Answers

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Answers Chapter 1

Exercise 1A

E 5

1a3 b9

c1

d -8 e 5

f2

g 3

-7 7

h

i

20

-10

j

k

14 l

23

3

3

5

L 2 a a + b

b a-b

b c

bc d ab e

a

a

3a 7

b5

c -3 d 14

7 e

14 f

2

3

g 48

3 h

i2

j3 k7

l2

2

4 4a

b -5 c 2

3

P 5a -1 b 18

6 c

d 23 e 0

f 10

5

g 12 h 8

i - 14 5

12 j

5

7 k

2

-b 6a

a

e-d b

c c -b

c

a

b d c-a

ab

b-d

e b+a f a+b g a-c

bd - c h

a

M 7 a -18 d 28

b -78.2 e 34

Exercise 1B

c 16.75 3

f 26

A1a x +2 =6,4

b 3x = 10, 10 3

c 3x + 6 = 22, 16 3

d 3x - 5 = 15, 20 3

e 6(x + 3) = 56, 19 f x + 5 = 23, 87

3

4

2 A = $8, B = $24, C = $16

S3 14 and 28 4 8 kg 5 1.3775 m2

2 a x = 8, y = -2

c x = 7, y = 1 2

3 a x = 2, y = -1

c m = 2, n = 3

e s = 2, t = 5

g

x

=

4 ,

y

=

7

3

2

i x = -1, y = 5 2

b x = -1, y = 4

b x = 2.5, y = -1 d x = 2, y = -1 f x = 10, y = 13 h p = 1, q = -1

Exercise 1D

1 25, 113

2 22.5, 13.5

3 a $70

b $12

c $3

4 a $168

b $45

c $15

5 17 and 28 6 44 and 12

7 5 pizzas, 25 hamburgers

8 Started with 60 and 50; finished with 30 each

9 $17 000

10 120 shirts and 300 ties

11 360 Outbacks and 300 Bush Walkers

12 Mydney = 2800; Selbourne = 3200

13 20 kg at $10, 40 kg at $11 and 40 kg at $12.

Exercise 1E

1 a x < 1 b x > 13 c x 3 d x 12

e x -6 f x > 3

g x > -2

h x -8 2a x>> Answers 705

e x1

1D 2D 3C 4A 5C 6 C 7 B 8 B 9 A 10 B

Short-answer questions

?2 ?1 0 1 2 3 4

(technology-free)

g

x

<

3

1 2

?2 ?1 0 1 2 3 4 h x3

?2 ?1 0 1 2 3 4

i

x

>

1 6

E 0 1

3 a x > -1

b x -5

5 87

L Exercise 1F

1 a 18 b 9

c3

d -18

e3

f 81 g 5

h 20

2 a S = a + b + c b P = xy c C = 5p

d T = dp + cq e T = 60a + b

3 a 15

b 31.4

c 1000

d 12

e 314

f 720

P 4aV = c p

ba= F m

cP= I rt

dr= w-H C

e t = S -lP Pr

f r = R(V - 2) V

5 a T = 48 b b = 8 c h = 3.82 d b = 10

M 6 a (4a + 3w) m

b (h + 2b) m

c 3w h m2

d (4ah + 8ab + 6w b) m2

7 a i T = 2( p + q) + 4h ii 88 + 112

b p= A -q h

8a D= 2 3

bb=2

Acn= 60 29

d r = 4.8

9

a

D

=

1 bc

(1

-

k2)

2

Sc k = 2 = 6

b k = 1 - 2D bc

1a 1

-3 b

2

-2 c

3

e 12

44 f

13

1 g

8

2a t =a-b

cd - b b

a

d -27

h 31 c d +c

a

cb - a d c-1

2b e c-a

1 - cd f

ad

3a x < 2 3

b

x

1 148

2

c x < 22 29

d x -7 17

4 x = 2(z + 3t), -10

5 a d = e2 + 2 f

d - e2 b f=

2

c f=1 2

6 x= a2 + b2 + 2ab = a + b

ac + bc

c

7

x

=

a

ab -b-

c

Extended-response questions

1 a c = -10 9

d x = -62.5

2

a

r

=

2uv u+v

b F = 86

e x = -160 13

bm= v u

c x = -40 fk=5

3 a T = 6w + 6l

b i T = 8w c i y = L - 6x

8 d x = 10, y = 5

ii

l

=

25 ,

w

=

1 12

6

2

ii y = 22

4 a distance that Tom travelled = ut km and

distance Julie travelled = vt km

b

it

=

d u+v

h ii

distance from A =

ud u+v

km

c t = 1.25 h, distance from town A

= 37.5 km

5

a

average

speed

=

uv u+v

b i uT

vT + uT ii

v

v

6a 3 + 3 ab

3 10 a P = 4b

3 b A = 2bc - c2

c b = A + c2 2c

c

i

c

=

2ab a+b

40 ii

3

11 a b = a2 - a 2

b x = -ay b

7a x, y 8 10

80(x + y) b 10x + 8y

c r = 3q - p2x2

x2 d v = u2 1 -

y2

c x = 320 , y = 310

9

9

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Answers

706 Essential Mathematical Methods 1 & 2 CAS

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Chapter 2

c

y

d

y

Exercise 2A

1

1a4 b2

1 c

d -4 e 1

f -1

4

5 g

4

h -2

-5 i

4

4 j

3

k0

2 Any line parallel to 3

y

the one shown y

(1, 6)

x 01

0

x

?1

4 a - 1 b - 5 c -2 d -8 e 0 f -1

4

2

E g 7 h 11 i -13 j 11 k 111 l 61

5 a -2

2

5

b

6 a 54

b

5

6

7a x =4

b y = 11

L Exercise 2B

1a

y

2

x

?2

0

P c

y

1

10

x

2

M 2a

y

1

1x 0

c

y

A1 x ?1 0

S3a

y

b

y

2

x 02

d

y

1

x 01

2

b

y

x 01 ?1

d

y

x ?1 0

?1

b

y

4 0

ey

8 x

x

0 21

3

?2

fy

3 0 g

6 y 6

2 x

0

4 x

h

y

x 03

x

?8

?4 0

4 Pairs which are parallel: a, b and c; Non-parallel: dy

4

3 4y ? 3x = 4

1 ?4 0

3

1 ?1 2

3y = 4x ? 3

x 34

5 c only 6 a -1 b 0

c -1 d 1

Exercise 2C

1 a y = 3x + 5 b y = -4x + 6 c y = 3x - 4

2 a y = 3x - 11

b y = -2x + 9

3a 2

b y = 2x + 6

4 a -2

b y = -2x + 6

5 a y = 2x + 4

b y = -2x + 8

6 a y = 4x + 4 b y = - 2 x c y = -x - 2 3

d

y

=

1 x

-

1

e

y

=

1 3

2

2

f x = -2

7 Some possible answers:

a y = 4x - 3 b y = - 2 x - 1 c y = -x - 1

3

d

y

=

1 x

+

1

e

y

=

4

f x = -1

2

Check with your teacher for other answers.

3

3

8

a

y

=

3 x

+

1 9

b y = -1x -1

42

2

x ?3 0

10 ?1 1

3

c y=3

d y = -3

x

9

a

y

=

4 x

+

3

b y = -1x +3

3

2

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Answers

Back to Menu >>> Answers 707

c y = -0.7x + 6.7

d

y

=

1 1

x

-

3

2

5 a x + 3y = 11 b 7x + 5y = 20 c 2x + y = 4 d ?11x + 3y = ?61

e y = -3x +6

f y = -x

6 a y = 2x - 9, m = 2

4

10 a y = - 2 x + 4 3

b y = -2x - 6

c y = -1x +4

2

11

a

y

=

2 x

+4

3

d y = -x + 8 b y = 2x - 2

33

c

y

=

1 x

+ 11

22

e y = x + 3.5

d y = -1x +2 2

f y = - 1 x + 0.25 2

12 Yes

13 AB: y = 2x + 1 BC: y = - 3x + 9

33

2

E AC:

y

=

1 x

-

3

84

Exercise 2D

1 a (0, 4), (4, 0)

L c (0, -6), (-6, 0)

2 a y = -2x + 6

c y=x

3a

y

b (0, -4) (4, 0) d (0, 8) (-8, 0)

1

b y = x-2

2

d y = -1 x + 3

2 b

y

P 2

x

x

01

?2 0

?1

c

y

Mx

02 ?4

4a y

by

0

A?4

c

x 6 y

S6

x

0

8

?2

d

y

10

b y = -3x + 5,m = -3

42

4

c y = - 1 x - 2, m = - 1

3

3

d

y

=

5 x

-

2, m

=

5

2

2

7a

y

b

y

5x 0

?10

c

y

d

10

x ?2 0

x 03 ?9 y 10

x 05

8 a = -4, b = 4 , d = -1, e = 14

3

3

Exercise 2E

1 a d = 50t

b d = 40t + 5

2 a V = 5t

b V = 10 + 5t

3 w = 20n + 350, possible values for

n = N {0}

4 a v = 500 - 2.5t

b domain: 0 t 200, range: 0 v 500

cv

500

0

200 t

5 C = 1.5n + 2.6

6 a C = 0.24x + 85

7 d = 200 - 5t

8a w (g)

b $145

51

x ?8 0

x ?4 0

ey

fy

50 0 1 2 3 4 5 6 x (cm)

x

0

15

?5

4

x 0 15

? 15

7

2

b w = 0.2x + 50 9 a C = 0.06n - 1 10 a C = 5n + 175

c x = 12.5 cm b $59 b Yes c $175

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708 Essential Mathematical Methods 1 & 2 CAS

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Exercise 2F

b d (km)

1a

b t =5

57

C

A

B 30

0

90 t (min)

20

c 10.30 am d Maureen 30 km, Anne 27 km

0 2a

d

15

t

dB dA

b 2.00 pm

5

E 0

t

3 a C1 = 210 + 1.6x C2 = 330

b C

300

L 75

200 0 20 40 60 80 x

c Fixed charge method is cheaper when x > 75.

4a d

P 500 B

A

C

0 5 10 15 20 25 t (hours)

b C wins the race

d

C,

leaving

5

hours

after

B,

overtakes

B

13

1 2

hours after B had started and then overtakes A

M 20 hours after A had started. C wins the race

with

a

total

handicap

time

of

22

1 2

hours

(12

1 2

hours for journey + 10 hours handicap) with A

and B deadheating for 2nd, each with a total

handicap time of 25 hours.

5 Both craft will pass over the point

5

1 3

,

-4

A6 a CT = 2.8x, CB = 54 + x

b $C

c >30 students

S54

8 a = 0.28 and b = 0.3, 10 m/s 3

Exercise 2G

1 a 135 b 45

c 26.57 d 135

2 a 45 b 135 c 45

d 135

e 63.43 (to 2 d.p.) f 116.57 (to 2 d.p.)

3 a 45 b 2634 c 16134

d 4924 e 16134 f 135

4 a 7134 b 135 c 45 d 16134

5

m BC

=

-

3 5

,

m

AB

=

5 3

m BC

?

m AB

=

-3 5

?

5 3

= -1

ABC is a right-angled triangle

6 mRS

=

-

1 2

,

m ST

=2

RS ST

m UT

=

-

1 2

,

m ST

=2

UT

ST

(Also need

to show S R = U T .)

RST U is a rectangle.

7 y = 2x + 2

8 a 2x ? 3y = 14

b 2y + 3x = 8

9 l = - 16 , m = 80

3

3

Exercise 2H

1 a 7.07 b 4.12 c 5.83

d 13

2 29.27 3 DN

Exercise 2I

1 a (5, 8) c (1.6, 0.7)

b

1 2

,

1 2

d (?0.7, 0.85)

2 MAB (3, 3). MBC

8,

3

1 2

. MAC

6,

1

1 2

3 Coordinates of C are (6, 8.8)

4 a PM = 12.04

b No, it passes through

0,

3

1 3

5 a (4, 4) b (2, ?0.2) c (?2, 5) d (?4, ?3)

6 1 + a , 4 + b ; a = 9, b = -6 22

0

30 x

7

a

dA

=

1 3

t

,

dM

=

57

-

3 t

10

Exercise 2J

1 a 3441 b 45 c 90 d 4924 e 2633

Multiple-choice questions

1A 2E 3C 4D 5B 6 E 7 D 8 C 9 E 10 E

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Back to Menu >>> Answers 709

Short-answer questions

5 a i C = 0.091n ii C = 1.65 + 0.058n

(technology-free)

1 a 9 b - 10 4 11

c undefined

d -1

b e

a

-b f

a

2 a y = 4x b y = 4x + 5 c y = 4x + 2

d y = 4x - 5

iii C = 6.13 + 0.0356n

b C ($) 15 10

(300, 16.81) (200, 13.25)

5 (50, 4.55)

3 a a = -2

20 b

0 100 200 300 n (kWh)

3 4 4y + 3x = -7 5 3y + 2x = -5

6 a = -3, b = 5, c = 14

7 a y = 11 b y = 6x - 10 c 3y + 2x = -3

8 a midpoint = (3, 2), length = 4

b midpoint =

-1,-9

, length = 74

22

c midpoint = 5, 5 , length = 5 2

E

9 3y - x = 3 3 - 2 10 y + x = 1 11 3752

Extended-response questions

1a

L S

8

1

220

279 l

P b Since the graph is a line of best fit answers

may vary according to the method used; e.g. if

the two end points are used then the rule is

S=

7 l - 25.1

or

l

=

59 S

+

1481

59

7

7

If a least squares method is used the rule is

l = 8.46S + 211.73.

c

M C

41

33 220

273 l

d Again this is a line of best fit. If the two end

Apoints are used then

C=

8

l

-

11

(or

l

=

53C

-

11 )

53 53

8

A least squares method gives

l = 6.65C + 0.6166.

S2 a C = 110+38n

b 12 days

i For 30 kWh, C = 2.73

ii For 90 kWh, C = 6.87

iii For 300 kWh, C = 16.81

c 389.61 kWh

6

a

y

=

-7x

+

2 14

3

3

1 b 20 km south

3

7 a s = 100 - 7x

b s (%)

100

0 5 c% 7

100 7

14.3

2 d 14 %

7

x (%)

e Probably not a realistic model at this value

of s

f

0

x

2 14

7

8 a AB, y = x + 2; CD, y = 2x - 6

b Intersection is at (8, 10), i.e. on the near bank.

128 9a

b y = - 199 x + 128

19

190 19

c No, since gradient of AB is 20 (1.053), 19

whereas the gradient of V C is ?1.047

10 a No

41 b 1 km to the east of H

71

11 a y = x - 38

b B (56, 18)

c y = -2x + 166 d (78, 10)

12 a L = 3n + 7

bL

61

10

01

18 n

13 a C = 40x + 30 000

b $45

c 5000

d R = 80x

e $'000

c Less than 5 days 3 a Cost of the plug

300

R = 80x

200

b Cost per metre of the cable

c 1.8

1 d 11 m

9

100

C = 40x + 30 000

0 1000 2000 3000 x

4 a The maximum profit (when x = 0)

f 751

g P = 40x - 30 000

b 43 seats

c The profit reduces by $24 for every seat empty.

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710 Essential Mathematical Methods 1 & 2 CAS

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14 a Method 1: Cost = $226.75; Method 2: Cost = $227; Method 1 cheaper

b

47 1 -2

=

x7 1 -2

if x = 4

0 1000 2000 3000 b Method 1 100 181.25 262.50 343.75

c

2 x4 -1 10 3

=

y 04 -1 10 3

Method 2 110 185 260 335

=

20 -1 10

4 3

if x = 0, y = 2

Cost the same for approx. 1600 units

c 110

cost ($) 100 300 200 100

343.75 335

0 1000 2000 3000 units 1600

E d C1 = 0.08125x + 100

(1)

C2 = 0.075x + 110

(2)

x = 1600

15 a (17, 12) b 3y = 2x + 2

16

a

PD:

y

=

2 x

+ 120;

DC:

y

=

2 x

+

136;

L 3

5

CB: y = - 5 x + 600

2

AB: y = 2 x + 20; AP: y = - 3 x + 120

5

5

b At B and C since product of gradients is -1

2

5

P e.g.

m DC

=

, 5

m CB

=

-; 2

2 ? - 5 = -1

52

17 a y = 3x + 2 b (0, 2) c y = 3x - 8

d (2, ?2)

e Area = 10 square units

f Area = 40 square units

Chapter 3

M Exercise 3A

1a 2?2 b 2?3 1 0 0 0 1

A2

a

0 0 0

1 0 1

0 1 0

1 0 1

0 0 0

10001

c 1?4 d 4?1 1 1 1 1 1

b

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

11111

1 0 0 0 0

S3

0 0 0

1 0 0

0 1 0

0 0 1

0 0 0

Only the seats for top-left to bottom-right diagonal

0 0 0 0 1 are occupied.

6 a x = 2, y = 3 c x = 4, y = -3

0310

7

3 1

0 2

2 0

1 1

0110

b x = 3, y = 2

d x = 3, y = -2 21 5 5

8

8 4 14

2 1 8

3 1 60

01 2

Exercise 3B

1X+Y=

4 -2

, 2X =

2 -4

,

4Y + X =

13 -2

,X-Y =

-2 -2

,

-3A =

-3 -6

3 -9

, -3A + B =

1 -7

3 -7

2a

6 12 4 8 42

5 1

c 3

3 7

0 13

6

33

b

51 0 18 7 13

3 2A =

2 0

-2 4

, -3A =

-3 3 0 -6

,

-6A =

-6 6 0 -12

4 a Yes

b Yes

5a

64 -4 -4

b

0 -9 12 3

c

6 -5 8 -1

d

-6 -13 16 7

6a

01 23

b

-2 3 63

-9

7X=

2 0

4 -3

,

Y

=

2 -1

c

33 -1 7

-23

2

11

2

4

200 110

180 135 117 98

110 89

56 28 53 33

8X+Y=

310 200

180 220 0 125

90 0

, representing

the total production at two factories in two

5 a [0 x] = [0 4] if x = 4

successive weeks.

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Answers 711

Exercise 3C

6.00

1 AX =

4 -5

, BX =

4 1

, AY =

-5 8

,

10

8.00 2.00 11.00

represents how much each student spends in a week on magazines.

6.50

IX =

2 -1

, AC =

0 -1 12

,

11 a SC =

s11c1 + s12c2 + s13c3 s21c1 + s22c2 + s23c3

CA =

1 0

-1 1

, (AC)X =

1 0

,

C(BX) =

9 5

, AI =

1 -1

-2 3

,

IB =

32 11

, AB =

10 01

,

BA =

10 01

, A2 =

3 -8 -4 11

,

E B2=

11 8 43

, A(CA) =

1 -3 -1 4

,

A2C =

-2 -5 37

L 2 a AY, CI are defined, YA, XY, X2, XI are not defined.

b AB =

00 00

3 No

4 LX = [7], XL =

4 -2 -6 3

P 5 AB and BA are not defined unless m = n.

6

10 01

7 One possible answer is

A=

12 34

,B =

-2 1 1.5 -0.5

M 8 One possible answer is A =

1 4

2 3

,

B=

01 23

,C =

-1 2 -2 1

,

A(B + C) =

-1 -4

11 24

,

AAB + AC =

-1 -4

11 24

,

(B + C)A =

11 7 16 12

S9

29 8.50

represents John spending 29 minutes

b SC represents the income from car sales for each showroom.

c SC = s11c1 +s12c2 +s13c3 s11u1 +s12u2 +s13u3 s21c1 +s22c2 +s23c3 s21u1 +s22u2 +s23u3

represents the income for each showroom for new car sales and used car sales. d CV gives the profit on each new car and each used car for the three models.

Exercise 3D

1a 1

b

2 -2 -3 2

c2

2a

-1 1 -4 3

1 d

2

22 -3 -2

2 -1

b

7 1

14 3

7 14

10

c

1 0

d

cos sin -sin cos

k

11

3 A-1 = 2 2 , B-1 = 0 -1

10 -3 1

,

1 1

AB =

5 -3

1 -1

,

(AB)-1

=

2 -3

2 -5

,

22

1

A-1B-1 = -1 2 ,

3 -1

11

B-1 A-1

=

2 -3

2 -5

,

(AB)-1

=

B-1 A-1

2

-1

3

4a 2 2

1 -2

5 -7

2

07 b

1 -8

consuming food which cost him $8.50.

29 8.50

22 8.00

12 3.00

John's friends spent

$8.00 and $3.00 and took 22 and 12 minutes respectively to consume their food.

c

2 11

2 -21

2 -3

5 a

8 1

2 11

8 7

-11 17

b

16 -1

16 3

16 16

44

Cambridge University Press ? Uncorrected Sample Pages ? 2008 ? Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

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