Topic 6 Answers

Cambridge IGCSE™ International Mathematics
6 Vectors and transformations

All answers were written by the authors. In examination, the way marks would be awarded to
answers like these may be different.
Exercise 6.1 page 354

1 a (24)
b (31)
c (−2
−4
)
3
d (−2 )
e (−6
1
)
6
f (−1 )
g (−3
−1
)
h (−5
−5
)
i (−1
3
)
2 a (44)
b (04)
c (−3
0
)
d (−2
6
)
e (−4
−2
)
0
f (−4 )
g (30)
2
h (−6 )
i (−4
−4
)
3
Cambridge IGCSE International Mathematics Online Teacher’s Guide 1

© Hodder & Stoughton Ltd 2018
Cambridge IGCSE International Mathematics answers

1
2 a + b = b + a, a + d = d + a, b + c = c + b
3
4 a (24)
0
b (−6 )
1
c (−1 )
d (10)
e (09)
f (52)

1
1 d = –c e = –a f = 2a g = 2c
1 1 3 3
h = 2 b i = – 2 b j = 2 b k=–2a
2 a (46)
b (−12
−3
)
2
c (−4 )
d (−2
2
)
e (−2
−5
)

f (−8
9
)
g (−10
−5
)
h (32)
10
i (−6 )
3 a 2a
b –b
c b+c
d a–b
e 2c
f 2c – a

1 |a| = 5.0 units, |b| = 4.1 units, |c| = 4.5 units, |d| = 7.0 units, |e| = 7.3 units, |f| = 6.4 units
2 a |AB| = 4.0 units
b |BC| = 5.4 units
c |CD| = 7.2 units
d |DE| = 13.0 units
e |2AB| = 8.0 units
f |–CD| = 7.2 units
3 a 4.1 units
b 18.4 units
c 15.5 units
d 17.7 units
e 31.8 units
f 19.6 units

1 a a
b a
c b
d –b
e 2b
f –2a
g a+b
h b–a
i b – 2a

2 a 2a
b b
c b–a
d b–a
e –a
f a – 2b
3 a –2a
b –a
c b
d b–a
e 2(b – a)
f 2b – a
g a – 2b
h b – 2a
i –a – b
4 a 5a
8
b 3
b
1
c (8b – 15a)
3
1
d 15
(8b – 15a)
e b – 2a
f 5a – 2b
8
g 5
b
1
h (10a + 8b)
5
1
i 5
(8b – 5a)

1 a i 2a
ii 2a – b
1
iii 4
(2a + 3b)
b Proof
2 a, b Proofs
3 a i 4a
ii 2a
iii 2(a + b)
3
iv 2
a

b Proof
1
4 a i 2
(2q – p)
2
ii 7
(p – q)
b Proof

1



1


1


1 a
b 180° clockwise/anti-clockwise

2 a
b 90° anti-clockwise
3 a
b 180° clockwise/anti-clockwise
4 a
5 a

6 a
b 90° clockwise
7 1 Rotation 180°
2 Rotation 90° clockwise
3 Rotation 180°
6 Rotation 90° anti-clockwise

1 A → B = (−6
0
), A → C = (36)
0
2 A → B = (−7 ), A → C = (−6
1
)
3 A → B = (06), 6
A → C = (−3 )
4 A → B = (50), A → C = (−3
−6
)

1


7 1 (−3
−5
)
2 (−5
4
)
4
3 (−6 )
4 (25)
5 (60)
6 (01)

1

1
6 1 scale factor
2
2
2 scale factor
3
1
3 scale factor
2
1
4 scale factor 3
5 scale factor 4


1


1

5 a and b
6 a and b

1 a
b y=2
2 a
b x = –1

3 a
b y = –1
4 a
b y=x–2
5 a
b y=–x–2
6 a
b y = 0, x = −3

7 a
b y = 1, x = −3, y = x + 4, y = –x – 2
8 a
b y = 1, x =−3, y = x + 4, y = –x – 2

1 a

2 a



1 2
2 1.5
3 3.5
4 2
5 1.5
6 4
7


1
Stretch factor is 2
2
Stretch factor is 2
3
Stretch factor is 1.5

4
Stretch factor is 3


1

A painted cube page 384

1 a 8
b 12
c 6
d 1
2 A: 8, B: 24, C: 24, D: 8
3 A: 8, B: 96, C: 384, D: 512
4 When (𝑛 − 2) ≥ 0,
A: 8, B: 12(𝑛 − 2), C: 6(𝑛 − 2)2 , D: (𝑛 − 2)3
5 When (𝑙 − 2) ≥ 0, (𝑤 − 2) ≥ 0 and (ℎ − 2) ≥ 0,
A: 8,
B: 4(𝑙 − 2) + 4(𝑤 − 2) + 4(ℎ − 2),
C: 2(𝑙 − 2)(𝑤 − 2) + 2(𝑙 − 2)(ℎ − 2) + 2(𝑤 − 2)(ℎ − 2),
D: (𝑙 − 2)(𝑤 − 2)(ℎ − 2)
Triangle count page 384

1 9
2 Student’s investigation and ordered table
Number of Number of
horizontal lines triangles
0 3
1 6
2 9
3 12
etc.
3 n = 3(h + 1)
4 12
5 Student’s investigation and ordered table
Number of Number of
horizontal lines triangles
0 6
1 12
2 18
3 24
etc.
6 n = 6(h + 1)

To swim or run? page 386

1 25 seconds
3 23.1 seconds
4 Proof
5 63.3m
6 21.3 seconds
Below is part of a suggested spreadsheet for modelling the data.
ICT Activity page 387

Student’s help sheet
Student assessment 1 page 388

1 a (−2
7
)
b (−5
−4
)
c (−1
−3
)
2 a (03)
b (−6
0
)
c (−1
−5
)
d (33)
4
e (−1 )

7
4 a (−11 )
b (−3
9
)
6
c (−6 )
d (−18
28
)

1 a ⃗⃗⃗⃗ | = 5 units
|FG
b │a│ = 6.1 units, │b│ = 12.4 units, │c│= 14.1 units
2 a 29.7 units
b 11.4 units
3 a 2a
b –b
c b–a

1 a a
b –b
c (1 + √2)𝐛
2 a i 5b
ii 5b – a
1
b 2
(𝐚 + 3𝐛)
1
3 a 2
𝐚
b b–a
c 2:3


1
0
3 a (−3 )
b (−8
2
)
4
5 a
b The scale factor of enlargement is –2.

c The inverse transformation is an enlargement of scale factor − 12 , with the same centre of
enlargement.

6 a 1.5
b 2
7 a A reflection in the line x = 0
b An enlargement by scale factor –0.5. Centre of enlargement (0, –1)
8 a and b


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Cambridge IGCSE™ International Mathematics

6 Vectors and transformations

Exercise 6.1 page 354

Cambridge IGCSE International Mathematics Online Teacher’s Guide 1

Exercise 6.2 page 355

Exercise 6.3 page 357

Cambridge IGCSE International Mathematics Online Teacher’s Guide 2

Exercise 6.4 page 359

Exercise 6.5 page 360

Cambridge IGCSE International Mathematics Online Teacher’s Guide 3

Exercise 6.6 page 361

Cambridge IGCSE International Mathematics Online Teacher’s Guide 4

Exercise 6.7 page 363

Cambridge IGCSE International Mathematics Online Teacher’s Guide 5

Cambridge IGCSE International Mathematics Online Teacher’s Guide 6

Exercise 6.8 page 364

Cambridge IGCSE International Mathematics Online Teacher’s Guide 7

Exercise 6.9 page 365

Cambridge IGCSE International Mathematics Online Teacher’s Guide 8

Exercise 6.10 page 366

Cambridge IGCSE International Mathematics Online Teacher’s Guide 9

Cambridge IGCSE International Mathematics Online Teacher’s Guide 10

Exercise 6.11 page 368

Exercise 6.12 page 368

Cambridge IGCSE International Mathematics Online Teacher’s Guide 11

Cambridge IGCSE International Mathematics Online Teacher’s Guide 12

Exercise 6.13 page 371

Cambridge IGCSE International Mathematics Online Teacher’s Guide 13

Cambridge IGCSE International Mathematics Online Teacher’s Guide 14

Exercise 6.14 page 373

Cambridge IGCSE International Mathematics Online Teacher’s Guide 15

Exercise 6.15 page 374

Cambridge IGCSE International Mathematics Online Teacher’s Guide 16

Exercise 6.16 page 377

Cambridge IGCSE International Mathematics Online Teacher’s Guide 17

Cambridge IGCSE International Mathematics Online Teacher’s Guide 18

Exercise 6.17 page 378

Cambridge IGCSE International Mathematics Online Teacher’s Guide 19

Cambridge IGCSE International Mathematics Online Teacher’s Guide 20

Cambridge IGCSE International Mathematics Online Teacher’s Guide 21

Exercise 6.18 page 380

Cambridge IGCSE International Mathematics Online Teacher’s Guide 22

Exercise 6.19 page 381

Stretch factor is 1.5

Cambridge IGCSE International Mathematics Online Teacher’s Guide 23

Exercise 6.20 page 383

Cambridge IGCSE International Mathematics Online Teacher’s Guide 24

A painted cube page 384

Triangle count page 384

Cambridge IGCSE International Mathematics Online Teacher’s Guide 25

To swim or run? page 386

ICT Activity page 387

Student assessment 1 page 388

Cambridge IGCSE International Mathematics Online Teacher’s Guide 26

Student assessment 2 page 389

Student assessment 3 page 389

Cambridge IGCSE International Mathematics Online Teacher’s Guide 27

Student assessment 4 page 390

b The scale factor of enlargement is –2.

Cambridge IGCSE International Mathematics Online Teacher’s Guide 28