 2x 6 

4 y 5x 2 

5 y 3x 10 

2 x 

4 e.g. (3, 2) 

5 a 1_ 

2 

Exercise 15F 

5 4 3 2 1 O 1 2 3 4 5 

1 

b (0, 4) 

2 

3 

4 

5 

1 a Gradient 40. This represents the extra time needed 

(in minutes) for each extra kilogram of chicken. 

b Cooking time 40 minutes per kilogram plus 20 minutes 

c You can’t have a negative weight of chicken. 

2 a 50°F 

x 


1 a 1_ 

2 , 2 b 1_ 

3 , 3 c 2_ 

3 , 2_ 

3 

In a and b the product of the gradients is 1. 

In c the gradients of the parallel lines are the same. 

Exercise 15H 

1 a 1_ 

c 5 d 

1_ 

3 e 6 

3 b 1_ 

4 

2 a y 2x c for any value of c except 5 

b y 1_ x c for any value of c except 1 

3 

c y c x for any value of c except 4 

3 a y c x for any value of c 

b y c 1_ x for any value of c 

3 

c y 2x c for any value of c 

4 y 4x 3 

5 2x y 0 

6 y 4x 

7 y x 3 

662 


7/30/12 5:20:55 PM


1 C 14:30, D 15:15, E 15:54 

Exercise 15I 

1 a 4 km b 75 minutes c 5 1_ 

3 km/h 

d 6 minutes each time 

e 10 km/h 

2 a £2.75 b 2 kg c £12.20 

3 a 

b 


1 a y 

5 

4 

3 

2 

1 

x 4 


Volume 

Volume 

5 4 3 2 1 O 1 2 3 4 5 x 

1 

y 2 

2 

3 

4 

y x 

Depth 

Depth 

5 

c 

d 

b 2 square units 

Volume 

Volume 

2 

16 

14 

12 

4 A b, B a, C c 

5 

15 

Depth 

Depth 

$ 

10 

8 

6 

4 

2 

Petrol consumption (km/l) 

14.5 

14 

13.5 

13 

0 

0 2 4 6 8 10 

£ 

3 a i 3 ii 1_ 

2 iii 0 

b i 1_ 

2 

ii 3 

4 

C 0 20 40 60 80 100 

F 30 70 110 150 190 230 

12.5 

12 

60 70 80 90 100 110 

Speed (km/h) 

a 13.5 km/l b 74 km/h, 97 km/h 

F 

250 

200 

150 

F 2C 30 

100 

50 

0 

0 10 20 30 40 50 60 70 80 90 100 C 

663 


7/30/12 5:21:05 PM


5 a y 

10 

8 

6 

4 

2x y 6 

2 

1 

O 

2 4 6 

x 

2 

4 

b i 2 ii 6 

6 a 120 km b 0.5 hour c 80 km/h 

d 18:52 e 48 km/h 

7 B (A gradient 2, B gradient 4, C gradient 2.5) 

8 a d b d 

V 

c d d d 

V 

9 a No b y 1_ 

2 x 5 c y 2x 9 

10 

y 

10 

V 

V 

13 

Cost (£) 

500 

400 

300 

200 

100 

Hermes 

Atlantis 

0 

0 10 20 30 40 50 60 70 80 

Number of sessions 

If Abbie plans to go to the health club more than 50 times 

a year, she should choose Atlantis. 

14 (3, 2) 

15 

Equation of line Gradient y-intercept 

y 2x 5 2 5 

y 7x 3 7 3 

y 6 x 1 6 

y 2_ 

3 x 1 2_ 

3 1 

y 3 4x 4 3 

16 A: The temperature stays constant. 

B: The temperature rises at a constant rate. 

C: The temperature rises at a constant rate and then falls 

at a faster constant rate. 

D: The temperature stays the same and then falls at a 

constant rate. 

E: The temperature rises at a constant rate, stays the 

same for a period of time and then continues to rise at 

the same constant rate. 

F: The temperature rises at a constant rate, stays the 

same for a period of time and then falls at the same 

rate at which it rose. 

17 a Perimeter 2 (2x y) 24, so 2x y 12 

b y 

12 

8 

10 

6 

4 

2 

1 O 1 

2 

y 7 2x 

2 3 4 

11 a 4 b ( 3_ , 0) c 4.5 

4 

12 a From the top on the right-hand side: B, A, C 

b £40 

c C would be cheapest. (A £32, B £30, C £25) 

x 

8 

2x y 12 

6 

4 

2 

0 

0 1 

2 3 4 5 6 

c x 3 

18 k 7. x 3 in y 3x 2 gives y 7, so (3, 7) also lies 

on y 3x 2 

19 y 5 2x 

20 a 1_ 

2 b y 2x 2 

x 

664 


7/30/12 5:21:15 PM




1 18 km 

Exercise 16A 

1 a 1 : 3 b 1 : 4 c 1 : 3.5 d 1 : 0.5 

e 1 : 0.3 f 1 : 0.6 g 1 : 8 h 1 : 8 

15 

2 1 : 5 

3 a 

4_ 

9 b 4 : 5 c 1 : 1.25 

4 1 : 375 

5 a 1 : 1_ 

6 

b 1 : 0.2 c 1 : 0.02 d 1 : 40 


1 a 2 : 3 b 1 : 5 c 8 : 7 d 9 : 200 

Exercise 16B 

1 a 10 g b 30 g c 250 g 

2 a 10 kg b 15 kg 

3 108.5 km 

4 £1080 

5 1.53 m 

6 250 


1 20.65 2 6 3 241.5 

Exercise 16C 

1 a £4.26 : £10.65 b 360 g : 240 g 

c £14.21 : £56.84 : £99.47 d 6.3 m : 12.6 m : 15.75 m 

2 60°, 50°, 70° 

3 £36.50 

7 

4 __ 

16 

5 18 

6 £52 


1 15 2 50p 3 80 mins 4 2.75 

Exercise 16D 

1 a 3 hours b 7 hours 

2 £64.05 3 £33.75 4 £4.50 

5 £3.24 6 70 cm 

Exercise 16E 

1 a 160 g b 150 g 


3 a $320 b £340 

4 £120 

5 America by $10.25 

6 $58.50 


1 a 52 b 16 c 60 hours 

Exercise 16F 

1 a 4 days b 5 days 

2 2 hours 3 9 hours 4 9 days 

5 192 cm 6 8 


1 3 : 2 

2 21 : 4 

3 5 : 2 

4 a 0.5 m b 16 m 

5 45 litres 

6 10 

7 110 

8 Local professional (local $652.50, USA $684) 

9 Small bottle (large 0.25 p/g, small 0.22 p/g) 

10 2.33 pm 

11 1.5 km 2 

12 a B b D 

c A and B, as they lie on the same line through the origin 

d D 

Multiplication 

1 £249.50 

2 Yes, her Nan will have to give her £35.80. 

3 £48.46 

Area 

1 e.g. 

2 £210 

3 £248 

4 e.g. 1 cm by 36 cm, perimeter 74 cm; 2 cm by 18 cm, 

perimeter 40 cm; 3 cm by 12 cm, perimeter 30 cm; 4 cm by 

9 cm, perimeter 26 cm; 6 cm by 6 cm, perimeter 24 cm 

5 £8 

Averages and range 

1 a Add 2 b the same c double 

d double e 1 2 3 4 5 8 19 

2 72 

3 The new mean is two larger. 

4 

English 

50 

Maths 

45 

Cumulative frequency 

40 

35 

30 

25 

20 

15 

10 

5 

0 

0 10 20 30 40 50 60 70 80 90 100 

Marks 

665 


7/30/12 5:21:19 PM


20 30 40 50 60 70 80 90 100 

Maths 

20 30 40 50 60 

English 

70 80 90 100 

The median for Maths (62) is larger than the Median for 

English (60). 

The Maths marks are more varied as The IQR is 20 

compared with the English IQR of 15. 

5 a May’s average hours of sunshine are higher than June 

so May had more hours of sunshine than June 

The number of hours of sunshine were more variable in 

June as the range was bigger. 

b i The mean and range take into account all of the data. 

ii The median and IQR are not affected by extreme 

values. 

Price comparisons 

1 Ahmed should use Cogas if he uses fewer than 3800 units 

per annum, otherwise he should use Ourgas. 

2 a Cable: C 30 5m; Broadband: C 6.5m 

b Broadband is cheaper for up to 20 months. 

3 a ACars: C 60 0.32x; BMotors: C 50 0.4x 

b ACars is cheaper for more than 125 miles a day. 

4 Quick Delivery for parcels up to 1.5 kg and Parcels Fly for 

parcels over 1.5 kg 

5 Pete’s Mix if more than 3.33 m 3 , otherwise Concrete Sue 

Intepreting and displaying data 

1 


100 

90 

80 

70 

60 

50 

40 

30 

20 

10 

0 

0 10 20 30 40 50 60 70 80 

Weight (g) 

3 Take a random stratified sample. 

Stratified by school 9 from Avon, 10 from Moorside, 

14 Heaton, 12 Moortop, 15 Brambell 

Students must be chosen randomly (names in a hat) 

Or 

Take a systematic sample ___ 5720 students 95 

60 

Put numbers 1–95 in a hat and draw out a number. 

Go systematically through each school numbering all 

students and selecting the students with numbers divisible 

by the number drawn. 

Or 

Put all names in a hat and draw out 60 students – this 

sample may not contain students from every school. 

4 75 

Probability 

1 1_ . 

, 33% or 0. 3 

3 

2 When Mona has eaten one sweet there are 19 sweets left 

in the bag. There are still x chocolates. So the probability of 

Sam eating a chocolate is __ x 

19 . 

3 The probability of the first sock picked being x, depends on 

the total number of socks in the drawer (i.e. the number of 

grey socks the number of black socks) and the number 

of black socks at the start. When one black sock has been 

removed, the number of black socks and the total number of 

socks will both be lower by one. Therefore the probability of 

picking a black sock the second time will not be x. 

4 3_ , 0.375 or 37.5% 

8 

5 20%, 1_ or 0.2 

5 

6 0.1, __ 1 or 10% 

10 

7 0.625, 5_ or 62.5% 

8 

Trigonometry 

1 14 cm 

2 12.0 cm 

3 From corner of shed to top of roof at opposite end 

√ __________ 

6 2 2 2 7 2 9.43 ft, so the bean poles will fit in the 

shed. 

Climbing Snowdon 

1 The 9 am train is cheaper by £17. (£35 compared to £52) 

2 Rashid: time 2.75 hours, distance 7 km, 

speed 2.55 km/h 

Chelsea: time 3 hours, distance 9 km, 

speed 3 km/h 

Rashid is not the faster walker. 

11 pebbles 

2 10 workers 

666 


7/30/12 5:21:24 PM

3 a 

Metres 

1400 

1200 

1000 

800 

600 


3 Suitable questions for a data collection sheet, for example: 

Are you male or female? Male Female 

How old are you? 

0–14 15–29 30–44 45–59 60–74 

75–89 90 or over 

How much do you spend on music each month? 

£0–4.99 £5–9.99 £10–14.99 £15–19.99 

£20 or more 

What kind of music do you buy most often? 

Pop Classical Jazz Folk Other 

400 

200 

0 

0 1000 2000 3000 4000 

Feet 

b Height of Snowdon 3560 0.3048 1085 m 

Ben Nevis is higher. 

Music sales 

1 Sales in £ millions: 

2006 2007 2008 2009 

Downloads 69 115 169 160 

CDs 134 135 98 101 

Total 203 250 267 261 

The total sales between 2006 and 2008 increase each year 

and then decrease slightly in 2009. 

CD sales stay approximately the same from 2006 to 2007 

and then decrease, whereas download sales increase 

significantly each year before steadying out in 2009. 

2 Various answers are possible, for example, two frequency 

polygons. 

CDs (millions) 

Downloads (millions) 

60 

50 

40 

30 

20 

10 

0 

0 10 20 30 40 50 60 70 80 90 

Age (years) 

60 

50 

40 

30 

20 

10 

0 

0 10 20 30 40 50 60 70 80 90 

Age (years) 

Communication 

1 Students’ comparisons of costs for different numbers of 

months. For more than 18 months, ‘pay as you go’ is 

cheaper, otherwise monthly contract is cheaper. 

2 860.16 seconds 

3 13 cm by 19 cm (285 ppi by 284 ppi) 

15 cm by 23 cm (247 ppi by 235 ppi) 

20 cm by 30 cm (185 ppi by 180 ppi) 

Ranji should print the photo at 13 cm by 19 cm. 

Energy efficiency 

1 100 mm: Space Combi (Economy roll £200, Easy Roll £175, 

Space Combi £40) 

150 mm: Space Blanket (medium) (Economy roll £300, 

Space Blanket (medium) £168) 

200 mm: Space Combi (Economy roll £400, Space Blanket 

(thick) £210, Space Combi £80) 

2 £18.66 

3 100 weeks 

Going on holiday 

1 155 points 

2 England (England €478.80, Malta €483) 

3 16:56 

All at sea 

1 12 664 m 

2 

65° 

3 12:29 

8000 m 



1 arc, mean 

a Start at square K, go 1 to the right, then 1 down and stop. 

Then go 2 up, and 1 to the left and stop. Then go 4 to the 

right, and 3 down and stop. 

50° 

667 


7/30/12 5:21:31 PM


b Start at square C, go 1 up, then 1 left and stop. Then go 2 

up and 1 to the right and stop. Then go 3 down and stop. 

Then go 1 to the left and 2 up and stop. Then go 3 to the 

right and 2 down and stop. 

c Start at square A, go 3 to the right and stop. Then go 1 up 

and then 1 down and stop. 

d Students' examples 

Exercise 17A 

1 a ( 5) 5 b 

2 

( c 

4) (0) 

4 

d (4) 0 e ( 2 f 

3) ( 2 

5) 

2 y 

6 

D 

5 

4 

3 

2 

1 

4 3 2 1 O 1 2 3 4 5 6 7 8 9 

1 

F 

2 

3 

4 

3 a y 

5 

b ( 5 3) 

4 

3 

2 

1 

5 4 3 2 1 O 1 2 3 4 5 

1 

2 

4 a, b y 

5 

A 

4 

3 

2 

1 

C 

5 4 3 2 1 O 

1 

1 2 3 4 5 

c ( 2 4) d ( 

2 

4) 

2 

3 

A 

4 

5 

A 

B 

B 

E 

x 

C 

x 

B 

x 


A: x 4, B: y 3, C: y x, D: y x 

Exercise 17B 

1 a, b y 

6 

Q 

5 

4 

3 

2 

1 

R 

6 5 4 3 2 1 O 1 2 3 4 5 6 

1 

T 

2 

3 

4 

5 

6 

c Reflection in the line y 1 

2 a, b y 

6 

5 

4 

3 

2 

1 

6 5 4 3 2 1 O 1 2 3 4 5 6 

1 

2 

3 

C 

4 

5 

6 

c Reflection in the line y x 

3 a y 

7 

Q 

6 

5 

4 

3 

2 

1 

3 

4 

5 

6 

7 

B 

x 1 

7 6 5 4 3 2 1 O 1 2 3 4 5 6 7 x 

1 

y 1 

2 

b Reflection in the line x 1 

A 

P 

P 

R 

x 

x 

668 


05/09/12 5:18 PM



1 a 9 b i 90° clockwise ii 180° iii 150° clockwise 

2 a 90° clockwise 

b 90° anticlockwise 

c 180° 

d 150° anticlockwise 

Exercise 17C 

1 y 

6 

5 

4 

D 

3 

2 

1 

6 5 4 3 2 1 O 

1 

1 2 3 4 5 6 

2 

B 

3 

4 

5 

6 

C 

2 y 

6 

5 

4 

S 

3 

2 

1 

6 5 4 3 2 1 O 1 2 3 4 5 6 

1 

Q 

2 

3 

R 

4 

5 

6 

3 a i 90° anticlockwise about the origin 

ii 180° about the origin 

iii 90° clockwise about the origin 

b 90° clockwise about the origin 

c 180° about the origin 

4 a i 180° about (3, 5) 

ii 90° clockwise about (2, 1) 

iii 180° about (0, 0) 

iv 90° anticlockwise about (0, 4) 

v 90° anticlockwise about (2, 2) 

b 90° clockwise rotation about (2, 8) 

c i Translation ( 6 

10) ii Translation ( 4 0) 

A 

P 

x 

x 


1,2 

y 

3 

2 

1 

O 

1 2 3 x 

3 They are both squares. 

They have a common vertex at (0, 0). 

The second square is twice as large as the first square. 

Exercise 17D 

1 a 48 cm, 52 cm, 20 cm 

b The perimeter is also 4 times as long. 

2 a, b 

A 

B 

c 6 

3 a P is 4 cm by 2 cm, Q is 8 cm by 4 cm, R is 12 cm by 6 cm. 

b i 12 cm ii 24 cm iii 36 cm 

c i 8 cm 2 ii 32 cm 2 iii 72 cm 2 

d i 2, same as scale factor 

ii 3, same as scale factor 

e i 4, same as scale factor squared 

ii 9, same as scale factor squared 

f 96 cm 

Exercise 17E 

1 a 

C 

669 


05/09/12 5:18 PM


b 

2 a, b 

iii 

ii 

i 

y 

7 

6 

5 

4 

3 

2 

1 

C 

A 

Exercise 17F 

1 a, b 

y 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

O 

F 

c Translation ( 9 4) 

d Translation ( 9 

4) 

G 

1 2 3 4 5 6 7 8 9 10 11 12 

H 

x 

109 

8 7 6 5 4 3 2 1 O 

1 

1 2 3 4 5 

2 

B 

3 

4 

5 

6 

7 

8 

c 6 

x 

2 a, b y 

6 

5 

4 

T 

3 

V 

2 

1 

6 5 4 3 2 1 O 1 2 3 4 5 6 

1 

U 

2 

3 

x 

3 a y 

5 

4 

3 

2 

P 

1 

5 4 3 2 1 O 

1 

1 2 3 4 5 

2 

3 

4 

5 

b Rotation 180° about (1, 2) 

x 

c Rotation 180° about (4, 3) 

3 a, b y 

4 

3 

2 

1 

2 1 O 1 2 3 4 

1 

2 

3 

4 

5 

U 

T 

V 

5 6 

x 

c Reflection in the x-axis 

4 Reflection in the line x 5 

5 Reflection in the x-axis 

670 


05/09/12 5:18 PM



1 

4 y 

7 

6 

5 

4 

B 

3 

2 

1 

A 

C 

6 5 4 3 2 1 O 

1 

2 

3 

4 

5 

1 2 3 4 5 6 7 

x 

5 Students’ tile designs 

6 90° clockwise rotation about (–2, 3) 

2 a y 

8 

6 

4 

2 

7 y 

5 

4 

3 

2 

8 6 4 2 O 2 4 6 8 

2 

b Translation ( 

3 

1) 

4 

6 

8 

x 

1 

5 4 3 2 1 O 

1 

B 

2 

3 

4 

5 

1 2 3 4 5 

A 

x 

3 a y 

5 

4 

A 

3 

2 

1 

5 4 3 2 1 O 

1 

2 

B 

3 

4 

5 

1 2 3 4 5 

x 

8 a Reflection in the line y x 

b 

y 

4 

C 

3 

A 

2 

B 

1 

4 3 2 1 O 

1 

2 

3 

4 

1 2 3 4 

x 

b 90° anticlockwise rotation about the origin 

671 


7/30/12 5:21:39 PM


9 a y 

6 

10 

6 5 4 3 2 1 O 

1 

b Translation ( 

5 

4) 

y 

6 

5 

4 

3 

2 

1 

5 

4 

3 

2 

1 

2 

3 

4 

5 

6 

1 2 3 4 5 6 

6 5 4 3 2 1 O 

1 

A 

1 2 3 4 5 6 7 8 9 

2 

3 

4 

5 

6 

11 180° rotation about (1, 0) 

12 180° rotation about the origin 



1 360 

2 90 

3 a 120 b 45 c 60 

Exercise 18A 

1 

Latte 

Coke 

Orange 

Chocolate 

Tea 

Black 

coffee 

P 

x 

x 

2 Chicken Cheese 

3 

Ham 

Prawn 

Motorbike 

Train 

Cycle 

Tuna 

BLT 

Walk 

Car 

Exercise 18B 

1 a Golf b Athletics c 45 d 30 

2 

Shop A 

Shop B 

Cappuccino 

Mocha 

Espresso 


a 50, 54, 65, 72 

b 4.0, 4.3, 4.4, 4.6 

c 0.01, 0.1, 0.11, 0.12 

Exercise 18C 

Americano 

Latte 

Cappuccino 

Mocha 

Espresso 

Americano 

Latte 

1 a 36 b 28 c 47 d 18, 36 e 18 

2 a 

0 4 5 6 7 7 8 9 9 9 9 

1 0 2 4 5 7 

2 1 4 8 

3 0 

Key 2 | 1 stands for 21 

b 9 minutes c 9 minutes d 26 minutes 

e 7 minutes, 17 minutes f 10 minutes 

3 a 

5 2 6 9 

6 3 3 4 5 5 8 8 8 9 

7 2 4 4 4 4 

8 2 3 5 8 

9 2 4 

Key 7 | 2 stands for 72 

b 74 km c 69 km d 42 km 

e 64 km, 82 km f 18 km 

672 


05/09/12 5:18 PM



a B occurs more frequently than A, which occurs more 

frequently than C. 

b The red category accounts for a higher proportion of the 

total than the green or blue categories. Red: 50%, green and 

blue: 25% each. 

Exercise 18D 

1 a 30°C b 33°C c G d C and F e A 

2 a Food b Pension c 25% 

3 a 70 g b 10 g c Fruitbix 

4 a Saturday b Thursday c 40 


a 3 b 16 c 25 

Exercise 18E 

1 

14 

Frequency 

12 

10 

8 

6 

4 

2 

0 

0 to 2 3 to 5 6 to 8 9 to 11 12 to 14 

Number of games 

2 a 135 w 140 b 140 w 145 

c 

16 

14 

12 

10 

8 

6 

4 

2 

0 

125 130 135 140 145 150 

Wingspan (cm) 

Frequency 

3 a 139 w 141 b 137 w 139 

c 

16 

14 

12 

10 

8 

6 

4 

2 

0 

135 137 139 141 143 145 

Weight (g) 

Frequency 


a 5 b 17.5 c 115.5 

Exercise 18F 

1 

Frequency 

25 

20 

15 

10 

5 

0 

2 3 4 5 6 7 8 

Number of peas 

2 a 70 d 80 

b, c 

3 

Frequency 

Frequency 

16 

14 

12 

10 

8 

6 

4 

2 

18 

16 

14 

12 

10 

8 

6 

4 

2 

0 

60 70 80 90 100 

Noise level (decibels) 

0 

24 25 26 27 28 29 

Trout length (cm) 

4 Girls, because their mode is 5 minutes and the boys’ mode 

is 9 minutes. 

Exercise 18G 

1 a Lifetime 

(l hours) 

Frequency 

Class 

width 

Frequency 

density 

10 l 15 4 5 0.8 

15 l 20 10 5 2 

20 l 25 20 5 4 

25 l 30 15 5 3 

30 l 40 6 10 0.6 

673 


05/09/12 5:19 PM


2 

b 

Frequency density 


25 

4.5 

4 

3.5 

3 

2.5 

2 

1.5 

1 

0.5 

0 

10 15 20 25 30 35 40 

Lifetime (hours) 

3.5 

3 

2.5 

2 

1.5 

1 

0.5 

0 

0 

5 10 15 20 25 30 35 40 

Distance (kilometres) 

3 a Age (y years) Frequency 

0 y 5 10 

5 y 10 28 

10 y 20 49 

20 y 40 48 

40 y 70 60 

Exercise 18H 

1 a Cumulative frequency: 3, 10, 20, 35, 43, 48, 50 

b 


60 

50 

40 

30 

20 

10 

0 

0 5 10 15 20 25 30 35 40 

Age (years) 

2 a 11 b 19 c 33 

3 a i 22 ii 74 

b 110 

c 18% 


a 9 and 12 

b 18 and 20 

Exercise 18I 

1 a 42 b 33, 50 c 17 d 70 

2 a £6000 b £4000, £7500 c £3500 

3 a Median £242 000, Q 1 £222 000, Q 3 £256 000 

b Range £120 000, interquartile range £34 000 


Median 15 

Lower quartile 8 

Upper quartile 25 

Interquartile range 17 

Exercise 18J 

1 


b 101 

6 

5 

4 

3 

2 

1 

0 

0 10 20 30 40 50 60 70 80 

Age (years) 

2 

4 5 6 

Height (m) 

29 30 31 32 33 34 35 36 37 38 39 40 41 

Height (m) 

3 a Male: min 10, Q 1 20, Q 2 40, Q 3 50, max 80 

Female: min 10, Q 1 30, Q 2 50, Q 3 60, max 80 

b 

Male 

Female 

0 10 20 30 40 50 60 70 80 90 

Age (years) 

674 


05/09/12 5:19 PM


The female members are older on average (higher 

median). The interquartile range is the same for males 

and females, but the range is slightly greater for females 

so their ages vary slightly more. 


1 

2 

Art 

Music 

108° 

72° 

60° 

120° French 

Tea 

History 

Hot 

chocolate 

120° 

80° 

60° 

100° 

Coffee 

Soup 

3 a 14 b 5 

c 

16 

Number of students 

14 

12 

10 

8 

6 

4 

2 

0 

d Tuesday 

4 a 4 6 8 

Monday 

5 1 2 8 

6 0 3 4 6 8 

7 4 7 8 9 

8 7 

Tuesday Wednesday Thursday 

Days of the week 

Key 5 | 1 means 51 kg 

b 5 

c 87 46 41 kg 

5 30 mm 

6 

30 

Frequency 

20 

10 

0 

100 110 120 130 140 150 

Weight (w grams) 

Key: 

Boys 

Girls 

Friday 

7 

Frequency 

20 

15 

10 

5 

0 

0 10 20 30 40 50 

Science mark 

8 James. The angle for the combined proportion must be 

between the Year 9 and Year 10 angles. 

9 On average, the prices are lower at Peter’s garage than at 

John’s garage, as the median is lower. The range and 

interquartile range are both smaller for Peter’s garage 

so the spread is less than for John’s garage. Both the 

cheapest and the most expensive cars come from John’s 

garage. 

10 a 88 people b 38 years c 57 21 36 years 

11 a 

1.30 1.40 1.50 1.60 

1.70 1.80 1.90 

Girls’ height (m) 

b On average, the girls are taller than the boys, as the 

median is higher, but the boys' heights are more 

variable as the range is larger. The interquartile ranges 

are the same so the variation in the middle of each 

group is similar. 

12 From a cumulative frequency chart, the median for the men 

is £240. On average, the men spent more than the women. 

13 a 23 kg is the value of Q 3 . The heaviest bag weighs 29 kg. 

b 17 kg c 13 kg d 60 bags 

14 a 21 b 10 

c Most of the high and low scores are scored by the 

boys. More of the girls scored an average mark. 

15 


4 

3 

2 

1 

0 

40 50 60 70 80 90 100 110 

16 a 

Time (t minutes) 

0 10 20 30 

Minutes 

40 50 60 

b The delays were greater on Saturday, as the median 

and the quartiles are all higher. There was also more 

variation in the delays on Saturday, since both the 

range and the interquartile range are larger. 

675 


7/30/12 5:21:49 PM


17 a 

100 

b 

5 


80 

60 

40 

20 

b 68 mph 

0 

40 50 60 70 80 90 100 

Speed (s mph) 

c 76 61 15 mph 


4 

3 

2 

1 

0 

0 10 20 30 40 

Weight (w kg) 

c 57.5 d 35 

18 a 

Distance (d km) Frequency 

0 d 5 15 

5 d 10 20 

10 d 20 25 

20 d 40 20 

40 d 60 10 

20 20 batteries 



1 a 2.8 b 0.4 c 0.8 d 2.6 

Exercise 19A 

1 a 

b 

4 

b 

5 4 3 2 1 0 1 2 3 4 5 

5 4 3 2 1 0 1 2 3 4 5 


3 

2 

c 

d 

5 4 3 2 1 0 1 2 3 4 5 

1 

5 4 3 2 1 0 1 2 3 4 5 

0 

0 10 20 30 40 50 60 

Distance (d km) 

e 

5 4 3 2 1 0 1 2 3 4 5 

19 a 

Weight (w kg) 

Frequency 

Frequency 

density 

0 w 5 20 4 

5 w 15 30 3 

15 w 25 15 1.5 

25 w 35 10 1 

35 w 40 5 1 

2 a x 4 b x 1 c x 5 and x 2 

d x 0 and x 3 

e x 3 and x 5 

f x 5 and x 1 


1 x 10 

2 x 1_ 

2 

3 

4 

1 2 3 4 5 6 7 8 9 10 

5 6 7 8 9 10 11 12 13 14 15 

676 


7/30/12 5:22:01 PM


Exercise 19B 

1 a x 4 

2 

y 

5 

4 

y 3x 4 

5 4 3 2 1 0 1 2 3 

4 5 

3 

2 

b x 1 

1 

5 4 3 2 1 0 1 2 3 4 5 

c x 2 

5 4 3 2 1 

O 

1 

2 

3 

1 

2 3 4 5 

x 

5 4 3 2 1 0 1 2 3 4 5 

d x 1.6 

4 

5 

5 4 3 2 1 0 1 2 3 4 5 

2 a x 4.5 b x 4 c x 2.5 d x 6.5 

3 a x 3.25 b x 23 c x 6 

11 d x __ 1 

19 


1 x 4 2 x 4 3 x 2 4 x 3 

Exercise 19C 

1 a 1, 0, 1, 2, 3, 4, 5 

b 4, 3, 2, 1, 0, 1 

c 1, 2, 3 

d 5, 4, 3, 2, 1, 0, 1, 2, 3, 4 

2 a 3, 2, 1, 0, 1, 2, 3 

b 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7 

c 0, 1, 2, 3, 4 

d 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 

3 a 2, 1, 0, 1, 2, 3 

b 1, 0, 1, 2, 3, 4 

c 1, 0, 1, 2, 3, 4 

d 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4 

4 a 2, 1, 0, 1, 2, 3, 4, 5, 6 

b 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7 

c 0, 1, 2, 3 

d 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 


1 

y 

5 

5 4 3 2 1 

O 

1 

4 

3 

2 

1 

2 

3 

4 

5 

1 

y 4x 1 

2 3 4 5 

x 

3 

y 

2y 3x 4 

5 

4 

3 

2 

1 

5 4 3 2 1 

O 

1 2 3 4 5 

x 

1 

2 

3 

4 

5 

Exercise 19D 

1 (1, 1), (1, 0), (1, 1), (0, 0), (0, 1), (1, 1) 

y 

4 

3 

2 

1 

4 3 2 1 O 1 2 3 4 5 x 

1 

2 

3 

4 

5 

677 


05/09/12 5:19 PM


2 a 

y 

6 

d 

y 

6 

4 

4 

2 

2 

6 

4 

2 

O 

2 4 6 

x 

6 

4 

2 

O 

2 4 6 

x 

2 

2 

4 

4 

6 

6 

b 

y 

6 

3 a 

y 

6 

4 

4 

2 

2 

6 

4 

2 

O 

2 4 6 

x 

6 

4 

2 

O 

2 4 6 

x 

2 

2 

4 

4 

6 

6 

c 

y 

6 

b 

y 

6 

4 

4 

2 

2 

6 

4 

2 

O 

2 4 6 

x 

6 

4 

2 

O 

2 4 6 

x 

2 

2 

4 

4 

6 

6 

678 


05/09/12 5:19 PM


c 

y 

6 

4 

2 

Exercise 19F 

1 a C 50d 90 b €790 

2 P 5x 3y z 

45m 75f 

3 T _________ _______ 3m 5f 

60 4 

4 P 4x 4y 

5 a T 2a 5c b P 2a 3c 

d 

6 

6 

4 

4 

2 

2 

O 

2 

4 

6 

y 

6 

2 

4 

6 

4 a y 3, y 2x 3, x y 1 

b y 3, y 2x 3, x y 1 

c (2, 2), (1, 2) and (1, 1) 

5 a y 2, x 3, y 1_ x 3, y x 

2 

b y 2, x 3, y 1_ x 3, y x 

2 

c 1 


1 a 5347.76 b 

Exercise 19E 

8_ 2_ 

, or 2 

3 3 

1 a 120 b 25 

2 a 65 b 5 

3 a 1570 b 120.576 

4 a 120 miles b 8 m/s 

5 a 5 hours 5 minutes b 18 lb 

6 a 12 b 9 


a 3200 b 9 

4 

2 

O 

2 4 6 

2 4 6 

x 

x 


1 60 miles 2 1.8 10 17 

Exercise 19G 

1 x _____ y 3 

5 

3 a _____ v u 

t 

5 m _____ 4 E 

3 

2 d _____ c 2 

5 

4 x P y _____ 

5y 

6 g ______ f 30 

3 

7 x 7T 2 8 n ________ 2Y 3m 

3 

3W 2y 

9 p ________ 

2y 

10 w 4 3A 


1 u v at 2 R V __ 

I 

Exercise 19H 

__ 

1 R √ 

__ A 

 

3 u √ ________ 

v 2 2as 

5 x √ _______ 

_______ 75 y 2 

50 

7 m _____ 1 

T 3 

9 c _____ a 3 

b 7 


3 m E __ 

c 2 

2 x P 2 y 

4 g 2s __ 

T 2 

6 g _____ f 4 

4 

8 p ______ q 12 

q 4 

10 T ________ 7 

2W 2 3 

1 9 

2 a b 3a b c a 5 

3 a £130 b C 90 0.5m 

4 0.6 

5 a 8 b p ______ S 3q 

4 

6 a 1 x 4 b 1 x 5 c x 2 

7 a 

b 

c 

d 

5 4 3 2 1 0 1 2 3 4 5 

5 4 3 2 1 0 1 2 3 4 5 

5 4 3 2 1 0 1 2 3 4 5 

5 4 3 2 1 0 1 2 3 4 5 

8 3, 2, 1, 0, 1 

679 


05/09/12 5:19 PM


9 a t 5.5 b 5 

10 3, 2, 1, 0, 1, 2 

11 p 3 

12 0 x 4 

13 a L x (x 4) 2(x 4) x x 4 2x 8 

4x 12 

b 4x 12 50 c 0 x 9.5 

14 

y 

6 

6 

4 

2 

4 

2 

O 

2 

4 

R 

x y 6 

2 4 6 8 

x 



1 a 144 b 54.76 

2 a 1681 b 276.93 

3 a 225 b 102.55 

4 a 25 b 12 

5 a 6.44 b 8.91 

Exercise 20A 

1 a 13 cm b 10 cm c 10.4 cm d 11.6 cm 

2 a 12.6 cm b 11.7 cm c 14.4 cm d 20.8 cm 

Exercise 20B 

1 a 7 cm b 35 cm c 12.3 cm d 8.03 cm 

2 a 15.2 cm b 5.77 cm 

c i D 

10.1 cm 

6 

10 

15 a __ 

7 b u _____ vf 

v f 

4y 15 

16 x _______ 

5 3y 

17 a 24.8 b r ______ P 2a 

2 

18 a Any two of: (1, 1), (1, 2), (2, 1) 

b y 

x 0 y 3x 

8 

7 

6 

5 

4 

3 

2 

1 

4x 3y 12 

y 0 

0 

0 1 2 3 4 5 6 7 8 

x 

19 a x 2, y 3, 2x 3y 20 

b y 

x 2 

8 

7 

6 

5 

4 

3 

2 

1 

0 

0 1 2 3 4 5 6 7 8 

y 3 

2x 3y 20 

x 

E 

ii 6.42 cm 


7.8 cm 

The rounded lengths are 21 cm and 30 cm. The diagonal is 

approximately 37 cm. 

Exercise 20C 

1 a 15.0 m b 6.92 cm c 18.7 cm 

2 3.47 m 

3 52.4 cm 2 

4 a 3.20 m 

b i 4.33 m ii 2.33 m 

c 8.66 m 

5 Triangles 1, 2, 4 and 6 are right-angled triangles. 

6 2.85 cm 

7 a 17 inches 

b Height 18 inches, width 32 inches 


a 26 b 19.3 c 10.3 

Exercise 20D 

1 a 10 b 26 c √ ___ 

212 14.6 

d √ ___ 

424 20.6 e 29 f √ ___ 

117 10.8 

2 a i 5 ii √ __ 

50 7.07 iii 5 

b 5 2 5 2 50, so triangle is right-angled at A. 

3 a 13 

b B, C, E lie on the circle. 


1 a p b r c q 

F 

680 


7/30/12 5:22:08 PM


2 a e b f c d 

3 a AC b BC c AB 

Exercise 20E 

1 a 0.3420 b 0.9542 c 0.5 d 0.9461 

e 1 f 15.8945 g 0.7408 h 0.0699 

i 0.7965 j 0.2538 k 0.3739 l 0.0471 

2 a 53.1 b 25.5 c 60 d 43.8 

e 58.4 f 63.8 g 2.7 h 60 

i 45 

Exercise 20F 

1 a sine b cosine c tangent d cosine 

Exercise 20G 

1 a 37.7° b 46.2° c 19.7° d 40.1° 

e 47.1° f 43.6° 

2 a 36.3° b 38.9° c 32.3° 

3 a 41.0° b 57.2° c 98° 


1 a 

N 


1 6.36 cm 

2 6.71 cm 

3 230 km 

4 19.4 cm 

5 20.6° 

6 a 51.3° b 10.5 cm 

7 21.7 m 

8 033° 

9 116 cm 

10 a Alan 

b 5 1_ seconds. Alan’s time _______ 60 80 

28 seconds, 

3 5 

Bhavana’s time _________ 

√________ 60 2 80 2 

33 1_ 

3 

3 seconds 

11 a 

b 

10 x 

x 2 x 2 10 2 , so x 7.07 cm 

x 

55° 

20 km 

12 

1 

x 10 cm 

5 2.24 m 

X 

2 

b 

2 cosine 

8 m 

3 m 

Exercise 20H 

1 a 5.47 cm b 17.1 cm c 11.6 cm 

d 26.1 cm e 10.4 cm f 11.3 cm 

2 a 18.1 cm b 13.4 m c 10.8 cm 

3 a i 9.96 cm ii 8.36 cm 

b 5.30 cm c 13.7 cm d 68 cm 2 

Exercise 20I 

1 a i 1.45 m ii 5.44 m iii 2.54 m 

b 65.0° 

2 a 16.8 m b 64.5° c 24.5° 

3 a 46.9 m b 88.3 m 

4 a 2.87 cm b 16.5° 

5 a i 13.0 km ii 10.9 km 

b i 7.73 km ii 23.8 km 

c i 20.7 km ii 34.7 km 

d 40.4 km, 059.2° 

6 a 3.84 cm b 16.8 cm c 43 cm 2 

a Area 4 2.24 8.96 m 2 , so he needs to buy two 

rolls of felt. 

b Two rolls of felt cost £24.00, two tins of adhesive cost 

£13.98, so total cost £37.98 

13 No. It depends on the diagonal of the desk: 

d 2 2.3 2 1.6 2 , d 2.80 m, but the room is only 2.75 m 

wide. 



1 

y 

5 

4 

3 

2 

1 

3 2 1 

O 

1 

2 

3 

1 

y 2x 3 

2 3 

2 a 4 b 4 

3 a 2 b 18 

x 

681 


05/09/12 5:20 PM


Exercise 21A 

1 a 

x 3 2 1 0 1 2 3 

y 6 1 2 3 2 1 6 

b 

y 

20 

18 

16 

b 

y 

8 

6 

y x 2 3 

14 

12 

10 

8 

6 

y 2x 2 2 

4 

4 

2 

2 

4 2 O 2 4 

x 

2 

4 

c x 0 d (0, 3) 

2 a 

x 3 2 1 0 1 2 3 

y 5 0 3 4 3 0 5 

4 2 

O 

2 4 

x 

c i y 6.5 ii x 2.1 and x 2.1 

4 a y 

12 

x 2 

10 

8 

6 

4 

2 

y x 2 4x 1 

2 

O 

2 4 6 

x 

2 

4 

6 

b 

y 

6 

4 

i x 4.2 and x 0.2 ii x 2 

b y 

x 1 

12 

10 

2 

8 

6 

4 2 O 2 

2 

4 

x 

y 4 x 2 

4 

y 2x 2 4x 3 

2 

2 

O 

2 4 

x 

2 

4 

4 

6 

6 

c (0, 4) d x 2 and x 2 

3 a 

x 3 2 1 0 1 2 3 

y 20 10 4 2 4 10 20 

i x 2.6 and x 0.6 ii x 1 

c 

x 2 

y 

20 

15 

10 

5 y (x 2) 2 

6 4 2 

O 

2 

x 

5 

i touches the x-axis at 2 

ii x 2 

682 


05/09/12 5:20 PM


d 

y 

10 

5 

2 

O 

2 4 

5 

10 

15 

20 

x 0.75 

y 5 3x 2x 2 

i x 2.5 and x 1 ii x 0.75 

Exercise 21B 

1 a x 0 and x 3 b x 1 and x 4 

c x 2.8 and x 1.3 d x 3.6 and x 1.6 

2 a x 0.4 and x 2.6 b x 0.4 and x 2.6 

c x 3.1 and x 1.6 d x 3.2 and x 1.2 

3 a 

x 2 1 0 1 2 3 4 

y 7 2 1 2 1 2 7 

b 

y 1 x 

y 

2 

x 

y 1 2x x 2 

Exercise 21C 

1 a 

x 3 2 1 0 1 2 3 

y 25 6 1 2 3 10 29 

b 

y 

30 

25 

20 

15 

10 

y x 3 2 

5 

3 2 1 O 1 2 3 x 

5 

10 

15 

20 

25 

30 

c y 17.6 

2 

O 

2 

2 

4 

x 

2 a 

x 4 3 2 1 0 1 2 3 4 

y 28 0 10 8 0 8 10 0 28 

4 

6 

8 

c x 0 and x 3 

4 a 

x 3 2 1 0 1 2 3 

y 32 16 6 2 4 12 26 

b 

y 2x 4 

y 

30 

25 

20 

15 

10 

2 O 2 

c x 1.5 and x 0.5 


5 

y 3x 2 x 2 

1 1, 8, 27, 64, 125 

2 a 1 000 000 b 1000 

x 

b 

y 

30 

y x 3 9x 

25 

20 

15 

10 

5 

4 2 O 2 4 x 

5 

10 

15 

20 

25 

30 

c x 3, x 0 and x 3 

3 a 

x 3 2 1 0 1 2 3 4 

y 45 4 7 0 13 20 9 32 

683 


05/09/12 5:20 PM


b 

y 

50 

40 

30 

20 

10 

y 12x 3x 2 2x 3 

3 a y 

50 

40 

30 

20 

10 

y 12 

(x 1) 

2 O 2 4 

y 2x 1 10 

20 

30 

c x 1.5, x 0.1 and x 3.1 

4 i C ii A iii D iv B 


1 a 

1_ 

b 4 c 0.4 d 2.5 

4 

2 As the value of x gets bigger, the value of __ 1 gets smaller. 

x 

Exercise 21D 

1 a 

b 

c 

x 0.2 0.4 0.5 1 2 4 5 10 20 

y 25 12.5 10 5 2.5 1.25 1 0.5 0.25 

x 20 10 5 4 2 1 0.5 0.4 0.2 

y 0.25 0.5 1 1.25 2.5 5 10 12.5 25 

y 

25 

20 

15 

10 

5 

y x 

5 

20 1510 5 O 

5 

5 10 15 20 

10 

15 

20 

25 

x 

x 

4 

b x 1 

2 O 2 

10 

20 

30 

40 

50 


1 a 81 b 1 c 

1_ 

9 

2 a x 4 b x 2 c x 3 

Exercise 21E 

1 a 

b 

x 3 2 1 0 1 2 3 

y 0.04 0.111 0.33 1 3 9 27 

y 

30 

25 

20 

15 

10 

5 

y 3 x 

3 2 1 

O 

1 2 3 

c i 5.2 ii 2.5 

2 A is y ( 1_ 

2 ) x , B is y 3 x , C is y 5 x , D is y 2 x 

3 a 10 b 320 c 17 minutes 

4 p 0.625, q 16 

x 

x 

2 y 

10 

8 

6 

4 

2 y x 

2 

10 8 6 4 2 O 2 4 6 8 10 

2 

4 

6 

8 

10 

x 


1 a 4.6 b 2.2 c 1.5 

2 When x 1, x __ 1 x 1 __ 1 1 2 

3 a 2 b 2 

Exercise 21F 

1 a 1 and 2 b 0 and 1 c 0 and 1, 1 and 0, 4 and 5 

d 2 and 3 

2 a 1.7 b 1.3 c 4.8 and 0.2 

3 a 3.04 b 2.17 c 4.72 

4 a x(x 2)(x 2) x 3 4x b x 8.6 

c 10.6 cm, 6.6 cm, 8.6 cm 

684 


7/30/12 5:22:17 PM



1 a 

b 

2 a 

y 

450 

400 

350 

300 

250 

200 

150 

100 

50 

t 0 2 4 6 8 10 

s 0 49 116 201 304 425 

O 

c i 80.25 m 

b 

3 a 

1 2 3 4 5 6 7 8 9 10 

ii 9.6 seconds 

y 2.25x 2 20x 

x 4 3 2 1 0 1 2 

y 8 3 0 1 0 3 8 

y x 2 2x 

x 1 

4 2 

O 

2 

y 

10 

8 

6 

4 

2 

c x 1 

d i y 1.25 ii y 1.6 and y 3.6 

b 

x 3 2 1 0 1 2 3 

y 10 4 0 2 2 0 4 

4 

2 

2 

O 

2 

2 

4 

6 

8 

10 

y 

y 2 x x 2 

c i x 1 and x 2 ii 2.8 and 1.8 

d (0.5, 2.25) 

4 a x 2 3x 2 x 2 

x 2 3x 2 x 2 0 

x 2 4x 0 

b x 0 and x 4 

c y 2 x 

x 

x 

x 

5 a When x 0.5, x 2 2 __ 

x 3 0.75 

When x 1, x 2 __ 2 x 3 2 

So there must be a solution to x 2 __ 2 x 3 0 

between x 0.5 and x 1. 

b 0.60 

6 a 

x 2 1 0 1 2 3 4 

b 

y 8 1 0 5 8 3 16 

y 

20 

15 

10 

5 

2 

O 

2 4 

5 

10 

x 

y x 3 2x 2 4x 

c i x 0, x 3.2 and x 1.2 

ii x 1.8, x 1 and x 2.8 

7 a 

x 3 2 1 0.5 0.1 0.1 1 2 3 

8 a 

b 

y 3.7 4 5 7 23 17 1 2 2.3 

c x 0, y 3 

b, c 

y 

25 

20 

15 

10 

5 

x 0 

2 

O 

2 

5 

2 

y 3 

10 

x 

15 

20 

y 3 

x 0.5 1 2 3 4 

y 8 4 2 1.3 1 

y 

8 

7 

6 

5 

4 

3 

2 

d x 2 

1 

y 4 x 

0 

0 1 2 3 4 

x 

x 

685 


7/30/12 5:22:28 PM


9 a 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

O 0.5 1 1.5 2 2.5 3 

b 1.5 seconds 

c 2.86 seconds 

10 a 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

O 10 20 30 40 50 60 70 80 

b i 22 metres 

ii 9 metres and 71 metres 

iii 82 metres 

11 a y 

40 

30 

20 

10 

0 

0 1 2 3 4 5 x 

b 3.2 cm 

12 7.62 cm 

13 i C ii D iii A iv E v B 

14 k 0.25 



1 a x 1.5 b y 4 

2 a 4 b 21 

y x 2 3x 

Exercise 22A 

1 x 4, y 1 2 x 5, y 3 

3 x 1, y 2 4 x 2, y 1 

5 x 1, y 4 6 x 1, y 2 

7 x 1, y 1 8 x 2, y 1 

9 x 0.5, y 2 10 x 0.5, y 4 

11 x 3, y 1 12 x 2, y 0.5 

13 x 3, y 5 14 x 1, y 3 

15 x 0.5, y 1 


1 x 2, y 3 

2 x 1, y 2 

3 x 3, y 1 

Exercise 22B 

1 7, 12 

2 £18.50 

3 £8.80 

4 Adult £4.50, child £3 

5 14 g 

6 a 4a 3b 1 or 4a 3b 1 

2a 3 5 b or 2a b 8 

b a 2.5 cm, b 3 cm 

7 Ann £9/day, Mary £12/day 

8 x 3, y 2 


1 4a 3b 4 

3a 24 3b 

2 a 4, b 4 

3 56 

Exercise 22C 

1 a B y 2x, A x y 3, C x 2y 3 

b i x 1, y 2 ii x 3, y 0 iii x 1, y 2 

2 a x 2, y 0 b x 2.625, y 2.25 c x 2, y 8 


1 a x(x 4) b y(2y 5) c (2x 1)(x 3) 

d (3y 4)(2y 5) 

2 a 8 b 1 c 149 

Exercise 22D 

1 a 0, 4 b 3, 5 c 0.5, 2.25 

d 0, 2 e 0, 1 f 0, 1.75 

2 a 2, 4 b 1, 6 c 3, 4 

d 3 e 9, 4 f 4, 4 

g 5 

h 10, 10 

3 a 5, 1_ 

5 

b 3, 

2_ 

3 c 1_ 

2 , 4 

d 3, 1_ 

5 

4 a 3, 2 b 5, 2 c 7, 3 

d 7, 5 e 3, 2.5 f 4 1_ 

2 , 2_ 

3 

g 8, 1 h 1, __ 9 

16 

686 


7/30/12 5:22:35 PM



1 x 2 6x 9 

2 x 2 10x 25 

3 x 2 2ax a 2 

Exercise 22E 

1 a (x 2) 2 4 b (x 5) 2 25 

c (x 6) 2 36 d (x 1) 2 1 

e (x 7) 2 49 f (x 12) 2 144 

g (x 0.5) 2 0.25 h (x 1.5) 2 2.25 

i (x 2) 2 3 j (x 4) 2 1 

k (x 5) 2 45 l (x 3) 2 2 

m (x 10) 2 20 n (x 13) 2 170 

o (x 0.5) 2 0.75 p (x 2.5) 2 11.25 

2 a 2(x 3) 2 18 b 2(x 1) 2 3 

c 3(x 2) 2 2 d 5(x 5) 2 25 

3 a p 4, q 8 b 8 

4 a (0, 10) b (3, 1) 

5 a r 5, s 2 b 5 c x 2 


1 x 2, x 2 

2 x 3, x 2 

3 x 7, x 4 

Exercise 22F 

1 a x 3 √ __ 

11 b x 2 √ __ 

3 

c x 5 √ __ 

37 d x 1 2 √ __ 

2 

e x 3 √ __ 

_______ 15 

f x 6 √ __ 

_________ 21 

2 

5 

2 a x 0.68, x 7.32 b x 8.27, x 0.73 

c x 2.37, x 3.37 d x 0.87, x 2.87 

e x 0.88, x 0.38 f x 0.93, x 0.43 


1 x √ __ 

14 2 

x 1.74, x 5.74 

2 x √ __ 

14 3 

x 0.74, x 6.74 

3 x √ __ 

8 1 

x 3.83, x 1.83 

Exercise 22G 

1 0.586, 3.41 2 0.807, 6.19 

3 0.606, 6.61 4 2.70, 3.70 

5 5.32, 1.32 6 4.30, 0.697 

7 0.293, 1.71 8 1.54, 0.260 

9 1.64, 0.811 10 0.322, 0.622 

11 1.39, 0.360 12 3.45, 1.45 

13 0.372, 5.37 14 5.11, 4.11 

15 1.72, 1.52 


1 2x 2 9x 18 

2 6x 2 18x 12 

3 2x 12 

Exercise 22H 

1 x 0, x 5 2 x 2, x 1.8 

3 x 1.5, x 2 4 x 1, x _ 5 

9 

5 x 0.5, x 0.6 6 x 2, x 0.75 

7 x 2.30, x 1.30 8 x 4.48, x 3.73 

9 x 5.32, x 1.32 10 x 3.27, x 4.27 

11 x 0.576, x 10.4 12 x 3.41, x 0.586 


1 4x 2x 2 

2 x 2 2x 1 

3 8 10x 3x 2 

Exercise 22I 

1 4, 6 

2 4 

3 12 and 48 

4 7.22 m 

5 a x 2 (x 1) 2 41 

2x 2 2x 40 0 

x 2 x 20 0 

b 4, 5 or 4, 5 

6 12.36 cm, 15.72 cm and 20 cm 

7 10 cm by 10 cm 

8 14 m or 16 m 

9 x 5 

10 54 km/h 


1 a, d and e 

2, 3 

Exercise 22J 

5 cm 

1 y 

10 

b 

8 

6 

e 

d 

4 c 

2 b 

a 

10 8 6 4 2 O 2 4 6 8 10 

2 

4 

6 

8 

10 

a 

x 

687 


7/30/12 5:22:42 PM


2 y 

5 

4 

3 

2 

1 

5 4 3 2 1 O 1 2 3 4 5 

1 

2 

3 

4 

5 

(2.83, 2.83) and (2.83, 2.83) 

3 a y 1, y 11 b y 9, y 1 

c x 2, x 8 d x 10, x 0 

4 y x 12.49, y x 4.49 


1 

2 

10 8 

x 3, y 7 

y 

10 

6 4 2 

O 

2 

y 3x 3 

10 8 

x 2, y 3 

8 

6 

4 

2 

4 

6 

8 

10 

y 

10 

8 

6 

4 

2 

6 4 2 

O 

2 

4 

6 

8 

10 

2 

2 

4 6 8 10 

y 2x 7 

y 2x 1 

x 

y x 10 

4 6 8 10 

x 

x 

Exercise 22K 

a graphs of the equations 

b : 

1 x 2, y 2; x 3, y 3 

2 x 4, y 7; x 2, y 1 

3 x 3, y 0.5; x 1, y 1.5 

4 x 1, y 1; x 1.67, y 6.33 

5 x 1, y 2; x 2, y 1 

6 x 3.5, y 21; x 0.5, y 3 


1 y 

10 

8 

6 

4 

2 

10 8 6 4 2 O 2 4 6 8 10 

2 

4 

6 

8 

10 

2 y 

10 

8 

6 

4 

2 

10 8 6 4 2 O 2 4 6 8 10 

2 

4 

6 

8 

10 

3 y 

10 

8 

6 

4 

2 

10 8 6 4 2 O 2 4 6 8 10 

2 

4 

6 

8 

10 

x 

x 

x 

688 


7/30/12 5:22:53 PM


Exercise 22L 

1 x 2.7, y 5.4; x 2.7, y 5.4 

2 x 1.2, y 4.2; x 4.2, y 1.2 

3 x 6.8, y 1.8; x 1.8, y 6.8 

4 a x 2, y 3; x 3, y 2 

b x 4, y 2; x 2, y 4 

c x 2.2, y 5.4; x 3, y 5 

5 a x 0.449, y 4.45; x 4.45, y 0.449 

b x 1.49, y 5.46; x 2.09, y 5.26 

c x 5.63, y 8.26; x 3.23, y 9.46 


1 a x 3 b x 6 c x 7.35 

2 a y 2 b t 2 c p 3 

3 3 cm 

4 a y 1 and y 6 b t 2 and t 2 

c p 1 and p 4 

5 a x 4_ 

3 , y 25 

3 b x 1, y 1 

c x 2, y 5 

6 a x 2, y 2 b x 0.8, y 2.4 

7 29.12 m 

8 a h 0 when t 0 or 6, so the ball was in the air for 6 

seconds. 

b By symmetry of graph, maximum height is when t 3, 

giving height of 45 m 

c h 25 when t 1 or t 5, so the ball was 25 m above 

the ground after 1 second and 5 seconds. 

9 a k 0.5 and k 5 b m 0.5 and m 1.5 

c n __ 13 6 

10 a p 4, q 11 b x 4 √ __ 

11 

11 x 6.27 and x 1.27 

12 a, b x 2.41 and x 0.414 

13 a (x 1) 2 2 b 2 c x 1 

d y 

3 

(1, 2) 

14 4(x 3) 2 36. a 4, p 3, q 36 

15 a x 2, y 2 b x 0, y 2 

c x 4, y 3 

16 F £12.50, g 5p 

17 y 2x 3 and 2y x 6 

18 x 5, y 8 

19 a p 4, q 7 b (4, 7) 

20 50p 

21 x 1, y 0 and x 1.33, y 9.33 

22 a 5(x 1) (4 3x)(x 2) 

5x 5 8 2x 3x 2 

3x 2 7x 13 0 

b x 1.22 and x 3.55 

23 a 5x x(2x 1) 95 

2x 2 6x 95 0 

b x 5.55 and x 8.55 

x 

24 a Time for first 10 km __ 10 

x 

Time for second 10 km _____ 10 

x 1 

Total time 4, so __ 10 

x _____ 10 

x 1 4 

b 10(x 1) 10x 4x(x 1) 

4x 2 24x 10 0 

c x 5.55 and x 0.450 

d 4.55 km/h 

25 a x 4.30, y 0.697 and x 0.697, y 4.30 

b x 0.350, y 6.70 and x 5.15, y 4.30 

26 x 3.41 and x 0.586 

27 x 2 y 2 16 is a circle with radius 4, centre the origin, 

so it intersects the x- and y-axes at 4. The point (1, 2) 

must lie inside this circle. Therefore any straight line that 

passes through (1, 2) must intersect the circle twice. 

4 

y 

4 

4 

(1, 2) 

28 x 1.83, y 4.65 and x 2.63, y 4.25 



1 a 6 b 

4 

x 

1_ 

6 c 1_ 

9 

Exercise 23A 

1 a 1.75 cm b 17.6 cm c 64.9 m 

2 a 55.4 cm 2 b 9.25 cm 2 c 230 cm 2 

3 a 19.3 cm b 41.6 m 

4 70° 

5 5 cm 

6 a 1710 m 2 b 48.9 m 

7 a 57.9 cm b 235 cm 2 


1 a 10a b 3b 

2 x 4.5 

Exercise 23B 

1 a 9 cm b m c 52 mm 

2 a 4 cm 2 b 100 m 2 c 400 m 2 

3 b 16 cm 

4 a 8 cm b 64 cm 2 

5 a 36 cm 2 b 4 cm c 36 4 cm 

6 Perimeter 1_ 1_ 1_ 

Y X (X Y) (X Y) 

2 2 2 

689 


22/08/12 12:00 PM



1 a 100 b 10 000 c 1000 000 

2 a 500 b 130 000 c 7 600 00 

3 a 70 b 0.35 c 0.049 

Exercise 23C 

1 a 45 239 cm 2 b 4.5239 m 2 

2 a 14 cm 2 b 1400 mm 2 

3 a 40 000 cm 2 b 69 000 cm 2 c 6 cm 2 

d 0.47 cm 2 

4 a 5 000 000 m 2 b 300 000 m 2 c 4 m 2 

d 0.056 m 2 

5 a 1000 b 1 000 000 c 8 300 000 mm 2 

6 a 44 800 cm 2 b 3.6 cm 2 


1 a 2.211 cm 2 b 145.6 mm 2 c 65.0 mm 2 (to 3 s.f.) 

Exercise 23D 

1 a 60 m 3 b 480 cm 3 c 320 cm 3 

2 16 400 000 m 3 

3 a 100 cm 3 b 125 cm 3 c 0.0804 cm 3 

4 53 1_ 

3 cm3 

5 1590 cm 3 


1 a 10.24 b 32.2 cm 2 (to 3 s.f.) 

2 a 64 b 216 c 1331 


1 15 cm 2 2 36.3 mm 2 (to 3 s.f.) 

Exercise 23H 

1 a 10 950 cm 2 b 11 700 cm 2 c 3524 cm 2 

d 3 m 2 e 684 cm 2 f 1392 cm 2 

2 £94 


1 3.2 cm 2 7.2 cm 3 64.2 mm 

Exercise 23I 

1 a 240.3 cm 2 b 3.3 m 2 

2 a 503 cm 2 b 5890 mm 2 c 302 cm 2 

3 a 100 cm 2 b 80 cm 2 c 356 cm 2 

4 a 192 cm 2 b 924 cm 2 


(1.5, 3) 

Exercise 23J 

1 O (0, 0, 0), A (4, 0, 0), B (4, 0, 3), C (0, 0, 3), D (0, 2, 3), E (0, 2, 0), 

F (4, 2, 0), G (4, 2, 3) 

2 y 

Exercise 23E 

1 268 cm 3 2 4090 m 3 3 8.31 cm 

4 17.9 cm 5 1940 cm 3 6 5.04 m 

C 

(1, 2, 3) 

A (1, 0, 0) 

x 


1 a 6.28 cm 3 (to 3 s.f.) b 576 cm 3 

Exercise 23F 

1 a 223 cm 3 b 177 m 3 c 52.5 m 3 d 812 cm 3 

2 120 cm 3 3 54 000 cm 3 4 1610 cm 3 

5 731.25 cm 3 6 4_ 3 r3 


1 a 1 000 000 000 b 4620 c 91.875 m 3 

Exercise 23G 

1 a 2 000 000 cm 3 b 6 750 000 cm 3 c 0.45 cm 3 

d 0.0068 cm 3 

2 a 7000 mm 3 b 3750 mm 3 c 25 mm 3 

3 a 0.075 m 3 b 0.0008 m 3 c 0.000 125 m 3 

4 a 0.83 l b 5.6 l c 1000 l d 0.003 54 l 

5 720 000 litres 

6 1277 cubes 

7 Volume A 12.5 m 3 and volume C 0.375 m 3 so shape A 

has the largest volume. 

z 

B (1, 0, 3) 

3 a O (0, 0, 0), A (4, 0, 0), B (4, 0, 3), C (0, 0, 3), D (0, 2, 0), 

E (4, 2, 0), F (4, 2, 3), G (0, 2, 3) 

b i (0, 2, 1.5) ii (4, 1, 1.5) 

4 (2, 3, 0), (2, 3, 1), (2, 0, 1) 


1 96 cm 2 

2 110 cm 2 

3 8 000 000 cm 3 

4 a 4.5 cm 2 b 60 000 cm 2 

5 6.54 km 2 

6 a 905 m 3 b 4.92 m 

7 9.3 cm 

8 h __ 10 

9 x3 3x 2 

9 10.9 cm 2 

10 12 4 

11 a ___ 7 

b 4 ___ 7 

2 

2 

12 vol 2.7 litres 

13 a (5, 2, 0) b (2.5, 1, 3) 

690 


7/30/12 5:23:03 PM


14 h 9x 

15 1700 cm 3 

16 Yes. Volume of a pyramid 1_ base area height and 

3 

Volume of a cone 1_ base area height, so if the 

3 

base areas and volumes are the same, then the heights 

must also be equal. 

17 18.3 cm 2 (3 s.f.) 



a 0.05 b 0.4 

Exercise 24A 

1 a 700 m 

b They stopped twice; once for 1 minute then again for 

2 minutes. 

c 9.5 minutes 

d 420 m 

2 a Month 1; 95 m b 80 m 

c Months 6 and 8. These correspond to June and August 

when it doesn’t rain much. 

3 a 18:00 hours, because people are making their evening 

meals 

b 102 000 kilowatts c 12:00, 16:36, 20:00 

d 22:00 


a A 63.5, B 64.25 b C 10.75 c D 54.9 d E 16.6 

Exercise 24B 

1 a 

14 

12 

10 

8 

6 

4 

2 

0 

0 0.5 1 1.5 2 2.5 3 3.5 

Engine size (litres) 

b Negative correlation 

c The greater the engine size, the lower the petrol 

consumption. 

2 a 

900 

800 

700 

600 

500 

400 

300 

200 

100 

0 

8 9 10 11 12 13 14 15 16 17 18 

Screen size (inches) 

Petrol consumption (mpl) 

Selling price (£) 

b No correlation 

c Screen size and selling price are not related. 

3 a 

Number of fish 

190 

180 

170 

160 

150 

140 

130 

120 

0 2 4 6 8 10 12 

Year after restocking 


c The more years have passed since restocking, the 

smaller the number of fish. 

4 a Related – the lighter the car, the faster it will go. 

b Related – the longer the motorway, the greater the 

number of petrol stations. 

c Unrelated 

d Related – the greater the number of bicycles sold, the 

greater the number of cycle helmets sold. 

Exercise 24C 

1 a, c 

800 

GNP (million £s) 

700 

600 

500 

400 

300 

200 

100 

0 5 10 15 20 25 30 35 40 45 50 

Energy consumption (kW/capital) 

b (20, 800) 

2 a, c 

Price (£1000) 

12 

10 

8 

6 

4 

2 

0 

0 2 4 6 8 10 12 14 

Age (years) 

691 


05/09/12 5:21 PM


b Car prices cannot go negative, so the relationship only 

holds for cars up to 9 years old. Cars older than this will 

still be worth a small amount. 

3 a, b, d 

Mean low temperature (°F) 

80 

75 

70 

65 

60 

55 

50 

45 

40 

35 

30 40 50 60 70 80 90 

Mean high temperature (°F) 

c Ottawa. The high and low temperatures are the wrong 

way round. 

Exercise 24D 

1 a 44 b 65 

2 a i 65°F ii 78°F 

b i 45 degrees ii 17 degrees 

3 a 

Length (metres) 

2.464 

2.4635 

2.463 

2.4625 

2.462 

2.4615 

2.461 

20 25 30 35 40 45 50 55 60 65 70 75 

Temperature (°C) 

b i 41.5°C ii 63°C 

c 2.461 45 m 


1 

No. of unemployed 

110 

100 

90 

80 

70 

60 

50 

40 

30 

20 

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 

Month 

b 65 

c August, because more people go to the seaside in the 

summer holidays. 

2 a, b 

Art mark 

75 

70 

65 

60 

55 

50 

45 

40 

30 40 50 60 70 80 90 100 

Science mark 

c Negative correlation 

d The higher the science mark achieved by a student, the 

lower the art mark. 

3 a, d 


18 

16 

14 

12 

10 

8 

6 

4 

2 

0 

0 2 4 6 8 10 12 14 16 18 20 

Height (100s metres) 


c The greater the height above sea level, the lower the 

temperature. 

4 a, c 

Length (mm) 

94 

93.5 

93 

92.5 

92 

91.5 

91 

90.5 

90 

55 60 65 70 75 80 85 


b Positive. As temperature rises so does the length. 

d 0.14 

e For every rise of 10°C, the length increases by 

approximately 1.4 mm. 

5 a The higher the price, the fewer the number of cameras 

sold. 

692 


8/3/12 11:00:40 AM


b 

Number of cameras sold 

70 

60 

50 

40 

30 

20 

10 

0 

60 65 70 75 80 85 90 

Price (£) 

6 a As a car gets older, its value decreases. 

b 

5000 

Value in pounds (£) 

4000 

3000 

2000 

1000 

c £2450 

d 1.9 years 

0 

0 1 2 3 4 5 6 

Age in years 

7 a The larger the weight of a child, the greater its height. 

b 

Height (cm) 

170 

165 

160 

155 

150 

145 

140 

35 40 45 50 

Weight (kg) 

c 155.5cm 

55 

8 a, c 

Weight (kg) 

90 

85 

80 

75 

70 

160 170 180 190 

Height (cm) 

200 

b Positive correlation 

c The taller the athlete, the greater his or her weight. 

Using the line of best fit above, 74.2 kg 

9 a, b, c 

Weight loss (lb) 

80 

70 

60 

50 

40 

30 

20 

10 

0 

0 2 4 6 8 10 12 14 16 18 20 22 24 

Months taking drug 

c This is unlikely to be a genuine piece of data as the 

weight loss is so high in only 6 months. 

d Positive correlation 

e The drug is effective as the longer that the drug was 

taken, the greater the weight loss. 

10 a Positive correlation 

b The greater the amount of fertiliser used, the higher 

the crop yield. 

c 20 200 kg 

d 2.5 kg per 80 m 2 

e No, because there will be a limit on how high the crop 

yield can be, regardless of how much fertiliser is used. 

11 a, b 

3500 

Number of fleas 

c y k x 

3000 

2500 

2000 

1500 

1000 

500 

0 

0 2 4 6 8 

Day 

693 


22/08/12 12:00 PM


12 a For every 100 m rise in height above sea level there is 

1° drop in temperature. 

OR For every 1 m rise in height above sea level there is 

0.01° drop in temperature. 

b At sea level the air temperature is 19°. 



1 

__ 1 

16 2 __ 9 

25 3 8 

Exercise 25A 

1 a 1 b 

1_ 

8 c 1_ 

5 d 1 

e 1_ 

8 f __ 1 

81 g ____ 1 

10 000 h 1 

i 

1_ 

9 j 1 k 1 l ______ 1 

1 000 000 

2 a 3 b 

7_ c 49 d 64 

2 

e 16 f 15 5_ 

8 g 1 h 5_ 

9 

25 

i __ 

49 j __ 27 

k 10 000 l 125 

64 


1 a 1000 b 

2 10 4 

3 23 500 

Exercise 25B 

___ 1 

100 

1 a 7 10 5 b 6 10 2 c 2 10 3 

d 9 10 8 e 8 10 4 

2 a 600 000 b 10 000 c 800 000 

d 300 000 000 e 70 

3 a 4.3 10 4 b 5.61 10 5 c 5.6 10 1 

d 3.47 10 1 e 6 10 1 

4 a 39 600 b 68 000 000 c 8020 

d 57 e 9.23 

5 7 10 9 

6 4 10 4 

Exercise 25C 

1 a 5 10 3 b 4 10 2 c 7 10 6 

d 9 10 1 e 8 10 4 

2 a 0.000 06 b 0.08 c 0.000 000 5 

d 0.3 e 0.000 000 01 

3 a 4.7 10 3 b 9.87 10 1 c 8.034 10 4 

d 1.5 10 4 e 6.01 10 1 

4 a 0.000 084 3 b 0.0201 c 0.000 000 42 

d 0.7854 e 0.000 94 

5 a 4.57 10 5 b 2.3 10 3 c 3 10 4 

d 2.356 10 6 e 7.82 10 1 f 8.9 10 4 

g 2 10 2 h 5.26 10 3 i 6.034 10 3 

j 8.73 10 6 

6 a 0.000 412 b 3000 c 20 650 000 

d 0.000 004 e 327 000 000 f 0.75 

g 156.23 h 0.000 000 512 i 270 000 

j 0.612 

7 1 10 6 m 

8 6.25 10 2 mm 

Exercise 25D 

1 a 4.5 10 4 b 9.8 10 1 

c 3.4 10 1 d 1.86 10 12 

2 a 9 10 2 b 4.5 10 4 

c 3.708 10 13 d 6 10 10 

3 a In standard form 

b 8.9 10 8 c 1.32 10 4 d 5.6 10 8 

e 6 10 4 f In standard form 

g 4.005 10 12 h 9.08 10 18 

i In standard form j 4.6 10 5 

k 6.7 10 1 l 4 10 0 

4 6 290 000, 6.3 10 6 , 63.4 10 5 , 0.637 10 7 

5 0.000 033, 3.35 10 5 , 0.034 10 2 , 37 10 4 

Exercise 25E 

1 a 8 10 11 b 9 10 8 c 1.2 10 1 

d 4.3 10 5 e 2.5 10 8 f 1 10 4 

2 a 4 10 10 b 2.5 10 9 c 1.6 10 13 

d 4.9 10 15 

3 a 3.6 10 8 b 3.2 10 8 c 3.5 10 3 

d 1 10 14 

4 3.2 10 0 mm 2 

5 Time ________ Distance ________ 1.5 108 

Speed 3 10 5 5 102 seconds 

500 seconds 8 mins 20 secs 

6 160 m 

Exercise 25F 

1 a 2.1 10 8 b 2.4 10 5 c 5.2 10 0 

d 4.416 10 3 e 2.684 10 4 f 9.84 10 3 

g 9.6 10 13 h 9.84 10 17 i 9.6 10 15 

2 a 5.38 10 4 b 6.20 10 14 c 5.38 10 16 

d 6.20 10 6 

3 a 7.45 10 3 b 1.36 10 19 c 4.24 10 5 

d 3.28 10 12 

4 a 6 10 5 b 1.99 10 5 c 1.6 10 5 

5 1 : 30.1 

6 6.28 10 24 


1 10 2 2 3 3 

Exercise 25G 

1 a 3 b 7 c 10 

d 2 

e 

1_ 

2 

2 a 3 b 10 c 4 

d 5 

e 

1_ 

2 

3 a 

1_ 

2 b 1_ 

2 c 1_ 

5 

d 2 

e 

3_ 

2 

4 a 9 b 100 c 16 

d 8 e 125 

1 

5 a __ 

25 b ___ 1 

1000 c 1_ 

3 

d 

1_ 

4 e ___ 1 

512 f ___ 1 

625 

2 

g __ 

25 

6 a n 1 b n 6 c n 1_ 

2 

d n 5_ 

2 e n __ 11 

3 

694 


7/30/12 5:23:26 PM



1 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 

2 a 6 b 10 

3 √ __ 

9 , √ __ 

64 

Exercise 25H 

1 a 2 b 3 c 5 d 4 

2 a 10 √ __ 

2 b 4 √ __ 

2 c 2 √ __ 

5 d 2 √ __ 

7 

3 x √ __ 

30 

4 a 3 2 √ __ 

3 b 5 3 √ __ 

3 c 3 √ __ 

5 

d 5 √ __ 

7 e 7 4 √ __ 

3 f 27 10 √ __ 

2 

5 2 √ __ 

10 cm 

6 a 12 cm b 4 cm 2 

7 3 2 √ __ 

2 cm 2 

Exercise 25I 

√ __ 

1 a ___ 2 

√ __ 

b ___ 5 

√ __ 

c ____ 10 

2 

5 

2 

d √ __ 

2 √ __ 

3 

2 e ____ 

3 

2 a 1 √ __ 

2 b 1 3 √ __ 

2 c 1 2 √ __ 

5 

d 1 4 √ __ 

3 e 1 2 √ __ 

7 

3 3 √ __ 

____ 6 

2 c m 2 

4 a x 3 √ __ 

7 b x 5 √ __ 

11 

5 a 1.5 cm 2 b √ __ 

22 cm 


1 a 1 b 

1_ 

4 c 1 d 1_ 

8 

2 a 1 b 1 c 

1_ 

3 d 1 

3 a 3 b 3 c 

1_ 

2 d 8 

4 a 3 b 10 c 2 d 4 

5 a 

1_ 

3 b 1_ 

7 c 1_ 

5 d 1_ 

2 

6 a 2 b 

1_ 

2 

7 a Mars b 1.4 10 9 km 

c Saturn d Neptune 

e Minimum distance from Earth to Mars 

2.3 10 8 1.5 10 8 8 10 7 km 

Speed ________ 8 107 9132 km/h 

365 24 

8 a i 7.9 10 3 ii 3.5 10 4 

b 5 10 7 

9 9.3 10 8 

10 n 5_ 

2 

11 a k 3_ 

2 b m 16 c √ __ 

___ 2 

32 

12 9.43 10 12 

13 x 5.8 10 4 

14 a 1 10 9 b 2 10 8 

15 a x 2 b x 4 c x 3 d x 1_ 

2 

16 Using a 2 b 2 (a b)(a b) 

______ 1 

√ __ 

2 1 ________ 1 

√ __ 

3 √ __ 

2 ________ 1 

√ __ 

4 √ __ 

3 … ________ 1 

10 √ __ 

99 

______ √__ 2 1 

2 1 √__ 3 √ __ 

________ 2 

√__ 4 √ __ 

________ 3 

… 10 √ __ 

________ 99 

3 2 4 3 100 99 

1 10 9 



1 a 9 b 16 c 90 

2 a 196 b 625 c 384.16 

Exercise 26A 

1 720 cm 2 2 270 cm 2 3 12 cm 2 

4 1125 cm 2 

Exercise 26B 

1 a 18 b 15 

2 8 cm 


1 a 7 b 10 

2 a 15.625 b 1331 

Exercise 26C 

1 270 cm 3 2 135 cm 3 3 1327 cm 3 4 2.25 cm 3 

Exercise 26D 

1 3 cm 

2 a 18 b 7 

3 a 28 b 1.2 

4 1.95 cm 

5 10 cm 


1 2 : 9 2 4 : 5 3 1 : 6 4 1 : 1 

Exercise 26E 

1 1.6 cm 2 2 3.91 cm 3 3 2950 cm 2 

4 444 cm 2 5 a 292 cm 2 b 208 cm 2 

6 0.101 m 2 


1 2.25 cm 

2 1.456 m 

3 12.96 cm 2 

4 a 20 cm b 6 cm 

5 a 320 cm 2 b 75 cm 3 

6 a 8 cm b 96 cm 3 

7 3800 cm 2 

8 a 1 : 4 b 1 : 8 

9 a All the edges of a cube are the same length, so the 

ratio between the edges of two cubes is a constant. 

b 1 : 6.25 

c The edges of a cuboid can be in any ratio, so two 

cuboids may not have the same ratio between every 

pair of edges. 

10 a 11.9 cm b 25.6 cm 2 

695 


7/30/12 5:23:30 PM




1 a 50p b 75p c £1 d £1.25 

2 a 20 seconds b 40 seconds 

c 80 seconds 

d 160 seconds 

Exercise 27A 

1 a 

Y 

16 

14 

12 

10 

8 

6 

4 

2 

0 

0 2 4 6 8 10 

b Yes, the graph is a straight line through the origin 

2 k 0.4 

3 a 

9_ 

4 ___ 22.5 

10 __ 36 

16 __ 45 

20 b k 2.25 

c 72 

4 a Y 

x 

b Y 1_ 

2 t 

c Y 50 

5 a i £3.60 ii £5.40 iii £7.20 

b C 0.45v c 13 cm 3 

27.2 Get ready 

1 a x 18 b x 72 c x 75 

Exercise 27B 

1 k 1.5 

2 k 0.7 

3 a P h, so P kh 

When P 40.5, h 18, giving k 9_ 

4 

So P 9_ 

4 h 

b 72 c 12 

4 a V c, so V kc 

When V 10, c 0.04, giving k 250 

So V 250c 

b 0.036 amps 

5 a E ___ 7 m b 21 mm c 1125 g 

150 

6 66 2_ 

3 cm3 


1 a V kT b P kT 

x 

Exercise 27C 

1 a i M n ii M kn 

b i L h 2 ii L kh 2 

c i P t 3 ii P kt 3 

d i Q √ __ 

y ii Q k √ __ 

y 

e i W √ 3 __ 

x ii W k √ 3 __ 

x 

f i A __ 1 b 

g i H __ 1 

g 2 

h i U 1 __ 

f 2 

i i E 1 ___ 

w 3 

j i V 1 ___ 

√ _ r 

ii A __ k b 

ii H __ k 

g 2 

ii U k __ 

f 2 

ii E k ___ 

w 3 

ii V k ___ 

√ _ r 

2 a N d 2 b N kd 2 

3 H kc 2 

4 Q kt 3 

5 F is inversely proportional to the square of r 

6 T is proportional to the square root of l 


1 a 225 b 512 c 1331 d 9 e 8 f 64 

Exercise 27D 

1 0.8 

2 16 

3 Z 1.28b 2 

4 L 1.875v 3 

5 4.5 

6 a 11.25 m b 14.1 m/s 

7 2.51 mm 

8 

Mass (grams) Length (cm) 

27.5 Get ready 

1 a P 1 __ 

T 

Exercise 27E 

20 5 

34.56 6 

43.94 6.5 

50 6.79 

1 45 

2 a 2.5 b 6 

3 a 3.6 b 2.67 

4 2.67 m 3 

5 a 384 hertz b 0.96 m 

6 a 3.27 b 6.32 

7 a S ____ 8000 

f 2 b 500 


1 0.75 0.5 2.25 1.5 5.25 3.5 6.75 

4.5 9.0 6.0 1.5 

2 a 960 b 560 

696 


7/30/12 5:23:36 PM


3 a T b b R is directly proportional to a 

c P m 2 d Z g 3 

e H is inversely proportional to y 

4 a S 1.5p 

b S 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

O 

1 2 3 4 5 6 7 8 9 10 

5 £70 125 

6 Yes, the pressure will be 6 bars. 

7 10.2 units 

8 a 

D__ 

t 2 __ 5 1 2 __ 20 

2 2 5 

9 

t 0 1 2 3 4 5 

D 0 5 20 45 80 125 

b 1.73 seconds 

Radius (r mm) 

p 

Resistance (R ohms) 

10 500 

15 222.22 

17.5 163.27 

14.14 250 

10 a X √ __ 

h , so X k √ __ 

h 

When X 3, h 16, giving k 0.75 

So X 0.75 √ __ 

h 

b 3.75 c 64 

11 a T ____ 1 

√ m , so T ____ k 

√ m 

When T 20, m 50, giving k 100 √ __ 

2 

__ 

So T 100 √ 

__ 2 

m 

b 88.9 g 

12 a A b C 

13 a 540 b 1.82 

14 a q ___ 136 b 5.44 

t 2 

15 51.6 

16 a y kz 

b u kv 2 v ___ l 

√ w 

v 2 __ l2 

w 

u k __ l2 

w 

uw kl 2 

kl 2 is a constant so uw is a constant. 



1 

0 1 1 

2 

Exercise 28A 

1 a 

2_ 

2 a 

d 

3 a 

4 a 

d 

1 

5 ___ 

500 

6 a 

7 a 

d 

8 a 

b i ___ 91 

181 

9 a 

1_ 

10 0.3 

2 

0 1 1 

2 

7 b 4_ 

7 c 1_ 

7 

1_ 

6 b 1_ 

2 c __ 1 

13 

1_ 

4 e __ 1 

26 

5_ 

9 b 4_ 

9 c 0 

1_ 

2 b 1_ 

2 c 1_ 

4 

1_ 

2 e 3_ 

8 

3 

0 1 1 

2 

1_ 

3 b 4_ 

9 c 2_ 

9 d 2_ 

3 

__ 2 

11 b __ 1 

11 c __ 3 

11 

__ 4 

11 e 0 

Left-handed 

Righthanded 

Total 

Boys 47 135 182 

Girls 61 119 180 

Total 108 254 362 

54 

ii ___ 

181 

__ 

12 

6 b 7 

119 

iii ___ 

362 

Exercise 28B 

1 a i 1 

36 ii 1_ 9 iii 1_ 6 b 1_ 

6 c 1_ 

9 

2 a 

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

Coin 

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

1 2 3 4 5 6 

Dice 

b i 1 

12 ii 1_ 2 iii 1_ 6 

3 a 

Spinner A 

b i 1_ 

4 

Spinner B 

1 2 3 4 

1 0 1 2 3 

2 1 0 1 2 

3 2 1 0 1 

4 3 2 1 0 

ii 

1_ 

8 iii 0 

697 


7/30/12 5:23:45 PM


4 a 

Spinner 

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

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

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

1 2 3 4 5 6 

Dice 

b i 1_ 6 ii 1_ 9 iii 1_ 3 

5 a R (R,R) (Bu,R) (Y,R) (Ba,R) 

Box B 

Y (R,Y) (Bu,Y) (Y,Y) (Ba,Y) 

6 3_ 5 

7 a 

Bu (R,Bu) (Bu,Bu) (Y,Bu) (Ba,Bu) 

R Bu Y Ba 

Box A 

b i __ 1 

12 ii 1_ 4 iii 3_ 4 

d 

1_ 

6 b __ 1 

10 c __ 1 

60 

__ 1 

20 e __ 1 

12 


1 a 0.4 b 0.65 c 

Exercise 28C 

3_ 

5 d 1_ 

6 

3 

1 a __ 

10 b 2_ 

5 c __ 7 

10 

2 0.9 

3 a 0.5 b 0.5 c 0.65 d 0.7 

3 

4 a __ 

13 b __ 3 

13 c __ 1 

13 

6 

d __ 

13 e __ 4 

13 f __ 4 

13 

5 a 

1_ 

2 b __ 7 

18 c 2_ 

9 d 2_ 

3 

1 

6 a __ 

26 b 1_ 

2 c __ 5 

52 

7 

d __ 

__ 

26 e 7 

26 

7 0.35 

8 The events are not mutually exclusive. The probability is __ 1 

Exercise 28D 

1 11 

72 

2 4_ 5 

3 0.35 

4 0.25 

1 

5 a __ 

13 b __ 12 

13 

6 a i 0.65 ii 0.85 

b 0.35 0.15 0.5 1 

7 a 

2_ 

9 b 7_ 

9 c 5_ 

9 

8 0.6 


1 a 

1_ 

2 b 3_ 

4 c 8_ 

Exercise 28E 

9 

1 __ 

20 

2 __ 13 

15 

9 d __ 3 

13 

36 . 

___ 77 

150 

3 a 

b No, red is almost three times as likely as orange. 

4 a Students’ own work 

b Drop the drawing pin more times, because the greater 

the number of trials, the more accurate the estimated 

probability. 

___ 33 

200 

5 , because the greater the number of trials, the more 

accurate the estimated probability. 

6 Students’ own work 


1 a 50 b 18 c 30 d 40.5 

Exercise 28F 

1 50 

2 10 

3 30 

4 

Colour of peg Red White Yellow 

Expected number 25 125 100 

5 30 

6 No, the draw is random so he could be unlucky. 

7 a 20 b 65 c 80 

8 The doctor’s estimate is a bit high, as the results from the 

240 patients suggests a probability of 0.083. 


1 a 

1_ 

6 b 2_ 

7 c 2_ 

5 d __ 3 

16 

Exercise 28G 

1 0.32 

2 1_ 2 

3 0.6 

4 a 0.65 b 0.85 c 0.5525 

5 a 

1_ 

4 b __ 1 

16 c ___ 3 

169 

1 

d ___ 

676 e ___ 1 

2704 

6 a __ 3 

__ 

__ 

44 b 2 

11 c 15 

22 


1 a 

3_ 

4 b __ 8 

15 c __ 13 

20 d __ 55 

63 

Exercise 28H 

1 a Bag A Bag B Outcomes 

b i __ 12 

3 

5 

2 

5 

white 

blue 

35 ii __ 6 

35 

4 

7 

3 

7 

4 

7 

3 

7 

white 

blue 

white 

blue 

18 

iii __ 

35 

ww 

wb 

bw 

bb 

12 

35 

9 

35 

8 

35 

6 

35 

698 


05/09/12 5:21 PM


2 a 

b 

3 a 

__ 21 

50 

First pencil Second pencil Outcomes 

3 

10 

7 

10 

HB 

not HB 

3 

10 

7 

10 

3 

10 

7 

10 

HB 

not HB 

HB 

not HB 

HB, HB 

HB, not HB 

not HB, HB 

not HB, not HB 

Ryan Ibrahim Outcomes 

3 

8 

5 

8 

orange 

red 

orange 

red 

orange 

red 

orange, orange 

orange, red 

red, orange 

red, red 

b i __ 3 ii __ 19 

20 40 

4 a i 0.6 ii 0.3 

b 1st lights 2nd lights Outcomes 

0.4 

0.6 

stopped 

not 

stopped 

0.7 

2 

5 

3 

5 

2 

5 

3 

5 

stopped 

stopped, 

stopped 

9 

100 

21 

100 

21 

100 

49 

100 

3 

20 

9 

40 

1 

4 

3 

8 

0.3 not stopped stopped, 0.12 

not stopped 

0.7 

0.3 

stopped 

not stopped 

not stopped, 

stopped 

not stopped, 

not stopped 

c i 0.28 ii 0.54 iii 0.82 

5 a Spinner A Spinner B Outcomes 

6 

12 

7 a 

0.3 

0.7 

6 

not 6 

0.45 

6 6, 6 

0.28 

0.42 

0.18 

0.55 not 6 6, not 6 0.165 

0.45 

0.55 

6 

not 6 

b i 0.385 ii 0.48 iii 0.315 

__ 5 

d 

___ 1 

169 b __ 9 

16 c 3_ 

8 

___ 51 

1352 e __ 7 

16 

8 a HHH HTH HHT HTT 

THH TTH THT TTT 

b i 1_ 

8 

9 0.243 

10 54 


1 3_ 9 1_ 

3 

ii 

3_ 

8 

not 6, 6 

not 6, not 6 

0.135 

0.315 

0.385 

Exercise 28I 

1 b i __ 7 

15 ii __ 8 

15 iii __ 7 

15 

__ 

33 

35 

ii 66 

15 

iii 22 

__ 3 

14 

___ 121 

b 4_ 

7 

2 b i 5 

3 a 

4 

315 

5 0.74 

6 0.1325 

7 a 

2_ 

7 b 2_ 

7 

1 

8 a ___ 

221 b __ 1 

17 c __ 13 

17 

37 

9 __ 

49 

10 0.7816 


4 

1 a __ 

11 b __ 4 

11 c __ 7 

11 

7 

d __ 

11 e __ 8 

11 f __ 3 

11 

23 

2 a __ 

45 b __ 3 

10 c __ 8 

45 

37 

d __ 

90 e __ 8 

45 

7 

3 __ 

18 

4 a 

1_ 

6 b 1_ 

4 c __ 11 

24 

5 a 97 or 98 b 7 or 8 c 30 

6 Roll the dice 100 times, recording the results. If the dice 

is fair, each number should occur approximately 17 times. 

If Amy wanted to be more confident in her results, she 

could roll the dice more times. 

7 90 

8 a If Megan bought all the tickets, she is definitely right. 

Otherwise, she could be right, but there is no way of 

knowing because the winning tickets are drawn at 

random. 

b The winning tickets are chosen at random, so she may 

just have been lucky. 

__ _____ 

9 a i b 

12 

10 

3_ 

4 

ii 

(12 b) 

12 

b 3 

11 0.5525 

12 a 96 

b The original set of marked rabbits came from 

throughout the warren. The marked rabbits spread 

throughout the warren after being returned. The dye 

did not come off any rabbits. 

___ 5 

144 

13 

14 a 

b 

__ 2 

15 

Sarah 

4 

9 

5 

9 

Daffodil 

Hyacinth 

Jim 

3 

10 

7 

10 

3 

10 

7 

10 

Daffodil 

Hyacinth 

Daffodil 

Hyacinth 

15 a No. Horse C is more likely to win, because of the four 

possible outcomes, two are a head and tail, but two 

heads and two tails are each only one outcome. 

699 


7/30/12 5:23:54 PM


b No. Horse 7 is more likely to win, because of the 36 

possible outcomes from rolling two dice, a score of 7 

occurs more often than any other score. 

16 Number of possible selections of six numbers 

_________________ 

40 39 38 37 36 35 

4 million 

6 5 4 3 2 1 

So P(choosing correct six numbers) 1 in 4 million 

17 ( 1_ ) 6 ____ 1 

2.1 105 

6 46 656 

18 

__ 

4 

19 

1_ 

5 

1 

20 a __ 

12 b __ 5 

18 c __ 13 

18 

21 Students’ investigations 



1 ABF, ABE, AEF, BEF, BCF, BCG, BFG, CFG, CDG, CDH, CHG, 

DHG, ADE, ADH, AEH, DEH, ABC, ABD, ACD, BCD, EFG, EFH, 

EGH, FGH 

Exercise 29A 

1 a i 8.94 cm ii 13.6 cm iii 15.3 cm iv 15.8 cm 

b i 58.4° ii 72.9° 

2 AG 2 AC 2 CG 2 AB 2 BC 2 CG 2 

3 No. The longest needle that could fit in the box is 14 cm. 

4 5 cm 

5 a i 14.1 cm ii 7.07 cm 

b 13.2 cm c 61.9° d 56.2° e 14.1 cm 

f 70.5° 


The diagrams could show the same pole viewed from different 

sides. 

Exercise 29B 

1 69.3° 

2 a i 51.3° ii 49.1° iii 38.7° b 90° 

3 a 470 m b 171 m c 985 m d 848 m 

4 a 15.6 m 

b i 26.6° ii 36.9° iii 22.6° 

5 a 86.5 cm b 63.9° 

6 a 

R 

P 

6 cm 

O 

6 cm 

b i 45° ii 45° 

c 1692 cm 3 


6 cm 

1 a (u, v) b (u, v) c (u, v) 

2 a 0.5 and 0.5 b 0.64 and 0.64 

Q 

c sin 30° and sin 150° give the same answer. 

cos 50° and cos 130° give the same value, but opposite 

signs. 

Exercise 29C 

1 a 

b 

y 

1 

0.5 

360 180 O 180 360 

0.5 

1 

y 

1 

0.5 

360 180 O 180 360 

0.5 

1 

2 a 30, 150 b 84.3, 275.7 

3 a sin ° 1_ so 19.5 

3 

b 19.5, 160.5, 379.5, 520.5 

4 a cos ° 0.3 so 107.5 

b 252.5, 107.5, 107.5, 252.5 


1 a 30 cm 2 b 54 cm 2 c 18.4 cm 2 

Exercise 29D 

1 a 21.9 cm 2 b 29.2 cm 2 c 15.4 cm 2 

d 30.0 m 2 

2 16.8 m 

3 33.3° 

4 a 11.4 cm 2 b 11.4 cm 2 

c The answers are the same because sin 25° sin 155°. 

5 a 45° b 12.7 cm 2 c 102 cm 2 

6 3.41 cm 2 


1 a 0.866 b 13.6 

Exercise 29E 

1 a 8.06 cm b 7.19 cm c 6.35 cm 

d 9.01 cm e 15.0 cm 

f f 6.06 cm, g 11.4 cm 


1 35.2° 

Exercise 29F 

1 a 45.0° b 63.6° c 23.6° 

d d 43.3°, e 63.7° 

2 a 13.3 cm b 39.3° c 154.7° 

θ 

θ 

700 


8/3/12 11:00:46 AM


3 a 9.72 cm b 9.89 cm c 37.9 cm 2 

4 a 66.0° b 7.63 cm 

5 a i 42° ii 55° 

b 9.79 km c 11.9 km d 6.55 km 


1 a 11 b 3 c 22.4 

2 3.82 

Exercise 29G 

1 a 8.80 cm b 12.6 cm c 5.01 cm 

d 8.42 cm e 15.3 cm f 16.1 cm 

2 12.6 cm 


1 a 50.8° b 72.3° c 54.8° 

Exercise 29H 

1 a 54.7° b 81.2° c 46.0° d 131.2° 

2 103.6° 

3 a 68° b 91.0 m 

4 a 22.4 cm b 127.7° c 161 cm 2 

5 16.7 km 

6 7.71 cm, 11.9 cm 


1 a cosine rule b sine rule 

c sine rule or cosine rule 

Exercise 29I 

1 a 70.5°, 59.0°, 50.5° b 42.4 cm 2 

2 a 9.68 cm b 42.9° c 17.7 cm 

3 a 7.32 cm b 6.65 cm c 78.7° 

4 a 6.76 cm b 75° c 13.1 cm d 19.9 cm 

5 a 039° (or 321°) b 148° (or 212°) 

6 50° 

7 a 70° b 4.43 m 

8 a 73.2° b 26.2 m c 67.3° d 22.6° 

9 20.5 km 


1 13 cm 

2 85.5° 

3 5.89 cm 

4 a 9.11 cm b 19.2° 

5 18.3 cm 

6 76.3° 

√ __ 

3 

7 a ___ b ___ √__ 3 

2 

2 

8 42° and 318° 

9 10.9 cm 2 

10 a 105.7° b 10.7 cm 2 

11 22.2° 

12 a 21.8 cm b 66.4 cm 2 

13 11.1 cm 

14 a 41.4° b 17.8 cm 

15 a 4.51 cm b 37° 



1 18 2 x __ 11 

4 

Exercise 30A 

3 y (t 1)2 

1 a 12 b 0 c 48 

d 1 e 4 f 8 

2 a 4 b 28 c 96 

d 6.4 

3 a 8 b 2 

4 a i 4 ii 3 iii 0 b 4, 4 

5 a i 18 ii 12 iii 30 b 3, 4 

6 a i 3 ii 4 iii 10 b 0, 4 c 1, 5 

7 a 16 b x 2 4 c (x 4) 2 

8 a 8 b 4(2x 1) c 12(x 1) 


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

Exercise 30B 

1 a 

b 

c 

y 

12 

10 

8 

6 

4 

O 

6 4 2 

2 

2 

y 

12 

10 

8 

6 

4 

2 

6 4 2 

O 

2 

4 

y 

10 

8 

6 

4 

2 

6 4 2 

O 

2 

4 

2 

2 

2 

4 6 

4 6 

4 6 

x 

x 

x 

701 


7/30/12 5:24:07 PM


d 

2 a i 

ii 

y 

10 

8 

6 

4 

2 

6 4 2 

O 

2 

4 

y 

4 

y f(x) 3 

2 

2 

2 

O 

2 

4 

y 

O 

2 

4 

b i (0, 3) ii (1, 0) 

3 a 

y 

y f(x) 2 

4 

2 

2 

b (2, 2.5) 

c i y f(x) 2 

d 

y 

y f(x 2) 

20 

15 

10 

5 

4 2 

O 

5 

10 

15 

20 

2 

4 6 

y f(x) 

2 

x 

x 

y f(x) 

y f(x 1) 

4 

2 

O 

2 

4 

2 

x 

y f(x) 2 

2 

x 

ii y 1 __ 

x 2 

y f(x) 

2 

4 

x 

7 a a 2, b 5 b Translation of ( 2 5) 

c, d 

y x 2 4x 9 

8 Translation by ( 

0 

6) 


1 

2 

y 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

(2, 5) 

0 

0 1 2 3 4 5 6 7 8 9 10 

y 

O 

y x 2 

The first triangle has been stretched by a factor of 2 parallel 

to the x-axis. 

y 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

0 1 2 3 4 5 6 7 8 9 10 

The first triangle has been stretched by a factor of 3 parallel 

to the y-axis. 

x 

x 

x 

1 

e p 1, q 4 f i y f(x 2) ii y ____ 

4 a (3, 2) b (1, 5) 

5 a Translation by 3 units parallel to the y-axis. 

b y f(x) 3 c y 4x x 2 3 

6 a Translation by 2 units parallel to the x-axis. 

b y f(x 2) 

c y (x 2) 2 

x 2 

702 


7/30/12 5:24:20 PM


Exercise 30C 

1 y f(2x) 

10 

b 

y 

4 

8 

2 

6 

y f(x) 

4 

2 

O 

2 4 

x 

2 

(2, 2) is mapped to the point (1, 2) 

2 a 

y y 4f(x) 

c 

O 

2 

y 

4 

90 

180 270 360 450 540 

x 

2 

y f(x) 

O 

x 

b y 4x 2 

c Stretch of magnitude 4 parallel to the y-axis 

Stretch of magnitude 1_ parallel to the x-axis 

2 

3 a 

2 

1 

b 

4 a 

O 

1 

2 

y 2 cos x ° 

2 

1 

O 

1 

2 

90 180 270 360 450 540 630 720 

90 180 270 360 450 540 630 720 

y cos(2x°) 

y 

20 

10 

2 1 

O 

10 

20 

b (0, 4) is mapped to (0, 12). ( 2_ 

3 

5 a y 

4 

2 

1 

2 

x 

x 

x 

, 0 ) is mapped to ( 

2_ 

3 , 0 ) 

6 a 

O 

2 

2 

1 

O 

1 

2 

90 180 270 360 450 540 

x 

90 180 270 360 450 540 630 720 810 900 990 1080 

b 1 solution 

7 a Stretch of magnitude 2 parallel to the x-axis. 

b y 

4 

O 

4 

2 4 6 8 

c i y f ( 

__ x 2 ) ii y __ x __ 

2 ( x 2 4 ) 

8 a p 4, q 11 

b (0, 5) 

c Stretch parallel to the y-axis of magnitude 0.6. 

0.6 (x 2 8x 5) 

Exercise 30D 

1 a y 

2 

1 

O 

1 

2 

90 180 270 360 x 

x 

x 

O 

2 

90 

180 270 360 450 540 

x 

b y 2 sin ( x __ 

2 ) 

703 


05/09/12 5:22 PM


2 a y 

14 

8 7 6 5 4 3 2 1 O 1 2 3 4 5 

2 

b y 2f ( x __ 

2 ) 

c 

12 

10 

8 

6 

4 

2 

4 

y 

14 

12 

x 


1 

y 

6 

5 

4 

3 

2 

1 

6 5 4 3 2 1 

O 

1 

2 

3 

4 

5 

5 

1 

2 3 4 5 6 

x 

10 

8 7 6 5 4 3 2 1 O 1 2 3 4 5 

2 

8 

6 

4 

2 

4 

x 

2 

y 

6 

5 

4 

3 

2 

1 

3 a, b 

y 

3 

2 

1 

O 

1 

90 180 270 360 450 540 630 720 810 900 990 1080 x 

6 5 4 3 2 1 

O 

1 

2 

3 

4 

5 

5 

1 

2 3 4 5 6 

x 

2 

3 

3 

y 

6 

4 a 4 __ 

( x2 

16 3 ) b 4 __ 

( 4 x 1 ) 

c 4 2 x_ 

4 

2 x_ 

4 2 d 8 sin ( x __ 

2 ) 

5 

4 

3 

2 

1 

6 5 4 3 2 1 

O 

1 

1 

2 3 4 5 6 

x 

2 

3 

4 

5 

5 

704 


05/09/12 5:22 PM


Exercise 30E 

1 a 

y 

4 a 

y 

6 

5 

y f(x) 2 

4 

O 

x 

3 

2 

1 

b y (x 2 3) 

2 a 

y 

y f(x) 

y f(x) 

5 4 3 2 1 O 

1 

2 

3 

4 

1 2 3 4 5 6 7 8 

x 

3 a 

O 

b p 2, q 8 c y x 3 

4 

y f(x) 

2 

b (r 1, t 2) 


y 

10 

5 

O 

5 

10 

x 

2 4 

c y 2 x 

y f(x) 

1 a 6 b 11 c a = 0 

2 f(x 1) (x 1) 2 3(x 1) x 2 x 2 

(x 2)(x 1) 

3 

y 

y x 2 

x 

b 

y 

6 

5 

4 

3 

2 

1 

5 4 3 2 1 O 

1 

1 2 3 4 5 6 7 8 

2 

y f(x) 

3 

4 

5 f(x 4) 

6 a, b y 

5 

b 

4 

3 

a 

2 

1 

x 

y x 2 

x 

2 1 O 1 2 3 

1 

c x 1 and x 2 

7 a (180, 0) 

b y 

x 

O 

A 

x 

705 


7/30/12 5:24:33 PM


8 y 

4 

3 

2 

1 

O 

1 

2 

3 

4 

90 180 270 360 x 

9 Translation by ( 

4 

5) 

10 a, b Stretch parallel to the y-axis of magnitude 4 

Stretch parallel to the x-axis of magnitude 0.5 

11 a p 2, q 12 

b (2, 12) 

c Stretch parallel to the y-axis of magnitude 2 

d y 2f(x) 



a 20° b 65° c 40° 

Exercise 31A 

1 a 25° 

2 b 33° 

3 c 40° 

4 d 19° (angles on a line, isosceles triangle in a circle) 

5 e 60°, f 40°, g 100° (equilateral triangle, isosceles 

triangle in a circle) 

6 h 54°, i 126°, j 27° (right-angled triangle, angles on a 

line, isosceles triangle in a circle) 


1 a 33°, b 57°, c 90° 

Exercise 31B 

1 a 50° 2 b 59° 

3 c 136° 4 d 66°, e 24° 

5 f 60° 6 g 107° 


1 a 28° b 49° c 57° 

Exercise 31C 

1 a 83° 

2 b 90°, c 52° 

3 d 64°, e 38° 

4 f 110°, g 55° 

5 h 30° 

6 i 72° 

7 j 49° 

The line drawn from the centre of a circle to the midpoint 

of a chord is perpendicular to that chord. 

8 k 68° 

The angle at the centre of a circle is twice the angle at the 

circumference, both subtended by the same arc. 

9 l 90° 

The angle in a semicircle is a right angle. 

10 a angle DAB 65° 


circumference, both subtended by the same arc. 

b reflex angle DOB 230° 

Angles about a point add up to 360° 

c angle BCD 115° 


circumference, both subtended by the same arc so reflex 

angle BOD is twice angle BCD. 


1 a 93° b 94° c 46° 

Exercise 31D 

1 a 40° 

2 b 82° 

3 c 38° 

4 d 50°, e 100°, f 40° 

5 g 58° 

6 h 74° 

7 i 71° 

Angles in the same segment are equal so angle SQR 31°, 

and angles in a triangle add up to 180°. 

8 j 35° 

Opposite angles of a cyclic quadrilateral add up to 180° and 

the base angles of an isosceles triangle are equal. 

9 k 57° 

The angle between a tangent and a chord is equal to the 

angle in the alternate segment and the angles on a straight 

line (or in a triangle) add up to 180°. 


1 angle OAD 90° (angle between tangent and radius) 

angle AOD 180 90 36 54° (angle sum in a 

triangle) 

angle ABC 1_ 54 27° (angle at centre is twice angle 

2 

at circumference) 

2 a i 140° 

ii The angle at the centre of a circle is twice the angle 

at the circumference, both subtended by the same 

arc. 

b i 110° 

ii The opposite angles of a cyclic quadrilateral add up 

to 180°. 

3 a 134°, b 42° 

4 d 90°, e 67°, f 67° 

5 g 33°, h 33° 

6 o 51° (angles in the same segment are equal) 

p 26° (angles in an isosceles triangle add up to 180°; 

angle between a tangent and a chord is equal to the angle 

in the alternate segment). 

7 s 94° (angles on a straight line add up to 180°; opposite 

angles of a cyclic quadrilateral add up to 180°) 

706 


7/30/12 5:24:37 PM


8 t 52° (angle between a tangent and a radius is a right 

angle; angles in a triangle add up to 180°) 

u 26° (angle at the centre of a circle is twice the angle 

at the circumference, both subtended by the same arc) 

9 angle CBT angle CBA (BC bisects angle ABT) 

angle CBT angle CAB (angle between tangent and 

chord is equal to angle in alternate segment) 

angle CBA CAB 

So triangle CAB is isosceles and CA CB 

10 angle PMA 1_ 132 66° (angle at the centre is twice 

2 

angle at circumference) 

angle MAT 66° (alternate angles, PM parallel to AT) 

angle OAT 90° (angle between tangent and radius) 

angle OAM 90 66 24° 

OA OM (radii), so triangle AOM is isosceles 

angle OMA 24° (base angles of isosceles triangle) 

11 OR OP (radii), so triangle ORP is isosceles 

angle OPR ________ 180 20 

80° (angle sum of isosceles 

2 

triangle) 

OQ OP (radii), so triangle OQP is isosceles 

angle OPQ _________ 180 100 

40° (angle sum of isosceles 

2 

triangle) 

So angle OPQ 1_ angle OPR and QP bisects angle OPR 

2 



1 a (x 4)(x 1) 

b (x 3)(x 2) 

c (2x 1)(x 3) 

Exercise 32A 

1 a 2x 3 b 

e 2x 

2 a _____ x 1 

x 2 

b 

3 a _____ x 1 

x 

b 

4 a ______ 2x 3 

3x 2 

b 

5 a _____ x 5 

x 5 

b 

d _____ 3 x 

x 3 

e 


___ x 

3y 

_____ x 1 

x 

_____ 4x 

x 6 

______ 5x 3 

3x 2 

_______ x 5 

2(x 5) 

2 x _____ 

x 2 

c _____ x 5 

2 

c _____ x 2 

x 4 

2(x 2) 

c _______ 

x 2 

c ______ 3x 1 

3x 1 

c ______ 2x 1 

2x 

f (x 4) 

d x 

1 a 30 b 12x c x(x 1) 

Exercise 32B 

1 a x b 2x c 

e 

x__ f ___ 5 

5 

3x 

2 a ___ 7x 

b 

x__ 

12 

3 

c 

e ___ 5 

f ___ 1 

6x 

4x 

3 a ______ 5x 2 

b ______ 9x 7 

6 

20 

c 

7 ___ 

10x 

3x ___ 

8 

x__ 

9 

d 

d 

d 

d 

d 

_____ x 4 

x 3 

_____ x 3 

x 

______ x 1 

2x 3 

4x ___ 

9 

5x ___ 

6 

d 

f 

____________ 2x 5 

(x 2)(x 3) 

______________ 4 

(2x 1)(2x 3) 

Exercise 32C 

e 

x 2 ____________ 

(x 2)(x 1) 

1 a i 2(x 1) ii 6(x 1) 

b 6(x 1) 

c _______ 2 

3(x 1) 

2 a 15x b (x 2)(x 3) 

c x(x 1) d (x 1)(x 2) 

e 2(x 3) f x(x 1) 

3 a (x 1)(x 2) b ____________ x 

(x 1)(x 2) 

4 a (x 2)(x 2) b ____________ 3x 4 

(x 2)(x 2) 

5 a (2x 1)(x 1) b ______ 1 

(x 1) 

6 _____________ x 1 

2(x 1)(x 3) 

7 ___________ 

2x 2 7x 4 

8x(x 1) 

8 ____________ 3x 

(3 x)(3 x) 

9 a i (x 4)(x 5) ii (x 5)(x 6) 

b _________________ 

3x 20 

(x 4)(x 5)(x 6) 

10 ____________________ 

4 

(2x 1)(2x 1)(2x 3) 


1 a a b 

Exercise 32D 

1 a __ x 2 

15 

xy 

2 a ___ 

2 

3 a 

5__ 

9 

4 a __ 27 

4 

(x 1) 

5 a _______ 

2 

2 

d 

1__ 

6 

b 

b 

b 

b 

__ 12 

y 2 

___ 2x 

9 

____ 5xy 

18 

___ 5x 

8 

1_ 

2 c _____ x 3 

x 1 

c ____ 15xy 

8 

c 

6__ 

y 

c x²y² 

c 

b (x 2)(x 1) 

e 3(3x 1) 

6 a (x 2)(x 2) b 

d 

d 

d 

_______ x(x 3) 

12 

_______ 2(x 1) 

x 1 

________ 2x(x 2) 

(x 1) 2 

___ 7x 

2y 2 d _____ 2x 

x 5 

c _____ x 

______ x 2 

x 2 4 

7 a i (x 1)(x 4) ii (x 2)(x 4) 

(x 2)(x 3) 

b ____________ 

(x 1) 2 

8 x 1 


f 

x 2 

_______ 3(x 4) 

4 

1 a even b either c odd d either 

e either 

Exercise 32E 

1 (2n 1) 2m 2(m n) 1 

707 


7/30/12 5:24:42 PM


2 1_ 1_ 

[n (n 1) (n 2) (n 3)] [4n 6] 2n 3 

2 2 

3 n (n 1) (n 2) 3n 3 3(n 1) 

4 a (2n 1)2m 2[m(2n 1)] 

b (2n 1)(2m 1) 4mn 2m 2n 1 

2(2mn m n) 1 

c (2n)(2m) 4mn 2(2mn) 

5 (m n)(m n) m² nm mn n² m² n² 

6 (n 4)² n² n² 8n 16 n² 8n 16 8(n 2) 


1 n (n 1) (n 2) (n 3) 4n 6 

(n 3)(n 2) n(n 1) (n 2 5n 6) (n 2 n) 

4n 6 

2 a 

e 

3 a 

d 

f 

__ x 2 

2 

_____ x 

x 3 

x__ 

2 

b 

f 

b 

_______ (x 1) 2 

3 

_____ x 5 

x 5 

13 ___ 

6x 

____________ 2x 7 

(x 1)(x 2) 

____________ 2 

(x 1)(x 5) 

4 a 2 b x c 

e 

____________ x 1 

(x 1)(x 3) 

c x 

g 

c 

e 

_______ x 1 

x(x 1) 

d 

h 

______________ 10x 

(5x 3)(5x 3) 

_______ x 2 

(x 1) 2 

2__ 

5 

f 1 g 

_____ x 1 

x 2 

x 4 ______ 

3x 2 

d ___ 2s 2 

3 

2_ 

3 h ______ 2x 1 

x 

5 a (n 1)(n 2) n(n 1) n² 3n 2 n² n 

2n² 4n 2 

2(n² 2n 1) 

2(n 1)² 

b 2(n 1)² is always even. 

6 (2n 1)² (2n 1)² (4n² 4n 1) (4n² 4n 1) 

8n 

100 (x 

7 __________________ 

2 16x 64) 

____________ 

36 16x x2 

_____________ 

(2 x)(18 x) 

4 

4 

4 

8 a 4 2 2, 6 4 2, 8 6 2,… 

The difference between each pair of consecutive even 

numbers is 2. Therefore if the nth even number is 2n, 

the next even number must be 2n 2. 

b 2n (2n 2) (2n 4) 6n 6 6(n 1), which 

must be a multiple of 6 

9 (3n 1) 2 (3n 1) 2 (9n 2 6n 1) (9n 2 6n 1) 

12n 4 3n, which must be a multiple of 4 

16(2x 1) 

10 _________ 

x(x 1) 



1 (2, 2) 

Exercise 33A 

1 

a b c d B e D 

a 

b 

c 

A 

C 

2 a i ( 5 6) ii ( 12) iii ( 4 6) 

3 a 

b They add up to ( 0 0) . 

A 

B 

b ( 9 3) 

c trapezium 

d They are parallel and the length of AD is 3 times the 

length of BC. 

4 a y 

A 

D 

B 

C 

O 

x 

b i (0) 

4 ii ( 2 

3) 

c i They are the same. 

ii They are parallel with the same length, but in opposite 

directions. 

5 a and c, b and h, d and g 


1 a 25 b 9.43 

Exercise 33B 

1 a 13 b 13 

e 17 f 4 √ __ 

5 

c √ __ 

10 d √ __ 

74 

2 a 25 

b length AC √ ________ 

(7 2 24 2 ) 25, so AB AC and the 

triangle is isosceles. 

3 rhombus 


y 

4 

3 

2 

1 

A 

1 O 1 2 

1 

C 

Translation by ( 1 2) 

B 

3 4 

x 

C 

D 

708 


7/30/12 5:24:51 PM


Exercise 33C 

1 

d 

a b 

a 

b 

a 

b 

2 a 

b 

1 

2 p 

a b 

2 a ( 6 8) b ( 4 8) c ( 2 4) d ( 

9 

5) e ( 8 

3) 

3 ( 

5) 

10 

4 a i ( 4 3) ii ( 4 3) b They are the same. 

c i ( 

10) 8 ii ( 

10) 8 

___ › 

___ › 

d They are the same. 

5 a ED is parallel to AB , in the same direction and has the 

same length. 

b i n m ii n m p c n m 


1 a ( 

10) 6 

b ( 9 

3) c (0) 

0 

Exercise 33D 

1 a 

c 

b 

a 

b 

b 

2p 

3 a i (15) 20 ii ( 12 iii 

6) ( 10 

b i 5 ii 10 

4 a ( 2 b i 

5 

5) ( 

iv 

30) ( 26 

48) 

ii 

20 

4) ( 

16) 

c They are parallel and the length of RS is 4 times the 

length of PQ. 

5 (16, 8) 

6 a b a b midpoint of OB 

___ › ___ › ___ › 

7 a AB , EF and GH 

b i 5p 3q ii 6m 14n 

8 a 2m b 2m n c m x d x m n 


1 a 

7 

( 

7) b ( 8 

5) c ( 15 2) 

Exercise 33E 

1 a i ( 3 9) ii ( 9 

27) 

b ABC is a straight line such that the length of AC is 3 

times the length of AB. 

2 a b a b 

1_ 

2 (b a) c 1_ (a b) 

2 

3 a 

1_ 

2 (a b) b a b c 1_ (a b) 

2 

d They are the same point. 

e The diagonals of a parallelogram bisect each other. 

4 a trapezium b n 2m k 

5 a i b a ii 1_ 

4 

1_ 

b iii a b iv 

1_ 

4 4 a b 

v 3_ (b a) 

4 

b EF and AB are parallel. The length of EF is 3_ 

6 a i 1_ 

2 

times the 

4 

length of AB. 

3_ 

(m n) ii (m n) iii 

1_ 

(3n m) 

4 

b 3n m 

c MQ and MR are parallel with the point M in common, so 

MQ and MR are part of the same straight line. 

MR ____ 

MQ 4 

4 

709 


05/09/12 5:22 PM


7 a 2b b 2a b c 4a 2b 2(2a b) 

d S is the midpoint of OT. 

e 30 


1 

P 

2 a (4, 5) b ( 3 c 5 

4) 

3 a b ___ a 

› 

b AP 3_ (b a) 

___ 5 

› ___ › ___ › 

OP OA AP a 3_ 

1_ 

( b a) (2a 3b) 

5 5 

4 a 2(a b) 

b 7a 6b 

5 a i ( 

3) 4 ii 5 

b ( 3 7) c (7, 4) 

6 a 2(a ___ 2c) 

› 

b OM 6c 3a 3(a 2c) 3_ ___ › 

OP 

2 

OP and OM are parallel with the point O in common, so 

OP and OM are part of the same straight line. 

7 a a ___ b 

› ___ › ___ › ___ › 

b FE FC CD DE a b a b 2a 

So FE is parallel to CD. 

c 2a 1_ 

___ 2 b 

› 

d FX 4_ 1_ 8_ 

(2a b) 

5 2 5 a 2_ 

___ 5 b 

› 

CX ( 8_ 

5 a 2_ 

3_ 

b) (a b) 

___ 5 5 ( a b) 

› 

CE a b 

CX and CE are parallel with the point C in common, so 

CX and CE are part of the same straight line. 

8 a i 5p ii 2q iii 4p q 

b 3p 2q 

9 a 

1_ (p q) 

___ 2 › ___ › ___ › 

b RS OR OS 1_ 

2 p 1_ 

1_ 

( p q) 

2 2 q 

So RS is parallel to OQ. 

10 a 2a ___ 2b 

› 

b QR 2a 2b 6a 4a 2b 

___ › 

XY 1_ ___ › ___ › 

MN MQ 1_ ___ › 

QR 1_ 

2 2 2 ( 2 a 2b) a 

1_ (4a 2b) 2a 

2 

So XY is parallel to OR. 

11 a ___ a b 

› 

b AX 2(a b) 

___ › ___ › ___ › 

DX AD AX 2b 2(a b) 2a 

So AB is parallel to DX. 

710 


7/30/12 5:24:55 PM

Answers - The Robert Napier School

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