Math AA SL

524 Answers

Chapter 1 Prior Knowledge 27 a 27u-6 b 32v-15
1
28 a 4 a10 b 1 27 b21

1 a 81 b 40 29 a 25x4 y6 b 81a8 b-8
b 8.56 × 10 -3 30 a 27x3
2 a 3.4271 × 102 b 25
3 26 x4

Exercise 1A 31 a 27 x6 b 2549ub2v86
8 y9

32 a 16v4 b 27 b6
9u2 8a9
1 a x6 b x12
2 a y6
3 a a7 b z10 33 a 2x - 7x6 b 3y5+y 3
4 a 517 b 5 a2b5 c a−c 4
5ax b a11 34 a 5u2 + 6uv2 b 2 s2t2−s3 2t+ 3 2
6 a y4 35 a 5 3p p− q −1 2 b 324
7 a b6 b 224 36 a 38 b 510
8 a 118 b 311
9 a x15 b x3 37 a 29 b 27
10 a y16
11 a c14 b z6 38 a 25
12 a 350 b b8
13 a 48x7 39 a 217
14 a 3a3 b 75
15 a 20x5 y3 z 40 a x = 4 bx= 3
16 a 2x5 b x32
41 a x = -1 bx= 6
b z25 7 9
42 a x = 3 b x = 2
b c14

b 1328 43 a x = -4 b x = -3

b 15x7 44 2x + 4x2

b 5b4 45 x5 y2
b 12x8 yz5
46 8ba63 n = 1000
D
b 3x6 47 a b 250 000
b 5n
17 a 1 2x3 b 1 c $10 million
4x
48 a kA = 8, kB = 40
18 a 5 b 3x2
3x2 4 c method B

19 a 7x2 y3 b 2x3 z 49 x = 2
y 50 x = 5
51 x = -1
20 a 1 b 17 52 x = -1
10 53 x = -2
54 a R = 3.2T2
21 a 1 27 b 125 55 a 3 v

22 a 4 3 b 7 c 40 km per hour b 250 K
5 56 x = 1, y = −3 b 1.5v

23 a 9 b 125 57 x = 4
4 8 58 x = 6
59 x = 1, 3 or -5
24 a 7 9 b 516 60 27000
61 7
25 a 6 x b 10x4
b 8164
26 a 8
9

Answers 525

Exercise 1B 16 a 4 log 3x b 1
log 2x
17 a 1
1 a 32 000 b 6 920 000 b 1
2 a 0.048 b 0.000 985 18 a -15 ln x
3 a 6.1207 10 b 3.07691 10 3 19 a 4.5
4 a 3.0617 10 2 b 2.219 10 −2 20 a 13 b 17
5 a 6.8 10 7 −3 b 9.6 10 11 21 a 3
6 a 1 10 0 b 1.2 10 −5 22 a 8 b -1.5
7 a 2.5 10 21 b 3.6 × 1013 23 a 25 b7
b 4 × 10-4 b7
8 a 2 × 10-2 b 2.5 × 102 b9
9 a 5 × 100 b 3.1 × 109 b 81
10 a 2.1 × 1011 b 3.02 × 108
11 a 2.1 × 105 b 8.91 10 14 24 a 1 3 b 16
12 a 7.6 10 5 b 6.13 10 13
13 a 4.01 10 4 b 400 25 a 1 b 164
32 b 3.15
b 1.77 × 10 -42 m3 b 2.88
b 7.41 × 108 26 a 2.10

ba+ b+ 1 27 a 5.50
ba- b- 1
20 a 1.22 × 108 28 a -0.301 b -1.40
c 4 × 102 b2 29 a log 5 b log 7
b6 30 a log 0.2 b log 0.06
21 6 10 −57 b -1 31 a ln 3 b ln 7
22 1.99 10 −23g b -4 32 a 2 lo+g 7 b log13 4−
b2 b log13 2−
23 a 1.5 × 10 -14 m b4 33 a 1 lo+g 7
24 a 2.98 b -1 b ln(2 1k) 5+ +
b -3 34 a ln( 2k) 2− −
c Europe b 1000
b 332 35 a 0.349 b -0.398
25 a 1.5 b -1.99 36 a 1.26 b 0.847
b e5
26 a 4 1 b e y2 37 10 000
27 r p=q+ + b 2e 2 y1+ + 12 38 332
b 5log x
Exercise 1C 39 a b+

40 0.699

1a1 41 0.824
2a5
3a0 42 0.845 b 0.0126
b 6.93 days
4 a -2 43 log 1.6
5a1 44 a 7.60
6a3 45 a 10
7a0
46 a i 1000 b 10ln3 = 11.0hours
8 a -2 ii 1220
9 a 100
10 a 50 002 47 4.62 years
11 a 1.55
12 a e2 48 a 57.0 decibels b 67.0 decibels
13 a ey + 1
14 a 1 2 ey−3 − 2 c Increases the noise level by 10 decibels.
15 a 5log x
d 10 W−3 m −2

49 a e ln 20 b lnln7 20
50 x = 100, 1y0=or x = −100, 1y0= −

51 1

526 Answers

Chapter 1 Mixed Practice 5a u = −37, d7 = b u = −4=3, d9
1 1

6 a 8 3( 1−)− n b 10 4(+1−) n

1 a 9.3 × 103 cm b 5 280 000 cm2 7 a 3 2( 1+) − n b 10 4(+1−) n

2 9y 8 a 1 5( +1)− n b 20 3(−1−) n
x
9 a 34 b 24
y6
3 9 x4 10 a 13 b 29

4 a 4 000 000 b 1 000 000 c 16 11 a 372 b 910

5x= 3 12 a -11 b -236
9 500 s 13 a 352 b 636
14 a 184 b 390
10 x = 99
11 x = 0.5e3 b0

12 x = 0.693 15 a -45 b 99
16 a 32 b 205
13 x = 0.531 17 a 105
( )14 y+1 18 a 806
x = ln 5 19 a 13
20 a 16
15 a m = 3, n = 4 b x = 7.5 b 354
b 10
16 a 2.8 ba+ b+ 1

17 a 1.2 ba-b b 16
b 2230
18 0.0259 to 0.0326 21 a 216 b 31
22 a 4 b 305
19 a 9.42 m s-1 b Yes b 112 c 465
c 480
20 a 3 23 a 8
b Strength increases by one.
24 a 5
c 3160 m (It would have had to be measured
using a special damped seismograph.) 25 a £312 b £420

( )21 x = log 23 ≈− 0.176 26 £300

22 x = 1, y = −2 27 14
23 x = 10, y = 0.1
28 a 387 b 731

29 140 b 798
b 945
Chapter 2 Prior Knowledge 30 a 21 b 390

31 a u = −7=, 7d
1

1 x = 4, y =− 1 32 a 296
2 a 80
b 36 33 632
3 x = ±2
34 x = 2
Exercise 2A 35 42
36 42
1 a 49 b 68 37 960 b 1679
2 a -22 b -40 38 a 876 b 7995 minutes
3a2 b7
4 a -0.5 b -2.5 39 408.5
40 53

41 -20
42 a Day 41

Answers 527

43 a 19 days b 45 days 26 x2 - n yn + 1
b 199 27 3069
44 a 3, 7
28 -1
45 2n + 3
46 0 29 88 572
47 70 336
48 735 30 a 0.246 m b 3.9 m
49 0.2 m
c The height is so small that measurement error
50 -1 and other inaccuracies would be overpowering.
52 b 150
Exercise 2C

Exercise 2B 1 a $2382.03 b $6077.45
2 a $580.65 b $141.48
1 a 20 971 520 b 1458 3 a $6416.79 b $1115.87
4 a 48 years b 5 years
2 a -15 625 b -1 441 792 5 a 173 months b 77 months
1 2 6 a 7.18% b 4.81%
3a 32 b− 729 ≈ 0.00274 7 a 14.9% b 7.18%
8 a $418.41 b $128.85
4a u = 7, =r 2 b u = 4 , r = 3 9 a £13 311.16 b £7119.14
1 1 3 10 a $97 b $294
11 a £737.42 b £2993.68
5a u = 3, r = ±2 b u = 5 , r = ±3 12 a $103 b $5130
1 1 9 13 a £94 b £2450
14 a €1051.14 b €579.64
6a u = −3, r = 2=or12u=o1 r 3, r =− 2 = − 1 15 a $598.74 b $4253.82
b 1 = 7168, r =u1 − 7168, r 2
16 £900.41 b £28.55
u 17 £14 071.00
1 18 $8839.90 Depreciation
19 €16 360.02 expense ($)
7a6 b8 20 £8874.11 6 000
21 a monthly 4 200
8 a 13 b8 22 £6960 2 940
2 058
9 a 5465 b 4095 23 Start-year 1 441
Year value ($) 1 008
10 a 190.5 b 242 1 20 000
2 14 000 706
11 a 153.75 b 14596 3 9 800 147
12 a 363 b 2800 4 6 860
5 4 802
13 a 76 560 b 324 753 6 3 361
7 2 353
14 a 68 796 b 488 280 000 8 1 647

15 a 9 b 24827
6 999

16 a 1 b 255 End-year
value ($)
17 a 2 b 96 c 3069 14 000
b 182.25
18 a 1.5 9 800
6 860
19 1920 cm2 4 802
3 361
20 0.0375 mg ml-1 2 353
1 647
21 15.5 1 500

22 0.671 > 5 b 11 days
23 a 3580 m3

24 255

25 a 9.22 × 1018 b 1.84 × 1019 24 15.4%
c 2450 years 25 12.7%

528 Answers

26 a 2.44% b 12.8% Chapter 3 Prior Knowledge

27 a 250 billion marks 1y

b 0.0024 marks c 0.0288 marks −4 3 x

Chapter 2 Mixed Practice

1 a €6847.26 b 7.18%
2a i dn
bi1 2
ii b n 3
ii − 256
3a 2
4a 4 b -4 c 300
58 c 43 690
6 10 b 512
7 a 48.8 cm c £43 800
8 a €563.50 b9 2x 3 − 12
9 a 8.04 billion b 6.25 years 2 y
10 a i 20 b 2033 30
b 7290
ii 10 4 x = −2.30
ii 16
11 a i 7 d 25 5
c2 f n = 36

e n = 498 1 x
1
12 x = 4
13 b 74 Exercise 3A
14 8190
15 16 months ii a 3 = 10 1a9 b 14
16 a 2bn 3−n ii 594 m 2a7 b 47
b 19
17 $11.95 million b A – 6th day 3 a −17
18 $10 450 b £1 402 500 b −44
19 8.63% 4 a −15 b Yes
20 $1212.27 5 a Yes b No
21 22 years 6 a No b No
7 a Yes b Yes
22 a a 1 = 6 8 a No b Yes
b i a2= 8 9 a Yes
c2
d i 22
e 28
f 16 m

23 b 32
24 a B – 12th day
25 a £68 500
26 3

Answers 529

10 a b 24 a y
11 a n ≠ 0 b x≠0 (−2, −2) y = f(x) = f −1(x)
12 a x − 5 b x 3.5
13 a x − 0.6 b x>4 (2, 2)
14 a x ≠ 2.5 b x ≠ −3
15 a f( x) − 2 b g( x) 7 x
16 a f( x) 18 b g(x) 4
17 a f( x) 0 b g( x) 0 by
18 a f(x) 2 b g(x) 3 y = f(x)
19 a 40
20 a 1 b -7 y = f −1(x)
b2 x
21 a -3
22 a b -1
y

−1 1 y = f −1(x)
y = f(x) x
1
−1

b y = f(x) 25 a (−1, π) y
4
π y = f − 1(x) x
y = f −1(x) 2 2
1
24
y = f(x)
1π

2

(π, −1)

by

23 a No inverse b No inverse

y = f(x)
y = f −1(x)

3

3x

530 Answers
y=x
26 a -13 b x=3 42 a 9 b 2.5
27 x = 41 cy y = f(x)
10
28 a 5.7 m s−1
8
b No, the car cannot accelerate uniformly for

29 a that long / The model predicts an unreasonable
30 a
speed of 114 m s−1. 1
3
x≠ 5 b 6
5
b q( x) −2
−4
4
c x = 4 or 4 − 2 y = f −1(x)

31 a 4.95 billion x
b E.g. smartphones are likely to get replaced by 2 4 6 8 10
another technology

32 a f(x) = 1.3x b 0.892
b Amount in pounds if x is the amount in dollars.
b f(x) 3 4 or f(x) 0
33 a 1 bx= 2 43 a 0.182 y
44 a x x 1, 1≠7 y=x
34 a x 2.5 b f (x) 0 c x=7 45 a and b

35 a 60 b 129.9; 130

c It predicts a non-integer number of fish after
15 months.

36 a 14 b4
37 y

8 y = f −1(x) x
−8

6

4 y = g−1 (t) y = f(x)
−8
2 c x= 2
y = g(t)
Exercise 3B
246 8t
1a y
38 a x > 0 b4 c 1
9 x
39 x > 5
40 a x 7 b x = −31
3 b xf( ) 4
41 a 19

c 1 is not in the range

x = −2

Answers 531
b
y by

y = 1
2

1x x

2a y 11 g(g)(xxl)n(4.8581)
3
4a f(x) b
b
5 a x f( ) ln(0.75)

y = 13 6 a − 1.17 f( ) x1.17 b −1.57 g( ) ex
−3
7 a (0.7, 0.55); 0.7 x =

x ( )b−2 , 379 ; x =− 2
7 7
−1

( )8 a−2±2
2 ,1.75 , ( −1,2); x =− 1

x=1 ( )b (0, 1−), (1, 1)−, 1 2 ,−34= ; x 1
by 9a y 2

1 y=1 y=1
x
2
− 1 x
2 x=3
by
x = −2
3a y x=1

y=3

−2 2 x −3 x

y = −2

532 Answers
10 a y
12 a (-1.49, -4.78) b (-0.661, 0.421)

13 a (-2, 7), (-1, 5), (1, 7)

b (-2, 23), (1, 5), (2, 3), (3, 3)

y=3 ( ) (14 a − −2−, 2, 2,2 2 2 −)

2 ( ) ( )b 1 2,−7 2 − , 1 2+,7 2 +

15 a x = 0.040, 1.78 b x = 0.213, 1.632

16 a x = 1, 6.71 b x = -3.48, 2.48

x 17 a x = 0.063, 1.59 b x = 0.288, 49.0

18 a x = -2.18, 0.580 b x = -1.67, 0.977

by 19 y

−2 x 1.61
1 x (−1.5, 1.01)

x

y=x+1 20 y
4
11 a y x=2 y=7
y=2

13 −1.69 x

by (−1.69, 0), (0, 4), 7 y = y
21 a x ≠ −2
x=0 x=4
b

13 x
(2, −2)
y=3

y = −8 − 21 31 x

x = −2
x = −2=, 3y

Answers 533

22 y 30 v
26

3 15
10
y=1
x

23 (1.84, 3.16) 0.5 1.0 t
24 (–1, 2.5)
25 235 31 p
600
26 x = -1
y
27

350

x 120 180 400 x
1 x
4

x = −2 x=2 − 200

x = −2, x = 2, y = 0 32 x = 0.755
28 y 33 x = 2−.20, 0.−714, 1.91
34 2−.41, 0.414, 2
y=3
35 1−.41, 1.41
36 0.920
37 y x = 2

−1 x
−3 x = 1
(6.32, 0.232)

29 N
80

40

38 x 2

15 30 45 m

534 Answers

Chapter 3 Mixed Practice 7 a 20.0 m s-1 b 0.630, 17.1 s c 5 s

1G 8 b 1.30 s

(20, 104) 9 a 100°C b 95.3°C c 8.82 km

10 a f(x) 1=.8 32x +

b Temperature in Celsius if x is temperature in
Fahrenheit.

11 a p = −2=, 4q
bi x≠2

ii g(x) 0
iii x = 2

12 a f(x) 20 b 12

13 x = -5.24, 3.24

G = 40 14 x = 1, 2.41
x 15 x ≠ ±3, h( x) 2 or−h( ) 0 x

16 −9.25 f( ) x3.

b $3538.09 17 a 4.89 b x = 28.9
x = −1=, 1x
2 x = 1.86, 4.54 18 a 19−g( ) 8 x b
3a x −5
b x = -2.38 c −2i0s not in the range of g

4 (-1.61, 0.399) and (0.361, 2.87) 19 y
5y

y=5

(2, 9)

3

5

−1 x

−1 x
6a y 5

20 T
90

1 y=3 T = 20
331 x t

x=3

b ≠x 3, f( x) 3≠

Answers 535
21 N
Chapter 4 Prior Knowledge
N = 2000

1 24 cm b ( 1−, 1)
2 a 10

Exercise 4A

600 1a 1 b2
2a
t 3− b −1
3a
22 a 1.10 m s-1 1 b 1
b e.g. The car will stop by then 2 −2

23 x = -2.50, -1.51, 0.440 4 a (2, 0), (0, -6) b (4, 0), (0, 8)
24 a y
5a (5, 0), (0 ,2.5) b (-3, 0), (0, 2)
x = −2 x = 2 6a
y= 1 7a (3, 0), (0, 4) b (4.5, 0), (0, -3)
8a b 5x −y + =1 0
12 x 9a 2x −y + = 3 0 b 3 x +y + =7 0
y = −1 10 a 2x +y − = 4 0 b x +y 3−9=0
x −y 2+ 7=0 b y x = 3+ 4; 3, 4
y =x 2+ 5; 2, 5
b y x = −52+ − 5; 5 2 , 5
11 a y = −32−x− −2; 2 3 , 2

12 a y = −0+.6−x1.4; 0.6, 1.4 b y − 2= 3−( 5)x
b y = −5−.5−x2.−5; 5.5, 2.5 b y + 1=2−( 2−) x
b y − 1= − −43( x3)
13 a y − 4= 2−( 1)x b 2 x −y + =7 0
14 a y − 3= 5−( 1+) x b 3 x +y + =2 0
b x +y 2− 4=0
b i (12, 0) 15 a y + 1=2 3 (x1−) b 8 5x1+y9 −0 =
16 a 2 x −y − = 5 0 b y x= − + 9
ii x = ±2, y = ±1 17 a x +y − =10 0 b y + 3= 0+.5( 1)x
18 a x +y 2− 7=0 b 2 x −y − = 7 0
25 x = 2.27,,4.47 26 −1.34 19 a 3 4x1+y3 0− = b x +y 5+8=0
20 a y x= 3+ 4 b 3 4x2+y0+ =
27 a x ≠ 0, 4 b y x= − 13+ 8
21 a y − 5=1.5( 5)x − b y − 4= 3−( 2)x
b f( x) −5.15 or f( x) 3.40 22 a x −y − = 6 0 b 2 x +y − =7 0
23 a x +y 3− 4=0 b x −y 3− 1=0 0
28 a x > 0, x ≠ e− 3 24 a 2 5x0−y = b 2 3x3+y0+ =

b g(x) 0 or g( ) 0x.271 b (5, 1)
29 a x 2, g( )x ∈
b (2, 5)
by
b (-3, -4)
25 a y x= − + 5

x 26 a y − 5= 4−( 1−) x
27 a 5 x +y + =3 0
c x = 2.12 28 a x −y 2+ 5=0
29 a 2 5x3−7y 0+ =
30 a (1, 2)

31 a (1, 3)

32 a (2, 5)

536 Answers

( )33 a 2 5 , 151 ( )b 115 , 25 17 a = 2.97 b y + 5= − −116 (x4)
18 a a = −2.2, b = −8.6 b 7 4x2+y0 = −
7 b y = − 4 x + 53 d 65
34 a 4 7 7 4 b 3.33 m
4 19 a 7 b (6, 5) and (0, 9)
35 a − 3 b 4 x + 3y = 36 b B(4, 5) D(5, 4)
c (0, -5)
36 a 5 b 17 b 74 8≈.60 m
−7 5 20 a 1.8 m
1
37 a b x − 5y = −8 21 a (3, 7)
38 a 5 22 a y +x = 9
− 7 b No
c 2 c5
2 51 23 a 50 7≈.07 m
y= 7 x 7
−

39 0.733 Chapter 4 Mixed Practice

40 a (8, 11) b8 2
d 56 3
c x=8 3 b x + 4y = −1 1 a i (2, -2) b k = −
2
41 a y = − x − 6 b 9.8 s ii 3 2
b 0.018 N
c ( 4.−6, 0.9) d 0.467 m iii 2− 3
b $6.80
42 6.10 m 2a s= 6, t = −2b 45 23x +y =
b 350 3b 3 c 2
43 a 0.5t + 0.1 d 400, fewer 2 d4
−3
44 a N/m b y x= + 3 c 4.5
c larger 4 a -1

45 a C = 0.01m + 5 5 a (2.5, -1) b 9.22
c 500 minutes c y x= 76− 47

12

46 a P = 10n - 2000 6 a (2.5, -1, 4) b 9.43
c P = 8n - 1200
7 a P(0, 3) Q(6, 0) b 45
48 1390 m
c (2, 2)
7
Exercise 4B 8a −4 bk= 2 cd= 1

11 ±20

1a5 b 13 12 a p = 1, q = −18 b 27.5
2 a 13 b 10
3 a 29 b 85 13 1.5
b 11
4a3 b9 14 4.37
5a6 b 65
6 a 98 15 b y = −2+x5 c S(1, 3)
b (4, 6)
7 a (4, -1) b ( 1−, 2, 1) − 16 10 m y
8 a (3, 4, 3−) b (−5.5, 3.5, 7) 17 a 6
b 171 5
9 a (5.5, 2, -3) B
10 a (1.5, 0.5, 5.5) 654321 4
11 (6, 3, −8) z 3
12 a = 3, b = 20, c = 4.5 2
1
13 196.5 0C 1 2 3 4 5 6 A
14 4.52 m s−1
15 30 x

16 k = 2± b (2.5, 2.5, 2) c 6.48

Answers 537

1 b (-1, 5) c 5.51 22 4.21 10 cm4 3 b 1100 cm2
18 a 2 23 a 12.0 cm b 36.9 cm
e 4.47 24 a 3.61 cm b 3.58 cm
c 2 x +y = 3 b 15.2 25 a 192 cm3 b 1330 cm2
26 a 15 cm b 474 cm2
19 a 42 b (6, 5) and (4, -1) 27 a 82.5 cm2
20 4.32 b 815 cm2
21 a y = 3−x13 c 594 cm3
b 12.9 mm
23 135 m 28 55.4 cm2 , 23.0 cm3 b 0.194 m3
24 4 m 29 1140 mm2 , 3170 mm3
30 1880 mm3 , 889 mm2 b 58.0°
Chapter 5 Prior Knowledge 31 a 13 cm b 39.5°
32 242 m2 b 53.1°
1 95° 33 a 151mm2 , 134mm3 b 3.86
2 12.4 cm 34 a 9.35 m3 b 11.5
3 60 b 14.1
4 α = 140°, =β ° 85 Exercise 5B b 26.3
5 050° b 25.7
b 4.23
6 Volume = 1570 cm3 , Surface area = 785 cm2 b 78.7°
b 15.9°
Exercise 5A 1a 29.7° b 12.5°
2a 53.1° b 10.6°
1 a 40.7 cm2 , 24.4 cm3 b 1580 m2 , 5880 m3 3a 40.1° b 45°
4a 2.5 b 14.5°
2 a 8.04 m2 , 2.14 m3 b 0.126m2 , 0.00419m3 5a 7.83 b 6.53 mm
101 km2 , 134 km3 6a 9.40 b 3.29 mm
3 a 170 m2 , 294 m3 b 7a 17.4 b 4.10 mm
36π m2 , 36π m3 8a 2.68 b 50.4°
4 a 47.9 cm2 , 38.6 cm3 9a 344 b 32.9°
10 a 71.6° b 51.2°
b 4.40 mm2 , 0.293 mm3 11 a 18.4° b 7 mm
12 a 21.8° b 6.40 mm
5 a 16 cmπ, 322 π cm3 b 13 a 7.13° b 6.43 mm
3 14 a 29.7° b 41.6°
15 a 10.5° b 62.8°
6 a 64 m ,π2562 3 π m3 b 288π m2 , 432π m3 16 a 8.49 cm
17 a 3.42 cm
7a3 π m 2, 6π m3 b 20 π mm2, 43π mm3 18 a 3.31 cm
4 9 19 a 37.9°
20 a 39.9°
8 a 90πcm2 , 100πcm3 b 36 cπm , 126 cmπ 3 21 a 20.7°
9 a 6 cm3 22 a
b 16 cm3 23 a 37 cm
24 a
10 a 20 cm3 b 18 cm3 25 a 5.06 cm
26 a 5.23 cm
11 a 26.7 cm3 b 2.67 cm3 39.4°
12 a 64 mm , 3144 mm 2 42.6°
13 a 2 mm ,133.8 mm 2 b 336 mm ,3365mm 2
14 a 5.33cm , 33 3.9 cm 2
15 a 75.4 cm ,3109 cm 2 b 320 mm ,3319 mm 2

b 816 cm ,3577 cm 2

b 20.9 cm ,346.4 cm 2

16 707 cm2
17 96.5 cm2
18 8.31 m3
19 134 cm3
20 3210 cm3
21 3400 cm 2, 8120 cm 3

538 Answers

27 a 53.8° b 97.3° 56 h= d tan 40
b 10.5 mm2 tan50 t−an 40
28 a 28.5 cm2 b 1.73 mm2
29 a 127 cm2 b 1710 mm2 57 a x = h, y = −t4an h
30 a 631 cm2 b 61.8° tan 30 10
31 a 52.2° b 5.23 mm
b 8.36 mm b 3.73

32 a 8.30 cm b 26.6° Exercise 5C
33 a 7.04 cm y

34 4.16 cm 1 a 35.5° b 59.0°
35 32.5° 2 a 16.7° b 51.3°
36 a (0, 3), (-6, 0) 3 a 55.6° b 18.9°
4 a 48.0° b 82.6°
37 a 5 a 68.9° b 59.8°

6 a 67.4° b 41.6°

7 a 49.3° b 48.2°

8 a 69.3° b 46.6°

9 a 2.73 m b 8.22 m

1 10 a 36.9° b 42.8°
**** − 13 x 11 a 36.6° b 46.5°
b 15.0°
12 a 31.2°

13 a 45.3° b 45.3°

14 a 50.6° b 61.5°

15 a 17.8° b 67.4°

16 a 54.2° b 43.4°

b 71.6° b 32.0 17 a 1.16 km, 243° b 2.71 km, 320°
18 a 4.00 km, 267° b 7.62 km, 168°
38 6.98 cm 19 a 3.97 km, 317° b 16.5 km, 331°
39 58.5°, 78.5°
40 87.4° 20 57.1 m b 18.4°
41 a 18.6° 21 a 15.8 E
42 19.0 22 a
43 x = 11.8cm, θ = 15.5°

44 38.7°

45 b 49.4°

46 82.2°

47 16.2 cm 10 cm 10 cm
48 5.69 cm

49 29.0°

50 47.0°

51 21.0°

52 5.63 cm A 7√ 2 cm C
53 15.2 c 59.3°
54 θ = 52.0°, AB = 8.70 b 8.69 cm
23 56.3°
55 9.98

Answers 539
24 a N
Tree Chapter 5 Mixed Practice

120 m 1 35.0 m G
56° 2 a AC = 22.6 cm, AG = 27.7 cm 16 cm

Tent b
b 67.1 m
25 a N Rock

0.8 km 27.7 cm
37°

A 22.6 cm C

c 35.3° b 24.1 cm c 29.1 cm
3 a 32.5 cm c 26.6°
1.2 km 4 45.2° b 1
5a 6 −2
6 a 12.9 cm
b 1.90 km
b 80.5°
26 13.8 m 7 12.3 cm
8 3.07 cm
27 42.3 m 9 21.8°
10 32.4°
28 3.18 m 11 37.4°
12 18.3 cm
29 11.4 m b 35.4° c 41.9° 13 55.4 m
30 a 12.1 cm b 3.11° 14 17.9 m
31 a 13.02 m b 45.2° 15 a y
32 a 7.81 cm
− 15 y = x13 + 5
33 191 m, 273° 10

34 2.92 km, 008.6° 5

35 146 m 10 x
y = 10 – x
36 a 9.51 m b 10.9 m c 48.3°

37 3.96 m

38 a 9 – assumes lengths given are internal or thin
glass or no large objects in the space

b 359 000 W
c yes – maximum angle is 51.8°

39 a 2.45 m b 42.9 m3

c 75.5% d 94.0%

40 38.9 cm2 b (3.75, 6.25)
c 63.4°
41 39.9°
42 92.3 m

540 Answers

16 23 900 cm3 b 21.0° 5 a all households in Germany
17 a 65.9° b 313 cm3
18 5.46 b 9.84 cm b convenience sampling
19 21 c e.g. Households in the city may have fewer pets
20 17.6 m V
21 a 131 cm3 than in the countryside.
22 a 5.12 cm
23 a 5 cm 6 a the number/proportion of pupils in each year
group
b 9.43 cm
c 58.0° c 31.4° b quota sampling
a 720 m c i keep
b 72 300 m2
c 88.3° ii discard
7 a convenience sampling
24 a i 22.5 m
ii b all residents of the village
c e.g. People using public transport may have

different views from those who drive.

d would need access to all the residents

8 Not necessarily correct, it could just be an extreme
sample.

22.5 m 22.5 m 9 a continuous
b They would all be destroyed.
A 53.1◦ c list in serial number order, select every 20th
C
c 27.0 m 10 a quota
e 41 600 m3 b more representative of the scarves sold
f 44 900 kg c 12 red, 12 green, 10 blue, 6 white
25 22.5
11 a quota
b more representative of the population
c difficult to compile a list of all the animals

12 a continuous
b Basketball team are likely to be taller than
average.

Chapter 6 Prior Knowledge c systematic sampling

1 a 4.75 b 5.5 c6 d9 d Some samples not possible, e.g. it is not possible
2 a 1.5 b (0, 7) to select two students adjacent on the list.

13 5 cats, 9 dogs and 6 fish

14 a Gender/age 12 13 14

Exercise 6A Boys 455

Girls 042

b no

1 a 9 lions, 21 tigers 15 a discrete
b 9 strawberry, 11 chocolate
b no
2 a 24 boys, 16 girls
b 18 HL, 27 SL c convenience
d e.g. Different species may live in different
3 a 10 football, 14 hockey, 16 basketball
b 6 cod, 9 haddock, 5 mackerel parts of the field.

4 a 24 chairs, 9 tables, 4 beds 16 a i possible
b 8 oak, 7 willow, 4 chestnut ii not necessarily true
iii possible/quite likely

Answers 541

b iii would be unlikely 18 a mean = 8, sd = 3
c discard –32, keep 155
b mean = 41, sd = 3
17 a 65%
b i 84 19 a 48 b 19 c 6.17, 1.31
ii 65% 20 a 31
b 1.4 <m 1.6 c 1.34
18 a 100 million
b e.g. No fish die or are born; captured cod mix 21 a 28
thoroughly with all other cod.
b 15.5 <t 17.5
Exercise 6B
c 17.3 s; used midpoints, actual times not
available

22 a 4.25
b 1.5

1 a 16.5 t < 18.5; 20.5 t < 22.5; 22.5 < t 24.5; c The second artist’s songs are longer on
3, 12, 3, 1, 1 average and their lengths are more consistent.
20 observations
23 a 4 b3 c yes (11)

b 14.5 t < 17.5; 20.5 t < 23.5; 10, 8, 2 24 a 67
20 observations b 23 cm

2 a 300 n 499; 500 n 699; c 24.7 cm
700 n 899; 11
40 observations d 20.5 l < 23.5 23.5 l < 26.5 26.5 l < 29.5

23 30 14

b 200 n 399; 400 n 599; e 44
600 n 899; 20
40 observations f no

3 a mean = 3.86, med = 3, mode = 1 25 a a = 12.5 b 13.4
b 30.9
26 a x = 5.5
b mean = 5.57, med = 6, mode = 6 27 63.8

4 a mean = 4.7, med = 4.5, mode = 3 28 62

b mean = 31.2, med = 24.5, mode = 24 29 a 161 +24a b a = 10
50
5 a x = 10 bx= 7
30 a x = 9.5
6 a x = 31 9 b x = 7.5 Q 5, Q 5, Q 6.5
b 1 = 2 = 3 =

7 a y = 10 b y = 52 c yes

8 a y = 4.6 b y = 2.2 31 a £440, £268 b median
c $576, $351
9 a i 13 b i 20.5
ii 0 <x 10 32 a mean = 4, sd = 2.94
ii 25 < x 35 b mean = 2012, sd = 8.83
10 a i 13.5 b i 10.2
33 10.6°C, 2°C
ii 12.5 <x 14.5 ii 11.5 < x 15.5
11 a 10.4 b 4.96 34 mean = 9.18 km, var = 11.9 km2

12 a 5.24 b 32.8 35 med = –75, IQR = 42 b m = 58
13 a x < 5 or x > 125 b x < 10 or x > 26 36 a med = 42, IQR = 8 b 0.968
37 74%
14 a x < 4.5, Q > 12.5 b x < 15.5 or x > 63.5
3 38 a p = 12
39 p = 7, q = 4
15 a mean = 46, sd = 8

b mean = 62, sd = 18

16 a median = 25, IQR = 13 40 c = 30
41 10
b median = 0.5, IQR = 2.1 42 (4, 8) or (5, 7)

17 a med = 360, IQR = 180

b med = 5, IQR = 2.6

542 bf Answers
t
Exercise 6C 60

You might find that different GDCs or programs give 50
slightly different values for the quartiles, resulting in
slightly different answers from those given here. In an 40
exam, all feasible answers would be allowed.
1a f 30

15

20

10 10

5 0 30 36 42 48 54 60

3 a 16 b 30
4 a 23 b4
5 a cf
00 5 10 15 20 25 x
bf
40
20
30

15 20 x
x
10 10

5 0 0 10 20
b cf
0 0 20 40 60 80 x
2a f 60
50
15 40
30
10 20
10
5 0 0 10 20 30 40 50 60 70 80

0 1.2 2.5 3.8 5.1 6.4 7.7 h

Answers 543
6 a cf
9 a 11 15 17 21 25 32
30
10 15 20 25 30 35 x
20 b
4 13 17 19
10 0
5 10 13.5 18.5 x
15 20

10 a f
20

01 2 3 4 56 78 h 15
b cf 10

60 5

40 00 4 8 12 16 20 l
m
b 25

20 11 a 105
bf

0 30 35 40 45 50 55 60 40
t
7 a i 13
ii 20 30
iii 6
iv 21 20

b i 4.6 10
ii 6.5
iii 2.3 99 0 80 100 120 140 160 180
iv 6.6 10 x c 46%

8a 3 5 6 24
25 x
0 5 21
b 11 15 17

10 15 20

544 Answers
12 a cf
15 a cf

350 140
120
300 100
80
250 60
40
200 20

150 00

100 b 40
c 45, 32
50 d 10

0 0 10 20 30 t 20 40 60 80 100 n
31 45 63 n
b i around 17°C 25 t
ii around 7°C < 30

13 a 45 4 110
b 22.2%
h
c 45 0 20 40 60 80 100 120

Time 5 t< 10 t 15 t 20 t e Overall, fewer candidates take History SL.
(min) 10 < 15 < 20 < 25 History has a larger spread of numbers

Freq 7 9 15 10 than maths (based on the IQR).

d 16.9 minutes 16 159 cm b 90%
14 a cf d 72, 18
17 a 160
c 75

100 40 62 72 80 100
90
80 40 60 80 100 n
70
60 e The second school has a higher median score,
50 but more variation in the scores.
40
30 18 a Q = 1, Q = 2, Q = 3
20 1 2 3
10
0 20 25 30 35 40 45 b2

c 7 is an outlier

d0 1 2 3 4 67

b median ≈ 31 cm, IQR ≈ 9 cm 0 4 8n
19 7 11 20 28 25 38
c 20 26.5 31 35.5

20 25 30 35 40 45 h 0 10 20 30 40 x

Answers 545
20 a f
2a y
80 10

60
40 5

20

0 0 20 40 60 80 100 x by 5 10 x
15 x
b 59.9
20 A2, B3, C1 15 20 25

Exercise 6D

1a y 10
10

5
10

5 3a y

25

5 10 15 x 20

by 15
5

10 x
10 15 20 25 30 35

by

5 10 x 25

20

15
−5

10 x
10 15 20 25 30 35

546 Answers

4 a weak positive b strong positive 15 a, d y

5 a strong negative b weak negative 250

6 a no correlation (circle)

b no correlation (V-shape)

7 a no correlation (slightly scattered around a
vertical line)

b no correlation (slightly scattered around a 200
horizontal line)

8 a i r = 0.828 b i r = −0.886

ii strong positive ii strong negative

iii yes iii yes

9 a i r = 0.542 150 x
0 5 10
ii weak/moderate positive

iii no b weak negative
c 3.6 km, $192 000
b i r = −0.595

ii weak/moderate negative e average price ≈ $161 000

iii yes 16 A1, B2, C3, D4

10 a y = 0.413x1+.57 b y = 0.690x − 2.60 17 a 0.688 b y = 0.418x + 18.1
11 a y = −0.589x + 6.72 b y = −0.632x21+.6 c 46.5 (or 47)
12 a i y = 21.9
ii reliable d no (correlation does not imply causation)

b i y = −4.26 18 a 0.745

ii not reliable (extrapolation) b The larger the spend on advertising, the larger
the profit.
13 a i y = 73.5

ii not reliable (no correlation) c y = 10.8x18+8

b i y = 87.9 d i $1270 ii $2350

ii not reliable (no correlation) e the first one (no extrapolation required)

14 a, c, d 19 a -0.0619
y b y = -0.370x + 51.6
c no (no correlation)
75
20 a y
70
4

65

3

60

55 x 2
145 150 155 160 165 170 175
1
b weak positive
c 156.7 cm, 64 cm 0x
e arm length ≈ 61.5 cm 0 2 4 6 8 10 12
f i appropriate
b -0.695
ii not appropriate (extrapolation)
iii not appropriate (different age from sample)

Answers 547

c It shows statistically significant negative c positive correlation in the summer, no
correlation. correlation in winter

21 a y d 39.5
26 a y
60
70

50 60
50

40 40

30
30

20

20 x 10 x
0 5 10 110 120 130 140 150 160 170 180

b weak negative c −0.480 b children and adults
d not significant c children (126, 24.3), adults (162, 56.8)

22 a moderate positive correlation y

b yes (there is correlation and value is within 70
range of data)
60
c 40.8 cm
23 a 0.820 b m = 0.631t5+.30 50

c For every extra minute practice he can expect 40
0.631 extra marks. With no practice he can
30
expect around 5 marks.

24 a Positive correlation: the larger the advertising, 20
budget, the larger the profit. 10 x

b no (correlation does not imply cause) 110 120 130 140 150 160 170 180

c i For each €1000 euros spent on advertising, d 19.0 kg LQ y =
the profit increases by €3250. 27 a LQ(x) 1=2, ( ) U24Q, (x) 1=1,

ii With no advertising the profit would be UQ(y) 1=9
€138 000.
c, d, e
25 a y y

50 30

40 25

30 20

20 15

10 10
0x
5
0 10 20 30 40
b summer and winter 00 x
5 10 15 20 25 30

548 Answers

Chapter 6 Mixed Practice c 2.5 kg, 0.9 kg
d

1 a i systematic sampling 1.2 2.5 3.2
ii e.g. Students may take books out on the
same day each week. 1.9 2.8

b i Each possible sample of 10 days has an 01 2 3 4m
equal chance of being selected. 4 8
4 a discrete
ii Representative of the population of all b0
days. c i 1.47
ii 1.5
c i 11 iii 1.25
ii 17.4
iii 3.17 5a 4
b
2a n
40 2

35 35

30 0 1 2 3 4 5 6 7 8 9 10

25 Number of books read

20 c 10

15 6 a 0.996
b a = 3.15, b =− 15.4
10 T
10 15 20 25 30 35 40 c 66.5

b strong positive correlation 7 a cf
c n = 1.56T − 10.9
d 30.0 50
3 a 2.38 kg
b cf 40

50 30

40

20

30

10

20

0 0 2 4 6 8 10 12 t

10 b i 6.9 minutes
ii 2.9 minutes
00 1 2 3 4m iii 9.3 minutes

Answers 549

c Time Freq Chapter 7 Prior Knowledge
4
0 t<2 5 1 a 2, 3{, 4, 5, 6, 7, 8, 9 }
2 t<4 7 b 5,{8}
4 t<6 11
6 t<8 8 2
8 t < 10 3
10 t < 12 P V
3
d 6.21 minutes 62

8 a med 46,= Q = 33.5, Q = 56
1 3

b yes (93) 9

c

26 46 61 94 3 Draw 1 Draw 2

5 R 6 5 30 1
R9 10 × 9 = 90 = 3
33.5 56
6
10
x 4 6 4 24 4
9 B 10 × 9 = 90 = 15
0 10 20 30 40 50 60 70 80 90 100
6 R 4 6 24 4
9 a 121 cm, 22.9 cm2 9 10 × 9 = 90 = 15
b 156 cm, 22.9 cm2 4
10 3 4 3 12 2
10 mean = $55.02, sd = $43.13
11 x = 6 B
12 a s
9 B 10 × 9 = 90 = 15
300
Exercise 7A

1a 3 b 2
10 15
250
3
2a1 5 b 5

200 3a 1 b 1
2 3

4a 1 b 1
4 13
150 w
5 6 7 8 9 10 5a 1 b 3
26 26
b w = 19.1s9+9.0
c 0.994 – strong positive correlation 6 a 0.94 b 0.55
d 252 g
e 137 g to 175 g; extrapolating from the data so 7 a 11 20 b 2340
8 a 0.85
not reliable. b 0.13

13 a i Athletes generally do better after the 9 a 47 120 b 41
programme. 10 a 0.73 48
11 a 0.44
ii Better athletes improve more. 12 a 4 b 0.66
b 11.6 miles 13 a 12
c i 0.84 14 a 4.8 b 0.11

ii Y = 1.2 X + 3.2 b 27

b6

b 7.5

550 Answers

15 a 1.6 b 1.5 4 H
16 a 0.0743 b 66.9 B 80 60
17 7.5 45
18 15 b 0.8
19 8 35
20 0.75
21 $1162.50 a 3744 b 7
22 a 1.5 5 11
23 116
C I
78 12 6
24 25 16

Exercise 7B

1 G 1
F
13 5 4 a 6 b 16
35 35

3 6

a 3 b 425 G S
25 0.2 0.5
0.2
0.1
2 b 0.5

A B
10 8
2 a 0.1
7
10
P
4 1 V
a 15 b 3 0.32 0.08 0.3

3 0.3
b 0.32
B H
45 45
35 a 0.3

25

a 7 b 930
30

Answers 551

8 Transport Late? 11 First Second

0.05 L 0.3 × 0.05 = 0 .015 11 R 12 11 22
C 29 30 × 29 = 145

R

0.3 0.95 L 0.3 × 0.95 = 0 .295 12 18
30
12 18 36
29 B 30 × 29 = 145

0.7 0.2 L 0.7 × 0.2 = 0.14 12 R 18 12 36

B 0.8 18 29 30 × 29 = 145
30
L 0.7 × 0.8 = 0.56
B 17 18 17 51

a 0.155 b 0.845 29 B 30 × 29 = 145

9 Pizza ? Fries? a 73 b 3
145 5
0.8 F 0.3 × 0.8 = 0.24
P
12 Revise? Pass?

0.3 0.2 F 0.3 × 0.2 = 0.06 0.9 P 0.8 × 0.9 = 0.72
R

0.7 0.5 F 0.7 × 0.5 = 0.35 0.8 0.1 P 0.8 × 0.1 = 0.08
P
0.5 F 0.7 × 0.5 = 0.35 0.2 0.7 P 0.2 × 0.7 = 0.14
a 0.41 b 0.59 R 0.3 P 0.2 × 0.3 = 0.06

10 Dice Coin

0.5 H1 1 a 0.72 b 0.14
6 6 × 0.5 = 12 13

1 1 First Dice
Dice 2 123456
6 0.5 1 1
T 6 × 0.5 = 12 3 234567
345678
H5 1 Second 456789
0.4 6 × 0.4 = 3 5 6 7 8 9 10
5 4 6 7 8 9 10 11
6 5 7 8 9 10 11 12
6
6 51
0.6 a 5 36
T 6 × 0.6 = 2

a 5 b 712 b 1
12 6

552 Answers

14 First Dice 26 a 1115 b 17
20
123456
19 30
1=>>>>> 27 a 49 b 49

2 < = > > > > 28 a 25 b 1
51 17
Dice

3<<=>>> 29 a 3 b 18
31 53
Second

4<<<=>>

5<<<<=> 16 4
39 9
6<<<<<= 30 a b

a 1 b 512 31 a 0.14 b 0.24
6
32 a 0.3 b 0.4

15 HT 33 a 1 2 3 4 56
H, H T, H 4 6 8 10 12
H H, T T, T 2 6 9 12 15 18
T 3 8 12 16 20 24
4 10 15 20 25 30
a 1 b 1 5 12 18 24 30 36
4 2 6

16 BG b 40
B, B G, B 13
B B, G G, G 34 16
G
35 a

a 1 b 1 L D
2 4
40 40 60
17 AILT IALT LAIT TAIL
IATL LATI TALI
AITL ILAT LIAT TIAL b 2 c 1 40
ALIT ILTA LITA TILA 9 2
ALTI ITAL LTAI TLAI G
ATIL ITLA LTIA TLIA 2
ATLI

1 36 0.496
b2
a 1 37 a i 1 ii 1
6 8 8

18 a 38109 b 55109 b yes

3 25 38 a
19 57
19 a b T

20 a 18 b 16 75
77 77

21 a 23 b 13
65 23
8
22 a 0.2 b 0.3

23 a 0.8 b 0.9 b 7 c57
22
24 a 0.5 b 0.7

25 a 0.2 b 0.1 39 a 47 b 83 c 33
130 71

Answers 553

40 0.582 c 4 d 11 e 1
15 15 4
41 0.75
1 b 0.88 1 b 1
52 a 3 2
42 a 15

43 1 Chapter 7 Mixed Practice
36
7 79 26
44 a 17 b 187 c 187

45 0.5 1 416 21 32 21
2 a 56115 115 59 32
46 0.5 b c d
2
1 b 11 3 a 15 b 3760
47 a 5 20 4a

13 b 52 c same T
48 a 23 161

49 a 0.226 b 0.001 73 F
50 a 15 5, 10, 20
3, 6, 9,
12, 18

M G 1, 2, 4, 7, 8, 11, 13, 14, 16, 17, 19

15 4 12 bi 1 ii 314
5
3
88 5 a 0.2 b 2
3

6 a 0.18 b yes
7 0.92
21 8a

S A B

29

b 0.21 c 11 0.1 0.6 0.2
51 a and b 40

0.1

PW b 0.75
x 11− x x − 4 c 0.667

x 9ai2 9
11− x 10− x
b 5
18

x−6 ii 118

T 0 10 a 0.143
b 0.111

c 0.238

554 Answers

11 a Toss 1 Toss 2 Toss 3 17 b 6 or 15
18 a 15%
1H 1 H1 1 1 1 b 60%
2 2× 2× 2= 8 c i 0.442 d 0.642
2
H 1 1 111 1 19 a, b
2
1 2T 1 T 2× 2× 2= 8 Mango Banana
2 1H 2 H1 1 1 1
x 7 2x
1 2 1 2× 2× 2= 8 15
2 1 2
1 111 1 18 12
T 2T 2
T 2× 2× 2= 8 x
1 H1 1 1 1
2 Kiwi fruit 8
1 2× 2× 2= 8
2
111 1
1
2 T 2× 2× 2= 8
H1 1 1 1

2× 2× 2= 8

111 1

T 2× 2× 2= 8

b 1 c 7 d 3 c x = 10
8 8 8
d i 50 ii 82 15
ii 0.37 82
( )48> 24 e i 0.08 iii
91 49
123 A0.s3h1e0r f 14
2475
8 13
14 a 23 b 23
Chapter 8 Prior Knowledge

15 a 1 4 b 7 c 25 1a 1 b 8
66 66 3 15

16 a 22 6 8.5 10 12

M C 0 10 20 x
11
5 Exercise 8A

6

10 1a x 0 12
72 2 41
P(X = x) 7 77

E bx 0 12
ii 56 4 81
P(X = x) 15 15 5

b 16 2ax 0 12
ci 3 1 11
d i 0.22 P(X = x) 4 24

ii 0.05 bx 0 12
iii 0.62 1 11
P(X = x) 4 24
iv 3319

Answers 555

3ax 012 3 16 a 1
1 17
P(X = x) 1 3 3 8
3
888 1 bh 0 1 2
8
bx 0 1 2 P(H = h) 19 13 1
3 34 34 17
P(X = x) 1 3 3 1
216
888 c 0.5

4 a P(X x= =) 1 6 for x 1=, 2, ..., 6 17 no

b P(X x= =) 1 8 for x 1=,2, , 8… 18 n = 1

5ax 012 19 $1.50
20 a To ss 1 Toss 2 Toss 3

P(X = x) 125 25 3 1 H1 1 1 1
216 72 72 2 2× 2× 2= 8
1H
bx 012 3 H 1 111 1
2 2
P(X = x) 1 1 T 2× 2× 2= 8
512 2 1
343 147 21 1 H1 1 1 1
2T 2 2× 2× 2= 8
512 512 512
1H
6ax 012 1 111 1
0.64 0.32 0.04 2 2
T 2× 2× 2= 8
1
P(X = x) 1 H1 1 1 1
2T 2 2× 2× 2= 8
bx
012 1 0123
P(X = x) 0.09 0.42 0.49 2 13 1 111 1
88 2
T T 2× 2× 2= 8

7 a k = 0.17 1 H1 1 1 1
i 0.77 2 2× 2× 2= 8

b k = 0.26 ii 0.416 1 111 1
i 0.49 2
ii 0.245 T 2× 2× 2= 8
8 a k = 0.2
i 0.4 ii 0.75 b 3
8
b k = 0.36 ii 0.814
i 0.43 cx
ii 0.833
9 a k = 0.125 P(X = x) 31
i 0.6 ii 0.667 88
b 1.9
b k = 0.2 b 2.24 d 1.5
i 0.9 b 9.2
b 0.7 21 a x 2345678
10 a 2.1 b 0.4
11 a 2.1 P(X = x) 1 1 3 1 3 1 1
12 a 5 12 16 8 16 4 16 8 16
48 15
13 a k = 0.4 91 91 c 2.9 b5 b 2.13 c 5.17
14 a k = 0.2 c 6.2 b 0.75 c 1.92
15 a x b 0.857 22 a 0.4 b 311
1
p 0
23 a k = 12
4
13 24 a c = 0.48
25 a = 0.3, b = 0.2

556 Answers

Exercise 8B 29 a 0.0296 b 0.008 76

( )1 a yes, ~XB 30,1 c e.g. A source of faulty components, such as
2
a defective machine, would affect several

components.

( )b yes, ~XB 45, 1 Exercise 8C
6

2 a no; number of trials not constant

b no; number of trials not constant 1 a 0.159 b 0.345

3 a no; probability not constant 2 a 0.726 b 0.274

b no; probability not constant 3 a 0.260 b 0.525

4 a yes, ~XB 50,(0.12 ) 4 a 0.523 b 0.244

b yes, ~XB 40,(0.23 ) 5 a 0.389 b 0.552

5 a no; trials not independent 6 a 0.246 b 0.252

b no; trials not independent 7 a 0.189 b 0.792

6 a 0.160 b 0.180 8 a 0.133 b 0.132

7 a 0.584 b 0.874 9 a 13.4 b 6.78

8 a 0.596 b 0.250 10 a 8.08 b 15.9

9 a 0.173 b 0.136 11 a 31.0 b 46.0

10 a 0.661 b 0.127 12 a 28.1 b 46.7

11 a 0.371 b 0.280 13 no; not symmetrical

12 a 0.571 b 0.280 14 a

13 a 0.0792 b 0.231

14 a 0.882 b 0.961

15 a 0.276 b 0.001 69

16 a 6.25, 2.17 b 10, 2.58

17 a 5, 1.58 b 10, 2.24

18 a 6, 2.24 b 3.33, 1.67

( )19 a B 10,1 b 0.291 c 0.225 5 10 x
6
8.7
20 a 0.292 b 0.736 c 10.2

21 a 0.160 b 0.872 b 0.660

c 0.121 d 4.8 15 a 0.465 b 0.0228
16 a 0.308
22 a 0.983 b5 c 0.383 17 a 0.274 b 0.328
18 a 0.0478
23 a All have same probability; employees 19 a 0.0831 b 10.4 c 0.282
independent of each other. 20 a 4.52
b 6.42 minutes

b e.g. May not be independent, as could infect b 6.34 hours c 62.3
each other.
b 6.47 m
c 0.0560
d 0.0159 21 12.8 s b 6.74
22 a 15.4
24 0.433
25 a 0.310 b 0.976 c 0.643 23 4.61
26 a 0.650 b 0.765
27 0.104 24 20.9
28 0.132
25 15.2

26 predicts 4% get a negative score

27 a symmetrical

Answers 557

b 13 30 36 42 59 17 a 10.8 cm b 0.698%

18 22.8% of times would be negative.

10 20 30 40 50 60 19 3201ml b 0.0426
Student mass (kg) 20 a 9

28 a 0.0228 e.g. anticipating the start gun 21 a 0.925 b k = 20.4
b
d 0.159 c 0.488 22 a 0.835 b k = 1006.58 c a = 6.58

The same distribution is true in all races. 23 a i 1 ii 0.0579
Unlikely to be true.
b ii a = 0.05, 0b.0=2

29 $728 c Bill (0.19)

Chapter 8 Mixed Practice 24 a i 0.845 ii 1.69

b That Josie’s second throw has the same

distribution as the first and is independent.

1 a k = 0.5 b 0.6 c 2.9 These seem unlikely.
2 a 0.296 b 0.323
25 n = 9
3 a 0.347 b 0.227 26 10
27 a 0.919
4 215 28 277 b 0.0561
29 b 6
5 0.346

6a 1 , 1 , 1
2 3 6

b no; expected outcome is not 0 Chapter 9 Prior Knowledge

7N= 7 1 1 x−1 − 1 x−2
8 a 0.933 4 2

bi 2 y x= 3+ 11

3 -2

Exercise 9A

0.85 1 a 0.6 b0

2 a 1.5 b3

3a2 b3

20 k x 4a1 b1
5 a 0.693

ii k = 23.1 b 0.153 c 0.0435 In further study you might learn how to show
1 b 0.360 that this is actually ln 2

9aP= 3 b 1.10 b0
10 a 0.206 6 a 0.2

11 a 0.356 7 a 0.5 b 0.5
8a1 b 10
b a box of six containing exactly one VL egg

12 a 0.354 b 0.740 b0
b 0.347
13 0.227 9 a −0.5
10 a 0.805
14 0.232
15 a 3.52 b 0.06 c 0.456 11 a 0 b4
12 a 6 b0
1
16 a = 6 13 a 0.5 b 0.0833

558 Answers

14 a -1 b -2 Exercise 9B
15 a 1
16 a ddvz b 0.693

b da 1a x>1 b x<0
db 2 a 1− < <x 1 b x < −1or 6x >
3 a 0 < <x 90 or 270 < <x 360
17 a dp b db
dt dx b 180 < <x 360

18 a dy b dt 4a y
dn df

19 a dh b dw 2
dt dv

20 a dw b dR
du dT

21 a dy = y b dy = x x
dx dx 2

22 a d s = kt2 b dq = k q
dt dp

23 a dP = kL(P) b dP = kA( P)
dt dt

24 a dh h= 1 b dr = k by
dx dθ 1
2
25 a s ( ʹ)=t7 b q (ʹ) x7 = x x

26 a d(pH) =k 5a y
dT
− 21
b dC = k(1000 − n) x
dn

Technically, the number of items produced is
not a continuous quantity so we should not
formally use derivatives. However, as long as
n is large, as is usually the case in economic
models, then approximating it as continuous is
reasonable.

27 a dV = V+ 1 b5
dt

28 57.3

29 1 x2 − 1
x−1
30 a

b 2, the gradient of the curve at x = 1

31 3

32 1
6

Answers 559
b y 7a y

6a 3 −2
x 2x

yb y

−4 2x
−1

−1 3 x

by 8a y

−1 2 −5 −1 3

x −3 1 5 x

560 Answers
b
y 12 a x < −1or 0x > b x < − 1 or x > 1
13 a 2 2

y

−3 3 3
x −1

x

9a y

by

3
x

3x

by

14 a y
x −2

x

10 a x > −1 b x<3
11 a x < −3 or x > 3 b 4− < <x 0

Answers y 16 a y 561
b 1x −1 x

15 a y by

−2 3 x
4x

by 17 a y

−1 1 2

4 x
x

562 y 19 Answers
b −1 2 y

x x
y
18 a y 20
1 x
x
by y
21 a x

3x

Answers y 563
b 23 a y

x −1 1 x

22 a y b x < -1 or x > 1
b x c x < - 0.707 or 0 < x < 0.707
24 a and e, b and d, f and c
y 25 a and c, f and b, d and e
x
Exercise 9C

1 a ʹ f ( x) 4=3 x b ʹ g ( x) 6x=5

2 a ʹ h (u) = −u− 2 b ʹ =z (−t)t−4 5

3a dy = 8x7 b dp = 1
dx dq

4a dz = −5−t6 b ds = −1−0r11
dt dr

5 a ʹ f ( x) 4= − b ʹ =g (x) x14

6a dy = 6 b dy = 15x4
dx dx

7a dy = 0 b dy = 0
dx dx

8a gʹ( x) = − x− 2 b hʹ( x) 3x=4 − −
9a
dz = −6−x3 b dy = −5−0t6
dx dt

10 a dy = − x− 5 b dy = − x− 6
dx dx

11 a ʹ f ( x) = 3 b ʹ f ( x) = 5
2x2 x4

12 a f ʹ( )x2 =4 −x b ʹ g ( x) 4x=5 −

c Gradient specifies shape but not vertical 13 a dy = 9−x1+20 7 x b dy = −4+x13−2x 2
position. dx dx

564 Answers

14 a dy = 2x3 b dy = − 9 x5 Exercise 9D
dx dx 2

15 a dy −3 3 x2 b dy = −6x2 + x3 1 a 16 b 20
dx dx
= 4 2a 1 b 16.75
8

16 a ʹ f ( x) 8= x3 − 15x2 3 a 32 b 7−

b ʹ =g (x) 3 x2 + 6−x 9 4a 23 b 100.4
9

17 a gʹ(x=) 2+ 2 x b ʹ =f (x) 2 1 x − 5 a 108 b6

18 a h ( ʹ) x= − − 2 2 6 a 24 b − 3
x2 x3 16

18 7 a1 b 2±
8 a2±
b g (ʹ)x8 = x −x 3 b ± 1
2
b 2ax +b 1
19 a a b 3x2 b2+ 9 a 1,−3− b ± 2
20 a 2 ax + 3− a b 3 2 a2 x
21 a 2x 10 a 4x - y = 1 by- x= 0
11 a 2x +y = 7
22 a 2a2x1 a − b − ax− a− 1 + bx−b− 1 b x7y + = 2
23 a
7a 6b 10b2x 3a 12 a x +y 3= 6 b 2x −y = 3
x3 cx2
+ b − 13 a y x 1=5− + b y x= −2 12

24 a 2 2 a x b 18 2a x 14 a y = -x b y = −x+ 3

25 a 2x a b+ + b 2abx b+ a+2 2 15 a y x= 1 1 b y =x 1+16 81
−3 3 16
10 5
26 a − x2 b x2 16 a y x= 2+−4x 3 b y x= 1+
17 a y = b = − 1+x6
27 a 41 b 3 4 1 y 3
x3 − x2 − x−4 x3 4
18 a y = − 4x+ 3 b y = − 3+x2 7
28 a 0.5 3 + x b 7 x3 + 4 x7 4 32
19 a y =x 1− b y =x 1−
29 a x < 1 b x < −4
20 a y ≈ 2.77 x1.−55 b y ≈ 1.10x1+
30 a x > 6 b x>0
21 a y = 1−4 4 x b y =x 1−
31 6 + 4
t2 22 a x 8y 5+47= b x4y11+ 1=26

32 −1 2 2 m− 23 a x +y 2≈.10 b x y+ 2≈13.4
3
33 2 k 24 a y x= − b y =x 2−

34 x> 1 25 a 0.−984 b 0.−894
2
26 a 2 b 0.693

35 x > − b 27 a -3, -1 b 0, -2
2 cyx =
28 a 4x13 − b -1 c y x= 2
36 -1 b2
37 −1 3 2x 29 a 3x2x− −2
11 9
k b– V2 c4 30 y = 4 x − 4
r k
38 a – 2 17
2
39 a dA = q+ 2qL 31 y x= +4
dL
32 x 0=
b q> 0 ( )33
− 3 , 94
2

( )34 − 81−, 8

Answers 565

35 y + 2x = −1 3 a ʹ =f+(x)+6 10x42 x
b f ( ʹ1−) 0=
36 y = 1 x + 15 cy= 2
4 4
37 y = −43 3orxy = − 43− 3 x
4 -0.5
38 (2, 6), (-2, -6) 5 y = 27x58−

39 (-3, -2), (1, 6) 6 a 12 10t+ dV (6) 7=2
b V(6) 3=02, dt
32(2o, r02.5)
40 After 6 minutes, there is 302 m3 of water in
41
the tank, and the volume of water in the tank

42 (1, 1), (3, 9) is increasing at a rate of 72 m3 per minute.

44 a = 20 c dV (10) 1=12 , so the volume is increasing
dt
Chapter 9 Mixed Practice
faster after 10 minutes than after 6 minutes.

1a81 x− b (2, 15) 7 a 2 2t − b i 0.75 ii 1
2a y c0 t
1
8a 6
c0 b x +x 8 −2
dy= 6
e, f

y

y= 1 2 − 8
2 x
x

1x

P (4, 6)

y = 6 (− 2, 6)

16 1 x
3

b x> 1
2

c y

g (4, 6)

9 (1, -3), (-1, 9) b y x= −8

10 a (0, 0), (4, -32)
11 a y x= y6 , =− −6x12

1 x b (-1, -6)
2 12 p = −2=, c− 3

−1 13 a 30 3t t− 2 b 72, 72−

c The profit is increasing after 6 months, but

decreasing after 12 months. 1
2
14 a i 2 ii 2 3x + b −

566 Answers
cy
16 a 3 b 2 1x −

c (2, 4) ( )d1 , 16
3 9

e y x= 5− 7 ( )f1 , 7 , gradient = 0
2 4

17 a 2 1 x + b 3 and 2 −
18 a = 2, b = −5
19 a = −8=, 1b3

20 (1.5, 6)

x 21 (0.701, 1.47)
You need to make good use of technology in this
question!

15 a i 0 x 4 Chapter 10
3 Prior Knowledge

ii y 1 x−2 − 4x−3 + 4x−4 2 ʹ =f (−x) +12 2 x−4

Exercise 10A

x 1 a f(x) = 1 x4c+ b f(x) x=c+61 6
4
4 b f(x) = x + c
3 2 a f( x) = −x+−c1 b y x= c− +5
b y x= c+83 8
y = f (x ) b f( x) = − 1 x−c2 + b y = − 61+x1c0
b i x 0 and 0 x 2 2 b y x= c− +−5
b y x= c2+ 2−
ii y 3 a f(x) = 1 b y = − 41+x−c7

2 4 a y =x c+3

5a y = − 7 x4c+
4

6a y = 1 x6c+
4

7 a y =x c+−3

8a y = − 3 x−c4 +
2

9a y = 2 x−c1 +
5

x 10 a y =x x−3x+c+2 52

b y = 7 x5x+x2−c2+3
11 a 5

b y = 1 x2 − 2 x6c+
2 3

y = f(x) y = 3 x4 − 85+x8c
2

Answers 567

12 a y = 1 x3 + 7 x2 + c 5 21.3
b 4 6
6 0.25

y = 4 x − 2 x5c+ 7 30
5 15
8 4.5

13 a y = − 1 x−4 + 3 x2 + c 9 32
2 2 3

b y = 5+x3 3 x− + c 10 18 b 18.2
11 a (2.5, 6.25)

14 a y = − 5 x−1 + 2 x−5 + c 12 12.7 litres
2 7
13 3600 g

b y = − 2 x−2 + 1 x−6 + c 14 a 58 g b 60 g
3 15
c It suggests it takes forever for all the sand to
1 5 fall through / sand is infinitely divisible.
4 3
15 a x4 + x3c+ b 3x2 − x+3c 15 18.75

1 x3 1 x2 6+x c 5 x2 1 x3 4x +c 16 12 x2 − 8
3 2 2 3 x3
16 a − − b − −
16
17 a 3x3 − 6−x x −1 + c 17 3

b 4 x3 − 4x−1 − 1 x−5 + c 18 afa( A) −
3 5

18 a 3 x2 − 2x +c b 5 x2 − 3+x c Chapter 10 Mixed Practice
2 2

19 a − x−2 + 7 x−3 + c b− 1 x−1 + 3 x−2 + c 1 4 x4 + 3+x c
6 2 8

20 a y x= +36 b y x= +55 4 x3 3 x2x+c5+
3 2
1 2 −
4
21 a y = x4 − 6x2 + 10 b y = 3−x2 x5 + 4

5 3 0.661
3
22 a y = −3x−2 + 2x2 − 4 y = x3 - 4x3 + 6

b y = 3x3 + 2x−1 − 4 5a= 8
6 5.7
4 1 +c 24 y x= −3 4x + 7 b 6.75
23 − +3t 2t4 7 a (2, 0), (5, 0) c 1.07
8 a y = 1.6x3.−2
25 y = − 4 +x3 10
x−
9 b 1.5
3x4 2x3 3x2
26 4 − 3 + 2 − 2x +c 10 1.5
11 1000 cm3
z4 z2 x3 2
27 4 + +2 c 28 9 + 3x + c 12 21.5 kg

29 a k = 0.2 b 3 kg 13 47
30 a 40 litres b 4 c no
14 a e.g. no energy lost to surroundings.

b 160 calories
5x3
Exercise 10B 15 y = 3 + 3

1 a 97.6 b 2.25 16 0

2 a -0.625 b -145.5 17 6x +x 2
3 a4 b 36
4 a 7.5 b 4.5 2

18 a (3, 0) b 18

568 Answers

Core SL content: b x = −4
Review Exercise d − ( 2.85078, 2−.35078)

OR (0.35078, 0.85078)

1a 4 b 83 c 900 e1
c 3x2+ y = 31
2a2 3 b2 f y =x − − 5
3a 11 x − 1+1y=112 0
y
(−1, e) 12 a 1000 b −2 −3
13 a 8x274− x−2 b 56x468x+

2 14 a 18 3 r b 18(1 −) r15
c 0.315 1− r

15 3.6%

16 60
41
−2
x 17 a 12 b 0.0733
c 0.764

b y=0 c y р 2.72 18 a −2 d (−3, 3)
b f( ) x0 c y = −2−x3 f 10
4 a a = 2.5; x 2.5 e 20 ii 6
c 18.5 b 95.5cm 2
b 0.305 19 a i 1295.12 cm3 iii 217 m
5 a 38.9° iii I 431cm3
b 15.6 II 4.31 × 1−0 m4 3
6 a 4.5 b2
c 28.2 b i I 73.5°
b 88.9 g II 55.8 m
7 a − ( 2, 0), (2, 0) e 30.9 cm 2
8 a 4.5 ii 55.0 m

c 7 10 20 a i 0.985
ii strong positive
9 a 9.56 cm3
d 52.8° b y = 260x6+99
c 4077 USD
10 a and c d e.g. 3952 4Ͻ077
e i 304x
y
ii 304x(2−60 699x)+
y=2 iii 16
x
21 a 0.159
b i 0.119
ii 0.394

x = −4 − 3 3 22 a 3
2 4 b 7 26

c 8 15
d 173225

Answers 569

Chapter 11 Prior Knowledge 5 4

9 a x2 b x3

5 3
b x2
1 2x2 + 7−x 4 10 a x3 5
2 2320 11 a 4 x−23 b 5x− 2

Exercise 11A 3 2

12 a 1 5 x2 b 1 3 x3

5 5

13 a x6 b x3

1 a = 4, =b 7 1
2 a = 6, =b 37 14 16
3 p = 4, =q ±5
15 2
8
8 a and c 16 27
9 a = 1, =b 3
10 p = 5, =q 2 11

13 a e.g. = x2 17 x12

, =y 1 b1 18 3x−13 + 1

2x2

Chapter 11 Mixed Practice 19 1 x− 3
2
3

20 1
4
1 b and c
2b k= 6 21 25
3 b p = 2, q = 5 4
4 a = 4, b = 16
6 A = 1, B = 2 7

12 a a = 2, b = 4 22 x6

3

23 x− 2

8

24 x3

Chapter 12 Prior Knowledge 25 1 x− 3
8 2

1 a 15 9 x5 y b 4 d c 9a8 b4 − 1
2 y = 91 2 c 26 x2x+ − 1

3 x = 35 31

27 x2x− 2

4 x = ln11 +1 ≈ 1.70 28 1 x− 3
2 3 2

Exercise 12A 29 1 x1 + 3 1
2 2
x− 2

1 a7 b5 8
2 a2 b3
3 a4 b5 30 y4 = 16x3
4 a4 b8
5 a 125 b 25 1

6 a 1 10 31 3 y x= 3 6

7 a1 4 3
8 a1 4
32 y x= 2

33 y = 8 3
27
x− 2

b 1 10 2

34 x =y 3

b19 35 ±27
36 64
b 1
1000

570 Answers

Exercise 12B 30 a 0.490 b 2.28

31 a 1.49 b −0.277

1 a4 b2 32 a ln 2+2ln 3 b 2ln 2 +ln 3
2 a −3 b −2 ln 3−ln 2 ln 3−ln 2

3 a1 2 b14 33 a 3ln 2 +5ln7 b 8ln 2 −ln 7
4 a −1 b −1 2ln7 l−n 2 3ln7 l−n 2
5 a0 b0 34 a x +y 4 b 2x +y z− 5

c 3+2 3+x y

6 a19 b 1 16 35 a 2 +1 2 x b y − 1−5 z

36 ln(a2 b6 )

7 a98 b 199 ln ⎛ 3 x⎞
⎜⎝ y⎟⎠
37

8 a 23 b 61

9 a3 b2 38 a − 1 2 b13
b 1.77
10 a 5 b −2 39 5
40 42 b log 2.24
11 a 2 p −q b 3q −p 41 a 1.43 b 4 e or 14 e
42 19.9
12 a 2 p −q 3 b 4q2− p 43 2
44 4
13 a 3 p + 1 q b 2 p + 3 q 45 a 2.4
2 2 2
1
14 a 2 + p− q 2 b +1 2 p −q 5 46 a ln x

15 a 500 b2

16 a 9.5 b4

17 a 177 b54

18 a 3 +5e 4 b e32+ 47 − 97
1 −e 4 e31−
3ln5 5+ln9
19 a ln l7n 2 b ln8ln5 48 ln 9−2ln5

log23 log35 49 a 10 b 6.58 days
log25 log37 ii 1210
20 a log54 b 50 a i 1000
21 a log5e b log12.1 7.2≈7
22 a log25 log47 hours
log4e
4 b 51 7.97 years

b log32 52 64 or 164
3 ln 96

23 a 2 b 3 53 ln 72
log52 log23
54 2log 120 years
24 a 4 or 1 4 b 25 or 1 25
55 a −ab b1

25 a 5 or 1 5 b 2 or 1 2 Chapter 12 Mixed Practice
b 256 or 1 256
26 a 8 or 1 8 b − 83 1 a 27 8 b −3
27 a 4 3 b9
2 ln 36
28 a − 97 b 0.528 35
29 a 2.58 4 log12.0514.≈2

Answers 571

5 2 Exercise 13A
8
65
1 a13 b1
7 e−4 2 a1
b59
8 a 2a b+ b a b− c− 3 c 1 c + 3 a b9
b2 2 2
b 2x3−3 −y c −1 3 a 16 3
9a 2 4 a13
b x+2y
10 a x − 21 y b14

11 a x +y c x − 2y 5 a 758 b 647

12 log 6250

13 1 5 6 a 1 Ͻ Ͻx 3 b − 4 Ͻ Ͻx −2
2
e3 1 1
2 3
1 7a x Ͻ b x Ͻ
9
14

15 − 133 8 a xϽ2 b xϽ5
16 x = 8, =y 9 9 a − 5 Ͻ Ͻx −3 b 0 Ͻ Ͻx 2
17 2
10 a x Ͻ 2 b x Ͻ 3
ln 2 3 4
18 2ln3 1−
11 4

ln5 +3ln 7 12 4
ln 7 −2ln5 3
19 13 2

log12 14 − 13
20 log 48

21 a 150 b 3397 15 1
c 1.82 hours 4

22 16 16 2, 32 , 29
23 a 3
24 a 5 b 125 17 1 , 32
25 a 12 b p = 5, =q −3 3
b −1
c8 18 a 1, 2 5 , 245 b53

26 − lnln23 19 3 b 277
27 2x 20 a x Ͻ 9
28 2y
29 a 2x3− 21 a 2 Ͻ Ͻx 4 b 5
30 210 ln x 4 −x
b2x 1 2
31 −4 22 a x Ͻ 2 b 1 −2x

Chapter 13 Prior 23 6 Ͻ Ͻx 1 b x
Knowledge 24 a 0 2 1 −4 2x

25 a 6 5 b xϽ2
26 9
320 ≈ 107
1 3

20
3 −1 Ͻ Ͻx 1
4 9 −24 16x +2 x

572 Answers

Exercise 13B 30 12 x
31 a 1024 15+360 103x6+80 2

1 a 64 +192 2x40+ x2 + 160x3 b 1039

b 128 +448 6x72+ x2 + 560x3 32 a 19 683 11−8 098 314x9+28 489 88x82 − x3

2 a 243 40−5 x + 270x2 − 90x3 b 18 533.0

b 81 −108 5x4+ x2 − 12x3 33 a 128 +448 6x72+2 x b 384 +1216 1x56+8 2 x

3 a 1+12 6x0+ x2 +160x3 34 a 32 −240 72x0+2 x b 240

b +1 14 8x4+ x2 + 280x3 35 1040
36 a 16x634+ x
4 a 1 −20 1x50+ x2 − 500x3 b 0.640 016

b 1 −25 2x50+ x2 − 1250x3 37 12
38 10
5 a 1024 +15 360 1x03+ 680 x2 + 414 720x3

b 512 69+12 41 x47+2 x2 +145152 x3 39 x4 + 8+x22+4 +32 16 x2x4

6 a 32 −240 72x0+ x2 − 1080x3 40 x10 + 15x990+270x480+5 243x7 + x6 + x5

b 64 −576 21x6+0 x2 − 4320x3

7 a a7 + 14a6b + 84a5b2 + 280a4b3 41 12 285

b a8 + 16a7b + 112a6b2 + 448ab5 3 42 −1 959 552
43 0
8 a 243a5 − 810a 4b + 1080ab3 2 − 720a2b3

b 729a6 − 2916ab5 + 4860a4b2 − 4320ab3 3 Chapter 13 Mixed Practice

9 a 1120 b 4032

10 a −10 206 b −61 236 1 16 +32+24x 8 x2x3+x4 +

11 a 13 440 b 5376 1
2
12 a 24 634368 b 7 185 024 2

13 a 760 b −684 3 9
2
14 a −24 634 368 b 3 421 440

15 a 1 +8 2+x4 x2 + 32x3 +16x4 43

b 16 +32 24x + x2 + 8x3x+ 4 5 14
6 a p = 5, 7q,=5 = r
16 a 27 +27 9x2+ x x+ 3 (or r= 7 )
b −24 634 368 b 262 440 3x
b +1+9 2x7 x2 + 27x3 b 1.02018
b59
17 a x5 + 10x4 + 40x3 + 80x2 + 80x3+2 7 a 11
b 108
b x5 + 5x4 +10x3 + 10x2 + 5+x1 8 a 1 +20+1x80 2 x ii 144

18 a 6 b8 9a 1 < <x 1
3
19 a 21 b 15
2
20 a 84 b 56 10 3

21 a 10 b 13 11 13 440
12 a 4
22 a 105 b 55

23 a 5, 9 b 7, 8 13 a n
bi9
24 a 3, 17 b 5, 12

25 a 12 b 18 14 ± 12

26 10 000 1−2 000 54x0+0 x2 − 1080x3 + 81x4 19
20
27 32x5 − 80x4 + 80x3 − 40x2 +10x1− 15

28 127 575 16 15
2
29 51 963120

Answers 573

e− x ( )c ln3 1 1
17 b 1 −e −x 2 7 a 3x3+ b 2x3+

18 1 8a 32 b 4 +3x 2
3 x −x 2 x

19 7 9 a x Ͼ −8 b x Ͼ −9

20 −96 10 a x Ͻ 1 b xϽ 0
1 5
21 a ii 2a iii 2 (a2d)n1 − 11 a x 2 b x 3
ii d Ͻ 0
bi 2a(n2d − 1) iv −1 12 a xр 7 b x 35
2d1− 2

2a 13 a x ≠ 41 b x≠1
iii −1 2 d 14 a x ≠ ln 3
15 a
n(n1−) 11 −9x b x ≠ ln 7
1
c k p= q n 2 b 3

Chapter 14 Prior 16 a x4 + 2+x22 bx= 1
Knowledge 17 16 9x b 193
18 a x −2
1 11
2 x ≠ −3 19 a fg(8) l=n3, gf(8) ln8=5−
3 a f(x) −3
b e85+
b Many-to-one
20 a i 2 ii 1
4 b4
y
21 a i 1 ii 1
b1
y = f(x) b e +3
22 a x > 3
3e3
c e4 d e31−

23 a x−3 b x ≠ 2.5, 3
2 x5−
x
y = f − 1(x) c7 3

24 k = 16 b y ≠ 0, 12
25 a x ≠ − 23−, 7 6

5a x = y +1 b x = ln (y)+ 2 c−53
y−3
b 4x2 + 2 Exercise 14B

Exercise 14A b ( 3x2− )2 1 a f ( )−12x 1 = x− b f ( )−15x 4 = x
4 −3
b 20x2 − 6x
1 a 3x2 − 1 b 4e3x+1 2 a f ( )−1x = +x 4 3 b f ( )−1x = −x 5 1
b 6ex5+
2 a ( 2x1+ )2 b e3x + 4ex 3 a f ( )−11x = 4 ln x b f ( )−11x = 3 ln x

3 a 12x2 + 4x ( )4 a f ( )−1lxn 3 = x + 2 ( )b f ( )−1lxn 2 = − x 3
4 a 3e2x+5
5 a 12ex1+
6 a e3x − 2ex