Math AI SL

Applications and interpretation SL: Practice Paper 2 425

e For Michele’s 15 shots in the first round, find the
i mean number of bullseyes
ii standard deviation of the number of bullseyes.

[2]

There are 4 further such rounds in the competition.
f Find the probability that Michele hits the bullseye at least 10 times in more than 3 of the 5 rounds.

[3]

3 [11 marks]

The surface area of a solid cuboid, S, can be represented by the function S(V ) = kV c , where V is
the volume of the cuboid and k and c are constants.

a Find the value of k and c.

[3]

b The domain of the function S is 0 ഛ V ഛ 125. [4]
i Sketch the graph of S. [2]
ii On the same axes sketch the graph of the inverse function S-1. [2]

c State the
i range of S-1
ii domain of S-1.

d i Calculate the value of S-1(54).
ii In the context of this question, explain the meaning of the value of S-1(54).

4 [16 marks]

A geographer proposes that the depth, d metres, of a river can be modelled by a function of the
form d = p sin qx, where x is the distance from the left bank of the river.

The river is 8 m wide and the depth halfway across the river is 2.3 m. [3]
a Find the value of p and q. [1]

b Calculate the depth the model predicts 2 m from the left bank.

The geographer checks the prediction by measuring the depth of the river 2 m from the left bank.

He finds the depth is 2.5 m.

c Calculate the percentage error in the model’s prediction.

[2]

In view of the extra measurement, the geographer decides that a model of the form [4]
d = ax3 + bx2 + cx would be more appropriate. [2]
d Find, to three significant figures, the values of a, b and c.
[4]
e Sketch the function found in part d for 0 ഛ x ഛ 8.

f Use this model to find
i the maximum depth of the river
ii the cross-sectional area of the river.

426 Applications and interpretation SL: Practice Paper 2

5 [15 marks]

The shipping costs a business pays vary according to the size of the shipment. The following data
is available:

Size of shipment, x Cost of shipment, y
(thousands of items) (thousands of euros)

76.4  6.0
102  6.9
125  8.2
130  8.5
154  9.2
180 10.6
226 11.8
253 12.3
288 12.7
315 13.2

The owner of the business believes the cost can be modelled by a linear function. [2]
a Calculate the value of the Pearson’s product-moment correlation coefficient, r. [2]

b Explain whether this value of r supports the use of a linear model.

The owner thinks the model can be improved by making it a piecewise linear model of the form

y = ax + b 0 ഛ x ഛ 200
cx + d x Ͼ 200

c Using linear regression of y on x, determine the values of a, b, c and d.

d Use this model to predict the cost of a shipment of [5]
[3]
i 90 000 items ii 200 000 items iii 330 000 items. [3]

e Comment on the reliability of the predictions in part d. [5]

6 [14 marks]

The point P with x-coordinate pϾ0 lies on the curve y = 1 .
x

a i Find, in terms of p, the gradient of the tangent to the curve at P.

ii Show that the equation of the tangent to the curve at P is x + p2 y − 2 p = 0.

The tangent to the curve at P intersects the x-axis at Q and the y-axis at R.
b i Find the coordinates of Q and R.
ii Find the area of the triangle OQR, where O is the origin.

c Show that the distance QR is given by 2 p2 + 1 . [4]
p2 [3]

d Find the minimum distance QR.

[2]

Answers

Toolkit: Using technology 5 a x = 6, y = 4 b x = 3, y = 1
c x = −1, y = 2 b 9
Computer algebra systems (CAS) 6 a -2
a 1 − x2 c 12 d 2
b 1 e 12.5 f 5

Spreadsheets 7 a x = y−4
b 370 m, 20 m s-1 2
3y
Dynamic geometry packages b x = 2− y
a AB = 2 MC
b 1:1 c x = 1− 2y
y −1
Programming
c Welcome to Mathematics for the International d x = by
y−a
Baccalaureate. We hope that you enjoy this book
and learn lots of fun maths! (This is an example of e x = ± a2 + 4
a ROT13 Caesar Shift Cipher)
f x = − 1
Toolkit: Algebra Practice 2y

8 a 2 − 2

b 2 15

c 4

1 a 11x + 15y b 10x2 + 7xy d 3 3
c 2x + 3
e xy2z d x + 4 e 3 − 2 2

g 8x3 f 63x2 y f -1

h x 9 a 2
6
j y b 2 + 1
i 1+ 2x 3z
y c 1+3 3
13
k x l −1
2 10 a x = −3 or − 2
2(x + 1)
m x2 n 5 b x = ± 3
15 2

o 2x c x = −1± 6
y
11 a b − a
2 a 2x2 − 6x b x2 + 6x + 9 ab
d x3 + 3x2 + 2x
c x2 + x − 20 f x3 + 9x2 + 23x + 15 b 2x + 5
b 3(x − 2 y) x2
e x2 + y2 + 2xy − 1 d 5xy(x + 2 y)
f (3x − 1)(x − 2)
3 a 4(3 − 2 y) c x2 − x + 13
x −1
c 7x(x − 2)
2
e (2z + 5)(x + y) d (1 − x)(1+ x)

4 a x = − 7 b x = 4 e 25 − 2a2
8 a(5 − a)

c x = −8 d x = 5

e x  −13 f x  5 f 2(x2 − 2)
2 3 (x − 2)(x − 1)

428 Answers

Chapter 1 Prior Knowledge 27 a 27u-6 b 32v-15

28 a 1 a10 b 1 b21
4 27
1 a 81 b 40
29 a 25x4y6 b 81a8b-8
2 a 3.4271 × 102 b 8.5 6 × 10-3
30 a x3 b 25
3 26 27 x4

Exercise 1A 31 a 27 x6 b 25u2v6
8 y9 49b8

1 a x6 b x12 32 a 16v4 b 27b6
9u2 8a9

2 a y6 b z10 33 a 2x - 7x6 b 3y + 5y3

3 a a7 b a11 34 a 5u2 + 6uv2 b 5a2b5c − 4ac2

4 a 517 b 224 35 a 5 p − 3p−1q2 b 2s2t−2 + 3s3t2

5 a x b x3 36 a 38 b 324
6 a y4
b z6 37 a 29 b 510
b b8
7 a b6 38 a 25 b 311

8 a 118 b 75 39 a 217 b 27

9 a x15 b x32 40 a x = 4 b x = 3

10 a y16 b z25 41 a x = -1 b x = 6

11 a c14 b c14 42 a x = 7 b x = 9
3 2

12 a 350 b 1328 43 a x = -4 b x = -3

13 a 48x7 b 15x7 44 2x + 4x2

14 a 3a3 b 5b4 45 x5y2
15 a 20x5y3z b 12x8yz5
46 b6
8a3
47 a n = 1000
16 a 2x5 b 3x6 D b 250 000

17 a 1 b 1 c $10 million
2x3 4x
48 a kA = 8, kB = 40 b n
18 a 5 b 3x2 c method B 5
3x2 4

19 a 7x2y3 b 2x3z 49 x = 2
y
50 x = 5
20 a 1 b 1
10 7 51 x = -1

21 a 1 b 1 52 x = -1
27 25
53 x = -2
7
22 a 4 b 5 54 a R = 3.2T 2 b 250 K
3 3 b 1.5v
55 a v
23 a 9 b 125
4 8 c 40 km per hour

24 a 7 b 5 56 x = 1, y = −3
9 16
57 x = 4

25 a 6 b 10 58 x = 6
x x4
59 x = 1, 3 or -5
26 a 8 b 81
9 64 60 27000

61 7

Answers 429

Exercise 1B 16 a 4 log 3x b 1
log 2x

1 a 32 000 b 6 920 000 17 a 1 b 1
2 a 0.048 b 0.000 985 18 a −15 ln x
3 a 6.1207 × 102 b 3.07691 × 103 b 17
4 a 3.0617 × 10−3 b 2.219 × 10−2
5 a 6.8 × 107 b 9.6 × 1011 19 a 4.5 b -1.5
6 a 1 × 100 b 1.2 × 10−5
7 a 2.5 × 1021 b 3.6 × 1013 20 a 13 b 7
8 a 2 × 10-2 b 4 × 10-4
9 a 5 × 100 b 2.5 × 102 21 a 3 b 7
10 a 2.1 × 1011 b 3.1 × 109
11 a 2.1 × 105 b 3.02 × 108 22 a 8 b 9
12 a 7.6 × 105 b 8.91 × 1014
13 a 4.01 × 104 b 6.13 × 1013 23 a 25 b 81
20 a 1.22 × 108 b 400
c 4 × 102 24 a 1 b 1
b 1.77 × 10-42 m3 3 6
21 6 × 10−57 b 7.41 × 108
25 a 1 b 1
22 1.99 × 10−23 g b a + b + 1 32 64
23 a 1.5 × 10-14 m b a - b - 1
24 a 2.98 26 a 2.10 b 3.15
c Europe
25 a 1.5 27 a 5.50 b 2.88
26 a 4
28 a -0.301 b -1.40
27 r = p + q + 1
29 a log 5 b log 7
Exercise 1C 30 a log 0.2 b log 0.06
31 a ln 3 b ln 7
32 a 2 + log 7 b log13 − 4

33 a 1 + log 7 b log13 − 2

34 a ln(k − 2) − 2 b ln(2k + 1) + 5

35 a 0.349 b -0.398

36 a 1.26 b 0.847

37 10 000

38 332

39 a + b

40 0.699

1 a 1 b 2 41 0.824
b 6
2 a 5 b −1 42 0.845
b −4
3 a 0 b 2 43 log 1.6 b 0.0126
b 4 44 a 7.60 b 6.93 days
4 a −2 b −1 45 a 10
b −3
5 a 1 b 1000 46 a i 1000 b 10 ln 3 = 11.0 hours
b 332 ii 1220
6 a 3 b −1.99
b e5 47 4.62 years
7 a 0 b ey2
b 2e2 y+1 + 12
8 a −2 48 a 57.0 decibels b 67.0 decibels
b 5log x
9 a 100 c Increases the noise level by 10 decibels.

10 a 50 002 d 10−3 W m−2
49 a e ln 20
11 a 1.55 b ln 7
ln 20
12 a e2
50 x = 100, y = 10 or x = −100, y = −10
13 a ey + 1
1 51 1
14 a 2 ey−3 − 2

15 a 5log x

430 Answers

Chapter 1 Mixed Practice 5 a u1 = −37, d = 7 b u1 = −43, d = 9
6 a 8 − 3(n − 1) b 10 + 4(n − 1)
1 a 9.3 × 103 cm b 5 280 000 cm2 7 a 3 + 2(n − 1) b 10 + 4(n − 1)

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

4 a 4 000 000 b 1 000 000 c 16 11 a 372 b 910
12 a -11 b -236
5 x=3

9 500 s 13 a 352 b 636
14 a 184 b 390
10 x = 99

11 x = 0.5e3

12 x = 0.693 15 a -45 b 0

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

16 a 2.8 b a + b + 1

17 a 1.2 b a - b 20 a 16 b 16

18 0.0259 to 0.0326 21 a 216 b 2230
22 a 4 b 31
19 a 9.42 m s−1 b Yes c 465

20 a 3 23 a 8 b 305

b Strength increases by one. 24 a 5 b 112

c 3160 m (It would have had to be measured 25 a £312 b £420

using a special damped seismograph.)

( )21 x = log2 ≈ −0.176 26 £300
3
27 14
22 x = 1, y = −2 28 a 387 b 731
b 798
23 x = 10, y = 0.1 29 140
30 a 21
Chapter 2 Prior Knowledge
31 a u1 = −7, d = 7 b 945

1 x = 4, y = −1 32 a 296 b 390

2 a 80 b 36 33 632
34 x = 2
3 x = ±2 35 42
36 42
Exercise 2A 37 960
38 a 876
1 a 49 b 68 b 1679 c 480
2 a -22 b -40
3 a 2 b 7 39 408.5
4 a -0.5 b -2.5 40 53
41 -20
42 a Day 41 b 7995 minutes

Answers 431

43 a 19 days b 45 days 26 x2 - n yn + 1
44 a 3, 7 b 199 27 3069
28 −1
45 2n + 3 29 88 572
46 0 30 a 0.246 m b 3.9 m
47 70 336 c The height is so small that measurement error
48 735
49 0.2 m and other inaccuracies would be overpowering.
50 -1
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 $1 115.87

4 a 48 years b 5 years

2 a -15 625 b -1 441 792 5 a 173 months b 77 months

3 a 1 b −2 ≈ 0.00274 6 a 7.18% b 4.81%
32 729
7 a 14.9% b 7.18%
4 a u1 = 7, r = 2 b u1 = 4 , r = 3
3 8 a $418.41 b $128.85

5 a u1 = 3, r = ± 2 b u1 = 5 , r = ±3 9 a £13 311.16 b £7119.14
9
10 a $97 b $294
6 a u1 = −3,r = 2 or u1 = 3,r = −2
1 1 11 a £737.42 b £2993.68
2 2
b u1 = 7168, r = or u1 = −7168, r = − 12 a $103 b $5130

7 a 6 b 8 13 a £94 b £2450

8 a 13 b 8 14 a €1051.14 b €579.64

9 a 5465 b 4095 15 a $598.74 b $4253.82

10 a 190.5 b 24 2 16 £900.41
11 a 153.75 b 14 56
17 £14 071.00
12 a 363 9
b 28 00 18 $8839.90

13 a 76 560 b 32 4 753 19 €16 360.02

14 a 68 796 b 48 8 280 000 20 £8874.11
21 a monthly b £28.55
9 b 24 827
15 a 6 999 22 £6960

16 a 1 b 25 5 23 Start-year Depreciation End-year
expense ($) value ($)
17 a 2   b 96 c 3069 Year value ($)

18 a 1.5 b 182.25 1 20 000 6 000 14 000

19 1920 cm2 2 14 000 4 200 9 800

20 0.0375 mg ml-1 3 9 800 2 940 6 860

21 15.5 4 6 860 2 058 4 802

22 0.671 > 5 5 4 802 1 441 3 361
23 a 3580 m3 b 11 days
6 3 361 1 008 2 353

24 255 7 2 353 706 1 647
25 a 9.22 × 1018 b 1.8 4 × 1019
c 2450 years 8 1 647 147 1 500

24 15.4%
25 12.7%

432 Answers

26 a 2.44% b 12.8% Chapter 3 Prior Knowledge

27 a 250 billion marks 1 y

b 0.0024 marks c 0.0288 marks −4 3 x

Chapter 2 Mixed Practice

1 a €6847.26 b 7.18%
2 a i dn
b i 1
ii bn 2
3 a 2 3
b  ii − 256 c 300
-4

4 a 4 b 512 c 43 690

5 8 − 12
y
6 10

7 a 48.8 cm b 9 2 x  3
3 0 2
8 a €563.50 b 6.25 years

9 a 8.04 billion b 2033 4 x = −2.30

10 a i 20 b 7290 5

ii 10

11 a i 7 ii 16

c 2 d 25

e n = 498 f n = 36

12 x = 4 1
1
13 b 74 x

14 8190

15 16 months

16 a2n b3−n

17 $11.95 million

18 $10 450

19 8.63% Exercise 3A

20 $1212.27

21 22 years

22 a a1 = 6 1 a 9 b 14
b i a2 = 8 ii a3 = 10 2 a 7 b 47
c 2 3 a -17 b 19
4 a -15 b -44
d i 22 ii 594 m 5 a Yes b Yes
6 a No b No
e 28 7 a Yes b No
8 a No b Yes
f 16 m 9 a Yes b Yes

23 b 32

24 a B – 12th day b A – 6th day

25 a £68 500 b £1 402 500 c £43 800

26 3

Answers 433

10 a  b  24 a y
(−2, −2) y = f (x) = f −1(x)
11 a n ≠ 0 b x ≠ 0
(2, 2)
12 a x− 5 b x3.5

13 a x− 0.6 b x > 4

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

18 a f(x)2 b g(x)3

19 a 40 b −7

20 a 1 b 2 b y
y = f(x)
21 a -3 b -1
y = f −1(x)
22 a y x

1 y = f −1(x)
x
−1 1
−1
y = f (x)

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

2

(π, −1)

b y

23 a No inverse b No inverse

y = f (x)
y = f −1(x)

3

3x

434 Answers

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

that long / The model predicts an unreasonable

speed of 114 m s−1. 1
3
29 a x ≠ 5 b 6

30 a − 5 b q(x)− 2
4
4
c x = 4 or − 4 2 y = f −1(x)

31 a 4.95 billion

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

32 a f(x) = 1.3x

b Amount in pounds if x is the amount in dollars. 43 a 0.182 b 0.892

33 a 1 b x = 2

34 a x2.5 b f (x)0 c x = 7 44 a x1, x ≠17 b f(x)  3 or f(x)0
4
35 a 60 b 129.9; 130 45 a and b
y
c It predicts a non-integer number of fish after
y=x
15 months.

36 a 14 b 4

37 y y = f −1(x)
−8
8 x

6

y = f(x)
−8

4 y = g−1 (t) c x = 2

2 Exercise 3B
y = g(t)

246 8t 1 a y

38 a x > 0 b 4 c 1
9
39 x > 5
7
40 a x 3 b x = −31
b f(x)4
41 a 19 x

c 1 is not in the range

x = −2

Answers 435
b
y b y

y= 1
2

1x x

2 a y f(x)131 81
f(x)ln(0.75) 8
4 a b g(x) ln(4.5)
5 a b g(x)

1 6 a −1.17f(x)1.17 b −1.57  g(x)  e
3
y = 7 a (0.7, 0.55); x = 0.7

−3 x ( ) b − 2 , 39 ; x = − 2
7 7 7
−1

( )8 a −2 ±
2 2 ,1.75 , (−1,2); x = −1

x=1 ( ) b 1 ,− 3 ;x = 1
y 2 4 2
9 a
b (0,−1), (1,−1),
y

1 y=1 y=1
x 2

− 1 x
2

x = −2 x=3

3 a y b y
x=1

y=3

−2 2 x −3 x

y = −2

436 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 − 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

b y 19 y

−2 x 1.61
1 x (−1.5, 1.01)

x

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

x=2

y=2

13 −1.69 x

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

13 x
(2, −2)
y=3

y = −8 1 1 x
2 3
−

x = −2

x = −2, y = 3

Answers 437
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
26 x = −1 600
27
y

350

x 120 180 400 x
1 x
4 − 200

x = −2 x=2 32 x = 0.755
33 x = −2.20, −0.714, 1.91
x = −2, x = 2, y = 0 34 −2.41, 0.414, 2
28 y 35 −1.41, 1.41
36 0.920
y=3 37 y

−1 x x=2
−3 x = 1
(6.32, 0.232)

29 N

80

40

38 x  2

15 30 45 m

438 Answers

Chapter 3 Mixed Practice 7 a 20.0 m s-1 b 0 .630, 17.1 s c 5 s
8 b 1.30 s b  95.3°C c 8.82 km
1 G 9 a 100°C

(20, 104) 10 a f(x) = 1.8x + 32
b Temperature in Celsius if x is temperature in

Fahrenheit.

11 a p = −2, q = 4

b i 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(x)  0
b $3538.09
2 x = 1.86, 4.54 16 −9.25  f(x)  3.
3 a x− 5 b x = -2.38
4 (−1.61, 0.399) and (0.361, 2.87) 17 a 4.89 b x = 28.9
5 y
18 a −19  g(x)  8 b x = −1, x = 1
(2, 9)
c −20 is not in the range of g
5
19 y

y=5

3

−1 x

−1 x
y 5

6 a 20 T
90

1313 y=3 T = 20
x t

x=3

b x ≠3, f(x)≠3

Answers N = 2000 439
21 N
Chapter 4 Prior Knowledge

1 24 cm b (−1, 1)
2 a 10

Exercise 4A

600 1 a 1 b 2
2 a −3
t b −1

3 a 1 b − 1
2 2
22 a 1.10 m s-1
4 a (2, 0), (0, −6) b (4, 0), (0, 8)
b e.g. The car will stop by then
5 a (5, 0), (0 ,2.5) b (−3, 0), (0, 2)
23 x = -2.50, -1.51, 0.440 6 a (3, 0), (0, 4) b (4.5, 0), (0, −3)
7 a 2x − y + 3 = 0 b 5x − y + 1 = 0
24 a y 8 a 2x + y − 4 = 0 b 3x + y + 7 = 0
x = −2 x = 2 9 a x − 2 y + 7 = 0 b x + 3y − 9 = 0
10 a y = 2x + 5; 2, 5 b y = 3x + 4; 3, 4
y=1

12 x 11 a y = − 2 x − 2; − 2 , −2 b y = − 5 x + 5; − 5 , 5
y = −1 33 2 2

12 a y = −0.6x + 1.4; −0.6, 1.4

b y = −5.5x − 2.5; −5.5, −2.5

13 a y − 4 = 2(x − 1) b y − 2 = 3(x − 5)

14 a y − 3 = −5(x + 1) b y + 1 = −2(x − 2)

b i (12, 0) 15 a y +1= 2 ( x − 1) b y − 1 = − 3 (x − 3)
3 4
ii x = ±2, y = ±1
16 a 2x − y − 5 = 0 b 2x − y + 7 = 0
25 x = 2.27, 4.47 26 −1.34
17 a x + y − 10 = 0 b 3x + y + 2 = 0
27 a x ≠ 0, 4
18 a x + 2 y − 7 = 0 b x + 2 y − 4 = 0
b f(x)  −5.15 or f(x)  3.40
19 a 3x + 4 y − 13 = 0 b 8x + 5y − 19 = 0
28 a x > 0, x ≠ e−3
20 a y = 3x + 4 b y = −x + 9
b g(x)0 or g(x)  0.271
21 a y − 5 = 1.5(x − 5) b y + 3 = 0.5(x + 1)
29 a x  2, g(x) ∈ 
22 a x − y − 6 = 0 b 2x − y − 7 = 0
b y
23 a x + 3y − 4 = 0 b x + 5y + 8 = 0

24 a 2x − 5y = 0 b 3x + 4 y + 2 = 0

25 a y = −x + 5 b y = − 1 x + 8
3
26 a y − 5 = −4(x − 1) b y − 4 = 3(x − 2)
27 a 5x + y + 3 = 0
28 a x − 2 y + 5 = 0 b 2x + y − 7 = 0
29 a 2x − 5y + 37 = 0
x 30 a (1, 2) b x − 3y − 10 = 0
31 a (1, 3)
32 a (2, 5) b 2x + 3y + 3 = 0

b (5, 1)

b (2, 5)

c x = 2.12 b (−3, −4)

440 Answers

( )33 a 2 ,11 ( ) b 11, 2 17 a = 2.97 1
55 55 16
18 a a = −2.2, b = −8.6 b y + 5 = − (x − 4)
7
34 a 4 b y = − 4 x + 53 19 a 4 b 7x + 4 y = −20
7 7 7
4 c (0, −5) d 65
35 a − 3 b 4x + 3y = 36 b 3.33 m

36 a − 5 b 17 20 a 1.8 m b (6, 5) and (0, 9)
7 5 b B(4, 5) D(5, 4)
21 a (3, 7)

37 a 1 b x − 5y = −8 22 a y + x = 9
5
7 c 5
38 a − 2 b No
23 a 50 ≈ 7.07 m b 74 ≈ 8.60 m
c y = 2 51
7 x − 7 Chapter 4 Mixed Practice

39 0.733

40 a (8, 11) b 8
d 56
c x = 8 3 b x + 4 y = −1 1 a i (2, -2) b k = − 2
2 3
41 a y = − x − 6 ii 3
2
c (−4.6, 0.9) iii 2
a s = − 3
42 6.10 m
2 6, t = −2  b 4x + 5y = 23
43 a 0.5t + 0.1 b 9.8 s
b 0.018 N 3 b 3 c − 2 d 4
44 a N/m d 0.467 m 2 3
c larger b $6.80 4 a -1 b y = x + 3 c 4.5
45 a C = 0.01m + 5
c 500 minutes b 350 5 a (2.5, −1) b 9.22
46 a P = 10n - 2000 d 400, fewer
c P = 8n - 1200 c y = 7 x − 47
6 12
6 a (2.5, −1, 4) b 9.43

7 a P(0, 3) Q(6, 0) b 45

48 1390 m c (2, 2)

Exercise 4B 8 a − 7 b k = 2 c d = 1
11 ±20 4

1 a 5 b 13 12 a p = 1, q = −18 b 27.5
2 a 13 b 10
3 a 29 b 85 13 1.5
4 a 3 b 11
5 a 6 b 9 14 4.37
6 a 98 b 65
7 a (4, -1) b (4, 6) 15 b y = −2x + 5 c S(1, 3)
8 a (3, −4, 3) b (−1, 2, −1)
9 a (5.5, 2, −3) b (−5.5, 3.5, 7) 16 10 m
10 a (1.5, 0.5, 5.5) b 171
17 a y

B 6
654321
z 5

4

3

2

1 A
56
11 (6, 3, −8) 0C 1 2 3
12 a = 3, b = 20, c = 4.5
4

13 196.5 x

14 4.52 m s−1

15 30 b (2.5, 2.5, 2) c 6.48

16 k = ±2

Answers 441

18 a 1        b (-1, 5) 22 4.21 × 104 cm3 b 1100 cm2
2 23 a 12.0 cm b 36.9 cm
c 2x + y = 3 e 4.47 24 a 3.61 cm b 3.58 cm
25 a 192 cm3 b 1330 cm2
19 a 42 b 15.2 c 5.51 26 a 15 cm b 474 cm2
27 a 82.5 cm2
20 4.32 b (6, 5) and (4, -1) c 594 cm3 b 815 cm2
21 a y = 3x − 13
28 55.4 cm2, 23.0 cm3 b 12.9 mm
23 135 m b 0.194 m3
29 1140 mm2, 3170 mm3
24 4 m
30 1880 mm3, 889 mm2
Chapter 5 Prior Knowledge 31 a 13 cm

1 95° 32 242 m2
2 12.4 cm 33 a 151 mm2, 134 mm3
3 60 34 a 9.35 m3
4 α = 140°, β = 85°
5 050° Exercise 5B
6 Volume = 1570 cm3, Surface area = 785 cm2
1 a 29.7° b 58.0°
Exercise 5A 2 a 53.1° b 39.5°
3 a 40.1° b 53.1°
1 a 40.7 cm2, 24.4 cm3 b 1580 m2, 5880 m3 4 a 2.5 b 3.86
5 a 7.83 b 11.5
2 a 8.04 m2, 2.14 m3 b 0.126 m2, 0.004 19 m3 6 a 9.40 b 14.1
7 a 17.4 b 26.3
3 a 170 m2, 294 m3 b 101 km2, 134 km3 8 a 2.68 b 25.7
9 a 344 b 4.23
4 a 47.9 cm2, 38.6 cm3 10 a 71.6° b 78.7°
11 a 18.4° b 15.9°
b 4.40 mm2, 0.293 mm3 12 a 21.8° b 12.5°
5 a 16π cm2, 32π cm3 b 36π m2, 36π m3 13 a 7.13° b 10.6°
14 a 29.7° b 45°
3 15 a 10.5° b 14.5°
16 a 8.49 cm b 6.53 mm
6 a 64π m2, 256π m3 b 288π m2, 432π m3 17 a 3.42 cm b 3.29 mm
3 18 a 3.31 cm b 4.10 mm
7 a 3π π b 20π mm2, 4π mm3 19 a 37.9° b 50.4°
4 m2 , 6 m3 93 20 a 39.9° b 32.9°
21 a 20.7° b 51.2°
8 a 90π cm2, 100π cm3 b 36π cm2, 16π cm3 22 a 37 cm b 7 mm
23 a 5.06 cm b 6.40 mm
9 a 6 cm3 b 16 cm3 24 a 5.23 cm b 6.43 mm
25 a 39.4° b 41.6°
10 a 20 cm3 b 18 cm3 26 a 42.6° b 62.8°

11 a 26.7 cm3 b 2.67 cm3

12 a 64 mm3, 144 mm2 b 336 mm3, 365 mm2

13 a 2 mm3, 13.8 mm2 b 320 mm3, 319 mm2

14 a 5.33cm3, 33.9 cm2 b 816 cm3, 577 cm2

15 a 75.4 cm3, 109 cm2 b 20.9 cm3, 46.4 cm2

16 707 cm2

17 96.5 cm2

18 8.31 m3

19 134 cm3

20 3210 cm3

21 3400 cm2, 8120 cm3

442 Answers

27 a 53.8° b 97.3° 56 h = d tan 40
28 a 28.5 cm2 b 10.5 mm2 tan 50 − tan 40
29 a 127 cm2 b 1.73 mm2
30 a 631 cm2 b 1710 mm2 57 a x = h , y = 4−h
31 a 52.2° b 61.8° tan 30 tan 10
32 a 8.30 cm b 5.23 mm b 3.73
33 a 7.04 cm b 8.36 mm
Exercise 5C
34 4.16 cm b 26.6°
y 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°
37 a 4 a 48.0° b 82.6°
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°
−1 3 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°

b 71.6° b 32.0 15 a 17.8° b 67.4°
38 6.98 cm 16 a 54.2° b 43.4°
39 58.5°, 78.5° 17 a 1.16 km, 243° b 2.71 km, 320°
40 87.4° 18 a 4.00 km, 267° b 7.62 km, 168°
41 a 18.6° 19 a 3.97 km, 317° b 16.5 km, 331°
42 19.0
20 57.1 m b 18.4°
21 a 15.8
22 a E

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

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

Answers 443

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
b

Tent Rock

b 67.1 m

25 a N 27.7 cm

0.8 km
37°

A 22.6 cm C
c 29.1 cm
c 35.3° b 24.1 cm c 26.6°
3 a 32.5 cm
1.2 km 4 45.2° b − 1
5 a 6 2

b 1.90 km 6 a 12.9 cm
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 15 a

32 a 7.81 cm b 45.2° y

33 191 m, 273°

34 2.92 km, 008.6° y = 1 x + 5
3
35 146 m 10
36 a 9.51 m b 10.9 m c 48.3°
5
37 3.96 m
− 15
38 a 9 – assumes lengths given are internal or thin 10 x
glass or no large objects in the space

b 359 000 W y = 10 – x
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

444 Answers

16 23 900 cm3 b 21.0° 5 a all households in Germany
17 a 65.9° b convenience sampling
c e.g. Households in the city may have fewer pets
18 5.46
than in the countryside.
19 21 6 a the number/proportion of pupils in each year

20 17.6 m group
b quota sampling
21 a 131 cm3 b 313 cm3 c i keep

22 a 5.12 cm b 9.84 cm c 31.4°
23 a 5 cm
ii discard
b 9.43 cm
c 58.0° 7 a convenience sampling
a 720 m
b all residents of the village
b 72 300 m2 c e.g. People using public transport may have
c 88.3°
different views from those who drive.
24 a i 22.5 m d would need access to all the residents
ii
V 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.
c list in serial number order, select every 20th
10 a quota
b more representative of the scarves sold

53.1◦ c 12 red, 12 green, 10 blue, 6 white
C
A 11 a quota

c 27.0 m b more representative of the population
e 41 600 m3 c difficult to compile a list of all the animals
f 44 900 kg 12 a continuous
25 22.5 b Basketball team are likely to be taller than

Chapter 6 Prior Knowledge average.
c systematic sampling
1 a 4.75 b 5.5 c 6 d 9 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.
Exercise 6A
13 5 cats, 9 dogs and 6 fish

14 a Gender/age 12 13 14
5 5
Boys 4 4 2

Girls 0

1 a 9 lions, 21 tigers b no
b 9 strawberry, 11 chocolate
2 a 24 boys, 16 girls 15 a discrete
b 18 HL, 27 SL
3 a 10 football, 14 hockey, 16 basketball b no
b 6 cod, 9 haddock, 5 mackerel
4 a 24 chairs, 9 tables, 4 beds c convenience
b 8 oak, 7 willow, 4 chestnut
d e.g. Different species may live in different
parts of the field.

16 a i possible
ii not necessarily true
iii possible/quite likely

Answers 445

b iii would be unlikely 18 a mean = 8, sd = 3
c discard –32, keep 155
17 a 65% b mean = 41, sd = 3
b i 84
19 a 48 b 19 c 6.17, 1.31
ii 65%
18 a 100 million 20 a 31 b 1.4  m < 1 .6 c 1.34
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
c 17.3 s; used midpoints, actual times not
Exercise 6B
available
1 a 16.5  t < 18.5; 20.5  t < 22.5; 22.5 < t 24.5; 22 a 4.25
b 1.5
c The second artist’s songs are longer on

average and their lengths are more consistent.

3, 12, 3, 1, 1 23 a 4 b 3 c yes (11)

20 observations 24 a 67

b 14.5  t < 17.5; 20.5  t < 23.5; 10, 8, 2 b 23 cm

20 observations c 24.7 cm

2 a 300  n  499; 500  n  699; d 20.5  l < 23.5 23.5  l < 26.5 26.5  l < 29.5
700  n  899; 11 23 30 14

40 observations e 44
f no
b 200  n  399; 400  n  599; 25 a a = 12.5
26 a x = 5.5
600  n  899; 20

40 observations b 13.4
b 30.9
3 a mean = 3.86, med = 3, mode = 1

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 b x = 7 30 a x = 9.5

6 a x = 31 b x = 7.5 b Q1 = 5, Q2 = 5, Q3 = 6.5
9 b y = 52 c yes
7 a y = 10

8 a y = 4.6 b y = 2.2 31 a £440, £268 b median

9 a i 13 b i 20.5 c $576, $351

ii 0  x < 10 ii 25  x < 35 32 a mean = 4, sd = 2.94

10 a i 13.5 b i 10.2 b mean = 2012, sd = 8.83

ii 12.5  x < 14.5 ii 11.5  x < 15.5 33 10.6°C, 2°C

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
36 a med = 42, IQR = 8 b m = 58
13 a x < 5 or x > 125 b x < 10 or x > 26

14 a x < 4.5, Q3 > 12.5 b x < 15.5 or x > 63.5 37 74% b 0.968
15 a mean = 46, sd = 8 38 a p = 12
b mean = 62, sd = 18 39 p = 7, q = 4
16 a median = 25, IQR = 13
b median = 0.5, IQR = 2.1 40 c = 30
17 a med = 360, IQR = 180
b med = 5, IQR = 2.6 41 10

42 (4, 8) or (5, 7)

446 Answers

Exercise 6C b f

You might find that different GDCs or programs give 60
slightly different values for the quartiles, resulting in 50
slightly different answers from those given here. In an 40
exam, all feasible answers would be allowed. 30
1 a f

15

20
10 10

0 30 36 42 48 54 60 t

5 3 a 16 b 30

4 a 23 b 4

00 5 10 15 20 25 x 5 a cf
b f
40

20 30

15 20

10 10

5 00 10 20 x
b cf
00 20 40 60 80 x
2 a f
60
15
50

40

10

30

5 20

10

0 1.2 2.5 3.8 5.1 6.4 7.7 h 00 10 20 30 40 50 60 70 80 x

Answers 447
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
10 a f 5 10 13.5 18.5 x
15 20

20

01 2 3 4 5 6 7 8 h 15
b cf 10

60 5

40 00 4 8 12 16 20 l
b 25

11 a 105

20 b f

0 30 35 40 45 50 55 60 t 40
30
7 a i 13 20
10
ii 20 0
c 46%
iii 6

iv 21

b i 4.6

ii 6.5

iii 2.3

iv 6.6 80 100 120 140 160 180 m

8 a 3 56 9

0 5 10 x
b 11 15 17 21 24

10 15 20 25 x

448 Answers
12 a
cf 15 a cf

350 140

300 120

250 100

200 80

150 60

100 40

50 20

00 10 20 30 t 00 20 40 60 80 100 n
b 40
b i around 17°C c 45, 32
d 10
ii around 7°C

13 a 45 31 45 63 110

b 22.2%

c 0 20 40 60 80 100 120 n

Time 5  t < 10  t 15  t 20  t 25  t e Overall, fewer candidates take History SL.
(min) History has a larger spread of numbers
10 < 15 < 20 < 25 < 30 than maths (based on the IQR).
Freq 79 15 10 4

d 16.9 minutes 16 159 cm b 90%
14 a cf 17 a 160 d 72, 18
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 18 a Q1 = 1, Q2 = 2, Q3 = 3
b 2
30

20 c 7 is an outlier

10 d 0 1 2 3 4 67

0 20 25 30 35 40 45 h 8n
38
b median ≈ 31 cm, IQR ≈ 9 cm 0 4
19 7 11 20 25 28
c 20 26.5 31 35.5 45

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

Answers 449
20 a f
2 a y
80 10

60
40 5

20

00 20 40 60 80 100 x b y 5 10 x
b 59.9 x
20 A2, B3, C1 15
15 20 25
Exercise 6D

1 a y 10
10

5
10

5 3 a y

5 10 15 x 25
20
b y
5 15

10 x
10 15 20 25 30 35

b y

5 10 x 25

20

15
−5

10 x
10 15 20 25 30 35

450 Answers

4 a weak positive b strong positive 15 a, d  y
250
5 a strong negative b weak negative

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

b i r = −0.595 c 3.6 km, $192 000
e average price ≈ $161 000
ii weak/moderate negative

iii yes 16 A1, B2, C3, D4

10 a y = 0.413x + 1.57 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 = −0.632x + 21.6 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

13 a i y = 73.5 the profit.

ii not reliable (no correlation) c y = 10.8x + 188

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
b y = -0.370x + 51.6
y c no (no correlation)
20 a y
75

70 4

65
3

60

55 x 2
145 150 155 160 165 170 175

b weak positive

c 156.7 cm, 64 cm 1

e arm length ≈ 61.5 cm

f i appropriate 0x
0 2 4 6 8 10 12
ii  not appropriate (extrapolation)

iii  not appropriate (different age from sample) b -0.695

Answers 451

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 5 x 10 x
0 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)
y
22 a moderate positive correlation

b yes (there is correlation and value is within 70

range of data) b m = 0.631t + 5.30 60
c 40.8 cm 50
23 a 0.820

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
27 a LQ(x) = 12, UQ(x) = 24, LQ( y) = 11,
the profit increases by €3250.
ii With no advertising the profit would be UQ( y) = 19
c, d, e
€138 000.
25 a y y

50 30

40 25

30 20

20 15
10
10 5
0x
0 30 x
0 10 20 40 0 5 10 15 20 25 30

b summer and winter

452 Answers

Chapter 6 Mixed Practice c 2.5 kg, 0.9 kg
d
1 a i systematic sampling
ii e.g. Students may take books out on the 1.2 2.5 3.2
same day each week.
1.9 2.8
b i Each possible sample of 10 days has an
equal chance of being selected. 0 1 2 3 4m
4 a discrete 4 8
ii Representative of the population of all b 0
days. c i 1.47

c i 11 ii 1.5
ii 17.4 iii 1.25
iii 3.17 5 a 4
b
2 a n
2
40

35

30

25 3 5

20 0 1 2 3 4 5 6 78 9 10

15 Number of books read
c 10
10 T 6 a 0.996
10 15 20 25 30 35 40 b a = 3.15, b = −15.4
c 66.5
b strong positive correlation
c n = 1.56T − 10.9 7 a cf
d 30.0
3 a 2.38 kg 50
b cf
40
50
30

40

20

30

10

20

10 0 0 2 4 6 8 10 12 t

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

Answers 453

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

d 6.21 minutes 623
8 a med = 46, Q1 = 33.5, Q3 = 56
b yes (93) 9
c

26 46 61 94 3 Draw 1 Draw 2 R

5 6 × 5 = 30 = 1
9 10 9 90 3
R
33.5 56
6

x 10 4 6 × 4 = 24 = 4
9 10 9 90 15
0 10 20 30 40 50 60 70 80 90 100 B

9 a 121 cm, 22.9 cm2 6 R 4 × 6 = 24 = 4
b 156 cm, 22.9 cm2 9 10 9 90 15
4
10 mean = $55.02, sd = $43.13 10 3 4 × 3 = 12 = 2
9 10 9 90 15
B

11 x = 6 B

12 a s Exercise 7A

300

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

200 3 a 1 b 1
2 3

4 a 1 b 1
4 13
150 w
5 6 7 8 9 10 5 a 1 b 3
26 26
b w = 19.1s + 99.0
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 b 23
20 40
not reliable.
13 a i Athletes generally do better after the 8 a 0.85 b 0.13

programme. 9 a 47 b 41
ii Better athletes improve more. 120 48
b 11.6 miles
c i 0.84 10 a 0.73 b 0.66
ii Y = 1.2 X + 3.2
11 a 0.44 b 0.11

12 a 4 b 27

13 a 12 b 6
14 a 4.8
b 7.5

454 Answers
4
15 a 1.6 b 1.5
b 66.9 BH
16 a 0.0743 b 0.8 45 80 60

17 7.5 54

18 15

19 8

20 0.75 35

21 $1162.50

22 a 1.5 a 37 b 7
44 11
23 1
16 5

24 78 C
25
I
Exercise 7B 16 12 6

1 G 1
F
13 a 6 b 16
35 35

3 6

a 3    b 4 G S
25 25 0.2 0.5
0.2
2

A B 0.1
10 8
2 a 0.1 b 0.5
7
10
P
a 4 b 1 0.32 V
15 3 0.08 0.3

3

B H 0.3
45 45
35 a 0.3 b 0.32

25

a 7 b 9
30 30

Answers 455

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
29
30 12 × 18 = 36
30 29 145
B

0.7 0.2 L 0.7 × 0.2 = 0.14 12 R 18 × 12 = 36
29 30 29 145
B 0.8 18
30
L 0.7 × 0.8 = 0.56
B 17 18 17 51
29 30 29 145
a 0.155 b 0.845 B × =

9 Pizza ? Fries? a 73 b 3
145 5
0.8 F 0.3 × 0.8 = 0.24
P
12 Revise? P ass?
0.9 P 0.8 × 0.9 = 0.72
0.3 0.2 F 0.3 × 0.2 = 0.06 R
0.8
0.7 0.5 F 0.7 × 0.5 = 0.35 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
R 0.3 P 0.2 × 0.3 = 0.06
a 0.41 b 0.59

10 Dice Coin

0.5 H 1 × 0.5 = 1 a 0.72 b 0.14
6 6 12

13 First Dice

1 0.5 123456
6 1234567
1 × 0.5 = 1 2345678
T 6 12 3456789
4 5 6 7 8 9 10
0.4 H 5 × 0.4 = 1 Second Dice 5 6 7 8 9 10 11
6 3 6 7 8 9 10 11 12
5
6

6 T 5 × 0.6 = 1
0.6 6 2

a 5 b 7 a 5 b 1
12 12 36 6

456 Answers

14 First Dice 26 a 11 b 17
123456 15 20

1=>>>>> 27 a 19 b 30
2<=>>>> 49 49
3<<=>>>
4<<<=>>Second Dice 28 a 25 b 1
5<<<<=> 51 17
6<<<<<=
29 a 3 b 18
31 53

30 a 16 b 4
39 9

a 1 b 5 31 a 0.14 b 0.24
6 12
32 a 0.3 b 0.4

15 H T 33 a 1 2 345 6
H, H T, H 4 6 8 10 12
H H, T T, T 2 6 9 12 15 18
T 8 12 16 20 24
3 10 15 20 25 30
12 18 24 30 36
4
D
a 1 b 1 5
4 2
6

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

a 1 b 1 L
2 4

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

ATLI 36 0.496

a 1 b 1 37 a i 1 ii 1
6 2 8 8

18 a 38 b 55 b yes
109 109
38 a

19 a 3 b 25 T
19 57

20 a 18 b 16 75
77 77

21 a 23 b 13
65 23
22 a 0.2 b 0.3 8
c 33
23 a 0.8 b 0.9 b 7 c 5
22 7 71
24 a 0.5 b 0.7
39 a 47 b 83
25 a 0.2 b 0.1 130

Answers 457

40 0.582 c 4 d 11 e 1
15 15 4
41 0.75

42 a 1 b 0.88 52 a 1 b 1
15 3 2

43 1 b 79 Chapter 7 Mixed Practice
36 187
44 a 7 c 26
17 187 1 416
2 a 56
45 0.5 21 32 21
115 115 59 32
46 0.5 b c d

47 a 1 b 11 3 a 2 b 37
5 20 15 60

48 a 13 b 52 c same 4 a
23 161

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

MG 1, 2, 4, 7, 8, 11, 13, 14, 16, 17, 19
15 4 12
b i 1 ii 3
3 5 14
88
5 a 0.2 b 2
3

6 a 0.18 b yes

21 7 0.92
8 a

S A B

29

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

0.1

P W b 0.75
c 0.667

x 11− x x − 4 9 a i 2
9
x
11− x 10− x b 5
18

ii 1
18
x−6
10 a 0.143

T b 0.111

0 c 0.238

458 Answers

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

2T 1 H 1 × 1 × 1 = 1 18 12
2 2 2 2 8
x
1 T 1 × 1 × 1 = 1
2 2 2 2 8

1 H 1 × 1 × 1 = 1
2 2 2 2 8

1 T 1 × 1 × 1 = 1 Kiwi fruit 8
2 2 2 2 8

b 1 c 7 d 3 c x = 10
8 8 8
d i 50 ii 82 iii 15
( )12 Asher e i 0.08 ii 0.37 82

13 0.310
48 > 24 f 14
91 49 2475

14 a 8 b 13 Chapter 8 Prior Knowledge
23 23

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

16 a 2 2 6 8.5 10 12

M C 0 10 20 x
11
5 Exercise 8A

6

10 1 a x 012
72

P(X = x) 2 4 1
777

E b x 01 2
1
P(X = x) 4 8 5

15 15 2
1
b 16 2 a x 01 4

c i 3 ii 56 P(X = x) 1 1 2
4 2 1
d i 0.22 4
b x 01
ii 0.05
P(X = x) 1 1
iii 0.62 4 2

iv 31
39

Answers 459

3 a x 012 3 16 a 1
1 17
P(X = x) 1 3 3 8
3
888 1 b h 01 2
8
b x 012 P(H = h) 19 13 1
3
P(X = x) 1 3 3 1 34 34 17
216
888 c 0.5
17 no
4 a P(X = x) = 1 for x = 1, 2, ..., 6 18 n = 1
6
1 19 $1.50
b P(X = x) = 8 for x = 1,2, …, 8 20 a To ss 1

5 a x 012 Toss 2 Toss 3

P(X = x) 125 25 3 1 H 1 × 1 × 1 = 1
216 72 72 2 2 2 2 8
1H
b x 0123 H 1 T 1 × 1 × 1 = 1
2 2 2 2 2 8
1
P(X = x) 343 147 21 1 2 1 1 H 1 × 1 × 1 = 1
2 2 2 2 8
512 512 512 512 2T

6 a x 012 1 T 1 × 1 × 1 = 1
2 2 2 2 8

P(X = x) 0.64 0.32 0.04 1 H 1 × 1 × 1 = 1
2 2 2 2 8
b x 1H
012 1 1 1 1 1
2 2 2 2 2 8
P(X = x) 0.09 0.42 0.49 1 1 T × × =
T 2
2T
7 a k = 0.17 1 H 1 × 1 × 1 = 1
2 2 2 2 8
i 0.77 ii 0.416
1 1 1 1
b k = 0.26 1 T 2 × 2 × 2 = 8
2

i 0.49 ii 0.245 b 3
8
8 a k = 0.2 c x

i 0.4 ii 0.75 01 23
31
b k = 0.36 P(X = x) 1 3 88
88
i 0.43 ii 0.814 45678
d 1.5 31311
9 a k = 0.125 16 4 16 8 16
21 a x
i 0.6 ii 0.833 23

b k = 0.2 P(X = x) 1 1
16 8
i 0.9 ii 0.667

10 a 2.1 b 1.9 b 5
22 a 0.4
11 a 2.1 b 2.24 b 2.13

12 a 5 b 9.2 23 a k = 1 b 0.75 c 5.17
12 c 1.92
13 a k = 0.4 b 0.7 c 2.9 3
c 6.2 24 a c = 0.48 b 11
14 a k = 0.2 b 0.4 b 0.857

15 a x 0 1 2 25 a = 0.3, b = 0.2

p4 48 15

13 91 91

460 Answers

Exercise 8B 29 a 0.0296 b 0.008 76
c e.g. A source of faulty components, such as
( )1 a yes,X ~ B 30, 1
2 a defective machine, would affect several
components.
( ) b yes, X ~ B 45, 1
6 Exercise 8C

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, X ~ B(50, 0.12) 4 a 0.523 b 0.244

b yes, X ~ B(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
15 a 0.465
c 0.121 d 4.8 16 a 0.308 b 0.0228
17 a 0.274
22 a 0.983 b 5 c 0.383 18 a 0.0478 b 0.328
19 a 0.0831
23 a All have same probability; employees 20 a 4.52 b 10.4 c 0.282
independent of each other.
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
21 12.8 s
d 0.0159 22 a 15.4 b 6.74

24 0.433 23 4.61

25 a 0.310 b 0.976 c 0.643 24 20.9

26 a 0.650 b 0.765 25 15.2

27 0.104 26 predicts 4% get a negative score
27 a symmetrical
28 0.132

Answers 461

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 19 320 ml
1
Student mass (kg) 20 a 9 b 0.0426
b k = 20.4
28 a 0.0228 e.g. anticipating the start gun 21 a 0.925

b 0.159 c 0.488 22 a 0.835 b k = 1006.58 c a = 6.58

d The same distribution is true in all races. 23 a i 1 ii 0 .0579

Unlikely to be true. b ii a = 0.05, b = 0.02

29 $728 c Bill (0.19)

Chapter 8 Mixed Practice 24 a i 0.845 ii 1.69

1 a k = 0.5 b 0.6 c 2.9 b That Josie’s second throw has the same
2 a 0.296 b 0.323 distribution as the first and is independent.
3 a 0.347 b 0.227 These seem unlikely.

25 n = 9

4 215 26 10 b 0.0561
27 a 0.919

5 0.346 28 277
29 b 6
6 a 1 , 13 , 1
2 6
Chapter 9 Prior Knowledge
b no; expected outcome is not 0

7 N=7 1 1 x−1 − 1 x−2
8 a 0.933 4 2

b i 2 y = 3x + 11

3 -2

Exercise 9A

0.85 1 a 0.6 b 0

2 a 1.5 b 3

3 a 2 b 3

20 k x 4 a 1 b 1

5 a 0.693

ii k = 23.1 In further study you might learn how to show

9 a P = 1 b 0.153 c 0.0435 that this is actually ln 2
3 b 0.360
10 a 0.206 b 1.10

11 a 0.356 6 a 0.2 b 0

b a box of six containing exactly one VL egg 7 a 0.5 b 0.5

12 a 0.354 b 0.740 8 a 1 b 10

13 0.227 9 a −0.5 b 0

14 0.232 10 a 0.805 b 0.347

15 a 3.52 b 0.06 c 0.456 11 a 0 b 4

16 a = 1 12 a 6 b 0
6
13 a 0.5 b 0.0833

462 Answers

14 a −1 b −2 Exercise 9B
15 a 1 b 0.693

16 a dz b da 1 a x > 1 b x < 0
dv db
2 a −1 < x < 1 b x < −1 or x > 6
dp
17 a dt b db 3 a 0 < x < 90 or 270 < x < 360
dx
b 180 < x < 360
dy dt
18 a dn b df 4 a y

19 a dh b dw 2
dt dv

20 a dw b dR
du dT
x
21 a dy = y b dy = x
dx dx 2

22 a ds = kt2 b dq = k q
dt dp

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

24 a dh = 1 b dr =k b y
dx h dθ
1
25 a s′(t) = 7 b q′(x) = 7x 2

26 a d(pH) = k x
dT

b dC = k (1000 − n)
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 5 a y

reasonable.

27 a dV =V +1 b 5
dt

28 57.3

29 1 1
2
30 a x2 − 1 −
x −1
x
b 2, the gradient of the curve at x = 1

31 3
32 1

6

Answers 463
b
y 7 a y
6 a
−2
2x

3
x

y b y

−4 2x
−1

b −1 3x y
y 8 a

−1 2 −5 −1 3
x −3 1 5 x

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

y

−3 3 3
x −1

x

9 a y

b y

3
x

3x

b y

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 465
b 1x −1 x

15 a b y

y

−2 3 x
4x

b 17 a y
y
−1 1 2 x
4
x

466 y 19 Answers
b −1 2 y

x x
y
18 a y
x
1 20
x y
b y
21 a

x

3x

Answers y 23 a 467
b y

x −1 1 x

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

Exercise 9C

1 a f ′(x) = 4x3 b g′(x) = 6x5
2 a h′(u) = −u−2 b z′(t) = −4t−5

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

4 a dz = −5t−6 b ds = −10r −11
dt dr

b y 5 a f ′(x) = −4 b g′(x) = 14x
x
6 a dy = 6 b dy = 15x4
dx dx

7 a dy = 0 b dy = 0
dx dx

8 a g′(x) = −x−2 b h′(x) = −3x−4

9 a dz = −6x−3 b dy = −50t−6
dx dt

10 a dy = − x−5 b dy = − x−6
dx dx

11 a f ′( x) = 3 b f ′( x) = 5
2x2 x4

12 a f ′(x) = 2x − 4 b g′(x) = 4x − 5

c Gradient specifies shape but not vertical 13 a dy = 9x2 − 10x + 7 b dy = −4x3 + 12x − 2
position. dx dx

468 Answers

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

15 a dy = 3− 3 x 2 b dy = −6x2 + x3 1 a 16 b 20
dx 4 dx
2 a 1 b 16.75
8 b −7
16 a f ′(x) = 8x3 − 15x2 3 a 32

b g′(x) = 3x2 + 6x − 9 4 a 23 b 100.4
9
5 a 108 b 6
17 a g′(x) = 2x + 2 b f ′(x) = 2x − 1

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

18 7 a 1 b ±2
x3 8 a ±2
b g′(x) = 8x − b ± 1
2
19 a a b 2ax + b
b 3x2 + b2 9 a −1, −3 b ± 1
20 a 2ax + 3 − a b 3a2x2 2

21 a 2x 10 a 4x - y = 1 b y - x = 0
11 a 2x + y = 7
22 a 2ax2a−1 b −ax−a−1 + bx−b−1 12 a x + 3y = 6 b 7x + y = 2
13 a y = −x + 15
23 a 6b b 10b2x 3a 14 a y = -x b 2x − y = 3
x3 cx2
7a + − b y = 2x − 12

24 a 2a2x b 18a2x b y = −x + 3

25 a 2x + a + b b 2abx + b2 + a2 15 a y = 1 x − 1 b y = 1 x + 81
3 3 16 16
26 a − 10 b 5 16 a y = 2x − 3 b y = x + 1
x2 x2
4 1 17 a y = x +1 b y = − x + 3
27 a x3 − x2 b − 3 − 4 4 16 4
x4 x3 x 3 x 7
18 a y = − 4 + 4 b y = − 32 + 32
28 a 0.5 + 3x b 7x3 + 4x7 19 a b
y = x−1 y = x−1
29 a x < 1 b x < −4
20 a y ≈ 2.77x − 1.55 b y ≈ 1.10x + 1
30 a x > 6 b x > 0
21 a y = 14 − 4x b y = 1 − x
31 6 + 4
t2 22 a 8x + 5y = 47 b 4x + 11y = 126

32 1 − 2m−2 23 a x + y ≈ 2.10 b x + 2 y ≈ 13.4

33 3 k 24 a y = −x b y = x − 2
2
25 a −0.984 b −0.894
34 x > 1
2 26 a 2 b 0.693

35 x > − b 27 a −3, −1 b 0, −2
2
28 a 4x3 − 1 b −1 c y = x
36 −1
29 a 3x2 − x−2 b 2 c y = 2x
37 1 − 3x2
30 y = 11 x − 9
38 a – k b V2 c 4 4 4
r2 – k
17
39 a dA = q + 2qL 31 y = 4x + 2
dL
32 x = 0
b q > 0
( )33 − 3 , 9
2 4

( )34 − 1 , −8
8

Answers 469

35 y + 2x = −1 3 a f ′(x) = 6x2 + 10x + 4

36 y = 1 x + 15 b f ′(−1) = 0 c y = 2
4 4
4 x 4 x 4 −0.5
37 y = 3 − 3 or y = − 3 − 3
5 y = 27x − 58
38 (2, 6), (−2, −6)
6 a 12 + 10t
39 (−3, −2), (1, 6)
b V (6) = 302, dV (6) = 72
40 (2, 0.5) dt

41 2 or 2 After 6 minutes, there is 302 m3 of water in
3
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) = 112 , so the volume is increasing
dt
Chapter 9 Mixed Practice
faster after 10 minutes than after 6 minutes.

1 a 8x − 1 b (2, 15) 7 a 2 − 2t b i 0.75 ii 1
2 a y c 0  t  1
8 a 6 b x + 8x−2
c 0 d y = 6
e, f
y

y= 1 x2 − 8
2 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)

10 a (0, 0), (4, −32) b y = −8x

11 a y = 6x, y = −6x − 12

1 b (−1, −6)
2 x 12 p = −2, c = −3

−1 13 a 30t − 3t2 b 72, −72

c The profit is increasing after 6 months, but

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

470 Answers
c
y 16 a 3 b 2x − 1

c (2, 4) ( )d 1 , 16
3 9

e y = 5x − 7 ( )f 12 , 7 , gradient = 0
4

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

19 a = −8, b = 13

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 f ′(x) = −12x−4 + 2

Exercise 10A

1 a f(x) = 1 x4 + c b f(x) = 1 x6 + c
4 6
4 x
3 2 a f(x) = −x−1 + c

b f(x) = − 1 x−2 + c
2
3 a f(x) = 1 b f(x) = x + c
b y = −x5 + c
4 a y = x3 + c

y = f (x) 5 a y = − 7 x4 + c b y = 3 x8 + c
4 8

b i  x  0 and 0  x  2 6 a y = 1 x6 + c b y = − 1 x10 + c
4 6

ii  y 7 a y = x−3 + c b y = −x−5 + c

8 a y = − 3 x−4 + c b y = 2x−2 + c
2

9 a y = 2 x−1 + c b y = − 1 x−7 + c
5 4
2
x 10 a y = x3 − 2x2 + 5x + c

b y = 7 x5 + 2x3 − 2x + c
5

11 a y = 1 x2 − 2 x6 + c
2 3

y = f(x) b y = 3 x 4 − 5 x8 + c
2 8

Answers 471

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

b y = 4 x − 2 x5 + c 7 30
5 15
8 4.5

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

b y = 5x + 3x−3 + 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 + x3 + c b 3x2 − x3 + c 15 18.75

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

17 a 3x3 − 6x − x−1 + c 17 16
3

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

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

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

20 a y = x3 + 6 b y = x5 + 5 2 4 x3 − 3 x2 + 5x + c
32
21 a y = 1 x4 − 6x2 + 10 b y = 3x − 2x5 + 4
4
3 0.661
5
22 a y = −3x−2 + 2x2 − 3 4 y = x3 − 4x3 + 6

b y = 3x3 + 2x−1 − 4 5 a=8

23 − 4 + 1 + c 24 y = x3 − 4x + 7 6 5.7 b 6.75
3t 2t4 7 a (2, 0), (5, 0) c 1.07
8 a y = 1.6x − 3.2
25 y = − 4 − x3 + 10
x 9 b 1.5

26 3x4 − 2x3 + 3x2 − 2x + c 10 1.5
4 3 2
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.

Exercise 10B b 160 calories

15 y = 5x3 + 3
3
16 0
1 a 97.6 b 2.25
2 a −0.625 b −145.5 17 6x + 2
3 a 4 b 36 x2
4 a 7.5 b 4.5 18 a (3, 0) b 18

472 Answers

Core SL content: b x = −4
Review Exercise d (−2.85078,−2.35078)
OR (0.35078, 0.85078)
1 a 4 b 83 c 900 e 1
2 a 2 c 3x + 2 y = 31 f y = −x − 5
b 2
3 11 x − 11y + 112 = 0 b −2
3 a y 12 a 1000 b 56x6 + 48x−3
(−1, e) 13 a 8x7 − 24x−2

2 14 a 18r3 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

18 a −2

b y = 0 c y  2.72 c y = −2x − 3 d (−3, 3)
b f(x)  0
4 a a = 2.5; x  2.5 e 20 f 10
b 95.5cm2
c 18.5 b 0.305 19 a i  1295.12 cm3 ii 6

5 a 38.9° b 15.6 iii I  431cm3
b 2
6 a 4.5 II  4.31× 10−4 m3
b 88.9 g
c 28.2 e 30.9 cm2 b i  I   73.5°

7 a (−2, 0), (2, 0) II  55.8 m

8 a 4.5 ii  55.0 m iii 217 m

c 7 20 a i  0.985
10
ii  strong positive

9 a 9.56 cm3 b y = 260x + 699

d 52.8° c 4077 USD

10 a and c d e.g. 3952  4077

y e i  304x

ii  304x − (260x + 699)

iii 16

21 a 0.159

b i  0.119

y=2 ii  0.394
x
22 a 3

x = −4 3 3 b 7
2 4 26
−

c 8
15

d 172
325

Answers 473

Chapter 11 Prior Knowledge 29 a i 128.4 ii 130
b 1.27%

1 a 1269 b 1360 30 a 329.54 ml b 0.140%
2 a 6420 b 16
31 84.5 cm  l  85.5 cm

Exercise 11A 32 777 g
33 a 11.292  AC  11.441
1 a 32.8 b 654.0 b 1.325%

2 a 0.2 b 0.1 34 12.7027  I  14

35 1.875  a  4.25

3 a 32.76 b 654.04 36 a 37.5% b 150%

4 a 0.25 b 0.05 37 a i 0.9

5 a 32.762 b 654.038 ii 1.2
iii −0.675
6 a 0.249 b 0.052
b ii and iii are wrong. Proportions must be
7 a 30 b 700 between 0 and 1 inclusive

8 a 0.2 b 0.05 38 a 1.445  a  1.455

9 a 33 b 650 b 27.86  10a  28.51

10 a 0.25 b 0.52 c 30 to one significant figure (maximum
significance that includes all possible values)
11 a 32.8 b 654

12 a 0.249 b 0.0517 39 Assuming limiting factor is weight rather than
e.g. volume or number of trucks available. 9 trucks
13 a 110 (2 s.f.) b 12.3 (3 s.f.)

14 a 0.16 (2 s.f.) b 3 (1 s.f.) 40 237.5 cm2  A  525 cm2

15 a 12.25  x  12.35 b 0.75  x  0.85 Exercise 11B

16 a 0.485  x  0.495 b 5.725  x  5.735

17 a 55  x  65 b 0.25  x  0.35 1 a $11 037.26 b $831.90

18 a 465  x  475 b 23.5  x  24.5 2 a $9648.31 b $10 598.34

19 a 11.675  a + b  12.685 3 a $94 841.43 b $87 410.24

b 3.575  c + d  3.685 4 a $89 654.86 b $98 150.01

20 a 0.75  pq  3.75

b 2.7225  rs  3.2825 5 a €5742.79 b €1347.86

21 a 3.315  a − b  4.325 6 a €5133.14 b €423.69

b 3.115  c − d  3.225 7 $4414.91

22 a 100  p  500 8 a 12 b $1013.20
q
9 $1203.38
r
b 0.1089  s  0.1313 10 a A: €42.03 B: €31.87

23 a 0.298% b 0.118% b €105.24

24 a 0.147% b 0.121% c It depends on the utility of the money used
in the payment holiday
25 a 6.25% b 3.66%
11 a 27 months b $186.65
26 a 5.14% b 3.34%
12 a B (€557.48 cf. €592.76)
27 a 46.512 cm3
b A (€177 828 cf. €200 693)
b i 46.51 cm3 ii 47 cm3
13 Yes: repayments of $300 in first two years leave
28 a r = 2.5 (2 s.f.) b r = 3 (1 s.f.) repayments of $270.29 in remaining three years

474 Answers

Chapter 11 Mixed Practice Exercise 12A

1 a 12.5 b 12 1 a x = 5, y = −2 b x = 34 , y = − 26
23 23
2 19.09%
2 a x = − 5 , y = − 65 b x = −3, y = 4
3 1.01% 19 38

4 £3719.95 3 a x = 17 , y = 16 , z = 39
37 37 37
5 a 1380 m b 743 m c 3.49%
b x = −11, y = 64, z = 88
6 a 1.75
4 a x = −2, y = 1, z = 0
b i x = 2, y = 1, z = 50
2 1 52
ii 1.98 b x= 5 , y = − 7 , z = 35

c 13.1% 5 a a = −3, b = 2, c = −5, d = 1

7 a 0.001 25 61 239 17
29 29 29
b i 0.0013 ii 0.001 b a = , b = −1, c = , d =

c 60% a = 121, b = 61, c = 13, d = −19

8 Age and height in metres 6 a 8 88 8

9 a 3.1031  π  3.1895 b a = 2, b = −1, c = 3, d = 4

b 0.135% 7 a b + g = 60; g = 2b b 40

c 1.52% 8 a 3w + 7g = 14.10; 5w + 4g = 12

10 1.360  t  1.474 b $2.70

11 9 9 a = 25, d = −1.8

12 a 27.2% b 300% 10 a = 3.4, d = 0.2

13 a i −0.5 11 t = $10.50, d = $5.25, p = $6

ii 0.149 12 A = $3900, B = $3600, C = $2500

iii 1.22 13 c = 4, f  = 5, s = 2

b i and iii are wrong. Probabilities must be 14 £301.75
between 0 and 1 inclusive
15 0.2
14 a 22.7% b $7.39
16 754
15 a 99 months b ¥202.81
17 x = ±1, y = ±3, z = ±4
16 Option B: £343 739 vs £351 486
18 5
17 a $648.96 b $12 500

18 a �66.61 b 33.2% c 72.4% Exercise 12B

19 604.75

20 a 2.55  a  2.65 1 a x = 0.438, 4.56 b x = −2.14, 1.64
2 a x = 1.5
b 354.8  10a  446.7 b x = 1
3
c 400 to one significant figure (maximum
significance that includes all possible values) 3 a x = −2.09, 0.797, 1.80

b x = −2.21, 0.539, 1.68

Chapter 12 Prior 4 a x = −1, 4 b x = −2, 2.5
Knowledge 3

5 a x = −4.72, −1.16, 0.364, 1.51

39 −1 b x = −0.798, 0.174, 0.872, 2.75 3
7 7 2
1 x = , y = 6 a x = −1.30, 2.30, 3 b x = −2,

2 a 3x2 + 11x − 11 = 0 b 3x2 − x − 4 = 0 7 a No real solutions b No real solutions