Cambridge IGCSE Mathematics Extended Practice Book

Exercise 18.5 (e)
50 y
1
x –5 –4 –3 –2 –1 0 1 2 3 4 5 40
–225 –128 –63 –24 –5 0 –3 –8 –9 0 25 30
(a) y x3 − 4x2
20

(b) y = x3 + 5 –120 –59 –22 –3 4 5 6 13 32 69 130 10 x
–10 –8 –6 –4 –2 0 2 4 6 8 10

–10

(c) y 2x3 + 5x2 + 5 380 213 104 41 12 5 8 9 –4 –43 –120 –20

(d) y x3 + 4x2 − 5 220 123 58 19 0 –5 –2 3 4 –5 –30 –30
y = x3 + 2x − 1–040

–50

(e) y = x3 + 2x −10 –145 –82 –43 –22 –13 –10 –7 2 23 62 125 (f )
50 y

(f) y 2x3 + 4x2 − 7 –157 –71 –25 –7 –5 –7 –1 25 83 185 343 40
30

(g) y x3 − 3x2 + 6 56 22 6 2 4 6 2 –14 –48 –106 –194 20

(h) y 3x3 + 5x 350 172 66 14 –2 0 2 –14 –66 –172 –350 10 x
–10 –8 –6 –4 –2 0 2 4 6 8 10

–10

–20

(a) (c) –30
50 y 50 y –40
y = 2x3 + 4x2 − 7
40 –50

30 40

20 30 (g)
50 y
10 20

–10 –8 –6 –4 –2 0 x 10 x 40
–10 2 4 6 8 10 –10 –8 –6 –4 ––1200 2 4 6 8 10 30
–20
20

–30 –20 10
–10 –8 –6 –4 –2 0
x3 4x3 –40 –30 x
–50 –10 2 4 6 8 10
y = − 2x3 5x2 –40
5
y = − + + –50

(b) –20
50 y
(d) –30
–40
40 50 y y = −x3 − 3x2 + 6
–50
30 40 (h)

20 30 50 y

10 x 20 40
2 4 6 8 10 30
–10 –8 –6 –4 –2 0 10 x
–10 –10 –8 –6 –4 –2 0 2 4 6 8 10 20
–20
–10
–30 10
–20 –10 –8 –6 –4 –2 0 x
y = x3 + 5 –40 2 4 6 8 10
–50 –30 –10

y = − x3 + 4x2 − 5–40 –20
–50
–30
–40
y = −3x3 + 5x
–50

Answers 193

2 (a) (d)
x −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 4 5 6
y –36.875 –18 –4.625 4 8.625 10 8.875 6 2.125 –2 –5.625 –8 –6 10 46 50 y
40
y = 2x + 3
x

30

(b) 20
50 y
40 (b) 50 y 1 –4 10 x
30 –4 x (e) –2 0 24
y = x3 +
20 40 –10
10
30 –20
–10 –8 –6 –4 –2 0 2 4 6
y = x − 5 –10 20 –30
–20
–30 x 10 –40
8 10
y = x3 − 5x2 + 1–040
–50 x –50
24
(c) (i) x = –1.3, 1.8 or 4.5 –2 0
(ii) x = 0 or 5 –10
(iii) x = –1.6, 2.1 or 4.5
–20 50 y

–30 40 y = x3 − 2
x
–40
30

–50 20

3 10

–4 –2 0 2 x
x –3 –2 –1 –0.5 –0.2 –0.1 0 0.1 0.2 0.5 1 2 3 –10 4

(a) y x − 1 –20
x
–2.67 –1.5 0 1.5 4.8 9.9 –9.9 –4.8 –1.5 0 1.5 2.67 –30
–27.33 –8.5 –2 –2.125 –5.008 –10.001 10.001 5.008 2.125 2 8.5 27.33
1 12.33 8 7 10.25 22.04 42.01 –37.99 –17.96 –5.75 –1 4 9. 67 –40
x
(b) y = x3 + –50

(c) y = x2 +2− 4
x

(d) y 2x + 3 –7 –5.5 –5 –7 –15.4 –30.2 30.2 15.4 7 5 5.5 7
(e) y x –26.33 –7 1 3.875 9.992 19.99 –19.99 –9.992 –3.875 –1 7 26.33

x3 − 2
x

(f) y x2 − x + 1 11. 67 5.5 1 –1.25 –4.76 –9.89 9.91 4.84 1.75 1 2.5 6.33
x

(a) (c) (f )
–4 –4
25 y 50 y 50 y
40
20 y = x2 + 2 − 4 40 1
x 30 x
y = x2 − x +
30
15 1
10 y = x − x 20 20

5 10 10
–2 0
–2 0 2 x –4 –2 0 x x
–5 4 –10 24 –10 24
–10
–15 –20 –20
–20
–25 –30 –30

–40 –40

–50 –50

194 Answers

4 (a) 2 (a)

x –4 –3 –2 –1 0 1 2 3 4 60 y
55 y = x3 − 1

y = 2x 0.0625 0.125 0.25 0.5 1 2 4 8 16 50

y = 2–x 16 8 4 2 1 0.5 0.25 0.125 0.0625 45
y = 2–
y 40
18
16 35
14
y=2 5 (a) & (b) (i) 30

300 y 25

20

250Number of organismsy = 3x 15
12
10

200 5 x
12 3456
10 150 –6 –5 –4 –3 –2 ––150
–10
8
100 –15

6 50 y = 12x + 1 –20
x
4 –25

2 –1 0 1 2 3 4 5 6 –30
Time (hours)
–35

–40

–6 –4 –2 0 2 46 (b) (ii) 12 per hour –45
(b) (c) (i) ≈ 3.4 hours
(ii) ≈ 42 –50

–55

x 0 0.2 0.4 0.6 0.8 1 Exercise 18.6 –60
y = 10x 1 1.58 2.51 3.98 6.31 10
y 2x −1 –1 –0.6 –0.2 0.2 0.6 1 1 (b) gradient = 12

12 y 10 y 3 answers should be close to: with light
9 y = x2 − 2x − 8 ≈ 1.4 cm per day, without light ≈ 1 cm
10 y = 10x per day
8 8

6 7 Mixed exercise

4 6

2 y = 2x − 1 5 1 (a) A: y 3x − 2
x B: y x2 + 3
4
0 0.2 0.4 0.6 0.8 1 1.2
–2 3

2 C: y = − x − 5 1

1 44
x
(b) (i) (–1, –5)
–5 –4 –3 –2 ––110 1 2 3 4 5 6 7 (ii) answer should be the point of
intersection of graphs A and C.
–2
(c) maximum value y = 3
–3

–4

–5

–6

–7

–8

–9

–10

(a) –2
(b) 6 and –6

Answers 195

2 (a) CD Exercise 19.4

10 y EF 1 (a) 15° (isosceles 6)
(b) 150° (angles in a 6)
y = x2 8 G (c) 35° (∠MON = 80°, and 6MNO in
6 isosceles, so ∠NMO = ∠NOM =
H has no line symmetry 50°, so ∠MPN = 35°)
4 (b) A = 0, B = 3, C = 4, D = 4, E = 5, (d) 105° (∠PON = 210° so ∠PMN =
F = 2, G = 2, H = 2 105° – half the angle at the centre)
2
x 2 (a) 2, student’s diagram 2 (a) 55° (angles in same segment)
–3 –2 –1 0 12 3 (b) 2 (b) 110° (angle at centre twice angle
–2 at circumference)
3 student’s own diagrams but as an (c) 25° (∠ABD = ∠ACD, opposite
y = x3 –4 example: angles of intersecting lines AC
and BD, so third angle same)
–6
3 ∠DAB = 65°, ∠ADC = 115°,
–8 ∠DCB = 115°, ∠CBA = 65°

–10

(b) (0, 0) and (1, 1) 4 35°

(c) (i) x = 2 or x = –2 5 59.5°

(ii) x = –2 6 144°
(d) gradient for y x2 when x = 2 is 4
7 (a) 22° (b) 116° (c) 42°
and gradient for y x3 when

x = 2 is 12 8 (a) 56° (b) 68° (c) 52°

3 (a) y = − 4 (b) y x
x
(c) y Exercise 19.5
8
y = −x 7 DIAGRAMS ARE NOT TO SCALE BUT
6 STUDENTS’ SHOULD BE WHERE
5 REQUESTED
4
y = − 4 3 1
x 2
1 (a) (b)
A B
A 3 cm
–8 –7 –6 –5 –4 –3 –2 –10 1 2 3 4 5 6 7 8x
Exercise 19.2
–2
–3 1 (a) 3 (b) 4 (c) A
–4
–5 (c) infinite number corresponding to
–6
–7 the number of diameters of the
–8
circle face

(d) as per part (c) (e) 2 BC
2
Chapter 19 (f) 3 (5 if face is a square)
4 cm
Exercise 19.1 (g) 1 O

1 (a) (h) infinite number corresponding to
A
the number of diameters of the

sphere

B 2 (a) 4 (b) 3 (c) 1
(f) 8
(d) infinite (e) 4

Exercise 19.3 3 1m

1 (a) x = 25° (b) x = 160°, y = 20°

2 6.5 cm

3 (a) 49.47 cm (b) 177.72 cm

196 Answers

4 (d) (i) (ii) eight Chapter 20

Road Exercise 20.1

5 4 km (e) (i) (ii) none 1 (a) English exam marks
A 60
3 km SCHOOL

1.5 km 50
School
2 (a) a hexagonal prism Frequency 40
(b) the axis of rotational symmetry
(c) 6 30
(d) 7
20
3 (a) w = 72° (base angle isosceles
ΔOCB), x = 90° (angle in semi- 10
circle), y = 62° (angles in a Δ)
B z = 18° (base angle isosceles 0
Δ ODC) 0 10 20 30 40 50 60 70 80 90 100
Mock exam score
(b) x = 100° (reflex ∠ADB = 200°,
6 NOT TO SCALE – note that on angle at circumference = half Frequency Maths exam marks
student drawing point X should be angle at centre) 60
2 cm (10 m) from B.
(c) x = 29° (∠ADB is angle in a semi- 50
D circle so ∠BDC = 90°, then angles
in a Δ) 40
A C
Cell Radio (d) x = 120° (angle at centre), y = 30° 30
tower X mast (base angle isosceles Δ)
20
B Flagpole 4 (a) x = 7.5 cm, y = 19.5 cm
(b) x = 277.3 mm, y = 250 mm 10

5 NOT TO SCALE 0
0 10 20 30 40 50 60 70 80 90 100
Mixed exercise Route 66 Mock exam score

1 Petrol station (b) 51−60 (c) 51−60
(a) (i) (d) students to compare based
(ii) none
on histograms, but possible
comments are: The modal value
for both subjects is the same
but the number in the English
mode is higher than the Maths.
Maths has more students scoring
between 40 and 70 marks. Maths
has more students scoring more
than 90 marks and fewer scoring
less than 20.

(b) (i) (ii) none 6 NOT TO SCALE 2 (a) bars are touching, scale on
X horizontal axis is continuous,
vertical axis shows frequency
AB
5 cm (b) 55 (c) 315 (d) 29−31
(e) scale does not start from 0

(c) (i) (ii) four X

Answers 197

3 Mixed exercise Chapter 21
Histogram of ages
1 (a) Exercise 21.1
one person
Heights of trees 1 (a) 3 : 4 (b) 6 : 1
12 (c) 7 : 8 (d) 1 : 5

10 (e) 1 : 4

Frequency 8 2 (a) x = 9 (b) x = 4
(c) x = 16 (d) x = 3
Frequency density 6 (e) x = 4 (f) x = 1.14
(g) x = 1.875 (h) x = 2.67
4 (i) x = 7 (j) x = 13.33

2

0 3 60 cm and 100 cm
0 2 4 6 8 10 12
Height in metres

(b) 19 (c) 6−8 metres 4 (a) 20 ml oil and 30 ml vinegar
(b) 240 ml oil and 360 ml vinegar
2 (a) (c) 300 ml oil and 450 ml vinegar

20 Masses of students 5 60°, 30° and 90°
18
15 20 25 30 35 40 45 50 55 Frequency 16 56 58 60 62 6 810 mg
Age (years) 14 Mass (kg)
12
4 (a) 300 (b) 480 (c) 100 10 Exercise 21.2 A
8
Exercise 20.2 6 1 (a) 1 : 2.25 (b) 1 : 3.25 (c) 1 : 1.8
4
1 (a) 2 64 2 (a) 1.5 : 1 (b) 5 : 1
0 (c) 5 : 1

54

150

140 (b) 60−62 kg

130 P80 (c) 7.4% Exercise 21.2 B
120 Q3
(d) 10 kg 1 240 km
110

Number of students 100 P60 3 2 30 m
90
Histogram of airtime minutes
80 Q2 3 (a) 5 cm
70 Frequency density 5 teenagers (b) 3.5 cm

60 4 (a) it means one unit on the map is
equivalent to 700 000 of the same
50 units in reality
40 Q1
(b)
30
Map
20 distance 10 71 50 80 1714 2143
(mm)
10 20 30 40 50 60 70 80 90 100 110 120130140150
0x Time (minutes) Actual
distance 7 50 35 56 1200 1500
0 10 20 30 40 50 60 70 80 90 100 4 (a) 6.5 cm (km)
Percentage

(b) Median = 57%, Q1 = 49% and (b) Cumulative frequency of plant heights
Q3 = 65%
Cumulative y
(c) IQR = 16 frequency 30
(d) 91%
(e) 60% of students scored at least 20

59%; 80% of the students scored 10 Q1 Q2 Q3 x 5 (a) 4 : 1 (b) 14.8 cm
at least 67% 0 (c) 120 mm or 12 cm
0 1 2 3 4 5 6 7 8 9 10 11 12
2 (a) 166 cm Exercise 21.3
(b) Q1 = 158, Q3 = 176 Height (cm)
(c) 18 1 25.6 l
(d) 12.5% median height = 6.8 cm
(c) IQR = 8.3 – 4.7 = 3.6 2 11.5 km/l

3 (a) 78.4 km/h
(b) 520 km/h
(c) 240 km/h

198 Answers

4 (a) 5 h (b) 9 h 28 min (b) (i) 2 1 days (b) (i) 0.35 h
(c) 40 h (d) 4.29 min 2 (ii) 4.7 h
(iii) 1.18 h
5 (a) 150 km (b) 300 km 1
(c) 3.75 km (d) 18 km 4 (a) 150 km
(ii) 2 day (b) after 2 hours; stopped for 1 hour
(c) 100 km/h
8 (a) 12 days (b) 5 days (d) 100 km/h
(e) 500 km
Exercise 21.4 9 5 h 30 min
5 (a) 20 seconds
1 (a) (i) 100 km 10 1200 km/h (b) 2 m/s2
(ii) 200 km (c) 200 m
(iii) 300 km Exercise 21.6 (d) 100 m

(b) 100 km/h 1 (a) k = 7 6 4.5 min
(c) vehicle stopped (b) a = 84
(d) 250 km 7 187.5 g
(e) 125 km/h 2 ratio of m to T is constant, m = 0.4587, 8 (a) P = k or PV = k
T
2 (a) 2 hours V
(b) 190 min = 3 h 10 min so m varies directly with T (b) P = 80
(c) 120 km/h
(d) (i) 120 km 3 (a) F = 40 9 (a) 7 : 4 (b) 54.86 mm
(ii) 80 km (b) m = 4.5
(e) 48 km/h
(f) 40 min 4 a = 2, b = 8, c = 11
(g) 50 min
(h) 53.3 − 48 = 5.3 km/h 3
(i) Pam 12 noon, Dabilo 11:30 a.m.
5 (a) y = 2 (b) x = 0.5

6 (a) y = 2x2 (b) y = 1250 Chapter 22
(c) x = 9

7 (a) y x = 80 Exercise 22.1 A
(b) y = 8
3 (a) (i) 40 km/h (c) x = 15.49 1 (a) x − 4
(ii) 120 km/h (b) P = 4x − 8
8 (a) b = 40 (b) a= 17 7 (c) A = x2 − 4x
(b) 3.5 min 9
(c) 1200 km/h2
(d) 6 km 9 (a) y = 2.5 (b) x = 2 2 (a) S = 5x + 2

4 (a) 0–30 s, 5 m/s2 10 (a) xy = 18 for all cases, so (b) M = 5x + 2
3
6 relationship is inversely
3 (a) x + 1, x + 2
(b) after 70 s, 0.5 m/s2 proportional
(c) 90 km/h (b) xy = 18 or y = 18 (b) S = 3x + 3
(d) 2 km
x 4 (a) x + 2 (b) x − 3
(c) y = 36 (c) S = 3x − 1

Exercise 21.5 Exercise 21.7

1 (a) Yes, A = 1 1 $300 Exercise 22.1 B
B 150
1 14
81 2 $72 000 2 9 cm
3 80 silver cars, 8 red cars
(b) No, 15 is not = 2 4 father = 35, mother = 33 and

(c) Yes, A = 10 3 (a) $51 000 (b) $34 000 Nadira = 10
B 1 5 breadth = 13 cm, length = 39 cm
6 X cost 90c, Y cost $1.80 and Z cost 30c
2 (a) $175 (b) $250 4 14 350 7 9 years
8 97 tickets
3 $12.50 Mixed exercise

4 60 m 1 (a) 90 mm, 150 mm and 120 mm
(b) Yes, (150)2 = (90)2 + (120)2
5 (a) 75 km (b) 375 km
(c) 3 h 20 min 2 1 : 50

6 (a) 15 litres (b) 540 km 3 (a) (i) 85 km
(ii) 382.5 km
7 (a) inversely proportional (iii) 21.25 km

Answers 199

Exercise 22.2 3 (a) h(x) 5 x Chapter 23

1 (a) V = U + T − W (b) (i) h(1) = ±2 Exercise 23.1 A
(b) V = U T 2 − W or (ii) h(–4) = ±3
3 1
4 (a) 4(x – 5)
(V 1 U −T W ) (b) 4x – 5 2
3
5 18
(c) B = C
A 6 (a) f–1(x) = x – 4
(b) f–1(x) = x + 9
(d) B = AC
(c) f −1 (x) = x
(e) Q P
5
(f) Q = ± P (d) f–1(x) = –2x 3
2 7 (a) x − 3
A P
(g) Q = ± P 2 P
R (b) x − 3 S
B R
(h) P = Q2 2 QQ
2 (c) 2(x + 3) A'
(d) 2x + 3 P' R
(i) P = Q2 (e) 2x + 3
R (f) 2(x + 3) Q'

(j) P = Q2 + R Mixed exercise
(k) Q = R2
1 10
P
2 41, 42, 43
2 (a) I = V
R 3 4 years

(b) 20 amps 4 Nathi has $67 and Cedric has $83

3 (a) r = A (note, radius cannot be 5 Sindi puts in $40, Jonas $20 and
π Mo $70

negative) 6 44 children
(b) r = 5.64 mm 7 (a) b = 9a − 26

4 (a) F 9 C + 32 8
(b) b = a2 − 4
5
17
(b) 80.6 °F
(c) 323 K 8 f −1 (x) = 5 + 3

Exercise 22.3 2 CS
S'
1 (a) 11 (b) –1 9 (a) f −1 (x) = x − 4 B'
(c) 5 (d) 2m + 5
3
2 (a) f(x) = 3x2 + 5 (b) 3 R'
(b) (i) 17 (c) a = 6 C'
(ii) 53 (d) 9x + 16
(iii) 113 (e) 37 4 A: y = 5
(c) f(2) + f(4) = 17 + 53 = 70 ≠ f(6) B: x = 0
which is = 113 C: y = −1.5
(d) (i) 3a2 + 5 D: x = −6
(ii) 3b2 + 5
(iii) 3(a + b)2 + 5
(e) a = ±3

200 Answers

5 (c) y (b) enlargement scale factor 2, using
(8, −1) as centre
10 y

8 A 3 (a) 10 y
D D' 6 B''
C'
4 A'
C'' x C x 8 F'
246 8 10 x 6
2 X
C' B 4 AB
–10 –8 –6 –4 –2 0 A'' (d)
2
–2
C –4 y DC x
–10 –8 –6 –4 –2 0 2 4 6 8 10
B' –6
D'' –8 –2

–4

–6

–10 –8

D' D

6 possible answers are: –10

for ABC: rotate 90° clockwise about B (b) rotation 180° about (4, 5)
then reflect in the line x = −6
Exercise 23.2 A
for MNOP: rotate 90° clockwise about
(2, 1) then reflect in the line x = 5.5 1 (a) ur = ⎛ 5⎞ (b) ur = ⎛ 4⎞
AB ⎝⎜ 0⎠⎟ BC ⎝⎜ 0⎠⎟

Exercise 23.1 B Exercise 23.1 C ur ⎛ 0⎞ ⎛ −1⎞
AE = ⎝⎜ −6⎟⎠ ⎝⎜ −6⎠⎟
1 A: centre (0, 2), scale factor 2 1 (a) (c) (d) ur
BD =

B: centre (1, 0), scale factor 2 y

C: centre (−4, −7), scale factor 2 A' 7 A ur ⎛ 1⎞ ur ⎛ 9⎞
6 DB = ⎜⎝ 6⎠⎟ EC ⎝⎜ 6⎠⎟
D: centre (9, −5), scale factor 1 (e) (f ) =
4
5
2 4 ur ⎛ −5⎞ ur = ⎛ −5⎞
C' B' 3 B C CD = ⎝⎜ −6⎠⎟ BE ⎜⎝ −6⎠⎟
2 (g) (h)
1
(a) y x (i) they are equal

–5 –4 –3 –2 –1 0 1 2 3 4 5 ⎛ 9⎞ ⎛ −5⎞
(j) ⎝⎜ 0⎟⎠ (k) ⎝⎜ −6⎠⎟
4 –2
A' 2 C'' B'' –3 (l) Yes

x –4 2
–5
–4 –2 0 A 2 4 –6
–2 X A'' –7
–4
(b) rotation 180° about (0, 0) (a)

2 (a)

y B
10

8 A

(b) y 6 B
4 C
X A
B' 2 4 x
2
–4 –2 0 2 4 x 2468
B –2 –4 –2 0
–2
–4

–4

Answers 201

(b) (g) b (h) –c (i) −7a + 7c Exercise 23.3
D
(j) b + 3c 1 (a)
C 2 y

(c) 3 (a) – (e) student’s own diagrams 10
F 9
4 (a) 6.40 cm 8
(b) 7.28 cm 7
(c) 15 cm 6
(d) 17.69 cm 5
4
5 (a) 5.10 3
(b) 5 2
(c) 8.06 1
(d) 9.22
x
6 (a) A(–6, 2), B (–2, –4), C (5, 1) –1–10 1 2 3 4 5 6 7 8 9 10

E (b) ur = ⎛ 4⎞ (b)
(d) AB ⎝⎜ −6⎟⎠ y

ur = ⎛ 7⎞ 10
BC ⎝⎜ 5⎠⎟ 9
8
ur = ⎛ −11⎞ 7
CA ⎜⎝ 1 ⎠⎟ 6
5
ur 4
7 (a) uXZr = x + y 3
G (b) ZX = –x – y 2
1
(c) ur = 1x+y
MZ 2 x
–1–10 1 2 3 4 5 6 7 8 9 10
H ⎛ 2⎞
8 (a) (i) x = ⎝⎜ 7⎠⎟ (c)
y
⎛ 8⎞ B ⎛ 2⎞ ⎛ 4⎞ (ii) y= ⎛ −3⎞
3 A ⎝⎜ 1⎠⎟ ⎝⎜ 3⎟⎠ C ⎝⎜ −3⎠⎟ ⎝⎜ −3⎟⎠ 10
9
⎛ −3⎞ ⎛ 9⎞ (iii) z= ⎛ 10⎞ 8
D ⎝⎜ −3⎟⎠ E ⎝⎜ 3⎠⎟ ⎝⎜ −4⎠⎟ 7
6
(b) (i) 7.28 5
4
Exercise 23.2 B (ii) 4.24 3
2
⎛ −8⎞ ⎛ 2⎞ (iii) 21.5 1
1 (a) ⎝⎜ 16⎟⎠ (b) ⎝⎜ 6⎠⎟ ur
⎛ 0⎞ 9 (a) (i) XY = b – a x
(c) ⎝⎜12⎠⎟ ur –1–10 1 2 3 4 5 6 7 8 9 10
(ii) AD = 1 (a + b)

⎛ −1⎞ ⎛ −2⎞ ⎛ −1⎞ ur 2
(d) ⎝⎜ 7 ⎠⎟ (e) ⎝⎜ 1 ⎠⎟ (f) ⎝⎜ 4 ⎠⎟ (iii) BC = 2(b – a)

⎛ −4⎞ ⎛ −8⎞ ⎛ 0 ⎞ ur ur
(g) ⎝⎜ 18⎟⎠ (h) ⎜⎝ 22⎠⎟ (i) ⎝⎜ −20⎠⎟ (b) XY = b – a and, BC = 2(b – a)

so they are both multiples of

⎛ 10 ⎞ (ub r– a), and henucer parallel, and
(j) ⎝⎜ −16⎟⎠ is double XY

2 (a) –a 10 28.2(3sf)
(d) 2c
(b) 2b (c) −a + c
(e) 2b (f) 2c

202 Answers

(d) ⎛3 9 0⎞ 2 (a) & (b)
(b) ⎝⎜12 6 3⎠⎟
y y
10
9 ⎛ −12 −6 6 ⎞ 7
8 (c) ⎝⎜ −14 4 −10⎠⎟ 6
7
6 5 (a) x = 3 1 ⎛ 4 −1⎞ 5 x
5 (b) 5 ⎝⎜ −3 2⎠⎟ 4
4 3
3
2 2
1 1

x Exercise 23.5 –5 –4 –3 –2 –1 12345
–1–10 1 2 3 4 5 6 7 8 9 10
1 (a) (i) reflection in the y-axis –2
–3
(ii) shear with x-axis invariant –4
–5
and shear factor 2 –6
–7
(b) (i) ⎛ −1 0⎞
⎜⎝ 0 1⎠⎟

(ii) ⎛1 2⎞ 3
⎝⎜ 0 1⎠⎟
2 (a) shear, x-axis invariant, shear 10 y (d)
factor 1.5 2 (a) R' (2, 8) 8
(b) a stretch parallel to the y-axis
(b) 2-way stretch: stretch of scale (x-axis invariant), scale factor 4 (c) 6
factor 4, invariant line x = 1 and
stretch of scale factor 2, invariant 3 (a) enlargement, scale factor 3, centre (a) 4
line y = 2 2
of enlargement the origin x
(c) shear, x = 0 (y-axis) invariant, –10 –8 –6 –4 –2 2 4 6 8 10
scale factor 0.75 (b) ⎛ −3 0⎞ –2
⎜⎝ 0 3⎠⎟ (b)
(d) shear, line y = 4 invariant, shear
factor 1 ⎛3 0⎞ –4
⎜⎝ 0 3⎠⎟ –6
4
Exercise 23.4 –8
–10
1 ⎛0 6 5⎞ Mixed exercise
(a) ⎝⎜ 7 2 −3⎟⎠
(a) B′ (−6, −6) (b) B′ (6, −2)
1 (a) (i) reflect in the line x = −1 (c) B′ (−1, 8) (d) B′ (3, 9)

⎛ −2 −2 1 ⎞ (ii) rotate 90° clockwise about ⎛⎞ ⎛ 0⎞
(b) ⎜⎝ 1 4 −3⎠⎟ the origin 4 (a) (i) ⎝⎜12⎠⎟ (ii) ⎝⎜ −8⎠⎟

⎛ 2 2 −1⎞ (iii) reflect in the line y = −1
(c) ⎜⎝ −1 4 3 ⎠⎟
(b) (i) rotate 90° anticlockwise ⎛ 1⎞ ⎛12⎞
−1 ⎛ 8 −5⎞ about (0, 0) then translate (iii) ⎝⎜10⎟⎠ (iv) ⎜⎝ 0 ⎠⎟
(b) 12 ⎝⎜ −4 1⎠⎟ (b) (i)
2 (a) –12 ⎛ 2⎞
⎜⎝ −1⎠⎟
⎛3 9⎞
3 (a) AB = ⎝⎜ 9 1⎠⎟ and BA = (ii) reflect in the line y = −1

then translate ⎛ −8⎞ a 2a
⎜⎝ 0 ⎠⎟
⎛6 6⎞ ; so AB ≠ BA
⎝⎜ −10 −4⎠⎟ (iii) rotate 180° about origin

(b) (i) –6 then translate ⎛ 6⎞
(ii) –84 ⎜⎝ 0⎟⎠

⎛ 7 0 −3⎞ (iv) reflect in the line x = 0
4 (a) ⎜⎝11 0 4 ⎠⎟ (y-axis) then translate

⎛ 0⎞
⎜⎝ −2⎠⎟

Answers 203

(ii) 7y 11 (a) & (c)
8
b c 10 y
b+c
8

6 6A
4

4 B 2 B Bx
A C 2 4 C6 8 10
(iii) –10 –8 –6 –4 –2–20
2

x –4 A
–2 D 2 4 6 C 8 –6

a a–b –2 –8
b –10 A
8y
(iv) b 10 ⎛1 0 ⎞
a 9 (b) ⎝⎜ 0 1⎠⎟
8
7 ⎛ 2 0⎞
6 (d) ⎜⎝ 0 2⎠⎟
5
4 ⎛2 0 ⎞
3 (e) ⎝⎜ 0 2⎠⎟
2
1 Chapter 24
x

–1–10 1 2 3 4 5 6 7 8 9 10

9 ⎛2 5⎞ Exercise 24.1
(a) ⎝⎜ 4 3⎟⎠
1 Card Coin

⎛ 0 1⎞ R H
(b) ⎜⎝ 0 1⎠⎟ T

2a + 3b ⎛ 5 7⎞ Y H
(c) ⎝⎜ 4 8⎠⎟ T
ur
5 (a) (i) u r = y G H
T
(ii) DuEr = –y ⎛ 7 5⎞
(iii) uFrB = x + y (d) ⎜⎝ 6 6⎠⎟ B H
(iv) uErF = x – y T
(v) FD = 2y – x
(b) 4. 47 (e) |P| = –3 2

6 (a) one-way stretch with y-axis ⎛ −1 2⎞ 1 G
invariant and scale factor of 2 ⎜ H
⎜ 3
(b) one-way stretch with x-axis (f ) P–1 = ⎜ 2 3 ⎟ 2 A
invariant and scale factor of 2 ⎟ B
−1 ⎟⎠
(c) a shear with x-axis invariant and ⎝3 3 C
shear factor = 3 3 D

(d) a shear with y-axis invariant, 10 a = 7, b = 2 and c = –9 4 E
shear factor = 2 F

3 (a) & (b)

7 Green
11

2 Green
3
4 Yellow

11

8 Green
11

1 Yellow
3

3 Yellow

11

204 Answers

Exercise 24.2 Mixed exercise

1 (a) 1 (a) & (b)

1 1 H 1 H
2 2 T
Blue T1 1 2 H
1 Yellow 1 T
3 Black 2 12 H1
1 H
(c) 2 H 2 H 1 1 2 T
1 T 6 2 1 H
1 H T
2 2 T 1 1 T2
1 1 6 1 H
2 H 22 H 2 T
T2 T 1 H
1 1 H 6 1 T
1 2 H 2 T 2 T1
6 H 1 2 10c
1 T 6 1 H 1
12 H
T T 1 32 2
6 1
1
(b) 1 (d) 5 1 2 T2
4 12 6 1
1 H 2

2 (a) 42 T1

1 1 1 H 2
2 2 1
H2
H 1 2
1 1
11 2 52
22 T2
1 1 H 1
2 2
T2
1 T
1
2 62

1 1
2
H2
11 (c) 1 (d) 1
22 1 8 12
2
T 2 (a) & (b)
1
1
2 T2

1 1 5c
2 6

(b) 1 (c) 1 2 5c 1
8 2 7
5
(d) 1 5 10c 5
2 6

(e) 0, not possible on three coin 5 10c
7

tosses (c) 5 (d) 1
7 21

(e) 1 (there are no 5c coins left)

Answers 205