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ANSWERS

Chapter 31

Exercise 31A

1 a e.g. b e.g.

c e.g.

d e

f e.g.

2

3 a

b 2 planes of symmetry

Exercise 31B

1

2

3 a

b

c

4 a b

Exercise 31C

1 1180 cm3 2 813 cm3 3 2.62 cm

Exercise 31D

1 245 cm3 2 60 m3 3 84 cm3

4 75.4 cm3 5 209 cm3 6 48cm3

7 73.6 cm3 8 289 cm3 9 288cm3

10 2 580 000 m3 11 221 cm3 12 25.5 cm3

13 264 cm3 14 253 cm3 15 454 cm3

16 9 cm2 17 8 cm 18 12.3 cm

19 4.21 cm 20 4.15 cm

Exercise 31E

1 a 3000 mm3 b 4.5 cm3 c 650 mm3

2 a 2 800 000 cm3 b 6 m3 c 3.2 m3

3 a 120 cm3

b 120 000 mm3

4 a 1.35 m3 b 1 350 000 cm3

5 8.48 litres 6 7.8 cm

Exercise 31F

1 a 325 cm2 b 383 cm2

2 598 cm2 3 131 cm2 4 223 cm2 5 152 cm2

6 a 33.9 cm2 b 32.2 cm2 c 36.2 cm2

7 36cm2 8 40cm2 9 2.19 cm

plan front elevation side elevation

plan front elevation side elevation

or

or

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ANSWERS

Chapter 32

Exercise 32A

1

2 a

b

c (0, 2) d 1.4, 1.4

3 a

b

4 a

b

c 0 and 1d ii See graph.

ii x 0.5

e 0.3

5 a

b

c x3 and x 1

d (1,4)

Exercise 32B

1 1, 2

2 b i

ii 2 and 2c approx. 2.6 and 2.6

3 a 0, 1.5b 1, 2.5

4 a approx. 1.4 and 2.9b 1 and 2.5c 0.5 and 2

O

2

4

6

8

9

1

3

5

7

y

3 2 1 1 2 3x

y 4

yx2

4

2

O

4

8

2

6

y

4 3 2 1 1 2 3x

yx2 2x 3

x 1x3

5

O

5

10

15

y

3 2 1 1 2 3x

yx2x

x 0.5

O

5

10

15

20

y

3 2 1 1 2 3x

y 2x2

5

6

7

8

4

3

2

1

1

O

2

3

y

4 3 2 1 1 2 3x

y 2 x2

5

4

3

2

1

1

O

2

3

4

5

6

7

8

y

3 2 1 1 2 3x

yx2 1

y 2x2 5

x 3 2 1 0 1 2 3

y 7 2 1 2 1 2 7

x 3 2 1 0 1 2 3

y 18 8 2 0 2 8 18

x 3 2 1 0 1 2 3

y 12 6 2 0 0 2 6

x 4 3 2 1 0 1 2

y 5 0 3 4 3 0 5

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ANSWERS

5 a

b i approx 3.3 ii 1.3 and 2.3

Exercise 32C

1 a y x 5 b y 2x 1

2 a

b

c i 2 and 3 ii x2 x 6 0

3 a 0.6, 1.6 b 2.4, 0.4 c 0.4, 2.6

4 a

b i 0.3, 3.7 ii 1.3, 2.3

5 a

b i 1.5, 1 ii 2, 1 iii 1, 1.5

6 a

b iii 2.6, 2.6

iii 2.7, 0.7

iii 0.4, 2.4

2

2

O

4

6

8

10

y

23 1 1 2 3x

yx2x 2

2

2

4

O

4

6

y

2 1 1 2x

y 5 2x2

2

2

O

4

6

8

10

y

2 1 1 2 3 4x

yx2 2x

2

2

O

4

6

8

10

y

23 1 1 2 3x

4

3

2

1

1

O

2

3

y

23 1 1 2 3x

y 3 xx2

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ANSWERS

Chapter 33

Exercise 33A

1 a A and Db i B iii A (0, 2) C (0, 1) D (0, 8)

2 a

b

3

4

5 a

b

c x1

6

7 p 0.5, q 16

Exercise 33B

2 a 0.4 b 0.5 c 0.9 d 0.73 1.76 4 2.6 5 b 4.6

6 b 3.73 c 573 cm

7 2.7 8 b 1.3

9 1.5 10 2.1 11 2.8

20

10

O

10

20

y

5 4 3 2 1 1 2 3 4 5 x

20

10

O

10

20

y

5 4 3 2 1 1 2 3 x

O

1

y

x

y 3xy 6x

O

10

15

20

25

5

1 2 3 4 5 6

(2, 18)

(3, 12)

yx3 10x2 25x(1, 16)

(0, 0)

(4, 4)

(6, 6)

(5, 0)

10

12

14

16

8

6

4

2

2

O

4

6

8

10

12

14

16

y

5 4 3 2 1 1 2 3 4 5 x

(2, 16)(4, 16)

(1, 11)

(3, 9)

(2, 16) (4, 16)

(3, 9)(1, 11)

(0, 0)

x 5 4 3 2 1.80.2 0 1 2 3

y 3 4 6 12 15 15 12 6 4 3

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ANSWERS

Chapter 34

Exercise 34A

1 2

3 4

5 6

7 8

9

10

11

12 13

Exercise 34B

1

2

3 4

5

6

7 8

9

2cm

2cm

2cm

A B

C

A

B

C

A

B

A

B

0

4

3

2

1

1 2 3 4 x

y

B

A

A B

2cm2cm

2cmAB

A

3cm

P

R

Q

P

R

Q

P

R

Q

PQ

P QR

P Q

R

P

R

Q

P

R

Q

P Q

P Q

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ANSWERS

9

10

11

12

13

Exercise 34E

1 a ABAC(given).Angle BAD angle CAD (given).AD is common.

Triangles are congruent (SAS).

b In the congruent triangles ABD andACD, angle B corresponds to angle C,and so angle B angle C.

2 a AB AD (given).Angle ABC angle ADB (given).ACis common.

Triangles are congruent

(RHS).b In the congruent triangles ABC

and ADC, angle BACcorresponds to angle DAC, and so

angle BAC angle DAC. i.e. CA bisects angle BAD.3 a AB AC(given).

BMCM(given, as Mis the midpointof BC).AMis common.

Triangles are congruent (SSS).

b In the congruent triangles ABMandACM, angle BAMcorresponds toangle CAM, and so angle BAM angle CAM.

4 a AE BE(given).CEDE(given).Angle AEC angle BED

(where two straight lines cross,opposite angles are equal).

Triangles are congruent

(SAS).b In the congruent triangles AECand BED, AC

corresponds to BD, and so ACBD.

5 a Angle ODA angle ODB 90 (given).OAOB (radii of circle).OD is common.

Triangles are congruent (RHS).

b In the congruent trianglesOAD and OBD, AD corresponds to BD,

and so AD BD. i.e. D is the midpoint of the chord AB.

6 a AB AC(given).angle BAD angle CAE(given).angle ABC angle ACB(in an isosceles triangle, theangles opposite the equal sidesare equal)

Triangles are congruent

(AAS).(AB and ACare corresponding sides, as they arebetween an equal pair of angles.)

b In the congruent triangles BAD and CAE, BD

corresponds to CE, (they are opposite equal angles) andso BDCE.

BAD

CAE

OAD

OBD

AEC

BED

ABM

ACM

ABC

ADC

ABD

ACD

Sea

Land

A

B

1

O

A2

3

4

5

y

3 2 1 1 2 3 4x

A

C

B

P

P

A

P

P

B

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ANSWERS

Chapter 35

Exercise 35A

1 a i 17.5 ii 11.5b i 18.5 ii 12.5c i 29 ii 5 iii 201.25 iv 1.4d i 31 ii 7 iii 231.25 iv 1.609

2 a i 21 ii 11.6 iii 76.0275 iv 3.442b i 21.2 ii 11.8 iii 78.1375 iv 3.5376

3 a i 14.6 ii 1.8 iii 52.3875 iv 1.2791b i 14.8 ii 2.0 iii 53.8575 iv 1.3150

4 a i 9.78 ii 0.4073 iii 5.978025 iv 1.5652b i 9.82 ii 0.4090 iii 6.027025 iv 1.5668

5 a i 7.065 ii 2.065 iii 11.30675 iv 0.52973b i 7.175 ii 2.175 iii 11.79375 iv 0.55255

6 a i 14.0625 ii 20.44 iii 1.8142b i 14.8225 ii 20.76 iii 1.83786

7 a 9.524 s

b 10.536 s

8 a 54 cmb 58 cmc 178.25 cm2

d 206.25 cm2

9 0.1583

10 1.08%

Exercise 35B

1 a 2 b 3 c 5 d 4 e 6

2 a 3 23 b 5 33 c 3 5d 12 e 7 43

3 a b c d e

4 a b c d 2 e 25

5 a 1 2 b 2 1 c 1 25d 5 1 e 2 7 1

6 4 cm

7 3 22 cm2

8 a i 14 cm ii 7 5 cm2

102

3

26

3

511

11

32

2

27

7

5

5

2

2