mathematics standard level 4eultimate2.shoppepro.com/~ibidcoma/samples/maths... · mathematics –...

807 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

ANSW

ERS

Chap

ter

2

Exer

cise

2.1

2a

[–2,

7]b

]9,

[c ]

0,5]

d ]–

,0]

e ]–4

,8[

f ]–

,–1[]2

,[

3a

b c

4a

4b

4 +

c 6d

31 +

12

5a

2 –

b +

2c

d

e f

6a

i ii

b i

ii

7a

{±3}

b {±

10}

c Ød

{–4,

2}e {

–12,

8}f {

0,4}

9a

]1,

[b

]4,

[c ]

4,6[

11a

b Ex

ercise

2.2

.1

1a

4b

3c –

6d

e f

2a

b c

d e

f

3a

b –3

9c

d –3

e 2f 4

4a

2b –

2b

c d

e a

bf

g

0

h

i a +

b

5a

–4, 4

b c –

6, 1

8d

e f

g

h i –

3, 0

j k

l

Exer

cise

2.2

.2

1a

b c

d e

f

2a

b

c

3a

x <

1b

x <

2 –

ac

d

4a

b c

d e

f g

h i

5a

b c

d

e f

g h

i 6

p <

3Ex

ercise

2.3

.1

1a

b

24

68

102

46

810

–20

24

6

c

d

24

68

10–2

02

46

–8–6

–4

ef

55

3–

3

62

3

37

23

15+

2–3

–4–

52–

15

36

1015

++

+2–

--------

--------

--------

--------

--------

--------

-3

62

15+

35

3+ 2

--------

--------

--------

103

15 2----

--------

-+

143

48+

13----

--------

--------

-----

1344

332

30+

169

--------

--------

--------

--------

-----

–40

4

-50

5–4

–20

24

–5–2

02

5

8a

b

cd

63

+2

22

–

11 2------

–1 10------

3 8---

17 5------

4 3---3 4---

–4 3---

35 2------

92 41------

44 5------

–1 7---

–

b1

b a---+

+ab a

b+

--------

----a

ab

+

a

ba

b–

--------

----

ab

+

a2b2

+----

--------

------

9 5---–3

11 2------

–17 2----

--

7 10------

–1 10------

5 8---

–3 8---

7 5---–

17 5------

4 3---20 3----

--

ab

– 2----

--------

ba

– 2----

--------

ab

b2

ab–

a

b

b a---–

2b a------

b0

x4–

x

1 5---–

x1

x

6–

x18 7----

--

x3 8---

x52 11----

--

x1

x

10 3------

x2

b3

a----

--

x2

a1

+

2----

--------

--------

2x

1

–

2x

3

–

3 2---x

5 2---

–

x1 2---

–=

7x

9

–

5x

3

–

4x

16

–

28x

44

–

5 12------

x1 12------

–

x3 2---

–

x5 2---

x3 2---

x

7 2---

x

12–

x16

x24–

x

6

x3 4---

x

9 4---

6

x14

–x

28–

x44

x5 12------

–

x1 12------

x4–

x

16

1a

bc

de

x

y

1-1

x

y -2

2x

y -3

3/2

x

y

2

2/3

x

y 1/2

-1

SL

Mat

hs 4

e re

prin

t.boo

k P

age

807

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

808

MATH

EMATICS – Standard Level

ANSW

ERS

2a 2

b 3c

3a y = 2x – 1

b y = 3x + 9c y = –x – 1

4a

b c

d –5

y = 2x 6

y = –x + 1 7

82

9a

b c

d

Exercise 2.3.21

a x = 1, y = 2b x = 3, y = 5

c x = –1, y = 2d x = 0, y = 1

e x = –2, y = –3f x = –5, y = 1

2a

b c x = 0, y = 0

d

e f

3a –3

b –5c –1.5

4a m

= 2, a = 8b m

= 10, a = 24c m

= –6, a = 9. 5

a b

c d

e f

Exercise 2.3.31

a b

c

d e

f Exercise 2.4.1

1a –5

b 4, 6c –3, 0

d 1, 3e –6, 3

f g 2

h –3, 6i –6, 1

j

2a –1

b –7, 5c

d –2, 1e –3, 1

f 4, 5

3a

b c

d e

f

4a

b c

d e

f –4, 2

g h

i j no real solutions

k

l no real solutionsm

n

o

5a –2 < p < 2

b p = ±2c p < –2 or p > 2

6a m

= 1b m

< 1c m

> 17

a b

c 8

a b

c Exercise 2.4.21

Graphs are show

n using the ZOO

M4 view

ing window

:

x

y3

–3/4x

y3

3

x

y2

4

x

y

6-2

x

y

2

5

x

y

1

-3

t

p

5-2/3

x

y

-1-4/5t

q2-4

x

y

1

1/2

fg

hi

j

kl

m n o

53 ---

12 ---–

13 ---32 ---

45 ---

yx

2+2

------------=y

52 ---x=

y32 ---

–x

3+

=y

56 ---x12 ---

–=

y2

x–

1+

=

10a

bc

d

x

f(x)

ba ---

–bx

f(x)b–a

2------

b

Note signs!

x

f(x)–a

a+1

x

f(x)2a

2a 2

–

x1311 ------

=y

1711 ------=

x914 ------

=y

314 ------=

x417 ------

=y

2217 ------–

=

x167 ------

–=

y787 ------

=x

542 ------=

y328 ------

–=

x1

=y

ab

–=

x1–

=y

ab

+=

x1a ---

=y

0=

xb

=y

0=

xa

b–

ab

+------------

=y

ab

–a

b+

------------=

xa

=y

ba

2–

=

x4

=y

5–z

1=

=x

0=

y4

z2–

==

x10

=y

7–z

2=

=x

1=

y2

z2–

==

x

2t

1–

=y

tz

t=

=

253 ---

–0

32 ---

25 ---–

3

1–

6

35

1

5

1–

338

------------------------9

73

4-------------------

185

6

-------------------

337

2

-------------------5

33

2-------------------

333

2

-------------------7

57

2-------------------

7–

652

------------------------

1–

22

5

–53

2------------------------

337

2

-------------------4

7

213

2

-------------------3

211

5

----------------------6

31

5-------------------

m2

2

=]

2–2

[

–]2

2

[

]

22

22[

–

k6

2

=]

6–2

[

–]6

2

[

]

62

62

[

–

ab

c

SL

Mat

hs 4

e re

prin

t.boo

k P

age

808

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

809 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

2G

raph

s are

show

n us

ing

the Z

OO

M6

view

ing

win

dow

:

(–2,

0),

(–1,

0),

(0, 2

)(–

2, 0

), (3

, 0),

(0, –

6)(–

0.5,

0),

(3, 0

), (0

, –3)

(–2,

0),

(2, 0

), (0

, –4)

(–2.

79, 0

), (1

.79,

0),

(0, –

5)(–

2, 0

), (3

, 0),

(0, 6

)

(–0.

62, 0

), (1

.62,

0),

(0, 1

)(–

2.5,

0),

(1, 0

), (0

, 5)

(–3,

0),

(0.5

, 0),

(0, –

3)

(3, 0

), (0

, 3)

(–2,

0),

(4, 0

), (0

, 4)

(–0.

87, 0

), (1

.54,

0),

(0, –

4)

3a

x =

1b

(1, –

1)c i

ii

(0, 1

)

4a

x =

1b

(1, 9

)c i

ii (0

, 7)

5a

b c

6a

b c

7a

b

c

8a

b c

d

9a

b c

d

Exer

cise

2.4

.31

a ]–

,–2[

[b

[–3,

2]c ]

–,0

]

[d

],3

[e ]

–,–

1.5]–

[f ]

0.75

,2.5

[

de

f

gh

i

jk

l

ab

c

de

f

gh

i

jk

l

22

2----

--------

----0

1

1

23

2

2----

--------

-------

0

1

7

k9 4---

=k

9 4---

k9 4---

k25 8----

--=

k25 8----

--

k25 8----

--

k1

=1

k1

–k

1k

1

–

y5 12------

x2

–

x

6–

=y

3 8---–

x4

+

2=

y3 4---

x2

–

21

+=

y3

x26

x–

7+

= y2 5---

xx

6–

–=

y3 4---

x3

–

2=

y7 9---

x2

+

23

+=

y7 3---

–x2

2x–

40 3------

+=

1 3---

SL

Mat

hs 4

e re

prin

t.boo

k P

age

809

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

810

MATH


ANSW

ERS

2a ]–

,–2[–

[b ]–2,3[

c ]–,–0.5]

3[

d [–2,2]e ]

[

f ]–,–2]

3[

g []

h [–2.5,1]i ]–

,–3[

[j ]1,3[

k ]–1,0.5[l Ø

m Ø

n [–1.5,5]o ]–

,–2[

[

3a –1 < k < 0

b c n –0.5

4a i ]–

,–1[

[ii [–1,2]

b i ]–,2[

[

ii [2,3]c i ]1,3[ ii ]–

,1]

[d i ]–

,1[ ii ]–,–

]

[ e i ]–

,–2[2

[ii [–2,2]

f i ][

ii ]–,

]

[

5]0,1[

6[–2,0.5]

7a {x: x < –3}

{x: x > 2}b {x: –1 < x < 4}

Exercise 2.4.4

1a (–2, –3) (2, 5)

b (–2, –1) (1, 2)c

d

e f

g h no real solutions

i j (–2, –3), (2, 1)

k no real solutions

2a (1, 4), (–7, 84)

b c (0, 2), (3, 23)

d

e Øf (2, 8)

g Øh

3a

b c

4

5

1.75 7

8

10a i (1, 3),

ii (–2, 12), c i A

(1,3), B(–2, 2)ii 4 sq. units

Chapter 3Exercise 3.1.11

82

4, 0.25 3

8, 18 4

8 and 11 or –8 and –11 5

6, –10 6

2 m

751

8

11, 13; –11, –13 9

25 days 10

30 11

a 30b $50 each.

126

13

16 14

6 15

3 hours 16

9 17

a 15 hrsb 10 hrs

18Chair-one: 20; Chair-tw

o: 2419

a 2 kmb 2.5 km

207.5 hrs, 10.5 hrs

Exercise 3.1.21

a i ii 0 < x < 50 (N

ote: if x = 0 or 50, A = 0 and so there is no enclosure)b i

ii 10 m by 80 m

or 40 m by 20 m

iii 1250 iv 25

m by 50 m

2a ii 0 < x < 12

b i 20 ii 32

iii 32 c

d 6 m by 6 m

3a

b $12900

1–

21–2

-----------------------1

–21

+2------------------------

15

–2----------------

15

+2----------------

13 ---

22

–k

22

23 ---23 ---

23

–2

3+

2

3–

23

+

x

y

1

1

13 ---–

2–

25

32 ---

–154 ------

–

1

0

92 ---–

194 ------–

12–

373

+4

-------------------3

–73

–8-----------------------

373

–4

-------------------3

–73

+8------------------------

113

–2

-------------------1

13–

1

13+

2-------------------

113

+

117

–2

-------------------5

317

–2

----------------------

117

+2

-------------------5

317

+2

----------------------

43 ---569 ------

–

34 ---

74 ---–

a

a2

––

a2 ---

a22 -----

12 ---234 ------

26

m

26

m2

6

–

2

6m

26

–

80

2312 ------–c

am ----=

143 ------–

1963 ---------

73 ---493 ------

kmh

1–

kmh

1–

1002

x–

A2

x50

x–

0x

50

=

m2

m2

m2

m2

0 126

36A

x

0 8

13600

C

t

12000

SL

Mat

hs 4

e re

prin

t.boo

k P

age

810

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

811 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

4a

b i $

9600

00ii

1837

7 or

816

22 (a

s an

swer

mus

t be i

nteg

er v

alue

s)iii

$10

0000

05

a b

i ii

0 <

x <

50c i

ii ; d

imen

sions

50

m b

y m

635

.83

7a

100%

b t =

0.2

29 (f

irst t

ime)

then

agai

n at

t =

13.1

04c

i 42.

26%

ii 1.

73 w

eeks

d A

s ti

me

incr

ease

s, o

xyge

n le

vel w

ill b

e 10

0%.

8a

b $9

5.26

9B(

3254

, 195

3), C

(614

6, 3

687)

uni

ts in

met

res

10a b

c $4.

70d

$1.8

8

11a

0 <

x <

4b

c i

ii3.

25iii

12

a i 2

00 m

ii 32

0 m

b i 0

.34

sec a

nd 1

1.66

sec

ii 11

.31

sc 1

2 s

d 36

0 m

13a

0.5

3 s (

on th

e way

up)

and

9.47

s (o

n th

e way

dow

n)b

10 s

c 500

md

12.0

7 s

e 750

m14

a $7

2500

b N

o. (l

oss o

f $20

000)

c 250

015

a b

i $70

000

ii $3

0000

c Fix

ed co

st (e

.g. s

alar

y, el

ectr

icity

, ...)

d Se

e gra

ph in

par

t ae $

7666

7 (to

nea

rest

dolla

r)f i

ii

$770

00iii

210

0g

i 0

x

580

or 3

620

x

6000

ii 58

1

x

3619

h Se

e gra

ph in

par

t a

Rx

xpx

400.

0004

x–

0x

100

000

==

y4 3---

50x

–

=

A8 3---

x50

x–

=50

00 3----

--------

m2

x25

y20

0 3---------

==

100 3----

-----

kmh

1–

[0, 1

0] b

y [0

, 1]

[0, 2

0] b

y [0

, 1]

usag

e ('1

00 M

J)

Cos

t

20

5

55

10

100

C

0.25

0x

50

0.65

50x

80

1.30

80x

160

1.

9516

0x

320

2.

5032

0x

500

3.

4050

0x

1000

=

50

80

16

0

3

20

50

0

100

0

0.25

0.65

1.30

1.95

2.50

3.40

$C

x (k

m)

Ax

3x

0.25

x20

x4

+

=

0

4

16

x

A

6–

48+

0

6

000

x

7000

0y($

)

Los

s

Los

s

Gai

n

Bre

ak-e

ven

poi

nts

300

000

yC

x

=

yR

x

=

3000

287

192

104

808

580

3

620 P

x

140

x1 30------ x

2–

7000

00

x60

00

–=

SL

Mat

hs 4

e re

prin

t.boo

k P

age

811

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

812

MATH


ANSW

ERS

16a b i

ii 0 x 6000d The com

pany will break even at

399 radios and 5421 radios. Provided the company sells betw

een 399 and 5421 radios they w

ill make a profit.

e 2910Exercise 3.1.31

a ii y = 0.4x + 7.2b ii y = 6 – 2x

c ii y = 0.5x + 3.22

Second difference = 0.643

b y = x2 + 4x + 2

4a &

c b y = 2x

2 – 3x + 7

5y = 2x

2 – x + 36

a c i p = –0.2q + 4ii p = 0.07q + 1.76

d Optim

ium scenario: dem

and = supply. This occurs w

hen p = 1.60, q = 12.7

a y = –0.6333x2 + 8.833x – 19.2

b –40°C at 11 p.m. The m

odel is not valid outside data range, therefore extrapolation w

ill not necessarily work.

8Equation of path:

. Greatest height: 21.17 m

.

9a

b linearc i ii M

= 0, x = 1, i.e. 1 m

10a

bc parabolad

11a

b 12

a i a = 45000, b = 500000ii

b

13a

b i parabolic

ii c i 6.75 m

ii 22.27 m14

a b [BE] :

, [BO

] : y = 5.5x, 0 x 1c

15a

b second difference is constant = –50

c d $22500 per car

e i $824750ii $19750

0 6000

x

72000 y($)

Loss

Loss

Gain

Break

-evenp

oints

yC

x=

yR

x=

3000399 5421

Px

130 ------x2

–194

x72000

–+

=

y31

2400------------x

2–

4948 ------x1

++

=

Px

2x

2–

x3

++

=P

x1

k–

x

2x

kx

k1

++

=k10

k2

4=

=(4, 680

000)

500000

S

t

hx

0.04694x

2–

0.96518x

1.7896+

+=

y29 ---x

2–

119 ------x92 ---

0x

5.5

++

=y

119 ------x–

12118---------

1x

5.5

+=

4936

y0.25

x2

–25

x580

++

=

SL

Mat

hs 4

e re

prin

t.boo

k P

age

812

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

813 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

16a

i & ii

hav

e a co

nsta

nt g

radi

ent

iii re

sults

impl

y qu

adra

tic fo

rmb

ii

, ,

c , m

ax. p

rofit

=

Chap

ter

4Ex

ercise

4.1

.11

a b

c d

e f

g h

i j

k l

m

n o

p q

r

Exer

cise

4.1

.21

a b

c d

e f

g 2

a –1

4000

00b

6000

c 540

d –2

40e 8

1648

f 40

31.

0406

0.0

004%

4a b

1975

0c 2

0.6

d 0.

1%5

196

–

7 8–

9–2

010 11 12 13

a 0

b –5

914 15

Chap

ter

5Ex

ercise

5.1

1a

dom

= {2

, 3, –

2}, r

an =

{4, –

9, 9

}b

dom

= {1

, 2, 3

, 5, 7

, 9},

ran

= {2

, 3, 4

, 6, 8

, 10}

c dom

= {0

, 1},

ran

= {1

, 2}

2a

]1,

[b

[0,

[c ]

9,

[d

]–

, 1]

e [–3

, 3]

f ]–

, [

g ]–

1, 0

]h

[0, 4

]i [

0,

[j [

1, 5

]k

]0, 4

[l

3

a r =

[–1,

[,

d =

[0, 2

[b

, d =

c r

= [0

, [ \

{3},

d =

[–4,

[ \

{0}

d r =

[–2,

0[,

d =

[–1,

2[

e r =

]–

, [ d

=f r

= [–

4,4]

, d =

[0,8

] 4

a on

e to

man

yb

man

y to

one

c man

y to

one

d on

e to

one

e man

y to

man

yf o

ne to

one

5a

\{–2

}b

]–

, 9[

c [–4

,4]

d e

\{0}

f g

\{–1

}h

[–a,

[

i [0,

[ \

{a2 }

j k

l \

6a

]–

,–a[

b ]0

,ab]

c ]

d [

,[

e \{

a}f ]

–,a

[

g [–

a,

[h

]–

, 0[

Exer

cise

5.2

Gra

phs w

ith g

raph

ics c

alcu

lato

r out

put h

ave s

tand

ard

view

ing

win

dow

unl

ess

othe

rwise

stat

ed.

1a

3, 5

b i 2

(x+a

) + 3

ii 2a

c 3

2a

0,

b c

3a

, b

c no

solu

tion

p10

0.00

1x

–=

Cx

2x

7000

+=

Rx

0.00

1x2

–10

x+

=

Px

0.00

1x2

–8

x70

00–

+=

P40

00

90

00=

b22

bcc2

++

a33

a2g

3a

g2g3

++

+1

3y3y

2y3

++

+

1632

x24

x28

x3x4

++

++

824

x24

x28

x3+

++

8x3

48x2

–96

x64

–+

1632 7----

-- x24 49----

-- x2

8 343

---------

++

+x3

124

01----

--------

x4+

8x3

60x2

–15

0x

125

–+

27x3

108

x2–

144x

64–

+27

x324

3x2

–72

9x72

9–

+

8x3

72x2

216x

216

++

+b3

9b2

d27

bd2

27d3

++

+

81x4

216

x3y

216x

2y2

96xy

316

y4+

++

+

x515

x4y

90x3

y227

0x2

y340

5xy

424

3y5

++

++

+12

5

p3---------

150 p----

-----60

p8p

3+

++

16 x4------

32 x------

–24

x28

x5–

x8+

+q5

10q4 p3----

--------

40q3 p6----

--------

80q2 p9----

--------

80q

p12

--------

-32 p1

5----

----+

++

++

x33

x3 x---

1 x3-----

++

+

160x

321

x5y2

448x

3–

810

x4–

216

p420

412p

2q5

–22

680p

– 64x6

960

x560

00x4

2000

0x3

3750

0x2

3750

0x

1562

5+

++

++

+

63 8------

231

16--------- 13

027---------

a3

=n

5=

n9

=

a3

n8

==

a2

b

1=

=

]

1–

1

[

–

ry:

y0

\{4

}=

]

3–

3

[

–

]

2–

2

[

–

]

a–

a

[

–

a1–

–

]

1 4---a3

–

1 4---a3

10 11------

5 4---–0

10 11------

1 2---–

x2x

–3 2---

+1 2---

–x2

x3 2---

++

2

4a

x =

0, 1

b

Win

dow

[–2,

2], [

–1,1

] Ra

nge:

[–12

, 4]

5a

iii

SL

Mat

hs 4

e re

prin

t.boo

k P

age

813

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

814

MATH


ANSW

ERS

b i ii {3, –2}

6b, c, d, e

8a, d, e, f

9a

Window

[–2,2], [–1,1]b [0, 1[

10a {y: y > 1}

{y: y –1.25}b 10

11b 1

12a only – it is the only one w

ith identical rules and domains

13a [–3,

[b [–3,0]

c [3,[

d [1.5,3[ ]3,

[14

a i ii

b i ii

Exercise 5.3.1

22

22

–

px

82

16x

2–

+0

x4

=

Ax

x16

x2

–0

x4

=

x

y

168

r = ]8,16[

4x

y8r =

]0,8]

4

x

y

3

–1 1

x

y2

x

y

(4, 2)

6

x

y2

1 a b c d

x

y

2

–2

2

2 a b

x

y

1–1

x

y1

–1

x

y

1

x

y

1

x

y2

–3

3

4

3 a b c d

4 a b

x

y

2–2

4

x

y

2–2

4

–4

5 a b c d

x

y

1

a

x

y3

–2ax

y

2

a = 4

4

x

y3

a = 1

6 a b c d

x

ya =

4

a(1, 5)

x

y

2

a = 4

4

x

y3

a = 1

x

y1a

2=

a2

–=

7 a b

x

ya–a

x

y

4a =

4

SL

Mat

hs 4

e re

prin

t.boo

k P

age

814

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

815 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

Exer

cise

5.3

.21

a

b

c

x

y

x

y 2x

y

–1

x

y

0.5

x

y 1

d

e

f

x

y

–4

g

h

i

x

y

3x

y

10

5

x

y

4

2

2 a

b

c

x

y

3–3

d

e

fx

y

2x

y

1–2

x

y

3x

y

2

–1x

y

8

g

h

i

x

y

2

8

x

y

2–2

8

x

y

(1, –

1)

3 a

b

c

x

y

2

1–1

[2,

[x

y

2–2

4[4

, [

d

e

f

x

y

[0,

[

x

y

]–

,0]

x

y

2–2

x

y

1–1

4 a

b

c

d

x

y

x

y 1

–1

x

y

–1x

y

1

1

5 a

b

c

x

y

2–2

x

y 7x

y

2

–8

SL

Mat

hs 4

e re

prin

t.boo

k P

age

815

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

816

MATH


ANSW

ERS

Exercise 5.3.3

3‘b’ has a dilation effect on f (x) = a x (along the y axis).

d e f

x

y

4

–4

x

y

1

–1

x

y2

g h i

x

y

4–4

x

y

1–1

–1

x

y

2–2

2

6 a b

x

y

4–4

8

x

y

1–1

1

c d

x

y

1–1

–1 2

x

y

1–1

–1

7 a

x

y

2–2

b i Ø

ii [–2,2]iii {±4}

8 a

x

y

3

1–3

x: x3–

x: x

1

1 a b c d

x

y1]0,

[

(1, 4)

x

y1]0,

[

(1, 3)

x

y1]0,

[

(1, 5)

x

y1]0,

[

(1, 2.5)

x

y1]0,

[

(1, 1.8)

x

y1]0,

[

(–1, 2)

x

y1]0,

[

(–1, 3)

x

y1]0,

[

(1, 3.2)

e f g h

i j k l

x

y1]0,

[

(–1, 5)

x

y1]0,

[

(–1, 4/3)

x

y1]0,

[

(–1, 8/5)

x

y1]0,

[

(–1, 10/7)

2 a b

x

y

–1 2

x

y

0.5

1.5

SL

Mat

hs 4

e re

prin

t.boo

k P

age

816

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

817 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

5a

[1,1

6]b

[3,2

7]c [

0.25

,16]

d [0

.5,4

]e [

0.12

5,0.

25]

f [0.

1,10

]

8a

]2, 2

+ e–1

[b

[–1,

1[

c [1

– e,

1 +

e–1]

9a

b 2

4 a

b

c

d

x

y

(–1,

3)

(1, 3

)

x

y

(–1,

5)

(1, 5

)

x

y

(–1,

10)

(1, 1

0)

x

y

(–1,

3)

(1, 3

)

6 a

b

c

d

x

y 1

]1,

[

x

y

]–

,3[

3

x

y

]–

,e[

e

x

y 2

]2,

[

7 a

–1. 5

b

c

i f =

g: x

= 1

ii f >

g: x

< 1

x

yf

g

x

y

1

10 a

b

c

x

y

1x

y

1

x

y

2

1

11 a

b

c

x

y 1

x

y

2–2

x

y

23

12 a

b

c

d

x

y

(1,2

)

(1,3

)

]0,2

[{3

}

x

y 23

]–

,3]

x

y(1

,1)

–1

]–

,1]

x

y

12–1

2

3

1

[0,1

1/3]

13 a

b

c

d

x

y 2[2

,[

x

y 2

[2,

[

x

y

x

y

[0,

[

14 a

b

c

x

y

(a, 1

)x

y

–a

x

y

–2a

x

y

–2a

d

e

f

x

ya

x

y –a

SL

Mat

hs 4

e re

prin

t.boo

k P

age

817

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

818

MATH


ANSW

ERS

Exercise 5.3.4

15 a b

x

y

423

x

y

a

16 a b c

x

y1]0,1]

d e f x

y1[1,

[x

y1

]0,[

x

y

1 2

[1,2[x

–2

]–,–2[

]0,[

y

x

y[0,

[

a > 1a <

1]0,

[

a = 1

{1}

1 a b c d

x

y

23]2,

[

x

y

–2

]–3,[

–3

x

y

]0,[

x

y27

e f g h

x

y

2]–

,2[x

y

]0,[

x

y

10 ]0,[

x

y

0.5

]0.5,[

2 a b c y

x

]0, [

x

y

105

]0,[

d e f

x

y

1]1,

[

x

y

1

]–,1[

x

y

–2/3]–2/3,

[

x

y

2]2,

[

3 a b c y

x

]0,[

y

x

]0,[

1

x

y

]e, [

e

ee

d e f

x

y

1/e]–

,1/e[

x

y

]0,[

e5

1]0, [

x

y

4 a b c

1]0, [

x

yy

x1

\{0}

y

x1

\{0}

y

x

]0,[

1

x

y]–1,1[

–11

d e f

x

y

2–2

]–,–2[

]2,[

SL

Mat

hs 4

e re

prin

t.boo

k P

age

818

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

819 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

5 a

b

c

x

y

1

]0,

[

x

y

2]1,

[

x

y

e

]0,

[

x

y(1

, 2)

]0,

[

d

e

f

x

y

–2

–1–3

\{–2

}

y

x1\{

0}

–1

6 a

b

x

y

1–1

i ii

x

y

x

y

1/e

c

d

x

y

2–2

7 a

b

0 <

x <

~ 4

.3

x

y

4

8 a

b

c

d

x

y

1

[0,

[

x

y

1–1

x

y

]–

,1]

x

y

1

1

[1,

[

9 a

b

c

y

x1

y

x3

2

y

x3

10 a

b

c

x

y

aa+

1

]a,

[

x

y

e a---]e

/a,

[x

y

10 a------

]–

,10/

a[

d

e

f

x

y

ae

\{ae

}x

y

ae

\{ae

}

x

y

a

]–

,a[

11 a

y

x

2/a

1/a

x: 1 a---

x1

1 a---+

12

x

y

]–

,0[

[e,

[

SL

Mat

hs 4

e re

prin

t.boo

k P

age

819

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

820

MATH


ANSW

ERS

Exercise 5.3.5

Exercise 5.4.11

a i [0,

[ ii

[1,[

iii ,

b i

ii

iii 2

a i ]–1,

[ ii

]–0.25,[

iii , [–4,4]

b i

ii

iii

3a i

, ii ]–

, [, ]–

, [

b i ,

ii [1, [, [1,

[

c i ,

ii [0, [, [–2,

[d i

, ii

, e i

, ii [0,

[, [0, [

f i , gof(x) does not exist.

ii ]–1, [

g i ,

ii ]0, [, ]0,

[

e f g h

y

1/2–2

xx

y

2–1

y

x

3

–1/3x

2

1.5

y

x

y

34/3

x

y

1–3/2

x

y

3–1

–1.5

0.5x

y

–2

–3

1 a b c d

2 a b c d

x

y1

x

y

–4

–1

x

y

2–2

4

x

y2

1

–2

3 a

2b

1=

=

x

y3

–1–1.5

4 a b c

x

y

1

iii

–1x

y

2–2

2

4–4

x

y

1

–1

2

d e f

x

y

2–2

x

y

1/2–1/2

x

y2

–1

fg

: 0

where

+

fg

+

x

x2

x+

=

fg

: 0

where

+f

g+

x1x ---

xln

+=

fg

: 3–

2–

23

w

here

+f

g+

x9

x2

–x

24

–+

=5

10

fg:

0

where

fg

x

x2

xx

52/

==

fg: 0

w

here

fg

x

xlnx-------------

=

fg:

3–2–

23

w

here

fg

x

9x

2–

x2

4–

=

fg–

: –

w

here

f

g–

x

2e

x1

–=

fg–:

1–

where

fg–

xx

1+

x1

+–

=

fg–: ]

–

[ where

f

g–

x

x2

–x

2+

–=

f/g: \

0

w

here

f/g

x

ex

1e

x–

--------------=

f/g: ]1–

where

f/g

x

x1

+=

f/g: \{–2} w

here f/g

x

x2

–x

2+

------------=

fog

xx 3

1+

=go

fx

x1

+

3=

fog

xx

1+

=x

0

go

fx

x 21

+=

fog

xx 2

=go

fx

x2

+

22

–=

fog

xx

x0

=g

of

xx

x0

=\{0}

\{0}

fog

xx

x0

=g

of

xx

=

fogx

1x 2 -----1

–=

x0

fog

xx 2

x0

=g

of

xx 2

x0

=

SL

Mat

hs 4

e re

prin

t.boo

k P

age

820

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

821 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

h i

, ii

[–4,

[,

[0,

[

i i

, ii

[–2,

[,

[0,

[j i

fog(

x) d

oes n

ot ex

ist,

ii [0

, [

k i

, ii

[0,1

[, [0

, [

l i

, ii

[1,

[,

m i

, ii

[1,

[, ]0

, [

n i f

og(x

) doe

s not

exist

, ii

o i f

og(x

) doe

s not

exist

, ii

]1,

[

p i

, ii

[1,

[, ]0

, [

4a

b c

5

6

a , ]

–,–

1] [3

,[

b go

f(x) d

oes n

ot ex

ist.

c , ]

– ,–

2.5]

[2

.5,

[

7a

9b

39

a b

10a

b

11

12a

b 13

a ra

nge =

]0,

[

b (=

x) r

ange

= ]–

,[

c ra

nge =

]e–1

,[

14a

hok

does

not

exist

.b

,

15a

S =

\]–3

,3[;

T =

b ; S

= ]–

,–3]

[3

,[

16 17a

Dom

f =

]0,

[, ra

n f =

]e,

[, D

om g

= ]0

,[,

ran

g =

b fo

g doe

s not

exist

:

go

f ex

ists a

s

c 18

; ran

ge =

[0,

[

23a

fog

x

x4

–=

go

fx

x4

–=

fog

x

x2

+3

2–

=g

of

x

x3=

go

fx

4x

–

x

4

=

fog

x

x2

x21

+----

--------

--=

go

fx

xx

1+

--------

----

2x

1–

=

fog

x

x2x

1+

+=

go

fx

x2x

1+

+=

0.75

[

fog

x

2x2

=g

of

x

22x

= go

fx

1x

1+

--------

----1

–x

1–

=

\{–1

}

go

fx

4x

1–

--------

---1

+=

fog

x

4x

=x

0

g

of

x

40.5

x=

fog

x

2x

3+

=x

g

of

x

2x

2+

=x

fo

fx

4x

3+

=x

gx

x21

x

+

=

fog

x

1 x---x

1x

\0

+

+=

go

gx

x1 x---

x

x21

+----

--------

--x

0

+

+=

x1

=x

13–

=1 x---

x–2

x1

+----

--------

---

ho

fx

x1

–

24

+x

2

5

x–

x2

=

(2, 3

)

y

x

rang

e =

]3,

[

4

r fd g

a

nd r

go

fd h

g

x

4x

1+

2

x

=

fog

x

xx

]0,

[

=

gof

x

1 2---e2

x1

–

1

+ln

x

=

fof

x

e2e2

x1

–

1

–x

=

koh

x

44x

1–

1x

1 4---

–

log

=

Tx

: x

6x

0

=

=

0

1

2 3

4

5

6 7

4 3 2 1

yfo

gx

=

5y

x

gof d

oes

not e

xist

0

1

2 3

4

5

6 7

8

9

4 3 2 15y

x

yg

of

x

=

yfo

gx

=

x

y

1+ln

2

r gd f

]0

,[

==

r f]e

[

d g

]0

,[

==

gof

: ]0,

[

, w

here

go

fx

x1

+

2

ln+

=

fog

x

xx

=

19 a

c

x

y

(1, 1

)

x

y

fof

x

x=

dom

=ra

n =

]0,

[

20 a

x

y 1

1

f

g

(1, 1

) in

f, n

ot g

b d g

of:

]1,

[

, w

here

go

fx

x=

fog

*: ]

1,

[

, w

here

go

fx

x=

rang

e =

]1,

[

d f

=\

a c---

r f

\=

a c---

r fd f

fof

x

x=

b c ,

rang

e =

d fog

2–

a2

a

=fo

g

2a

x2 a-----

–=

d go

f21

4/–

a21

4/a

=

fog

1 a---

2a4

x4–

=

02

a

x

y 2a

2a

2–

a

SL

Mat

hs 4

e re

prin

t.boo

k P

age

821

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

822

MATH


ANSW

ERS

Exercise 5.4.2

1a

b c

d

e f

g h

6a

b

c d

e f

8a

b

c d

e f

12 ---x

1–

x

x

3x

3

x3

+

x

52 ---

x2

–

x

x2

1–

x0

x1

–

2x

1

1x ---

1–

x0

1

x1

+

2-------------------

x1–

1

012 ---

–

x

y

x

y9

–3x

y

2

–5 y

x

1–1 y

x

1

y

x

e f g h

1

–1x

y

–1

(0, 1)

x

y

2 a b c d

3 a b x

3+

x3–

x3

+–

x3–

–3

03

x

y–30

3–

x

y

4 x

1x

2–

------------------1

x1

–

5 a b c d

x

y

1/2

–1

x

y

1

–1x

y4

4

x

y

2

4

x

y

–1

x

y

23

x

y

2

(4,–2)

e f g h

x

y

24

(8,2)

f1–

xx

1–

3x

1

log

=f

1–x

x5

+

2

x5–

log=

f1–

x12 ---

x1

–3

log

x

0

=

g1–

x1

3x

–

10

x3

log+

=

h1–

x1

2x ---+

3x

\[–2,0]

log

=g

1–x

1x

1+

------------

2

x1–

log=

7 a b c

x

y

21

1

inverse

inversex

y–4

–5

–4–5

x

y

3

inverse

inverse

x

y

inverse

d e f

x

y

(1, 1)

(–2, –2)

inverse

x

y–1

f1–

x2

x1

x

–

=f

1–x

12 ---10

x

x

=

f1–

x2

1x

–,x

=f

1–x

3x

1+

1+

x

=

f1–

x5

x2/

5+

x

=

f1–

x1

103

2x

–

–

x

=

9 f

1–x

1–

x1

++

x1–

=

x

y

(–1, –1)

dom =

[–1, [, ran =

[–1, [

10 a b

c f

1–x

ax

–=

f1–

x2

xa

–-----------

a+

=f

1–x

a2

x2

–=

SL

Mat

hs 4

e re

prin

t.boo

k P

age

822

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

823 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

12[2

, [

13\{

1.5}

18go

f exi

sts as

. I

t is o

ne:o

ne so

the i

nver

se ex

ists:

19a

i ii

b

i

ii

c i &

ii n

eith

er ex

ist

d A

djus

ting

dom

ains

so th

at th

e fun

ctio

ns in

par

t c ex

ist, w

e hav

e:

a

nd

e Yes

as

rule

s of

com

posi

tion

OK

.

x

y

(1, 1

)

f1–

x

2x

–3

=11

+

14 a

Inve

rse

exis

ts a

s f i

s on

e:on

e

x

y

1

b C

ase

1: S

= ]

0,

[

Cas

e 2:

S =

]–

, 0[

g1–

x

xx2

4+

+2

--------

--------

--------

----=

x

y

–1g

1–x

xx2

4+

–2

--------

--------

--------

---=

15f

1–x

ax2

1+

x0

=

x

y

a

a

x: f

x

f1–

x

=

=

16 a

b

f1–

x

2x

1+

–

x1–

x3

–

x1–

= x

y

1

1f

1–x

x

ln1

–

0x

e

x

e–

xe

= x

y

c

d

f1–

x

1e

x–+

x

0

2x

–

x0

= x

y

2

f1–

x

x4

–

2

x4

x

4–

0

x4

= x

y 4

4

–4

–4

17 a

b

x

y

2–2

aa+

1

f 1/f

x

y

a+1

a

f1–

:

, f

1–x

aea

x+

=

x

y

(5, 4

)

6

6

r fd g

19 a

b

i

i

i

x

y

x

y 1

–11

–1

f is

one:

onef

x

1 2---x

1–

x1–

x3 –

1 2---x

1+

1x

1

–

x1

=

x

y

1

1 –1

iii

i

v {–

1, 0

, 1}

tom

x

ex

x0

=m

otx

exx

=

tom

1–x

x

ln2

x1

=

mot

1–x

x2ln

x0

=

t1–

om

1–x

mot

1–x

=m

1–o

t1–

x

tom

1–x

=

SL

Mat

hs 4

e re

prin

t.boo

k P

age

823

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

824

MATH


ANSW

ERS

20a 1

b 0.206

21a

b fog exists but is not one:one

c i B =

[ln2, [

ii iii

Chapter 6

Exercise 6.11

a b

c d

e f

g h

i j

k l

x

y

0.25

x

y

x

y

1 (tom) –1

(mot) –1

x

y

–2

f

g

fog

1–: [0,[ w

here, fog

1–

xx

2+

ln=

x

y

ln2

ln2

yx

4–

2

=y

x2

+

2=

yx

25

+=

x2

–

2y

+2

=

x2

y+

4=

x2

y+

0=

y8

x4

–-----------

x4

=y

8x ---1

–x

0

=

x1

+

2y

2+

4=

y2

9x

3–

-----------x

3

=

y3

+

29x ---

x0

=x

y2

+8

=

–2 –1 1 2 3 4 5 6

4321

y

x

y

x–5 –4 –3 –2 –1 1 2 3 4 5

4321

–1–2

a i b i

–2 –1 1 2 3 4 5 6

4321

y

x

y

x–5 –4 –3 –2 –1 1 2 3 4 5

4321

–1–2

a ii b ii

2

–2 –1 1 2 3 4 5 6

1–1–2

yx

y

x–5 –4 –3 –2 –1 1 2 3 4 5

–1–2

a iii b iii

–2 –1 1 2 3 4 5 6

4321

y

x

y

x–5 –4 –3 –2 –1 1 2 3 4 5

4321

–1–2

a iv b iv

3 a b c

x

y

4

x

y

4

–2x

y(2,3) (6,5)

4 a b c

x

y

–1x

y–4

x

y

–2

–3

5 a b c

x

y–1x

y

1x

y

1.5

2

SL

Mat

hs 4

e re

prin

t.boo

k P

age

824

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

825 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

8a

b c

d e

f g

h i

j k

l m

n

o

9a

b c

6 F

irst

fun

ctio

n in

bla

ck, s

econ

d fu

nctio

n in

mar

oon

x

y 2–2

4x

y

5

–5

a

b

c

x

y

2

–2

x

y

1.5

–1.5

2.25

d

e

f

x

y

8–8

x

y 1–1

g

h

i

x

y

2

–2

x

y 3

–3

x

y 2

–2

4

j

x

y 2

–4

4

x

y

(2, 3

)

7

a

b

c

d

x

y 2

–1

3

x

y

2

(1, –

1)–2

x

y

2

0.5

e

f

g

h

x

y

(–2,

–8)

x

y

(–3,

–9)

x

y (2, 2

)x

y

4

(–4,

2)

x

y

–2

(–1,

–1)

m

n

o

x

y 4/3

–4–3

x

y

1

–1

x

y

2

–2

x

y 2

3x

y

(2, 8

)x

y

(4, –

2)8

i

j

k

l

7 N

ote:

coor

dina

tes w

ere a

sked

for.

We h

ave l

abell

ed m

ost o

f the

se w

ith si

ngle

num

bers

.

0 4

0 2–

1– 0

2 0

2– 0

0 4–

2 2–

2– 3

4 2

2 3

3 1–

k– h

2– 4

1 1–

1– 2

gx

fx

1–

1+

=g

x

fx

2+

4–

=g

x

fx

2–

=

SL

Mat

hs 4

e re

prin

t.boo

k P

age

825

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

826

MATH


ANSW

ERS

d e

11

Exercise 6.2g

xf

x1

–

1

+=

gx

fx

1–

3+

=

10 a i ii iii y

x

yx

y

x

y

x

(–0.5, 0)

(2, –4)(3, 3)

(–1, 1)–2 –1 0

–3 y

x–3

–5

y

x

4

2

y

x

–2 –1 0 1 2 3

1

–1

y

x

2

3

b i ii

(–2, –2)

iv

iii iv c i

ii

y

x

–4

y

x

–6

–2

iii iv y

x

1

x

–1

y

y

x

2

(–4, 4)y

x

7

(1, 9)

x

5

(–3, 7)y

x

2

d i ii iii

iv yf

x2

+

2

3x

1–

–

+

fx

4+

25

x3–

–+

=

1 a b c d

x

yy

x

y

x

y

x

y

2 a b c d

x

yy

x

y

x

y

x

y

–2 –1 1 2 3 4 5 6

4321

y

x

y

x–5 –4 –3 –2 –1 1 2 3 4 5

4321

–1–2

3a i b i

–2 –1 1 2 3 4 5 6

4321

y

x

y

x–5 –4 –3 –2 –1 1 2 3 4 5

4321

–1

ii ii

–2 –1 1 2 3 4 5 6

4321

y

x

y

x–5 –4 –3 –2 –1 1 2 3 4 5

4321

–1

iii iii–2

SL

Mat

hs 4

e re

prin

t.boo

k P

age

826

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

827 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

5a

b

c d

e f

–2 –

1

1

2

3

4

5

6

4 3 2 1

y

x

y

x–5

–4

–3

–2

–1

1

2

3

4

5

4 3 2 1

–1 –2 –3–4

iv

iv

i

ii2

–2

9 2---–

y

x–3

0

3

1

–1y

x

y

x

(0, 3

) (3, 0

)

–2

2

(–3,

6)

y

x

iii

i

v

yf

x

=

yf

x

=

yf

2 3---x

=

9 2---9 2---

4 a

i

iiy

x–3

0

3

4

–4y

x

y

x

(0, 1

2)

(3, 0

)

–8

8

(–2,

24)

y

x

iii

iv

yf

x

=

yf

x

=

9 2---

y4

fx

=

–3

0

3

b8

–8

fx

x=

yf

2x

1+

=f

x

x2=

y1 2---

fx

2–

3–

=

fx

1 x---=

y1 2---

fx

1 2---–

=f

x

x3=

y27

fx

2 3---–

=

fx

x4=

y12

8f

x1 2---

–

2

–=

fx

x=

y2

fx

2+

=

–6

–5 –

4 –3

–2

–1

0

1

2

3

4

5

(0, –

5)

(0, 7

)

x

x

(0.5

, –2)

(0.5

, 4)

x

(–2,

3)

(4,–

1)

(–6,

–5)

(–1,

4)

(–5,

–4)

(1, 8

)

5

(–1,

–4)

(1, –

4)

y

2.5

–2.5

(–0.

5, 2

)

yy

6 a

b

(0.5

, –2)

(0.5

, 4)

x

y

2.5

–2.5

(–0.

5, 2

)

c

d

7f

x

x2

+

2

if

x0

4

x

if x

0

–

=

a

b

c

x

y 4

hx

x23

–

i

f x

2

3x

if

x2

–

=

x

y

(2, 1

)3

hx

2x2

if

x2

12

2x

if

x2

–

=

x

y

(2, 8

)

d

e

f

kx

4x2

if

x1

6

2x

if

x1

–

=

kx

2x

1–

2

if x

3 2---

72

x

if x

3 2---

–

=f

x

1 2---4

x2

+

2

if

x0

22

x

if x

0

–

=

x

y (1, 4

)

x

y (1.5

, 4)

x

y

(0, 4

)

SL

Mat

hs 4

e re

prin

t.boo

k P

age

827

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

828

MATH


ANSW

ERS

10Exercise 6.3

8

x

y

x

y

x

y

(4, 0)

(2, 2)

(2, –2)

(2, 1)

(4, 0)(8, 0)

(4, 2)y

12 ---fx

=y

f12 ---x

=

yf

x=

ab

x

y

x

y

(1, a)(1/a, 1)

9 a b

x

y

x

y

(–b, 0)

(1, 1)b

b

(1, –1)

1a 2 ------1a ---

1a 2 ------1a ---

a < 0

a > 0

c d

x

y

x

y

b–

a----------

0

ba -------

0

ba ---

0

ba ---

–0

ab ---–

a–

ab

1 a i ii

–3 0 3

2

–2

32 ---–

32 ---

y

x–3 0 3

2

–2

32 ---–

32 ---

y

x

b i ii

–2 –1 0 1 2 3

1

–1 y

x–3

–2 –1 0 1 2 3

1

–1 y

x

c i iiy

x

(0, 3)

(–2, 0)

–2

y

x(0, –3)

2

–2 –1 0 1 2 3

2 y

x

1

(–1, 2)

(1, –1)

d i ii

2

(2,6)y

x

e i ii

–2 –1 0 1 2 3

2 y

x1

(1, –2)

(–1, 1)

–2

(–2, –6)

y

x

SL

Mat

hs 4

e re

prin

t.boo

k P

age

828

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

829 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

2a

b c

d e

3

ab

c

d

ef

gh

ij

kl

mn

op

qr

4a

b

cd

ef

gh

ij

5a

i ii

6b

i ii

3

y

x–2

–11

f i

ii

–3y2

1–1

yf–

x

=y

fx–

=y

fx

1+

=y

f2

x

=

y2

fx

=

x

y9

3–

0

30

x

y4

22

0

22

–0

x

y

21

x

y(–

2, 3

)

–1

x

y

x =

1

2

x

y

–2

1x

y (2, –

2)

6

x

y

2

x

y4

2 2-------

2 2-------

–

x

y 11

3

x

y1

–1

x

y

9

12

xy –2

0.5

x

y

8

2 2-------

x

y

2

2x

y(2

, 2)

3

x

y

(2, 4

)

–4

x

y 14

x

y(2

, 2)

(0, –

2)

3–1

x

y(1

, 1)

(–1,

–3)

2–2

x

y(0

, 2)

(–2,

–2)

1–3

x

y

(1, –

1)

(–1,

3)

2–2

x

y

(1, –

1)

(–1,

3)

2–2

1

x

y(1

, 4)

(–1,

0)

2(–

2, 2

)(2

, 2)

x

y

(1, –

1)

(–1,

1)

2–2

x

y

(0.5

, 2)

(–0.

5, –

2)

1–1

x

y

(2, 0

)

(–2,

4)

(4, 2

)

(–4,

2)

x

y

(2, –

4)

(3, –

2)(–

1, –

2)

x

y b

1 ab

------

–0

x

y

a

0b a---

xy

–a

x

y

–a

b a----

---b a

-------

–

SL

Mat

hs 4

e re

prin

t.boo

k P

age

829

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

830

MATH


ANSW

ERS

c i ii

7a

b c

d e

f

Exercise 6.41

a b

c

d e

f

2a

b c

d e

f

g h

i

j k

l

3a

b

4Exercise 6.5 Miscellaneous questions

1a i

ii

x

y

a

a2

x

ya

–a2

x

y(3, 2)

x

y

(–3, 2)

x

y2(2, 2)

(1, 3)

5

(–6, 2)(–2, 2)

(–4, 3)

x

y

x

y5(1, 2)

x

y1

(1, –2)

x

y

3–3

0.5–0.5x

y

–2–1 1

–11

23

x

y

2–0.5

13 ---

x

y

1/2x

y

3–2

–1–1

1

x

y

2

13 ---

1

23 ---

y

4

–2

x2 1

x

y

–2–4

–4

x

y2

1/2

x

y

–1

x

y

1x

y

1–1

x

y

21

3x

y

1

10.5 2

x

y1

–1

x

y2(1,1)

1/2

0.5

x

y

2–2

4

1x

y

23

x

y

13

–1

x

y

13

–2

x

y

35

–4

x

y

2–1

1

(1, –2)x

y

2(1, –0.5)

SL

Mat

hs 4

e re

prin

t.boo

k P

age

830

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

831 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

b i

ii

c

i ii

2a

b

c d

3 a

b

i ii

c i

ii iii

iv

v vi

4a

b c

5a

b

c i

ii

6a

b i

ii

7 8a

b

x

y

1–1

(0, –

4)2

x

y

1(0

, –1)

–1

x

y

1/2

–1/2

(0, –

4)2

x

y

1/2

(0, –

1)–1

/2

x

y

24

x

y

3

x

y

3

1

x

y

3

2(3

, 1)

x

y 1

fg

x

y 1

fg

x

y

1

x

y1

x

y

x

y1

x

y

x

y1

x

y

x

y

21

3

2

x

21

3

y

–3–2

0.5

x2

13

y

1

x

y

a

ax

y

a

1

x

y

a–a

x

ya

–a

x

y

2–2

12

12–

x

y

2–2

1212

–x

y

1212

–

x

y

aa+

1

x

yf

gx

yf

g–a

a

SL

Mat

hs 4

e re

prin

t.boo

k P

age

831

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

832

MATH


ANSW

ERS


1a

b c

d e

f g

h i

2a

b c)

d e

f

3a

b c

d 9e

f g

h i

4

5a

b c

d e

f

6a

b c

d e

f

7a

b 1c

d e

f g

h

8a

b c

d e

f

Exercise 7.1.2

1a 2

b –2c

d 5e 6

f –2.5g 2

h 1.25i

2a –6

b –c –3

d 1.5e 0.25

f 0.25g

h i –1.25

Exercise 7.1.3

1a 3.5

b 3.5c –3

d 1.5e 3.5

f 1.5g 1.8

h i 0

2a –0.75

b –1, 4c 0, 1

d 3, 4e –1, 4

f 0, 2

3a –1, 1, 2

b –3, 1, 3, 4c

, , 2

d –1, 1, 2e 3, 7,

,

Exercise 7.1.41

a i 5.32ii 9.99

iii 2.58b i 2.26

ii 3.99iii 5.66

c i 3.32ii –4.32

iii –6.32d i –1.43

ii 1.68iii –2.86

2a 0

b 0.54c –0.21

d–0.75, 0e 1.13

f 0, 0.16Exercise 7.1.51

a 2b –1

c 0.5d 0.5

2a 1

b 0.6c 0

3a 0

b

4a –1, 2

b –2, 3c –1

d –6, 1e 0, 1

f 1 5

a 1.3863b 2.1972

c 3.2189d Ø

6

a 0.4236b 0.4055

c 0.3054d –0.4176

7a 0

b –0.6733c 0

99. 36

10

Exercise 7.21

a 1000b 1516

c 2000d 10 days

2a 0.0013

b 2.061 kgc 231.56 years

d

3a 0.01398

b 52.53%c 51.53 m

d 21.53 me

4a i 157

ii 165iii 191

b 14.2 yearsc 20.1 years

d

5a 50

b 0.0222c 17.99 kg

d e

6a 15 000°C

b i 11 900°Cii 1500°C

c 3.01 million years

d

27y

15

8x

3--------------

91

216a

6---------------

2n

2+

8x

11

27y

2------------

3x

2y2

8---------------

3n

1+

3+

4n

1+

4–

24

n1

+4

–

1

b6

–

16b

4------------

6423 ---

x2

2y

1+

1b2

x--------

y2 ---

692 ---

n2

+

z 2xy -----3

7n

2–

5n

1+

26

n1

+2

13

n–

x2

4n

n2

–+

x3

n2

n1

++

27

y2

m2

–

xm

-----------------

81–

9x

8

8y

4---------

–y

x–

2x

1+

x1

+---------------

1–b–

1

x2y

2-----------

1x4

-----1

xx

h+

--------------------–

1x

1–

-----------1

x1

+

x

1–

5

------------------------------------1x2

-----

1185

n2

–

b7

a4

-----a

mn

pq

+pq

------------2

aa

1–

------------78 ---

a7

8/

x11

12/

2a

3n

2–

b2

n2

–2

n7

mn

–8--------------

–6

5n

5

n5

+---------------

x1

+

23 ---13 ---

23 ---18 ---

–114 ------

–47 ---–

43 ---53 ---

1–

2332

---------------------------13 ---

23 ---

a2

ek

2

ln

==

W

t

2

x

I0

I

N

t

150

50

W

t

50W

t

150 000T

t

SL

Mat

hs 4

e re

prin

t.boo

k P

age

832

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

833 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

7a

0.01

51b

12.5

0 gm

c 20

year

sd

8a

$2 m

illio

nb

$1.5

89 m

illio

nc 3

0.1

year

sd

9b

0.01

761

c 199

230

d 22

.6 y

ears

10a

20 c

m2

b 19

.72

cm2

c 10

0 da

ysd

332

days

11a

1b

i 512

170

ii 51

7217

c 54.

1 ea

rly

2014

12a

i $93

3.55

ii $9

35.5

0b

11.9

5 ye

ars

c

13a

99b

c 684

14a

b 38

.85°

C a

t ~ m

idni

ght

15a

19b

2.63

c 10

016

a 18

cm

b 4

cmc 1

.28

md

36 m

e i 2

1.7

year

sii

27.6

yea

rsiii

34.

5 ye

ars

f 36

g

17a

5 m

g/m

inb

13.5

1 m

inc i

2.1

ii 13

.9iii

68

min

d 19

.6 m

ge

f No

18a

i $49

9ii

$496

iii $

467

c 155

37d

i $49

9000

ii $2

.48

mill

ion

iii $

4.67

mill

ion

f 123

58g

$5.1

4 m

illio

nb

& e

Exer

cise

7.3

1a

2b

2c 5

d 3

e –3

f –2

g 0

h 0

i –1

j –2

k 0.

5l –

2 2

a b

c d

e f

3a

b c

d e

f

4a

16b

2c 2

d 9

e f 1

25g

4h

9i

j 21

k 3

l 13

5a

54.5

982

b 1.

3863

c 1.6

487

d 7.

3891

e 1.6

487

f 0.3

679

g 52

.598

2h

4.71

83i 0

.606

5Ex

ercise

7.4

1a

5b

2c 2

d 1

e 2f 1

2

a b

c

d e

f

3a

0.18

b 0.

045

c –0.

094

a b

c d

e f

5a

b c

d e

f no

real

soln

g 3,

7h

i 4

j k

l

6a

b c

d e

f

7a

1b

–2c 3

d 9

e 2f 9

8

a 1,

4b

1,

c 1,

d 1,

200

Q

t

2 00

0 00

0V

t

700

A

t99

20.1

394

t

t

T(5

.03,

1.8

5)

36h

t

R

A98

t

x

R

p

1000

010

log

4=

0.00

110

log

3–=

x1

+

10

log

y=

p10

log

7=

x1

–

2

log

y=

y2

–

2

log

4x=

29x

=bx

y=

bax

t=

10x2

z=

101

x–

y=

2ya

xb

–=

24

1 3---3

alo

gb

log

clo

g+

=a

log

2b

log

clo

g+

=a

log

2c

log

–=

alo

gb

log

0.5

clo

g+

=a

log

3b

log

4c

log

+=

alo

g2

blo

g0.

5c

log

–=

xyz

=y

x2=

yx

1+ x

--------

----=

x2y

1+

=y

x=

y2x

1+

3

=

1 2---1 2---

17 15------

3 2---1 3---

331

–2

--------

--------

---

103

+64 63----

--2 15------

2w

x3

log

x 7y

------

4lo

gx2

x1

+

3

a

log

x5

x1

+

3

2x

3–

--------

--------

--------

------

alo

gy2 x----

-

10

log

y x--

2lo

g

33

4

43

55

4

SL

Mat

hs 4

e re

prin

t.boo

k P

age

833

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

834

MATH


ANSW

ERS

9a

b c

d e

f 5.11g

h 7.37i 0.93

j no real solutionk

l

10a 0.5, 4

b 3c –1, 4

d 10, 10 10e 5

f 3 11

a b 100, 10

c 2, 1

12a

b c

13a

b c

d Ø

14a

b c

d

e f

g h

i j Ø

k l

15

a b

c d 0

e f

16

a 4.5222b 0.2643

c 0, 0.2619d –1, 0.3219

e –1.2925, 0.6610f 0, 1.8928

g 0.25, 2h 1

i 121.5j 2

17a –3.1831

b 1.3098c 0.1422, 0.5574

d 2.6692e 1.8960

f 1.7162Exercise 7.51

a 10b 30

c 402

a 31.64 kgb 1.65

c

3a 4.75

b

4a [0,1[

b i 2.22ii 1.11

iii 0.74 yearsc A

s c increases, reliability reduces.d

e

5a

6a 0.10

b c 16.82%

d


i b 4c

ii b –3c

iii b –5 c

iv b 0.5c

v b 2c

vi b –2c

2–28

39, 17

4–43

57

67

7–5

80

9a 41 b 31st

10

11

a i 2ii –3

b i 4ii 11

12

13

14a –1

b 0Exercise 8.1.21

a 145b 300

c –170 2

a –18b 690

c 70.4 3

a –105b 507

c 224 4

a 126b 3900

c 14th week

5855

6a 420

b –210 7Exercise 8.1.3 M

iscellaneous questions1

123 2

–3, –0.5, 2, 4.5, 7, 9.5, 12 3

3.25 4

a = 3 d = –0.05 5

10 000 6

330 7

–20 8

328 9

$725, 37 weeks

10a $55

b 2750 11

a i 8 mii 40 m

b 84 mc D

ist = d 8

e 26 players, 1300 m

12a 5050

b 10200c 4233

14log

2log--------------

3.81=

8log10log--------------

0.90=

125log

3log

-----------------4.39

=

12

log-----------

113 ------

log

2

–0.13

–=

103

log–

log4

3log

-------------------------------0.27

=2

log–2

10log

-----------------0.15

–=

3log

2log-----------

2–

0.42–

=1.5

log3

log---------------

0.37=

411

4log

yxz

=y

x3

=x

ey

1–

=

1

e4

1–

--------------13 ---

51

–2----------------

21ln

3.0445=

10ln

2.3026=

7ln

–1.9459

–=

2ln

0.6931=

3ln

1.0986=

2149 ------

ln

0.8837=

e3

20.0855=

13 ---e2

2.4630=

e9

90.0171

=

e2

4–

3.3891=

e9

320.0855

=0

2ln

5

ln2

ln3

ln

05

ln

10ln

W2.4

100.8

h

=

W

hW

h

2.42.4

d e

LL

010

6m

–2.5-------------

=

m

L

L

m

6

L0

6

L0

c d

x1

10ct

––

=y

t

x

y1

1

Iank

-----=

010

kx–

=

k1x ---

0

------

log

–=

tn4

n2

–=

tn3

n–

23+

=tn

5n

–6

+=

tn0.5

n=

tny

2n1

–+

=tn

x2

n–

4+

=

23

x8y

–

tn5

103 ------n

1–

+=

a9

b7

==

2n

22n

–2n

n1

–

=

SL

Mat

hs 4

e re

prin

t.boo

k P

age

834

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

835 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

13a

145

b 39

0c –

1845

14

b Ex

ercise

8.2

.1

1a

b

c d

e f

2a

b

3a

b 15

th

4a

b c n

= 5

4 ti

mes

5 6a

i $40

96ii

$209

7.15

b 6.

2 ye

ars

7

82.

5, 5

, 10

or 1

0, 5

, 2.5

9

5375

7 10

108

952

11

a $5

6156

b $2

9928

4

Exer

cise

8.2

.2

1a

3b

c –1

d e 1

.25

f

2a

2165

13b

c

d e

3a

11; 3

5429

2b

7; 4

73c 8

; 90.

9090

9d

8; 1

72.7

78e 5

; 2.2

56f 1

3; 1

11.1

1111

1111

1

4a

b c

d 60

e 5

4; 1

1809

66

$210

9.50

79.

28 cm

8a

b 7

9

54

1053

.5 g

ms;

50 w

eeks

. 11

7

129

13–0

.5, –

0.77

97

14r =

5,

15

$840

7.35

16

or a

bout

200

bill

ion

tonn

es.

Exer

cise

8.2

.31

Term

9 A

P =

180,

GP

= 25

6. S

um to

11

term

s AP

= 16

50, G

P =

2047

. 2

183

12

47,

12

58

wee

ks K

en $

220

and

Bo-Y

oun

$255

) 6

a w

eek

8 b

wee

k 12

7

a 1

.618

b 1

2137

9(~

1214

00, d

epen

ds o

n ro

undi

ng er

rors

)Ex

ercise

8.2

.4

1a

b c 5

000

d

2

366

67 fi

sh. (

Not

e:

. If w

e use

n =

43

then

ans i

s 666

0 fis

h); 2

0 00

0 fis

h.

Ove

rfish

ing

mea

ns th

at fe

wer

fish

are c

augh

t in

the l

ong

run.

4

275

48, 1

2, 3

or 1

6, 1

2, 9

6

a b

c

712

8 cm

8

9

10

11

Exer

cise

8.2

.5 M

isce

llane

ous

ques

tion

s1

3, –

0.2

2 3 4a

b c

599

006

3275

73

3n2

–

r2

u 5

48u n

32n

1–

=

==

r1 3---

u 5

1 27------

u n

31 3---

n

1–

=

==

r1 5---

u 5

2 625

---------

u n

21 5---

n

1–

=

==

r4–

u 5

256

–u n

1–4–n

1–

=

==

r1 b---

u 5

a b3-----

u n

ab1 b---

n

1–

=

==

rb a---

u 5

b4 a2-----

u n

a2b a---

n

1–

=

==

125

2

--------

--

96 u n10

5 6---

n

1–

=

1562

538

88----

--------

---4.

02

24 3---

– u n10

0016

9----

--------

12 5------

n

1–

=

1990

656

4225

--------

--------

-----47

1.16

1 3---1 3---–

2 3---–

1.63

8410

10–

256

729

---------

729

2401

--------

----81 1024

--------

----–

127

128

---------

63 8------

130

81---------

63 64------

Vn

V0

0.7n

=

1.8

1010

1.8

1019

81 2------

10 13------

30 11------

2323 99----

--

t 431

11 30------

37 99------

191

90---------

121 9----

-----

24 3---

3+

1t–n

– 1t

+----

--------

--------

-1

1t

+----

-------

1t2 –n

– 1t2

+----

--------

--------

----1

1t2

+----

--------

--

2560 93----

--------

10 3------ 43 18----

--45

899---------

413

990

---------

SL

Mat

hs 4

e re

prin

t.boo

k P

age

835

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

836

MATH


ANSW

ERS

8

96

10–

11

a 12 b 26 12

9, 12 13

±2 14

(5, 5, 5), (5, –10, 20) 15

a 2, 7b 2, 5, 8

c 3n – 1 16

a 5b 2 m

Exercise 8.31

$2773.08 2

$4377.63 3

$1781.94 4

$12216 5

$35816.956

$40349.377

$64006.80 8

$276971.93, $281325.419

$63762.25 10

$98.62, $9467.14, interest $4467.14. Flat interest = $600011

$134.41, $3790.44, 0.602% /m

onth (or 7.22% p.a.)

Chapter 9Exercise 9.11

a cmb cm

c cmA

BC

a3.8

4.11.6

67°90°

23°b

81.598.3

55.056°

90°34°

c32.7

47.133.9

44°90°

46°d

1.6130.7

30.73°

90°87°

e2.3

2.741.49

57°90°

33°f

48.577

59.839°

90°51°

g44.4

81.668.4

33°90°

57°h

2.9313.0

12.713°

90°77°

i74.4

94.458.1

52°90°

38°j

71.896.5

64.648°

90°42°

k23.3

34.124.9

43°90°

47°l

43.143.2

2.387°

90°3°

m71.5

80.236.4

63°90°

27°n

33.534.1

6.579°

90°11°

o6.1

7.23.82

58°90°

32°p

29.130

7.376°

90°14°

q29.0

29.12.0

86°90°

4°r

34.588.2

81.223°

90°67°

s24.0

29.717.5

54°90°

36°t

41.246.2

21.063°

90°27°

u59.6

72.941.8

55°90°

35°

v5.43

6.84.09

53°90°

37°w

13.019.8

14.941°

90°49°

x14.0

21.316.1

41°90°

49°y

82.488.9

33.368°

90°22°

2a

b c 4

d e

f

4a

b

Exercise 9.21

a i 030°Tii 330°T

iii 195°Tiv 200°T

b i N25°E

ii Siii S40°W

iv N10°W

2

37.49m

318.94m

4

37° 18' 5

m/s

6N

58° 33’W, 37.23 km

7

199.82 m

810.58 m

972.25 m

10

25.39 km

1115.76 m

12

a 3.01 km N

, 3.99 km E

b 2.87 km E 0.88 km

Sc 6.86 km

E 2.13 km N

d 7.19 km 253°T

13524m

Exercise 9.31

a 39°48'b 64°46'

2a 12.81 cm

b 61.35 cmc 77°57'

d 60.83 cme 80° 32'

3a21°48'

b 42°2'c 26°34'

4a 2274

b 12.7° 5

251.29 m

6a 103.5 m

b 35.26°c 39.23°

7b 53.43

c 155.16 md 145.68 m

8

b 48.54 m

9a

b c

d

1082.80 m

11a 40.61 m

b 49.46 m

12a 10.61 cm

b 75° 58'c 93° 22'

13a 1.44 m

b 73° 13'c 62° 11'

Exercise 9.41

a 1999.2cm

2b 756.8

cm2

c 3854.8cm

2d 2704.9

cm2

e 538.0cm

2

tn6

n14

–=16 ---

23

51

3+

21

3+

43 ---3

3+

1065

–

251

3+

403

3-------------

269 ------

bc

–

2h

2+

ha ---

1–tan

hb

c–

-----------

1–

tan

2b

c+

h2

a2

+2

ab

c–

2

h2

++

SL

Mat

hs 4

e re

prin

t.boo

k P

age

836

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

837 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

f 417

.5cm

2g

549.

4cm

2h

14.2

cm2

i 516

.2cm

2j 2

81.5

cm2

k 91

8.8

cm2

l 387

.2cm

2m

139

.0cm

2n

853.

7cm

2o

314.

6cm

2

269

345

m2

310

0 –

cm

2

417

.34

cm

5a

36.7

7 sq

uni

tsb

14.7

0 sq

uni

tsc 6

2.53

sq u

nits

6

52.1

6 cm

2

77°

2'

8

9A

rea o

f =

101

.78

cm2 ,

Are

a of

= 6

1.38

cm2

Exer

cise

9.5

.11

a cm

b cm

c cm

AB

Ca

13.3

37.1

48.2

10°

29°

141°

b2.

71.

22.

874

°25

°81

°c

11.0

0.7

11.3

60°

3°11

7°d

31.9

39.1

51.7

38°

49°

93°

e18

.511

.419

.568

°35

°77

°f

14.6

15.0

5.3

75°

84°

21°

g26

.07.

326

.479

°16

°85

°h

21.6

10.1

28.5

39°

17°

124°

i0.

80.

20.

882

°16

°82

°j

27.7

7.4

33.3

36°

9°13

5°k

16.4

20.7

14.5

52°

84°

44°

l21

.445

.664

.311

°24

°14

5°m

30.9

27.7

22.6

75°

60°

45°

n29

.345

.659

.129

°49

°10

2°o

9.7

9.8

7.9

65°

67°

48°

p21

.536

.654

.216

°28

°13

6°q

14.8

29.3

27.2

30°

83°

67°

r10

.50.

710

.952

°3°

125°

s11

.26.

917

.025

°15

°14

0°t

25.8

18.5

40.1

30°

21°

129°

Exer

cise

9.5

.21

ab

cA

°B°

C°c*

B*°

C*°

a7.

4018

.10

21.0

620

.00

56.7

810

3.22

12.9

512

3.22

36.7

8b

13.3

019

.50

31.3

614

.00

20.7

714

5.23

6.49

159.

236.

77c

13.5

017

.00

25.9

028

.00

36.2

411

5.76

4.12

143.

768.

24d

10.2

017

.00

25.6

215

.00

25.5

513

9.45

7.22

154.

4510

.55

e7.

4015

.20

19.5

520

.00

44.6

311

5.37

9.02

135.

3724

.63

f10

.70

14.1

021

.41

26.0

035

.29

118.

713.

9414

4.71

9.29

g11

.50

12.6

022

.94

17.0

018

.68

144.

321.

1616

1.32

1.68

h8.

3013

.70

18.6

724

.00

42.1

711

3.83

6.36

137.

8318

.17

i13

.70

17.8

030

.28

14.0

018

.32

147.

684.

2716

1.68

4.32

j13

.40

17.8

026

.19

28.0

038

.58

113.

425.

2414

1.42

10.5

8k

12.1

016

.80

25.6

323

.00

32.8

512

4.15

5.30

147.

159.

85l

12.0

014

.50

24.3

521

.00

25.6

613

3.34

2.72

154.

344.

66m

12.1

019

.20

29.3

416

.00

25.9

413

8.06

7.57

154.

069.

94n

7.20

13.1

019

.01

15.0

028

.09

136.

916.

3015

1.91

13.0

9o

12.2

017

.70

23.7

330

.00

46.5

010

3.50

6.93

133.

5016

.50

p9.

2020

.90

27.9

714

.00

33.3

413

2.66

12.5

914

6.66

19.3

4q

10.5

013

.30

21.9

620

.00

25.6

713

4.33

3.03

154.

335.

67r

9.20

19.2

026

.29

15.0

032

.69

132.

3110

.80

147.

3117

.69

s7.

2013

.30

18.3

319

.00

36.9

712

4.03

6.82

143.

0317

.97

t13

.50

20.4

025

.96

31.0

051

.10

97.9

09.

0112

8.90

20.1

02

a–d

no tr

iang

les e

xist.

Exer

cise

9.5

.31

30.6

4km

2

4.57

m

347

6.4

m

420

1°47

'T

522

2.9

m

6a

3.40

m b

3.1

1 m

7

b 1.

000

m c

1.71

5m

8

a 51

.19

min

b1 h

r 15.

96 m

inc 1

4.08

km

9

$488

6 10

906

m

Exer

cise

9.5

.41

a cm

b cm

c cm

AB

Ca

13.5

9.8

16.7

54°

36°

90°

b8.

910

.815

.235

°44

°10

1°c

22.8

25.6

12.8

63°

87°

30°

d21

.14.

421

.085

°12

°83

°e

15.9

10.6

15.1

74°

40°

66°

f8.

813

.620

.320

°32

°12

8°7

9.2

9.5

13.2

44°

46°

90°

g23

.462

.558

.422

°89

°69

°h

10.5

9.6

15.7

41°

37°

102°

i21

.736

.036

.235

°72

°73

°j

7.6

3.4

9.4

49°

20°

111°

k7.

215

.214

.328

°83

°69

°l

9.1

12.5

15.8

35°

52°

93°

m14

.911

.216

.263

°42

°75

°n

2.0

0.7

2.5

38°

13°

129°

o7.

63.

79.

056

°24

°10

0°p

18.5

9.8

24.1

45°

22°

113°

q20

.716

.313

.687

°52

°41

°r

14.6

22.4

29.9

28°

46°

106°

s7.

06.

69.

945

°42

°93

°t

21.8

20.8

23.8

58°

54°

68°

u1.

11.

71.

341

°89

°50

°

691

ba

ta

n

+

2

2

tan

--------

--------

--------

--------

-----

A

CD

A

BC

SL

Mat

hs 4

e re

prin

t.boo

k P

age

837

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

838

MATH


ANSW

ERS

v1.2

1.20.4

85°76°

19°w

23.727.2

29.749°

60°71°

x3.4

4.65.2

40°60°

80°Exercise 9.5.51

a 10.14km

b 121°T 2

7° 33' 3

4.12cm

4

57.32m

5

315.5m

6

124.3km

b W28° 47' S

Exercise 9.5.6 Miscellaneous questions

139.60 m

52.84 m

230.2 m

3

54°,42°, 84° 4

37° 5

028°T. 6

108.1 cm

7a 135°

b 136 cm8

41°, 56°, 83° 9

a 158° leftb 43.22 km

10

264 m

1153.33 cm

12

186 m

1350.12 cm

145.17 cm

15

a 5950 mb 13341 m

c 160°d 243°

17a 20.70°

b 2.578m

c 1.994m

3

18a 4243

m2

b 86m

c 101m

Exercise 9.61

5.36 cm

212.3 m

3

24 m

440.3 m

, 48.2° 5

16.5 min, 8.9°

6~10:49 am

7a i

ii b

or c

Exercise 9.7

1a

, b

,

c ,

d ,

e ,

f ,

g ,

h ,

i ,

j ,

k ,

l ,

m

,

n ,

o ,

2, 36°

30.0942 m

3

4

579 cm

65.25 cm

2

7a 31.83 m

b 406.28 mc 11°

8

9

10a

b i 37.09 cmii 88.57 cm

c 370.92 cm2

1126.57

12

193.5 cm

13a 105.22 cm

b 118.83 cm

14a 9 cm

b 12 cmc

15b

c 0.49

161439.16 cm

2


a 120°b 108°

c 216°d 50°

2a

b c

d 3

d

sin

–

sin-------------------------

d

sin

–

sin-------------------------

d

tan

sin

–

sin--------------------------

d

tan

sin

–

sin--------------------------

d

cos

sin

–

sin-------------------------

1–

169150

------------cm2

5.21315

---------+

cm529

32------------cm

223

238 ---------+

cm

242cm

288

11cm

+1156

75---------------m

213.6

6815

---------+

m

96625---------cm

21.28

1225

---------+

cm361

15------------cm

215.2

193 ---------cm+

5248.8m

2648

32.4cm

+12943

300------------------cm

217.2

30130

------------cm+

192275

---------------cm2

12.4124

15------------

+cm

158843

------------------cm2

152418

3------------

+cm

12cm

224

2cm

+983 ---------cm

228

143 ---------+

cm196

75------------cm

25.6

2815

---------cm+

1153225

------------------cm2

49.6186

5------------

+cm

350 ------cm2

2.410 ------cm

+

0.63c

1.64c

1.11c

0.75c

1.85ccm

2

3652

y1

tan------------

=

y5.5

–

–

=5.5

0.49

yc

32 ------ c79 ------ c

169 --------- c

SL

Mat

hs 4

e re

prin

t.boo

k P

age

838

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

839 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

3a

b

c d

–2e

f g

h i

j

k 1

l m

n

o –1

p q

0r 1

s 0t u

ndef

ined

4a

0b

–1c 0

d –1

e f

g –1

h i

j k

l m

n

o p

2q

r s –

1t

5a

b c 1

1d

e f

g h

6a

b

c d

–2e 1

f g

h i

j

k l

7a

b

c d

8a

0b

c d

10a

b c

11a

b c

12a

b c

13a

b c

14a

b c

15a

b c

16a

b c

17a

b c

18a

b c 1

d 1

e f

19a

b c

d e

f

Exer

cise

10.

2.1

1a

b

c d

e

2a

i ii

b i

ii

3a

b c

d

9a

b

10a

i 1ii

1b

1

11a

b c

12a

i 6ii

iii

b i 5

ii 1

iii –

2

13a

b o

r

14a

i 25

ii

b i 2

7ii

15a

b

16a

b i

ii

17a

b

18 Exer

cise

10.

2.2

1a

b c

d e

f

2a

b c

d

e f

g h

i

3 2-------

1 2---–3

–1 2---

–3 2-------

–1 3

-------

31 2

-------

–1 2

-------

–

2–

1 2----

---–

1 2----

---2

1 2----

---1 2

-------

–2

1 2---–

3 2-------

–1 3

-------

33 2-------

–1 2---

3–

1 2----

---–

1 2----

---2

–

1 2---3 2-------

1 2---1 3

-------

–1 2---

–2

–2 3

-------

–

1 2---–1 2

-------

–3

1 2---1 3

-------

–3 2-------

–2 3

-------

–1 3

-------

2 3----

---3 2-------

–

1 2---3 2-------

1 2---

–3 2-------

1 2

-------

–1 2

-------

–

3 2-------

1 2---–

3 2-------

1 3----

---1

3+ 2

2----

--------

----

2 3---–2 3---–

2 3---–

2 5---–5 2---

2 5---

k1 k---–

k–

5 3-------

3 5----

---5 3-------

–

3 5---–3 4---

4 5---

4 5---3 4---

5 3---–

k–1

k2–

–k

1k2

–----

--------

------

–

1k2

––

k

1k2

–----

--------

------

1

1k2

–----

--------

------

–

si

n

cot

co

t

tan

3---2 3------

3---

5 3------

3---

4 3------

5 6------

7 6------

5 6----

--11 6----

-----

7 6------

11 6----

-----

x2y2

+k2

=k

xk

–x2 b2----

-y2 a2----

-+

1=

bx

b

–

x1

–

22

y–

2

+1

=0

x2

1

x–

2

b2----

--------

-------

y2

–

2

a2----

--------

-------

+1

=

5x2

5y2

6xy

++

16=

4 5---–

5 3---–

4 7----

---7 3-------

–

3---2 3------

4 3------

5 3------

2---

7 6------

11 6----

-----

0 6---

5 6------

2

2---3 2------

2a

a21

+----

--------

---a2

1–

a21

+----

--------

---

1x2

1–

–x

--------

--------

--------

---1

x21

–+

x----

--------

--------

--------

2 x2-----

1–

5 2---9 8---

2 6---

2k

k

+

7 6------

2k

k

+

1 54-----

1 3---

12k

+1

k–

12k

+

1a

–

2a

--------

----2

2aa2

–+

2aa2

––

1a

–----

--------

--------

-----

2 3---0

22

3----

------

0 3---

2 3------

sin

cos

+co

ssi

n3

2

32

sin

sin

–co

sco

s2x

y2

xy

sin

cos

–co

ssi

n

2

2

sin

sin

+co

sco

s2

ta

n–

tan

12

ta

nta

n+

--------

--------

--------

--------

----

3

tan

–ta

n1

3

tan

tan

+----

--------

--------

--------

--------

2

3–

sin

25

+

co

sx

2y

+

si

nx

3y

–

co

s

2

–

tan

xta

n 4---

–

tan

4---

+

+

si

n2

xsi

n

SL

Mat

hs 4

e re

prin

t.boo

k P

age

839

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

840

MATH


ANSW

ERS

3a

b c

4a

b c

5a

b c

d

6a

b c

d

7a

b c

d

8a

b c

d

12

14a

b c

15a

b 10

16a

b –11

18Exercise 10.3

1a 4

b c 3

d 4e 2

f

2a 5

b 3c 5

d 0.5 3

a b

c d

e f

g h

i j

5665 ------–

3365 ------1663 ------

–

1665 ------6365 ------

5633 ------

511

18-------------

–718 ------

–5

117

-------------35

11162

----------------

35 ---–

45 ---–

34 ---247 ------

13

+22

----------------1

3+2

2----------------

13

+22

----------------–

32

–

2ab

a2

b2

+------------------

a2

b2

+2ab

------------------a

46a

2b2

–b

4+

a2

b2

+

2----------------------------------------

2ab

b2

a2

–-----------------

21

–

03 ---

53 ------2

6 ---

56 ------32 ------

0

2

2

–

12-------

1–

tan=

Ra

2b

2+

tan

ba ---

==

Ra

2b

2+

tan

ba ---

==

23

–

23 ------2 ---

22

63

4

3

623 ------

14 ---

383 ------

23 ---

4 a b c d

x

y

2–3 3

x

y1–1

–

x

y2–2

3x

y0.5

–0.5

e f g h

x

y

2

x

y

– x

y

– /3

/3

1/3

–1/3

x

y

2

3

–3

–

5 a b c d

x

y3

2

x

y

–2

–

–1

x

y3

–4x

y2

1.5

2.5

e f g h

x

y

x

y

–

x

y2/3

– /3

/3

x

y

–5

–

1

–1

6 a b c d

x

y

2

–3

3

x

y1–1

–

y

x

–2

x

–1/2

e f g h

x

y

2x

y

–x

y1/3

– /3

/3

x

y3

–3

–

–4

7 a b c d

x

y

2–2

1

–1

x

y–1

–22

–2

1

x

y

2 –2

x

y2

2 –2

–2

SL

Mat

hs 4

e re

prin

t.boo

k P

age

840

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

841 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

Exer

cise

10.

4

1a

b c

d e

f g

1.10

71c

h –0

.775

4ci 0

.099

7c

j 1.2

661c

k –0

.643

5cl 1

.373

4cm

und

efin

edn

–1.5

375c

o 1.

0654

c

2a

–1b

c

4

5a

b c

d e

f –1

6a

1b

c d

unde

fined

e f

g h

9a

b

Exer

cise

10.

5

1a

b c

d

e f

2a

b c

d

e f

3a

b c

d e

f 3

4a

90°,

330°

b 18

0°, 2

40°

c 90°

, 270

°d

65°,

335°

e f 0

, , 2

g h

x

y

–22

–2

3 –1

x

y2

–2

2–2

–2

x

y

–22

–2

x

y

–22

–2

3

–3

i

j

k

l

x

y1

–2

–3

2x

y

2–2

x

y

2–2

4 –4x

y 1

2 –2

3

m

n

x

y

–22

2

–2

x

y2

–2

2–2

e

f

g

h

8 a

b

c

x

y

cose

cx

2 x

yse

cx 2 x

y

cotx

4--- 2---

3---

4--- 3---–

3 4-------

1

32

--------

--–

1 3---1 2--- 2 3---

1 3---1 2---

3 4---3

24

--------

--

7 25------

–63 65----

--4

59

--------

--3 5---

4 3---1 2---

1k2

– k----

--------

------

1

1k2

+----

--------

-------

10 a

b

x

y

2–2

22

–

x

y

0.5

–0.5 1 2---

1 2--- –

02

31–

–

c

d

x

y

2x

y

–2–3

–1

x

y

1–1

/2

Cos

–1 Sin

–1

12 a

b

c

i

ii

2--- 2---

x

y

1–1

/2

13

4---1

n1

+----

--------

1–ta

n– 4---

3 4------

7 6------

11 6----

-----

3---

2 3------

18------

5 18------

13

18--------

-17

18--------

-25

18--------

-29

18--------

-

3---5 3----

--

5 4---7 4---

13 4------

15 4------

21 4------

23 4------

4---7 4------

2 3------

4 3------

6---

11 6----

-----

6---

5 6------

7 6------

11 6----

-----

3 2---

5 2---11 2----

--

6---7 6------

3 4------

7 4------

3---

4 3------

4

21–ta

n 3---

5 6------

4 3------

11 6----

-----

12------

5 12------

13

12--------

-17

12--------

-

3---2 3----

--4 3------

5 3------

3 8------

7 8------

11 8----

-----

15 8----

-----

SL

Mat

hs 4

e re

prin

t.boo

k P

age

841

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

842

MATH


ANSW

ERS

5a 60°, 300°

b c

d 23°35’, 156°25'e

f

g h 3.3559 c, 5.2105 c

i j

k

l 68°12', 248°12'm

n

o Ø

6a

b c

d e

f

g h

7a

b

8a

b c 0,1,2,3,4,5,6

9a

,b

c

d

10a

b

11a

b

12

13a b

14a ii

b ii

16a i

ii

b c

17a

b

18c

19

21a 90°,199°28',340°32'

b (199°28',340°32')

24

Exercise 10.6

1a 5, 24, 11, 19

b c 23.6°

2a 3, 4.2, 2, 7

b

3a 5, 11, 0, 7

b

4a 1, 11, 1, 12

b

5a 2.6, 7, 2, 6

b

6a 0.6, 3.5, 0, 11

b

7a 0.8, 4.6, 2.7, 11

b

8a 3000

b 1000, 5000c

9a 6.5 m

, 7.5 mb 1.58 sec, 3.42 sec

10a 750, 1850

b 3.44c m

id-April to end of August11

a 15000b 12 m

onthsc

d 4 months

12a , –2, 2

b m

c m

43 ------53 ------

6 ---76 ------

3 ---23 ------

43 ------53 ------

23 ------

53 ------

56 ------96 ------

3 ---43 ------

3 ---23 ------

43 ------53 ------

6 ---

23 ------76 ------

53 ------

3 ---53 ------

4 ---34 ------

54 ------74 ------

34 ------–

4 ---

3 ---

78 ------–

38 ------–

8 ---58 ------

2 ---–

2 ---

8 ---78 ------

98 ------158 ---------

2 ---32 ------

2 ---32 ------

34 ------74 ------

23 ---

1–tan

23 ---

1–tan

+

3 ---23 ------

34 ------43 ------

53 ------74 ------

12 ------512 ------

712 ------1112

---------1312

---------1712

---------1912

---------2312

---------

23 ------43 ------

3 ---53 ------

cos

14 ---

1–

34 ------74 ------

31–

tan

31–

tan+

6 ---

76 ------2 ---

32 ------

32 ---

1–tan

2

1–tan

–

32 ---

1–tan

+2

21–

tan–

2x

6 ---+

sin0

23 ------2

2x

3 ---–

sin6 ---

32 ------

3 ---23 ------

6 ---56 ------

136 ---------176 ---------

13

-------

1–

sin+

213

-------

1–

sin–

3

13-------

1–

sin+

413

-------

1–

sin–

04 ---

54 ------

2

06 ---

2 ---56 ------

32 ------2

xx

k

1–

k+

k

=

x

2k

+

x2

k1

+

–

k

xx

2k

1+

5 ---

=

x

x2

k=

k

xx

2k5 ---------

10 ------+

=

x

x2k

2 ---–

=

k

03 ---

53 ------2

2

22 -------

29 ---

cos2

59 ------cos

279 ------

cos

4 ---

23 ------34 ------

xy

x2

k2 ---

+=

y2k

=

xy

x2k

2 ---–

=y

2k

+=

k

T5

t

12 ------3

–

19

+sin

=

L3

t

2.1-------

3–

7+

sin=

V5

2t

11--------

sin

7+

=

P211 ------

t1

–

sin

12+

=

S2.6

27 ------t

2–

sin6

+=P

0.64

t7 --------

sin11

+=D

0.82.3

-------t

2.7–

sin11

+=

49 ---

t

R9 15

12

d 4 m

onths

13 ---43 ---

SL

Mat

hs 4

e re

prin

t.boo

k P

age

842

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

843 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

13a b

c 3d

38.4

%

14a

b i 7

, 11,

19,

23

ii c 1

4.9

m

Chap

ter

11Ex

ercise

11.

11

vect

or

2sc

alar

3

scal

ar

4ve

ctor

5

vect

or

6ve

ctor

7

scal

ar

8sc

alar

Ex

ercise

11.

2

1a

b c

d

2d

3a

{a,b

,e,g,

u}; {

d,f}

b {d

,f}; {

a,c}

; {b,

e}c {

a,g},{

c,g}

d {d

,f}, {

b,e}

e {d,

f}, {

b,e}

, {a,c

,g}

5a

ACb

AB

c AD

d BA

e 0

6a

Yb

Nc Y

d Y

e N

iv

v

872

.11

N, E

N

927

19 N

alon

g riv

er10

b i 2

00 k

ph N

ii 2

13.6

kph

, N

W11

b i 2

00 i

i 369

.32

Exer

cise

11.

3

1a

b c

2a

b c

d

3a

0b

PSc A

Yd

6OC

t0

0.5

11.

52

2.5

33.

54

F(t)

68

64

68

64

6G

(t)4

4.06

254.

254.

5625

55.

5625

6.25

7.06

258

t

d 915

12

07

1119

2324

a b c

4 a

b

c

d

a

b

ab

ab

a b

e

f

g

ab

ab

a

b

7 a

i

b

i

c

i

20 k

m

20 k

m

A

B

C

20°

A

B

C

15 k

m10

km

45°

45°

N

W

E

S

10 m

/s60

°

20 m

/s

80°

ii32

520

21

110

cos

–

10

54

110

cos

–

334

1

73

7

ca

–b

c–

1 2---b

a+

ba

–b

2a

–2

b3

a–

1 2---b

2a

+

SL

Mat

hs 4

e re

prin

t.boo

k P

age

843

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

844

MATH


ANSW

ERS

4a

b c

7a

b c

8a

b

1516m

=

Exercise 11.41

a b

c d

2a

b c

d

3a

b c

d

45, (–2, 3)

6a

b c

d

7a

b c

d

8A = –4, B = –7

9a (2, –5)

b (–4, 3)c (–6, –5)

10D

epends on basis used. Here w

e used: East as i, North j and vertically up k

b c

Exercise 11.51

a b

c d 3

e f

g h

2a

b c

d

e f

g h

3a D

epends on the basis: or

b

4a

b

5

6Exercise 11.61

a 4b –11.49

c 25 2

a 12b 27

c –8d –49

f 4g –21

h 6i –4

j –10 3

a 79°b 108°

c 55°d 50°

e 74°f 172°

g 80°h 58°

4a –8

b 0.55

a –6b 2

c Not possible

d 5e N

ot possiblef 0

6a

b c

d Not possible

71

8105.2°

9

10

12a

b e.g.

14 if

or

15a

b

16a

b 131.8°, 48.2°, 70.5°

18a

b

1925a U

se as a 1 km

eastward vector and

as a 1 km northw

ard vector. b

, and

c d

e

12 ---b

a+

13 ---2

ba

+

14 ---

ab

2c

++

cb

–c

a+

ac

2b–

+

221

226

m1323 ------

n5023 ------

==

43 ---

4i

28j

4k

–+

12i

21j

15k+

+2

i–

7j7

k–

+6

i–

12k

–3

i4

j–

2k

+8

i–

24j13

k+

+18

i32

j–

k+

15i–

36j

12k

++

1108

27

–122–

3–6–12

161–14

5–3

2–3

8i

4j

–28k

–19

i–

7j

–16

k–

17i

–j

22k

++

40i

4j

20k

–+

20125

12216

4–38–32–

20–22–40–

D600i

800j

60k

+A

–1200i

–300j

–60

k+

==

1800i

500j

–

105

230

5341

1417

12-------

ij

+

141

----------4

i5

j+

15-------

i–

2j

–

146

----------i

6j3

k–

+

15-------

i2k

+

117

----------2i

2j

–3

k–

13 ---212

1

33

----------1–51

3i

–4j

k+

+4

i–

3j

–k

+26

3i

j–

k+

14 ---3

ij

–2

k+

11134

23

–2

34

–14

23

–

x167 ------

–=

y447 ------

–=

111----------

i–

j3

k+

+

16i

–10j

–k

+

i

j37 ---k

++

ab

c–

b

c

bc

=

35 ---45 ---

22 -------12 ---

12 ---–

23 ---–

23 ---13 ---

13 ---13

-------

6

–32

6

b i ii

c 81.87°

u110

----------3i

j–

=

v15

-------i

2j+

=19 a

u

v

y

x

12 ---i

–j

2k

++

ij

WD

4i

8j

+=

WS

13i

j+

=D

S9

i7

j–

=180

----------4

i8

j+

d80----------

4i

8j

+

3i

6j

+

SL

Mat

hs 4

e re

prin

t.boo

k P

age

844

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

845 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

Exer

cise

11.

7.1

1a

i i

i i

ii b

line j

oins

(1, 2

) and

(5, –

4)2

a b

c

d e

or

f o

r

3a

b c

4a

b

c d

5a

b c

d

6a

b c

d

e f

7

a b

c

8a

b

9 11a

(4, –

2), (

–1, 1

), (9

, –5)

b –2

d e i

ii

12

13a

b

14b

ii an

d iii

15

(–83

, –21

5)16

17a

b Ø

c Lin

es ar

e coi

ncid

ent,

all p

oint

s are

com

mon

.

Exer

cise

11.

7.2

1a

No

b 52

.5 m

ins a

fter A

2a

i ii

b N

oc i

ii

11 a.

m.

Exer

cise

11.

7.3

1a

b

2a

b c

3a

b , y

= 3

c

4

5 6a

b c

d

7a

b x

= 2,

9a

b , y

= 1

10a

b

12. L

ine p

asse

s thr

ough

(1, 0

.5, 2

) and

is p

aral

lel t

o th

e

vect

or13

a 54

.74°

b 82

.25°

c 57.

69°

14

a b

Doe

s not

inte

rsec

t.

15a

b Ø

c 84.

92°

d i

ii

18

ri

2j

+=

r5

i–

11j

+=

r5i

4j–

=r

2i

5j

3

i4

j–

++

=r

3i

–4j

i

–5

j+

++

=r

j

7i

8j

+

+

=

ri

6j

–

2i

3j

+

+

=r

1– 1–

2– 10

+=

ri

–j

–

2i

–10

j+

+=

r1 2

5 1

+=

ri

2j

5i

j+

++

=

r2

i3

j

2i

5j

+

+

+=

ri

5j

3

i–

4j

–

+

+=

r4

i3

j–

5–

ij

+

+

=r

9i

5j

i

3j–

++

=r

6i

6j

–t

4i

–2

j–

+=

ri

–3

j

4–i

8j

+

+

+=

ri

2j

1 2---

i1 3---

j–

++

=

x8

–2

+=

y10

+

=

x7

3

–=

y4

2

–=

x5

2.5

+=

y3

0.5

+=

x0.

50.

1t–

=

y0.

40.

2t

+=

x1

– 3----

-------

y3

–=

x2

– 7–--------

---y

4– 5–--------

---=

x2

+y

4+ 8

--------

----=

x0.

5–

y0.

2– 11–

--------

--------

=

x7

=y

6=

r2

jt

3i

j+

+=

r5

it

ij

+

+

=r

6–i

t2i

j+

+=

6i

13j

+16 3----

-- i–

28 3------ j

–

r2

i7

jt

4i

3j+

++

=r

4i2

j–

5i

–3

j+

+=

ML

M

L=

4x

3y

+11

=3– 13

--------

--2 13

--------

--

4 5---3 5---

rk 7---

19i

20j

+

= 92 11----

--31 11----

--

r A5 1–

t3 4

+=

r B4 5

t2 1–

+=

4 5

t1

–

2 1–

+

r2i

j3k

ti

2j

–3

k+

++

+=

r2

i3

j–

k–

t2

i–

k+

+=

r2

i5

kt

i4

j3k

++

++

=r

3i

4j

–7

kt

4i

9j

5k

–+

++

=r

4i

4j

4k

t7

i7k

+

+

++

=x 3---

y2

– 4----

-------

z3

– 5----

-------

==

x2

+ 5----

--------

z1

+ 2–--------

---=

xy

z=

=

x5

7t

–=

y2

2t

+=

z6

4t

–=

r

5 2 6

t

7– 2 4–

+=

x5

– 7–--------

---y

2– 2

--------

---z

6– 4–--------

---=

=

13 5------

23 5------

0

x2

3t

+=

y5

t+

=

z4

0.5t

+=

x1

1.5

t+

=

yt

=

z4

2t

–=

x3

t–

=

y2

3t

–=

z4

2t

+=

x1

2t

+=

y3

2t

+=

z2

0.5t

+=

x4

– 3----

-------

y1

– 4–--------

---z

2+ 2–--------

---=

=y

z1

– 3–--------

---=

x1

+ 2----

--------

y3

–z

5– 1–--------

---=

=x

2– 2

--------

---z

1– 2–--------

---=

11–

0

a

15b

11–=

=11

a

b

–2

z

x

y0

5

5

z= –

2 pl

ane

3z

x

y0

z =

3 p

lane

x +

y =

5

1x =

2 +

2t

x2

2t

+=

y1

=

z3

=

x1

t+

=

y4

t–

=

z2–

=

r

1 0.5 2

t

2 1.5

–

1

+

=

2i

3 2---j

–k

+

410

.515

L:

xy

2– 2

--------

---z 5---

==

M:

x1

+ 2----

--------

y1

+ 3----

--------

z1

– 2–--------

---=

=

02

0

01 2---

0

x 4---y 9---

z 3---=

=

SL

Mat

hs 4

e re

prin

t.boo

k P

age

845

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

846

MATH


ANSW

ERS

19

2064°

213 or –2

22 (or any m

ultiple thereof) 23

Not parallel. D

o not intersect. Lines are skew.Chapter 12Exercise 12.11

a i 14 500ii 2 000

b 305 (304.5) 2

Sample size is large but m

ay be biased by factors such as the location of the catch. Population estim

ate is 5000. 3

a i 1500ii 120

b 100c 1 000

4a, c num

erical, b, d, e categorical 5

a, d discrete, b, c, e continuousExercise 12.2123

Set A M

ode = 29.1 Mean = 27.2 M

edian = 27.85Set B M

ode = 9 Mean = 26.6 M

edian = 9. Set B is much m

ore spread out than set A

and although the two sets have a sim

ilar mean, they have very different m

ode and m

edian.Exercise 12.31

Mode = 236–238 g, M

ean = 234 g, Median = 235 g

2M

ode = 1.8–1.9 g, Mean = 1.69 g, M

edian = 1.80 g3

Set A M

ode = 29.1, Mean = 27.2, M

edian = 27.85; Set B Mode = 9, M

ean = 26.6, M

edian = 9.4

a $27522b $21025

c Median

5a $233 300

b $169 000c M

edian 6

a 14.375b 14.354

Exercise 12.41

a Sample A

Mean = 1.99 kg; Sam

ple B Mean = 2.00 kg

b Sample A

Sample std = 0.0552 kg; Sam

ple B Sample std = 0.1877 kg

c Sample A

Population std = 0.0547 kg; Sample B Population std = 0.1858 kg

2a 16.41

b 6.84 3

Mean = 49.97, Std = 1.365

Exercise 12.51

a Med = 5, Q

1 = 2, Q3 = 7, IQ

R = 5b M

ed = 3.3, Q1 = 2.8, Q

3 = 5.1, IQR = 2.3

c Med = 163.5, Q

1 = 143, Q3 =182, IQ

R = 39d M

ed = 1.055, Q1 = 0.46, Q

3 = 1.67, IQR = 1.21

e Med = 5143.5, Q

1 = 2046, Q3 = 6252, IQ

R = 42062

a Med = 3, Q

1 = 2, Q3 = 4, IQ

R = 2b M

ed = 13, Q1 = 12, Q

3 = 13, IQR = 1

c Med = 2, Q

1 = 2, Q3 = 2.5, IQ

R = 0.5d M

ed = 40, Q1 = 30, Q

3 = 50, IQR = 20

a $84.67b $147.8

c $11d Q

1 = $4.50, Q3 = $65 IQ

R = $60.50e M

edian and IQR.

3a 2.35

b 1.25c 2

d Q1 = 1, Q

3 = 3, IQR = 2

4a $232

b $83c–e

5

218–220

221–223

224–226

227–229

230–232

233–235

236–238

239–241

242–244

245–247

14

43

68

95

71

1.1–1.2

1.2–1.3

1.3–1.4

1.4–1.5

1.5–1.6

1.6–1.7

1.7–1.8

1.8–1.9

1.9–2.0

2.0–2.1

51

22

76

112

75

k72 ---

–=

12i

6j

7k

–+220

2 4 6 8

230

240

1.2

2 4 6 8

1.5

2.0

100200

300

10 20 30

Sales

40

400

Med =

$220 Q

1 = $160

Q3 =

$310 IQ

R =

$150

1020

30

10 20 30

x

40

40 Med =

14 Q

1 = 10

Q3 =

19 IQ

R =

9

SL

Mat

hs 4

e re

prin

t.boo

k P

age

846

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

847 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

Exer

cise

12.

6 M

isce

llane

ous

ques

tion

s1

a Sa

mpl

e–10

0 ra

ndom

ly se

lect

ed p

atie

nts,

popu

latio

n –

all s

uffe

ring

from

AID

S b

Sam

ple–

1000

wor

king

aged

peo

ple i

n N

.S.W

, pop

ulat

ion

– al

l wor

king

aged

pe

ople

in N

.S.W

.c S

ampl

e – Jo

hn’s

I.B. H

ighe

r Mat

hs cl

ass,

popu

latio

n –

all s

enio

rs at

Nap

pa V

alle

y H

igh

Scho

ol.

2D

iscre

te: a

, b, d

; Con

tinuo

us: c

, e, f

, g.

3b

4su

gges

ted

answ

ers o

nly:

a 2

00–2

24; 2

25–2

49; 2

50–2

74; .

. . 5

75–5

99b

100–

119;

120

–139

; . .

. 400

–419

c 440

–459

; 460

–479

; . .

. 780

–799

.5

Mak

e use

of y

our g

raph

ics c

alcu

lato

r.6

a 16

b gr

aphi

cs ca

lcul

ator

c 15.

23d

1.98

92

7a

30–3

4b

grap

hics

calc

ulat

orc 3

0.4

d 8.

9205

8

b 21

5.5

c 216

.2d

18.8

0 se

c9

48.1

7, 1

4.14

10a

Q1~

35,

Q3~

95

b ~

105

c 61%

d 67

.15

11ra

nge =

19,

s =

5.49

125.

8; 1

.50

1317

.4;

14

a 6.

15b

1.61

15,

Chap

ter

13Ex

ercise

13.

21

a i I

ncre

asin

g, p

ositi

veii

Appr

ox. l

inea

riii

Mild

(to

wea

k)b

i No

asso

ciat

ion

ii–iii

0b

i Inc

reas

ing,

pos

itive

ii Li

near

iii V

ery

stro

ng

d i I

ncre

asin

g, p

ositi

veii

Squa

re ro

otiii

Mild

(stre

ngth

not

appr

opria

te as

it is

a no

n-lin

ear r

elat

ions

hip!

)e i

Dec

reas

ing,

neg

ativ

eii

Expo

nent

ial

iii M

ild (s

teng

th n

ot ap

prop

riate

as it

is a

non-

linea

r rel

atio

nshi

p!)

f i D

ecre

asin

gii

Appr

ox. l

inea

riii

Mild

2a

b Po

sitiv

e ass

ocia

tion,

line

ar, s

treng

th: v

ery

stro

ng

3a

b Po

sitiv

e ass

ocia

tion,

line

ar, s

treng

th: v

ery

stro

ng

4 5

Exer

cise

13.

31

a r =

0.9

6b

2

3

4

5 6

7

8

481216

Fre

quen

cy

scor

e 2

3

4

5

6

7

8

204060 103050

Cum

ulat

ive

freq

uenc

y

scor

e

s n3.

12=

s n1

–3.

18=

s n18

.8=

s n1

–19

.1=

Dat

a disp

lays

a st

rong

pos

itive

asso

ciat

ion.

In

crea

se in

lead

cont

ent c

an b

e attr

ibut

ed

to in

crea

se in

traf

fic fl

ow.

Wor

ksaf

ety

polic

y ha

s ha

d de

sired

effe

ct, i

.e.

num

ber o

f acc

iden

ts h

as

decr

ease

d. D

ata d

ispla

ys a

stro

ng n

egat

ive

asso

ciat

ion.

SL

Mat

hs 4

e re

prin

t.boo

k P

age

847

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

848

MATH


ANSW

ERS

2a

b r = 0.70 (aasumed linear)

3a

b r = 0. No, not linear!

4N

o. The relationship is not linear. 5

a i 64%ii 81%

b i 51%ii 64%

6a

b r = 0.457

38

± 0.9229

82%10

b r = 0.7715 There is strong evidence to suggest that a student who does w

ell in M

aths will also do w

ell in Biology.c Values of r are the sam

e.Exercise 13.41

a i ii y = –1.33x + 21.11

b i ii y = 0.64x + 6.94

2a i ii y = 20.6 – 1.26x

iii

b i ii y = 1.6 +

0.8x iii

c i ii y = 14.8 +

3.44x iii

d i ii y = 0.6 +

0.8x iii

3a

b r = 0.891 c 79.4%

d y = 29.76 + 2.15x

e

SL

Mat

hs 4

e re

prin

t.boo

k P

age

848

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

849 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

4a

5a

6a

c i y

= 89

.50

+ 1.

02x

ii

d i 1

35.6

ii 17

6.5

iii

x =

85 is

a fa

ir w

ay o

ut fr

om th

e set

of v

alue

s use

d to

obt

ain

the

regr

essio

n lin

e7

a b

c i y

= 4.

74 +

0.6

xii

d i 8

.63

ii 10

.73

8a

b Li

near

tren

d ex

ists,

r = 0

.96

c i y

= 2.

68x

+ 16

.86

ii d

i 27.

57ii

57.0

3

9a

r = 0

.838

4b

70.2

9%c y

= 1

.20x

+ 8

.9d

(11,

22.

1), (

24.4

5, 3

8), (

27, 4

1.3)

,(6

0.08

, 81)

. The

equa

tion

is us

ed to

pre

dict

y fro

m x

-val

ues,

not x

from

y-va

lues

. W

e wou

ld n

eed

to fi

nd th

e reg

ress

ion

of x

on

y.10

a b

i r =

0.9

7ii

222

c M =

0.2

967T

+ 4

8.28

11Re

mai

ns th

e sam

e.12

a b

–0.9

3

c i y

= –

0.37

x +

74.4

4ii

d y(

40) =

59.7

22 (i

nter

pola

tion)

; y(1

20) =

30.

278

(ext

rapo

latio

n)

b r =

0.5

53

c i y

= 40

+ 0

.5x

ii

x =

24.1

+ 0

.61y

b Ba

sed

on th

e sca

tter d

iagr

am, t

here

is a

defin

ite

linea

r rel

atio

nshi

p. T

here

fore

, ow

ner i

s jus

tifie

d.c i

r =

0.99

ii C

= 4.

19 +

1.8

2wd

i 20.

57, i

.e. 2

1ii

95.1

9, i.

e. 95

iii

Fro

m ii

, ser

ving

95

peop

le p

er h

our i

s un

real

istic

.

b Sc

atte

r dia

gram

show

s a li

near

rela

tions

hip.

Th

eref

ore s

tatis

tic is

appr

opria

te, r

= 0

.877

.

Scat

ter d

iagr

am sh

ows a

line

ar re

latio

nshi

p.

Ther

efor

e sta

tistic

is ap

prop

riate

, r =

0.9

45

SL

Mat

hs 4

e re

prin

t.boo

k P

age

849

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

850

MATH


ANSW

ERS

Exercise 13.5 Miscellaneous questions

1a

b y = 0.57x – 26.2c 0.9388

Because of the strong positive linear association, and the high r value, we

can say that the taller the student the greater their weight.

20.057

3B

4a 0.8

b Strong positive relationship 5

1.5 6

a 0.78b i P = 1.07M

– 12.91ii 73%

c i M = 0.77E + 27.14

ii 100iii Extrapolated. C

ontinued linear trend highly likely. Therefore confidentd Find regression equation of E on M

, then use M = 90 into this new

equation. 7

a Positiveb Linear

c Very strong 8

a b 0.9645

c y = 1.68x – 2.79

a y = –1.75x + 64.67b 22.67

c –11.1210

a i

d e x = 37, y = 46.35; Expenditure is $4635

11a i 4.4; 2.02

ii 14.06; 2.92b b = 0.4895; r = 0.34

c d Regression equation is

12a i

ii r = 0.9629b y = 0.635x – 33.815

c d


15 2

a 25b 625

3a 24

b 256 4

a 24b 48

515

6270

7120

8336

960

10a 362880

b 80640c 1728

1120

12a 10!

b 2 8!c i 2 9!

ii 8 9! 13

34650 14

4200 15

4 Exercise 14.21

792 2

a 1140b 171

31050

470

52688

6a 210

b 420 7

24000 8

8 9

155 10

5 Exercise 14.3 M

iscellaneous questions1

a 120b 325

25040

x20.57

y,31.86

==

ii = 0.8908b r 2 = 1.7935, that is, 79.35%c y = 2.15x – 33.28

r 20.3397

20.1154

==

y14.06

–

0.4895

x4.4

–

w

hen x\

3.5y,

13.63=

==

When x = 1040, y = 626.59. The carcass

weighs 626.59 lbs.

SL

Mat

hs 4

e re

prin

t.boo

k P

age

850

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

851 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

3a

144

b 14

40

4a

720

b 24

0 5

1176

0 6

7056

; 460

67

a 84

0b

1680

8

190

910

080

1022

6800

11

a 71

b 31

5c 6

65

13

14

15b

92

1625

2 17

a 12

87b

560

18 2

56

19 2

88

20a

1008

0b

3024

0c 1

4400

21

1008

0, 1

080

2235

2800

023

720;

240

24

1036

8025

a 12

b 12

8

2628

80

27a

3003

0b

3731

0 28

7705

5 29

a 48

b 72

Ch

apte

r 15

Exer

cise

15.

1

1a

b c

2a

b

3a

b

4{H

H, H

T, T

H, T

T}a

b

5{H

HH

,HH

T,H

TH,T

HH

,TTT

,TTH

,TH

T,H

TT}

a b

c

6a

b c

d

7a

b c

8a

b c

d

9{G

GG

, GG

B. G

BG, B

GG

, BBB

, BBG

, BG

B, G

BB}

a b

c

10a

b c

11a

b c

d

12a

{(1,

H),(

2, H

),(3,

H),(

4,H

),(5,

H),(

6, H

),(1,

T),(

2, T

),(3,

T),(

4, T

),(5,

T),(

6,T)

}b

13a

b c

Exer

cise

15.

2

1a

b c

2a

b c

d

3 4a

1.0

b 0.

3c 0

.55

a 0.

65b

0.70

c 0.6

56

a 0.

95b

0.05

c 0.8

07

a {T

TT,T

TH,T

HT,

HTT

,HH

H,H

HT,

HTH

,TH

H}

b i

ii iii

iv

8a

b c

9b

c d

e

10a

b c

d 11

a 0.

1399

b i 0

.879

7ii

0.6

12b

c d

Exer

cise

15.

31

a 0.

7b

0.75

c 0.5

0d

0.5

2a

0.5

b 0.

83c 0

.10

d 0.

903

a b

c

d

Cn2

Cn4 2 5---

3 5---2 5---

2 7---5 7---

5 26------

21 26------

1 4---3 4---

3 8---1 2---

1 4---

2 9---2 9---

2 3---1 3---

1 2---3 10------

9 20------

11 36------

1 18------

1 6---5 36------

1 8---3 8---

1 2---

1 2---1 4---

1 4---

3 8---1 4---

3 8---3 4---

1 4--- 1 216

---------

1 8---3 8---

1 4---5 8---

3 4---

1 13------

1 2---1 26------

7 13------

9 26------

3 8---1 2---

1 4---3 8---

6 25------

6 25------

13 25------

3 4---1 2---

1 6---7 12------

1 4---1 2---

8 13------

7 13------

4 15------

4 15------

11 15------

4/9

5/9

3/5

2/5

3/5

2/5

R R_

R_ R_R R

8 45------

22 45------

6 11------

SL

Mat

hs 4

e re

prin

t.boo

k P

age

851

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

852

MATH


ANSW

ERS

4a 0.5

b 0.30c 0.25

5a

b c

67a

b c

89a 0.88

b 0.42c 0.6

d 0.28 10

a 0.33b 0.49

c 0.82 d 0.55111

a 0.22b 0.985

c 0.8629 12

a 0.44b 0.733

14a 0.512

b 0.128c 0.8571

15a 0.2625

b 0.75c 0.4875 d 0.7123

16a 0.027

b 0.441c 0.453

Exercise 15.41

a 0.042b 0.7143

2a 0.4667

b 0.38683

a b

4

5b i

ii 0.2

6a i

ii b

78a 0.07

b 0.3429c 0.30

d 0.02829

a 0.8008b 0.9767

c 0.0003 10

a 0.0464b 0.5819

c 0.9969 11

0.2b 0.08

c 0.72

Exercise 15.5

1a

b c

2a

b c

d

3a

b c

4

5a

b

6a

b c

78a

b

9a

b

10a

b c

11a

b

12

13a

b

14a

b

15a

b 0.6

16Chapter 16Exercise 16.11

0.3 2

a 0.1bi 0.2

ii 0.7

2H2T

F

HT0 1

1/2

HT

01

1/2

HT

12 ---23 ---

13 ---

5/103/10

2/10

3/9

4/92/9

5/92/92/9

5/93/9

1/9

YBGYBGYBG

3145 ------29 ---

23 ---

57 ---913 ------

59 ---

140 ------

2Nm

–2N

-----------------2

Nm

–

2N

m–

----------------------m

mN

m–

2

n+

------------------------------------

919 ------

5126---------

518 ------1126

---------

15 ---110 ------

25 ---35 ---

725525------------

15525------------

11201------------

25 ---

63143---------

133143---------

512 ------533 ------

56 ---

311 ------413 ------913 ------

6791 ------2291 ------

14 ---128 ------

514 ------

528 ------128 ------

613 ------16 ---14 ---

1210---------

79 ---

71938------------

1121 ------

SL

Mat

hs 4

e re

prin

t.boo

k P

age

852

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

853 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

3a

b c

4a

{2, 3

, 4, 5

, 6, 7

, 8, 9

, 10,

11,

12}

b

5a

b c

6a

bi

ii

c

7a

i 0.9

048

ii 0.

0904

8b

0.00

028

0.37

12

9a

b i

ii

10

11 a

b

12a

0.81

b 0.

2439

Exer

cise

16.

21

a 2.

8b

1.86

2

a 3

b i 1

ii 1

c i 6

ii 0.

43

a i 1

.3ii

2.5

iii –

0.1

b i 0

.9ii

7.29

c i

ii 0.

3222

4

5a

7b

5.83

33

6 7a

b 2.

8c 1

.166

8a

0.1

b i 0

.3ii

1c i

0ii

1iii

29

5.56

10b

i 0.9

ii 0.

49

c W =

3N

– 3

, E(W

) = –

0.3

11

a $

–1.0

0b

both

the s

ame

12a

50b

18c 2

13a

11b

c –4

14a

0.75

b 0.

6339

15

a E(

X) =

1 –

2p,

Var

(X) =

4p(

1 –

p)b

i n(1

– 2

p)ii

4np(

1 –

p)

16a

b W

= 2

1.43

x

23

45

67

89

1011

12

p(

x)

p0

6 15------

=p

1

8 15------

p2

1 15------

==

12

x2468

15p

x

14 15----

--

1 36------

2 36------

3 36------

4 36------

5 36------

6 36------

5 36------

4 36------

3 36------

2 36------

1 36------

2 36------4 36------6 36------

24

68

1012

p(y)

y

c5 36------

d

H T

T H T

H T H T H TH T

12

x

p(y)

1/8

3/8

3

p0

1 8---=

p1

3 8---=

p2

3 8---=

p3

1 8---=

4 7---

1 35------

p0

1 35------

=

p1

4 35------

=

p2

7 35------

p

3

10 35------

==

p4

13 35------

=

2x

35p(y)

32468101214

14

6 7---

p0

11 30------

p1–

1 2---

p3

2 15------

==

=11 30----

--13 15----

--

n1

23

4P(

N =

n)

n0

12

P(N

= n

)6 15------

8 15------

1 15------

1 4---1 4---

1 4---1 4---

s2

34

56

78

P(S

= s)

1 16------

2 16------

3 16------

4 16------

3 16------

2 16------

1 16------

31 60------

2 3---

2

0.35

56=

=

np

31 2---

1.

5=

=

1 25------

p0

35 120

---------

=p

1

63 12

0----

-----p

2

21 12

0----

-----p

3

1 120

---------

==

=

3 3-------

n0

12

P(N

= n

)

28 45------

16 45------

1 45------

SL

Mat

hs 4

e re

prin

t.boo

k P

age

853

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

854

MATH


ANSW

ERS

17a

b

18a E(X) = 4, Var(X) = 20

Exercise 16.31

a 0.2322b 0.1737

c 0.59412

a 0.3292b 0.8683

c 0.2099d 0.1317

3a 0.1526

b 0.4812c 0.5678

4a 0.7738

b c 0.9988

d

5a 0.2787

b0.4059 6

a 0.2610b 0.9923

7a 0.2786

b 0.7064c 0.1061

8a 0.1318

b 0.8484c 0.054

d 0.326 9

a 0.238b 0.6531

c 0.0027d 0.726

e 12.86 10

a 0.003b 0.2734

c 0.6367d 0.648

11a 0.3125

b 0.0156c 0.3438

d 312

a 0.2785b 0.3417

c 120 13

a 0.0331b 0.565

14a 0.4305

b 0.61c $720

d 0.2059 15

a i 1.4ii 1

iii 1.058iv 0.0795

v 0.0047b i 3.04

ii 3iii 1.373

iv 0.2670 v 0.139016

38.23 19

a i 0.1074ii

iii 0.3758b at least 6

20a

b c

d

21a 20

b 3.4641 22

a 102.6b 0.000254

23a i 6

ii 2.4b i 6

ii 3.6 24

0.1797 25

1.6, 1.472 26

a 0.1841b $11.93

27a $8

b $160 28

a 0.0702c

29b

30b 0.8035

c 39.3 Chapter 17Exercise 17.11

a 0.6915b 0.9671

c 0.9474d 0.9965

e 0.9756f 0.0054

g 0.0287h 0.0594

i 0.0073j 0.8289

k 0.6443l 0.0823

2a 0.0360

b 0.3759c 0.0623

d 0.0564e 0.0111

f 0.2902g 0.7614

h 0.0343i 0.6014

j 0.1450k 0.9206

l 0.2668m

0.7020n 0.9132

o 0.5203p 0.8160

q 0.9388r 0.7258

Exercise 17.21

a 0.0228b 0.9332

c 0.3085d 0.8849

e 0.0668f 0.9772

2a 0.9772

b 0.0668c 0.6915

d 0.1151e 0.9332

f 0.0228 3

a 0.3413b 0.1359

c 0.0489 4

a 0.6827b 0.1359

c 0.3934 5

a 0.8413b 0.4332

c 0.7734 6

a 0.1151b 0.1039

c 0.1587 7

a 0.1587b 0.6827

c 0.1359 8

a 0.1908b 0.4754

c 16.88 9

a 0.1434b 0.6595

10a 0.2425

b 0.8413c 0.5050

11a –1.2816

b 0.2533 12

a 58.2243b 41.7757

c 59.80 13

39.11 14

9.1660 15

42%

160.7021

17a 0.2903

b 0.4583c 0.2514

1823%

19

0.5 20

11%

215%

22

14%

231.8

24252

250.1517

260.3821

270.22

28322

290.1545

307

3187

32a i 0.0062

ii 0.0478iii 0.9460

b 0.058533

a $5.11b $7.39

34a 0.0062

b i 0.7887ii 0.0324

c $1472 35

a μ = 66.86, = 10.25b $0.38S

36a μ = 37.2, = 28.2

b 20 (19.9) 37

a i 0.3446ii 0.2347

b i 0.3339ii 0.3852

c 0.9995

a23 ---

=0

b1

E

X

b1

+3------------

=V

ar

X

19 ---

27

bb

2–

+

=

3.12510

7–

310

5–

7.910

4–

43 ---109 ------

16 ---5288

---------

0.6

1

p

1

1

p

SL

Mat

hs 4

e re

prin

t.boo

k P

age

854

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

855 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

Chap

ter

18Ex

ercise

18.

1

1a

b c

d 1

e f 0

2a

4b

0.2

c 0.0

27d

0.43

3e –

0.01

f 6.3

4g

6.2

h 0

3a

6 m

/sb

30 m

/sc 1

1 +

6h +

h2 m

/s

412

m/s

5

8 +

2h

6–3

.49°

C/se

c 7

a 12

7 cm

3 /cm

b i 1

9.66

67

cm3 /c

mii

1.99

67

cm3 /c

miii

0.2

000

cm3 /c

m

81.

115

9

a –7

.5°C

/min

b t =

2 to

t =

6

10a

28 m

b 14

m/s

c ave

rage

spee

dd

49 m

e 49

m/s

11

a $1

160,

$13

45.6

, $15

60.9

0, $

1810

.64,

$ 2

100.

34b

$220

.07

per y

ear

Exer

cise

18.

2

Exer

cise

18.

3

1a

h +

2b

4 +

hc

d 3

– 3h

+h2

2a

2b

4c –

1d

3

3a

2a +

hb

–(2a

+ h

)c (

2a +

2) +

hd

3a2 +

1 +

3ah

+ h

2

e –(3

a2 + 3

ah +

h2 )

f 3a2 –

2a

+ (3

a –1

)h +

h2

g

h i

4a

1 ; 1

b 2a

+ h

; 2a

c 3a2 +

3ah

+ h

2 ; 3a

2d

4a3 +

6a2 h

+ 4a

h2 + h

3 ; 4a

3

5a

b i 3

ms–1

ii 2

ms–1

iii 1

.2 m

s–1

6a

b i 2

0 cm

2ii

17.4

1 cm

2iii

2.5

9 cm

2

iv –

1.29

cm2 /d

ayc

cm2 /d

ayd

i –1.

3863

cm2 /d

ayii

–1.2

935

cm2 /d

ayEx

ercise

18.

4

1a

15b

8c 0

d 1

e 0f 6

g h

ei 6

j 2

a 4

b –3

c 0.5

d 0

e 33

a 0

b un

defin

edc 1

d 1

e und

efin

ed4

a i 0

ii 0

b i –

8ii

unde

fined

5a

2b

1c 0

.5d

1e 3

6a

1b

–1c u

ndef

ined

7a

b c 6

d 12

e

8a

0b

4c 4

d 0

e 19

a 3

b –2

c 2.5

d –1

e 010

a i 2

ii iii

2iv

0

Exer

cise

18.

5

1a

3b

8c

d 1.

39e –

1f

24.

9 m

b m

c 9.8

m/s

3

a 8x

b 10

xc 1

2x2

d 15

x2e 1

6x3

f 20x

3 4

a 4x

b –1

c –1

+ 3x

2d

e f

5a

1 m

s–1b

(2 –

a) m

s–1

3 4---3

a4

b----

--1–

15 8------

–

t

h1

a

b

t

h

2 a

b

c

t

h

t

h

t

h

d

e

f

t

h

t

h

t

h

1– 1h

+----

--------

2–a

ah

+

----

--------

--------

1a

1–

a1

–h

+

----

--------

--------

--------

--------

-------

–1

ah

+a

+----

--------

--------

--------

--

1

1

3 4---1 8---

–

x(t)

t

d Fi

nd (l

imit)

as

e 4

t – 3

h0

20s(

t)

t20

12

0.1

h–

–

2 ---5

1 3---3

x22

+1 x2-----

–

b a---

1 9---–

17 16------

4.9

h22

h+

x2–

–2

x1

+

2–

–0.

5x1

2/–

SL

Mat

hs 4

e re

prin

t.boo

k P

age

855

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

856

MATH


ANSW

ERS

6a

b i 5 ms –1

ii 4 ms –1

c m

s –1d

sec


a 5x 4b 9x 8

c 25x 24d 27x 2

e –28x 6f 2x 7

g 2xh 20x 3 + 2

i –15x 4 + 18x 2 –1j

k 9x 2 – 12xl

2a

b c

d e

f g

h

i j

k l

3a

b c

d e

f g –7

h i

j k

l

Exercise 19.2.11

;

2;

;

3–12

4a 3

b c 12

d 4e 4

f g

h

5

6a 2x – 12

b –18c (8, –32)

7a –3x 2 + 3

b 0c

8a

,(0, 0)b

9

10a –2, 6, 3

b –211

a = 1 b = –8 1213

a b

14–56

Exercise 19.2.21

ab

c

23

4

83 ---25627---------

x(t)

t

8t

3t 2

–83 ---

43 ---–

x3

10+

325 ---x

4x3

++

3x4

-----–

32 ---x

52 ---x

31

3x

23

--------------2x

------9

x1x

------3x2

-----+

32 ---x

1

2x

3-------------

–

10

3x

3-----------

9–

51

2x

----------85x

3---------

––

4x------

15x6

------12 ---

+–

1

2x

3-------------

–1x

------–

x2

+

32 ---x

1x------

+4

x3

3x

21

–+

3x

21

+1x2

-----1x

3---------

12 ---1

4x

3-------------

–

2x

8x3

-----–

2x

2x2

-----4x5

-----–

–12 ---

3x ---1

6x

3-------------

+2

x125 ------

x5

2

5x

35

--------------+

–

3

2x

----------1x ---

1+

1x------

x–

2–

mP

Q4

h+

=m

PQ

h0

lim4

=

P1

1

Q

1h

22

h+

------------

+

mP

Q1

2h

+------------

–=

mP

Qh

0 lim

12 ---–

=

14 ---–

76 ---112 ------

–5316 ------

83 ---

22

22 -------

116 ------–

x:1–2

-------x

0

x:x

12-------

x13 ---

=1–

f'

ab

+

2

ab

+

2

a2

b+

==

4a

22

aa

0

–

41a ---

a0

–

034 ---

y'

x

y'

x4

–1 1y'

x5

y'

x4

–1

y'

x

y'

x5

y'

x

y'

x

2

–1 1 23

5–5

a b

y'

xa b c d

gh

i

de

f

y

x

1

1

y

x1 2

42

SL

Mat

hs 4

e re

prin

t.boo

k P

age

856

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

857 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

Exer

cise

19.

2.3

1a

b c

d

e f

g h

i

2a

b c

d e

f

Exer

cise

19.

3

1a

b c

d

2a

b c

d e

f

3a

b c

d

e f

g

h i

4a

b c

d e

f g

h i

5a

b c

d

e f

g h

0

i j

k l

6a

b c

d

e f

g h

i

j k

l

7a

b c

d e

f

g h

i j

k l

m

n

o p

8a

b c

d e

f

g h

i

j k

l

9a

b c

d e

f

g h

i j

k

l m

n

o p

q r

s t

u v

w

x

10x

= 1

110

12

0

131

14 –

2e

15a

b c

16b

i 2xs

inxc

osx

+ x2 co

s2 x - x

2 sin2 x

ii

48t3

1 2t

---------

–2n

2 n2-----

–4 n5-----

–3 2---

r5

6r

6----

------

1 r----

--–

+2

9 2---

–3

1

2

--------

--–

+

403L

2–

100

v3---------

–1

–6l

25

+2

8h

+4n

31

3n2

3----

--------

--

+–

8 3t3

--------

2

r20 r2----

--–

5 2---s3

2/3 s2-----

+6 t4----

–2 t3----

1 t2----–

+4 b2-----

–1

2b3

2/----

--------

--+

3m

24

m–

4–

3x2

5x4

–2

x2

++

6x5

10x4

4x3

3x2

–2x

–+

+4 x5-----

–6x

58

x32

x+

+

2

x1

–

2----

--------

-------

–1

x1

+

2----

--------

-------

1x2

–2

x–

x21

+

2----

--------

--------

------

x43

x22

x+

+

–

x31

–

2----

--------

--------

--------

--------

-----

2x2

2x

+

2x

1+

2

--------

--------

-------

1

12

x–

2

--------

--------

------

xsi

nx

cos

+

ex

xln

1+

ex2x

36

x24

x4

++

+

4x

3x

cos

x4x

sin

–

sin2

x–

cos2

x+

2xx

tan

1x2

+

sec

2x

+4 x3-----

xx

cos

2x

sin

–

exx

xx

xx

sin

+si

n+

cos

xln

1x

xln

++

e

x

xsi

nx

xco

s– sin2

x----

--------

--------

--------

---x

sin

x1

+

x

cos

+

–

x1

+

2----

--------

--------

--------

--------

--------

--------

--ex

ex1

+

2----

--------

--------

--2

xx

cos

xsi

n–

2x

x----

--------

--------

--------

-------

xln

1– x

ln2

--------

---------

x1

+

x

xln

–

xx

1+

2

--------

--------

--------

--------

--xe

x1

+

x1

+

2----

--------

-------

2–

xsi

nx

cos

–

2----

--------

--------

--------

--------

x2x

–2x

xln

+

xx

ln+

2

--------

--------

--------

--------

---

5e

5x

––

1+

44x

cos

36x

sin

+1 3---–

e

1 3---x

–1 x---

–18

x+

255x

cos

6e2

x+

4se

c24

x2

e2x

+4–

4x

sin

3e

3x

–+

44

x1

+----

--------

---1

–

1 2---x 2---

cos

22

xsi

n–

77x

2–

cos

1

2x

--------

--1 x---

–1 x---

66

xsi

n+

2xx2

cos

2x

xco

ssi

n+

2sec

22

co

s

sin2

--------

-----

–1

2x

--------

--x

cos

1 x2-----

1 x---

sin

3

cos2

si

n–

exex

cos

1 x---se

c2x e

log

2si

nx

–

2x

cos

--------

--------

---

cos

–

sin

sin

4

sin

sec2

5

5x

cos

–cs

c25x

6c

sc2

2x

–

2e2

x1

+6

e43

x–

–12

xe4

3x2

––

1 2---ex

1

2x

--------

-- ex

e2x

4+

2xe2

x24

+6

e3x

1+

--------

--------

–6

x6

–

e3

x26

x–

1+

e

si

nco

s

22

e

2

cos

–si

n2

x2e

x–

ex–

1+

2

--------

--------

--------

-3

exe

x–+

exe

x––

2

ex2

+2

x–

9+

e

x2–

9x

2–

+

2x

x21

+----

--------

---

cos

1+

si

n

+----

--------

--------

-ex

ex–

+

exe

x––

--------

--------

---1

x1

+----

--------

–3 x---

xln

21

2xx

ln----

--------

------

12

x1

–

----

--------

-------

3x2

– 1x3

–----

--------

--1

2x

2+

--------

--------

----–

2x

xco

ssi

n– co

s2x

1+

--------

--------

--------

-----1 x---

xco

t+

1 x---x

tan

+

x32

+

ln

3x3

x32

+----

--------

---+

sin2

x

2x

--------

-----2

xx

xco

ssi

n+

1 ----

---

co

s

sin

–

3x2

4x4

–

e2

x23

+–

xln

1+

–x

xln

sin

1x

xln

--------

---

2x

4–

x2

sin

2

xx2

x24

x–

cos

–

x2si

n

2----

--------

--------

--------

--------

--------

--------

--------

--------

--------

--------

--------

--------

--10

10x

1+

ln1

–

10

x1

+

ln

2----

--------

--------

--------

--------

--------

-----

2x

22x

sin

–co

s

ex

1–

2x4

sin

x

ln

4x2

4x

cot

+x

cos

xsi

n–

1

2x

--------

-- ex

–

2x

sin

2x

xco

s+

–2

xx

sin

sin

e5

x2

+9

20x

–

1

4x–

2

--------

--------

--------

--------

--------

cos2

sin2

si

n

ln

+ sinc

os2

--------

--------

--------

--------

--------

--------

--------

x2

+

2x

1+

x1

+----

--------

--------

--------

--------

2x2

2+

x22

+----

--------

-------

10x3

9x2

4x

3+

++

3x

1+

2

3/----

--------

--------

--------

--------

--------

-----3x

23

x31

+

2x3

1+

--------

--------

--------

--------

2

x21

+----

--------

---1 x2-----

x21

+

ln

–2

xx

2+

--------

--------

---2

x–

2x2

x1

–----

--------

--------

-----x2

–x

9–

+ x29

+----

--------

--------

--------

ex–

7x3

12x2

–8

–

22

x–

--------

--------

--------

--------

----n

xn1

–xn

1–

lnnx

2n

1–

xn1

–----

--------

--------

+

cos2

xsi

n2x

– 18

0----

-----x

cos

180

---------

xsi

n–

ex3

–2

2x

xco

sln

cos

3x2

–2

xx

cos

lnsi

n2

sin

xx

tan

–

SL

Mat

hs 4

e re

prin

t.boo

k P

age

857

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

858

MATH


ANSW

ERS

17a i

ii b i

ii

18

192021a

b c

d

e f

22a

b c

d e

f g

h i

23a

b c

d e

f

g h

i 0Exercise 19.41

a b

c d

e f

g h

i j

k l

m

n o

p q

r s

t

234a

b

5–1

6[0,1.0768[

]3.6436,2]


a y = 7x – 10b y = –4x + 4

c 4y = x + 5d 16y = –x + 21

e 4y = x + 1 f 4y = x + 2

g y = 28x – 48h y = 4

2a 7y = –x + 30

b 4y = x – 1c y = –4x + 14

d y = 16x – 79e 2y = 9 – 8x

f y = –4x + 9g 28y = –x + 226

h x = 2 3

a y = 2ex – eb y = e

c y = d y = –x

e y = xf

g y = ex

h y = 2x + 1 4

a 2ey = –x + 2e 2 + 1b x = 1

c x = d y = x – 2

e y = –x + f

g ey = –xh 2y = –x + 2

5A

: y = 28x – 44, B: y = –28x – 44, Isosceles.

62 sq. units, y = 2 x = 1

74y = 3x

8

9y = 4x – 9

10y = loge 4

11;

12A

: y = –8x + 32, B: y = 6x + 25,

13y = –x, Tangents:

tangent and norm

al meet

at (0.5, –0.5) 14

a y = 3x – 7b

15

m = –2, n = 5

Exercise 20.21

ab

cd

2a m

ax at (1, 4)b m

in at c m

in at (3, –45) max (–3, 63)

d max at (0, 8), m

in at (4, -24)e m

ax at (1, 8), min at (–3, –24)

f min at

, max at

g min at (1, –1)

3x ---–

xln

23

x2

1x

3–

--------------–

2e

2x

–e

2x

–

cos

–

2x

x2

cos–

ex

2sin

–

15 ---–

k

xa

bm

bn

a+

mn

+--------------------

=:n

m

tan

ntan

m

mn

–=

44

x

csc

–2

2x

sec2

x

tan

33

x

cot

3x

csc3

3x

sin–

4 ---x

–

2

csc2–

2x

sec

2x

tan

2x

x2

secx

2

tan

sec2x

xtan

3cot 2xcsc

2x–

xx

xsin

+cos

2xcsc

2xcot

–4

x3

4x

csc

4x

44x

cot

4x

csc–

2xsec

22

x

cot

csc2x

2x

tan

–x

xx

sin–

tansec

2x

xsec

+cos

---------------------------------------

ex

secx

xtan

sece

xe

x

sec

ex

tane

xx

sece

xx

secx

tan+

csc2

xlog

–

x------------------------------

55

x

csc

–5

x

sec

xcotx---------------

x2

cscx

log–

xx

sin

cot

cos–

xsin

csc

xcsc

cos

–x

xcsc

cot

20x

348

12

x+

2

2x3

-----2

1x

+

3-------------------

26

x2

–

3-------------------

42x8

------24

12

x–

1x2

-----–

2x

21

+

1

x2

–

2-----------------------

–16

4sin

–2

xx

xsin

–cos

6x

2x

6x

xx

3x

sin–

sin+

cos

1x ---10

2x

3+

3

-----------------------6

xe2

x12

x2e

2x

4x

3e2

x+

+8

4x

154

xcos

–sin

ex

---------------------------------------------

2x

2cos

4x

2x

2sin

–48

x2

2x

5+

–

4x

31

–

3-----------------------------------

10

x3

–

3-------------------

6x

5–

lnx4

--------------------n

2x

lnn

xln

2n

–1

–+

xn

2+

------------------------------------------------------

fx

x1

+x

1–

------------

n

f''

x

4n

nx

+

x

21

–

2-----------------------

x1

+x

1–

------------

n

==

21

82

----------+

3

+2------------

ey2

e1

–

xe

2–

2e

1–

+=

2e

1–

y

ex–

3e

24

e–

1+

+=

z0

a2

3a4

–

by

a2

b2

–x

=

8y

4

2+

x

2

–=

4

2+

y

8x

–4

2

++

=

12 ---28

y12 ---

y12 ---

–=

=12 ---

–12 ---

12 ---12 ---

–

Q2

1–

(1, 2)

(3, –2)

y

x(2, 0)

(0, –4) y

x(4, 0)

y

x3

y

x

(4, 4)

92 ---–

814 ------–

113

+3

-------------------70

2613

–27

----------------------------

113

–3

-------------------70

2613

+27

----------------------------

SL

Mat

hs 4

e re

prin

t.boo

k P

age

858

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

859 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

h m

ax at

(0, 1

6), m

in at

(2, 0

), m

in at

(–2,

0)

i min

at (1

, 0) m

ax at

j min

at

k m

in at

(2, 4

), m

ax at

(–2,

–4)

l min

at (1

, 2),

min

at (–

1, 2

)

4m

in at

(1, –

3), m

ax at

(–3,

29)

, non

-sta

tiona

ry in

fl (–

1, 1

3)

5 6a

i (co

sx -

sinx)

e–xii

–2co

sx.e–x

b i

ii

c Inf

.

d

7a

i ex (s

inx

+ co

sx)

ii 2e

x cosx

b i

ii

c St.

pts.

, In

fl. p

ts.

,

d

8a

i ex (c

osx

– sin

x)ii

–2sin

x.ex

b i

ii 0,

, 2

c St.p

ts.,

Inf.

pts.

(0, 1

), (

, –e

), (2,

e2)

d

9a

i (1

– x)

e–xii

(x –

2)e

–xb

i x =

1ii

x =

2c S

t. pt

. (1,

e–1) I

nf. p

t. (2

, 2e–2

)

10a

8b

0c 4

d

11a

min

val

ue –

82b

max

val

ue 2

612

a pt

A: i

Yes

ii no

n-st

atio

nary

pt o

f inf

lect

; pt B

: i Y

esii

Stat

iona

ry p

oint

(loc

al/

glob

al m

in);

pt C

: i Y

esii

non-

stat

iona

ry p

t of i

nfle

ct.

b pt

A: i

No

ii. L

ocal

/glo

bal m

ax; p

t B: i

No

ii Lo

cal/g

loba

l min

; pt

C: i

Yes

ii St

atio

nary

poi

nt (l

ocal

max

)c p

t A: i

Yes

ii St

atio

nary

poi

nt (l

ocal

/glo

bal m

ax);

pt B

: i Y

esii

Stat

iona

ry

poin

t (lo

cal m

in);

pt C

i Ye

sii

non-

stat

iona

ry p

t of i

nfle

ct.

d pt

A: i

Yes

ii St

atio

nary

pt (

loca

l/glo

bal m

ax);

pt B

: i N

oii

Loca

l min

; pt

C: i

Yes

ii St

atio

nary

poi

nt (l

ocal

max

)

1 3---–32 27----

--

4 9---4 27------

–

5

–1.5

y

x

(–1.

5, 7

.25)

y

x

1 4---11 16----

--

–

1(–

3, 4

)

y

x

4

–1–4

y

x

(–1.

15, 3

.08)

(1.1

5, –

3.08

)

y

x

4

28 3---

(4, 0

)

y

x3(3

, 27)

(2, 0

)

(0, –

8)

y

x

y

x–2

2

–16

3 a

b

c

d

e

f

g

h

x

y1 36------

1 108

---------

1 16------

i

j

(1,–

1)

4x

y

2–

4–

2

4–

x

y

–2

2

4---5 4------

2---

3 2------

2---e

2---–

3 2------

e

3 2------

–

–

/4

2

x

y

x3 4------

7 4------

=

x 2---

3 2------

=

3 4------

1 2----

---e3

4------

7 4----

--1 2

-------

–e7

4------

2---

e 2---

3 2----

--e3

2------

–

/2

2

x

y

4---5 4----

--

4---1 2

-------

e 4---

5 4------

1 2----

---–

e5 4------

¼/2

3¼/2

1

y

x

(1, e

–1)

(2, 2

e–2)

y

x

279

356

.16

SL

Mat

hs 4

e re

prin

t.boo

k P

age

859

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

860

MATH


ANSW

ERS

e pt A: i N

o ii Cusp (local min); pt B: i Yes

ii Stationary pt of inflect; pt C: i Yes ii Stationary point (local m

ax)f pt A

: i Yesii Stationary point (local/global m

ax); pt B: i Yesii Stationary point

(local/global min); pt C: i N

oii Tangent parallel to y–axis.

13a i A

ii Biii C

b i Cii B

iii A14

a b c

1516

171819 m

= –0.5, n = 1.520

a i ii

b i ii

21a = 2, b = –3, c = 0

22Stationary points: local m

in at (–1, 0) and local max at

.Inflection pts are:

and

23Absolute m

in at ~ , local m

ax at ~Inflection pts at ~ (–0.4384, –1.4489) and (–4.5615, 0.1488)

24–27 are left as questions for classroom

discussion.

28 a = 1, b = –12, c = 45, d = –34

29b b = 1

c d

30a 2.7983, 6.1212, 9.3179

b Use a graphics calculator to verify your sketch.

Exercise 20.31

a Local min. at

, local max at

b Local max. at x = 0, local m

in. at x = ±1c Local m

ax. at x = 0.25d Local m

ax. at x = 1e none

f Local max. at x = 0.5, local m

in. at x = 1, 0g Local m

ax. at x = 1, local min. at x = –1

h none 2

a max. = 120, m

in. = b m

ax. = 224, min. = –1

c max. = 0.5, m

in. = 0d m

ax. = 1, min. = 0.

34

f'

x

f''

x

y

x–a a

y

x

f'

x

f''

x

f'

xf

''x

y

x

yx

36

x2

9x

4+

++

=

(–3, 4)

y

x

4

–1–4

fx

13 ---x3

x2

–3

x–

6–

=

fx

3x

520

x3

–=

0.05x

y

120 ------120 ------

12 ---20

ln–

x4

y

x4

32 ---x

4–

3x

10–

2x

4–

-------------------

14

e1–

12

64

2+

e

12

+

–

+

12

64

2–

e

12

–

–

–

3–

13+2

------------------------2.1733

–

3

–13

–2-----------------------

0.2062

y

x

(3, 20)

(5, 16)

–34

a12

-------=

fx

12-------xe

x2

–=

y

x

x43

-------=

x43

-------–

=

128

33

----------–

(2/3,–3/4)

(4/3,–3/4)

(,–1/2)

1.5(2

, 1.5)

3 ---3

34

----------

53 ------3

34

----------–

¼

SL

Mat

hs 4

e re

prin

t.boo

k P

age

860

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

861 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

5St

atio

nary

poi

nts o

ccur

whe

re ta

nx =

–x

6a

Loca

l min

. at (

1, 2

); in

fl. p

t. at

b Lo

cal m

in. a

t (1,

2);

loca

l max

. at (

–3, –

6)c n

one

7 8 –

11 V

erify

you

r gra

phs w

ith g

raph

ics c

alcu

lato

r.8

a G

loba

l min

. at (

0, 0

); lo

cal m

ax. a

t In

fl. p

ts.

b G

loba

l max

. at

, inf

l. pt

. at

c Loc

al m

ax. a

t

9a

Glo

bal m

ax. a

t . I

nfl.

pt. a

t b

Glo

bal m

in. a

t

c Glo

bal m

in. a

t (2,

1 +

ln2)

; Inf

l. pt

. at (

4, 2

+ ln

4)d

none

10

a b

i ; n

one

ii ; l

ocal

max

. at

; loc

al m

in. a

t (2,

0)

iii

; loc

al m

in. a

t (±2

, 0),

loca

l max

. at (

0, 1

6).

11a

Glo

bal m

in. a

t (1,

c –

1); c

1

b

12 13G

loba

l max

. at

; inf

l. pt

. at

.

Exer

cise

20.

4

1a

y =

2,x

= –1

b y =

1, x

=

c y =

, x

=

d y =

–1,

x =

–3

e y =

3, x

= 0

f y =

5, x

= 2

4a

= 2

, c =

4

33

1 3---3

+

2 3------

5 4---

4 3------

5 4---

(, 1

)

24

e2–

2

26

42

–

e2

2–

–

–

22

64

2+

e

22

+

–

+

0e4

1 2----

---e3

.5

21 2---

e–

–

ee

1–

e1.5

1.5

e1.

5–

1 2

-------

21 2---

2ln

+

f'

x

x2

–

a1

–x

2+

b

1–

ab

+

x2

ab

–

+

=f

x

x2

–x

2+

--------

----=

fx

x2

–

2x

2+

=2 3---

256

27---------

–

fx

x2

–

2x

2+

2

=

4

(1, 3

)

y

x

0

1

e1

2/–

1.5

e1–

e3

2/–

9 2---e

3–

y

x

e0.5

0.5

e1–

e5

6/5 6---

e5

3/–

1 3---–

1 2---1 4---

–

x

y

3

x =

–0.

5

x

y

y =

1

-1

x =

–2

x

y

y =

3

d

e

f

xy

x =

3

y =

–2

x

y

y =

1

x =

1.5

x

y -5

5

x =

0.5

y =

–0.

5

3 a

b

c

5 a

b

x =

1

y =

2

y

x

x =

–1

y

x

1

SL

Mat

hs 4

e re

prin

t.boo

k P

age

861

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

862

MATH


ANSW

ERS

6a i (0, 1), (2, 0)

ii y = –1, x = –2iii

iv d = \{–2}

b c

7a y =

8, x = 3

b


a i x < 0ii x > 4

iii 0 x 4b i –1 < x < 2

ii x < –1, 2 < x < 5iii

c i –1 < x < 1ii x < –1

iii x 1d i 0 < x < 1

ii 2 < x < 3iii x < 0, 1 x < 2

e i ii –2 < x < 4

iii f i –4 < x < –1, 2 < x < 5

ii –1 < x < 2, 5 < x < 8iii

Exercise 21.21

4.4 (4 deer per year, to nearest integer) 2

a 200 cm3

b 73.5 cm3/day

3a 75

b No

4a $207.66

b $ 40.79 per yearc $41.54 per year

5a 2.50

b 3.33c 2.50

6a 1230 < x < 48770 approx.

b i 0 < x < 25000ii 25000 < x < 50000

766667 to nearest integer, 1446992 to nearest integer

8b 133.33

d 46.67e 0 < x < 5700

9a

b 22.22 22 items/dollar

10a

b i ii

11a i 0 m

m/s

ii ~ 90.69 mm

/sb 0.6 sec

12a 8.53 cm

/sb never

c never13

–e–1 m

s –2

Exercise 21.31

a i ii

b i

ii c i

ii

d i ii

e i ii

f i ii

2a 8 m

s –1b never at rest

c i 5m from

O in negative direction

ii 4 ms –1

d 40 me

3a 1 m

s –1b never

c d 20 m

s –2

4a

; b ~ 141 sec

c onced use graphics calculator

5a 3 m

in positive directionb i 5 m

ii 2 mc 5 m

s –1

e oscillation about origin with am

plitude 5 m and period 2 seconds

7a 100 m

, in negative directionb 3 tim

esc i 80 m

s –1ii –34 m

s –2d 1481 m

8

a max. = 5 units, m

in. = –1 unitb

sc i

ii

9a 0318 m

aboveb i

ii c 0322 m

d

10a 0 < t < 05 or t > 1

b t > 05c t = 1 or 168 t 5

11a This question is best done using a graphics calculator:b From

the graph the particles pass each other three tim

esc 045 s; 285 s; 387 s

d i m

s –1ii

ms –1

e Yes, on two occassions.

12a 2m

in positive directionb i 2 s

ii neverc 0026 m

s –2

13a

b 0295

x

y

y = –1

x = –2

f1–: \{-1}

where

f

1–x

21

x–

1x

+

-------------------

=

x

y

y = 1

x = –2

x

y

y = 8

Range = \{8}

D'

x40000

–2

x12

+

x

212x

20+

+

2-----------------------------------------

5x

18

=

3000

x32

+

2----------------------

x0

x

v1

t1

–

2------------------

t1

–=

a2

t1

–

3------------------

t1

=v

2e

2t

e2

t–

–

t

0

=

a4

e2

te

2t

–+

t0

=v

24t 2

–-----------------

0t

2

=a

2t

4t 2

–

32/

--------------------------0

t2

=

vt

t1

+

10

ln----------------------------

t1

+

10

t0

log+

=a

110ln -----------

1

t1

+

2------------------

1t

1+ -----------

+t

0

=

va

2b

tet 2

–t

0

–

=a

2b

et 2

–2

t 21

–

t

0

=

v2

ln

2

t1

+3

ln

–

3

t

t0

=v

2ln

22

t1

+3

ln

2–

3

t

t0

=

s

t1

–5

t13 --- or t

1=

=

v6

t 2–

12+

=a

12t

–=

2 ---a

122

t

–

cos

–=

a4

x2

–

–

=

v3.75

e0.25

t–

3–

=a

0.9375e

0.25t

––

=a

0.25v

3+

–=

B

AvA

0.3e

0.3t

–=

vB10

et–

1t

–

=

window

: [0, 3] by [0, 14]

SL

Mat

hs 4

e re

prin

t.boo

k P

age

862

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

863 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

Exer

cise

21.

41

22.6

m

2a

1.5

mh–1

b $1

9.55

per

km

3

a 40

0b

$464

0000

0

4$2

73.8

65

$0.4

0

61.

97 m

7

0.45

m3

85

m b

y 5

m

912

8

10, d

im o

f rec

t. i.

e. ap

rrox

7.0

0 m

by

7.00

m

11 12a

b u

nits

c At i

nfl.

pts.

whe

n .

1364

8 m

2

14a

10.5

b 5.

25

1572

16

a y =

100

– 2

xb

A =

x(1

00 –

2x)

, 0 <

x <

50

c x =

25,

y =

50

17a

b cm

3

18a

400

mLs

–1b

40 s

c

19a

b 8.

38, 7

1.62

c 9

x

71

d 80

x –

x2 – 6

00, $

1000

20 &

21 b

y

224

by

23 ~

243

.7 cm

2

242

25ra

dius

=

cm, h

eigh

t =

cm

26 275

cm

28a

b c r

=

, h =

48

29r :

h =

1 :

2

30~

(0.5

5, 1

.31)

31

b 2.

5 m

32

altit

ude =

h

eigh

t of c

one

33~

1.64

0 m

wid

e and

1.0

40 m

hig

h

34

35w

here

XP

: PY

= b

: a

365

km37

r : h

= 1

:1

38 cm

392

: 1

40

410.

873

km fr

om P

42

b ,

43b

whe

n , i

.e. ap

prox

. 6.0

30 k

m fr

om P.

44a

b

45c i

f k <

c, sw

imm

er sh

ould

row

dire

ctly

to Q

. 46

a i

ii c r

: h

= 1:

1

47

48b

4 km

alon

g th

e bea

chc r

ow d

irect

ly to

des

tinat

ion

r50

4

+----

--------

7.00

=

504

+

--------

----50

4

+----

--------

6---

=

3---3

34

--------

--

5 3------

33

4----

------

–

¼

33

2----

------

xco

s1 4---

–=

100 x----

-----1 2---

x0

x10

2

–20

00 9----

--------

654

4.3

t

y(8

0,28

,444

.44)

120

x

y(5

0, 2

500)

C R

600 11 2------

7 2---

11 2----

--–

7 2---

52

5 2---2

8 3---

348

817

0–

10 3------

210 3----

--

15 ------

3

h24

r2

r214

4–

--------

--------

----=

8r4

r214

4–

--------

--------

----12

2

1 3---

22 3

--------

--

4 3--- 10 3

--------

-- r3

2=

h6

2=

ar

csin

5 6---

=

ta

nxl

x2k

lk

+

+

--------

--------

--------

------

=x

k2kl

+=

r2

h2 3---r

3+

3r

22r

h+

a23/

b23/

+

32/

SL

Mat

hs 4

e re

prin

t.boo

k P

age

863

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

864

MATH


ANSW

ERS


1a

b c

d e

f g

h

2a

b c

d e

f g

h

3a

b c

d e

f g

h i

4a

b c

d e

f

5a

b c

d

e f

6a

b

c d

e f

8a

b

Exercise 22.2

1a

b c

d e

f g

h

23$3835.03

49.5

5 cm

3

6292

781, –8

910

11a

b

12

13

14Vol ~ 43202 cm

3

15110 cm

2

Exercise 22.3

1a

b c

d e

f

g h

i j

k l

2a

b c

d e

f

g h

3a

b c

d

4a

b c

d e

f g

h i

j k

l

m

14 ---x4

c+

18 ---x8

c+

16 ---x6

c+

19 ---x9

c+

43 ---x3

c+

76 ---x6

c+

x9

c+

18 ---x4

c+

5x

c+

3x

c+

10x

c+

23 ---xc

+4

x–

c+

6x–

c+

32 ---x–

c+

x–

c+

x12 ---x

2c

+–

2x13 ---

+x

3c

+14 ---x

49x

–c

+25 ---x

19 ---x3

c+

+13 ---x

32/

1x ---c

++

x5

2/4

x 2c

++

13 ---x3

x2

c+

+x

3x

2–

c+

x13 ---x

3–

c+

13 ---x3

12 ---x2

6x–

c+

–14 ---x

423 ---x

3–

32 ---x2

–c

+14 ---

x3

–

4c

+

25 ---x5

12 ---+

x4

13 ---x3

12 ---x2

c+

++

x12 ---x

223 ---x

32/

25 ---x5

2/–

c+

–+

27 ---x7

2/45 ---x

52/

23 ---x3

2/2x

–c

++

+

12 ---x2

3x

–c

+2

u2

5u

1u ---c

++

+1x ---

–2x2

-----–

43x3

--------–

c+

12 ---x2

3xc

++

12 ---x2

4x–

c+

13 ---t 32t

1t ---–

c+

+

47 ---x

74

2x

5x

–c

++

13 ---x3

12 ---x2

47 ---x7

2/–

45 ---x5

2/–

c+

+

12z 2

--------–

2z ---2

z 2z

c+

++

+12 ---t 4

tc

++

25 ---t 5

2t 3

c+

–13 ---u

32

u2

4u

c+

++

18 ---2

x3

+

4c

+3

x2

4+

c+

x2

x3

++

2x

13 ---x3

1+

–83 ---

x3

12 ---x2

403 ------–

–12 ---x

21x ---

2x

32 ---–

++

x2

+

3

34 ---x

43

14 ---x4

x+

+13 ---x

31

+x

4x

3–

2x

3+

+

12 ---x2

1x ---52 ---

++

2513 ---------

57 ---x

3237 ------

+

Px

255

x–

13 ---x2

+=

N20000

201---------------t 2.01

500t

0

+

=y25 ---x

2–

4x

+=

y16 ---x

354 ---x

22

x+

+=

y2

x3

x2

x+

+

=

fx

310 ------x3

–4910 ------x

135 ------–

+=

15 ---e5

xc

+13 ---e

3x

c+

12 ---e2

xc

+10e

0.1x

c+

14 ---–

e4

x–

c+

e4

x–

–c

+

0.2e0.5

x–

c+

–2

e1

x–

c+

–5

ex

1+

c+

e2

2x

–c

+

3e

x3/

c+

2e

xc

+

4xe

cx

0

+

log3

xelog

–c

+x

0

25 ---

xelog

cx

0

+

x1

+

e

logc

x1–

+12 ---

xelog

cx

0

+

x2

xelog

1x ---c

x0

+–

–

12 ---x2

2x

–xe

logc

x0

++

3x

2+

c+

ln

13 ---–

3x

cosc

+12 ---

2x

sin

c+

15 ---5

x

tan

c+

xcos

c+

12 ---–

2x

cos12 ---x

2c

++

2x

314 ---

4x

sin

–c

+15 ---e

5x

c+

43 ---–

e3

x–

212 ---x

cos

–c

+3

x3 ---

sin13 ---

3x

cos

c+

+

12 ---e 2x

4xe

logx

–c

x0

++

12 ---e 2x

2e x

xc

++

+

54 ---4x

cos

xxe

logc

x0

+–

+13 ---

3x

tan2

xelog

2e

x2/

cx

0

+

+–

12 ---e 2x

2x

–12 ---e

2x

––

c+

12 ---e 2x

3+

c+

12 ---2

x

+

cos

–c

+

x

–

sin

c+

SL

Mat

hs 4

e re

prin

t.boo

k P

age

864

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

865 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

5n

o

6a

b c

d

e f

g h

i j

k

l m

n

o

7a

b c

d e

f

g h

i

8a

b

c d

914

334

1013

.19m

s–1 or

1.1

9ms–1

112.

66 cm

12 13

a ,

b

14a

0.25

ab

15b

666

g

16a

b 73

.23%

c ~25

.24

litre

s

17a

b 70

00c 1

.16

day

d 2

days

Exer

cise

22.

4.1

1a

b

c d

e f

g h

i j

k

l m

n

o

p q

r

s t

u

v w

x

2a

b c

d e

f g

h i

j k

l

3a

b c

d

41 4---

x 2---

+

co

s–

c+

2ex

2+ ex

--------

------

c+

1 16------

4x1

–

4c

+1 21------

3x

5+

7

c+

1 5---2

x–

5

–c

+1 12------

2x

3+

6

c+

1 27------

73

x–

9

–c

+1 5---

1 2---x

2–

1

0c

+1 25------

5x

2+

5––

c+

1 4---9

4x

–

1–

c+

1 2---x

3+

2––

c+

x1

+

ln

cx

1–

+

2x1

+

ln

cx

1 2---–

+

23

2x

–

ln

–c

x3 2---

+3

5x

–

ln

cx

5

+

3 2---3

6x–

ln–

cx

1 2---

+

5 3---3x

2+

lnc

x2 3---

–

+

1 2---2

x3

–

x2

–c

+co

s–

62

1 2---x

+

5

xc

++

sin

3 2---1 3---

x2

–

2

x1

+

c

+ln

+si

n

100.

1x

5–

2x

–c

+ta

n2

2x

3+

2e

1 2---x

–2

+c

++

ln2

2x

3+

--------

-------

–1 2---

e2x

1 2---–

–c

+

xx

1+

4x

2+

c+

ln–

ln+

2x

3x

2+

1 2---2

x1

+

c

+ln

+ln

–

12

x1

+----

--------

---–

2x

1+

c+

ln+

fx

1 6---4

x5

+

3=

fx

24

x3

–

2

+ln

=

fx

1 2---2

x3

+

1

+si

n=

fx

2x

1 2---e

2x

–1

+1 2---

e+

+=

2ex

2/1 2---

2x

sin

–2

–

pa

a2b

2+

--------

--------

--=

qb

a2b2

+----

--------

------

–=

1 13------ e

2x

23

x3

3x

cos

–si

n

c

+

a1 2---

8

3/

0.15

75a

12

13.5

10.5

2412

618

6am

12

noon

6pm

1

2pm

6a

m

V'

t

t

57 3

V'

t

t

B A

4

2 3---5x

22

+

32/

c+

1

3x3

4+

--------

--------

-------

–c

+3 8---

12

x2–

4

c+

1 5---9

2x3

2/+

5

c+

9 4---x2

4+

4

3/c

+1–

2x2

3x1

++

2

--------

--------

--------

--------

------

c+

4x2

2+

c+

1

121

x4–

3

--------

--------

--------

----c

+

2 3---1

e3x

+

32/

c+

1–

2x2

2x

1–

+

----

--------

--------

--------

-------

c+

2 3---x3

3x

1+

+c

+

1 12------

34

x2+

3

2/c

+2

ex2

+c

+1 4---

1e

2x

––

2––

c+

2 3---x3

1+

5

c+

1 24------

x48

x3

–+

6

c+

1 5---x4

5+

5

2/c

+1

2x

sin

––

c+

2 9---4

3x

sin

+

32/

c+

112

13

4x

tan

+

----

--------

--------

--------

--------

--–

c+

3 2---x

xco

s+

2

3/c

+

1 2---x 2---

c+

4co

s–

21

xx

sin

+c

+4 3---

x12/

1+

3

2/c

+

ex21

+c

+6

ex

c+

1 3---e

3x

tan

c+

e–a

x2b

x+

–c

+6e

x 2---co

s–

c+

4e

4x

1–+

–c

+1 2---

2ex

c+

cos

–1

21

e2x

–

----

--------

--------

-----

c+

1e

x–+

c+

ln–

5 2---1

2ex

+

c

+ln

2 3a------

–4

ea

x–

+

32/

c+

1e2

x+

ln2

4----

--------

--------

--------

--------

c+

x21

+

c

+co

s–

10x

c+

cos

–2

21 x---

+

c

+si

n–

2 3---x

cos

3

2/–

c+

SL

Mat

hs 4

e re

prin

t.boo

k P

age

865

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

866

MATH


ANSW

ERS

e f

g

h i

j k

l m

n

o

4a

b c

d

e f

g h

i j 0

k l

m

n

Exercise 22.4.2

1a

b c

d

e f

g h

i j

k

2a

b c

d

e f

3a 0

b c

d e

f

4a

b c

d e 1

f

5a

b c

d

6a

b c

d e 3 + 2ln4

7a

b c

d e

f g

Exercise 22.5

1a

b c

d –8

2a

b c –2

d 0e

f g

h i

j 0

k l

4a e b

c 0d

e f

g h

i

6a

b 2ln5c 4 + 4ln3

d e

f 2ln2 g h 4ln2 – 2

i

8a 1

b c

d –2e

f 0g 0

h i 0

9a

b c 0

d e

f 1 – ln2

10

11; 0

12a

b

c d

e

13a

; b i 99 accidents

ii 14

a 1612 subscribersb 46220

15b ~524 flies

Exercise 22.6

1a 4 sq.units

b sq.units

c 4 sq.unitsd 36 sq.units

e sq.units

2a e sq.units

b sq.units

c sq.units

d sq.units

3a

sq.unitsb 2ln5 sq.units

c 3ln3 sq.unitsd 0.5 sq.units

4a 2 sq.units

b sq.units

c sq.units

d sq. units

e sq.units

612 sq. units

7sq.units.

8ln2 + 1.5 sq.units.

92 sq.units.

10 sq. units

13 ---3

xcos

c

+log

–43 ---

13

xtan

+

c

+log

4–3

3x

1+

tan

------------------------------------

c+

2x

ln

c

+sin

16 ---1

2x

cos+

3

2/–

c+

ex

c+

sine

x3

–2

+

–

c+

12 ---xsin

ln

2c

+x

c+

sec14 ---

12

ex

+

ln

2c

+13 ---x

33

x–

c+

tan

5313779

------------------2

2–

21

e+

+3

22

2+

lne

e1–

sin–

sin

23 ---1

2 ---

32/

cos–

23 ---e

e1–

–2

ln7

73

----------35 ---

164 ------13 ---

160 ------–23 ---

x2

1+

3

2/c

+23 ---

x3

1+

3

2/c

+13 ---

4x

4–

1.5

–c

+x

31

+

c

+ln

1

183

x2

9+

3

--------------------------------–

c+

ex

24

+

c

+z 2

4z

5–

+

c

+ln

38 ---2

t 2–

4

3/–

c+

ex

sinc

+e

x1

+

c

+ln

15 ---sin5x

c+

ex

tanc

+1

2x2

–

c

+ln

–1

12

x2

–------------------

c+

12 ---x

ln

2c

+

1e

x–+

c+

ln–

xln

c

+ln

22

ln3-----------

7754 ------ln

2ln

13 ---2

ln14 ---

77

3----------

83 ---–

38 ---

2cos

1–

10425

------------4

ln54 ---

e5

e1–

–

14 ---2

23 ---3

–3180 ------

42

2–

25 ---–

325 ---

3263 ------

43 ---–

3 ---8

Sin1–

23 ---

4 ---12 ---S

in1–

12

22

–2 ---

–4 ---

2

Tan

1–13 ---

–

152 ------383 ------

536 ------

3524 ------85 ---

22

–120 ------

43 ---–

76 ---56 ---

203 ------

203 ------23 -------

–

2e

2–e

4––

2e

e1–

–

e

24

e2–

–+

12 ---e

e5

–

2e

3–

14 ---16e

14/

e4

–15

–

12 ---

e1–

e3

–

32

ln1717

4------------

32 ---3

ln34 ---

2ln

33

2----------

32 -------

2

32 ------1

–32 -------

12 ---–

315 ------7

73

----------3

–572 ------

32

332 ---

–

215 ------

ln

2x

2x

2x

cos+

sin2m

n–

ma

b–

+3

n–

m2

ab

–

n

a2

e0.1

x0.1

xe0.1

x+

10xe

0.1x

100e

0.1x

–c

+

N12

t10

te0.1

t100

e0.1

t–

978+

+=

323 ------16 ---

12 ---e

42

–e

2–

2e

e1–

2–

+

2e

22

–e

–

54 ---

ln

2 ---38 ---

22

2–

+2

43

43

13 ---–

3712 ------

SL

Mat

hs 4

e re

prin

t.boo

k P

age

866

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

867 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

11a

0.5

sq. u

nits

b 1

sq. u

nit

c sq

. uni

ts

12

13–2

tan2

x;

sq.u

nits

14a

sq. u

nits

b 3

sq. u

nits

15a1

sq.u

nit

b 10

sq. u

nits

16

a xl

nx –

x +

cb

1 sq

. uni

t 17

sq. u

nits

18a

sq. u

nits

b sq

. uni

ts

19a

i sq

. uni

tsii

sq. u

nits

20 sq

. uni

ts

21b

i sq

. uni

tsii

1 sq

. uni

tiii

2ln

(2) s

q. u

nits

22b

3.05

sq. u

nits

23a

b sq

. uni

ts

24a

sq. u

nits

b sq

. uni

tsc

~ 0

.100

66 sq

. uni

ts25

a =

16Ex

ercise

22.

7

1a

b c

2a

b 10

0c

m

3a

b 6.

92 m

4 m

5 s;

63.

8 m

6a

sb

m7

80.3

7 m

8a

b 86

.94

mc –

6.33

md

116.

78 m

9a

b k

= 2

c 52.

2 m

10b

0.08

93 m

Exer

cise

22.

8A

ll va

lues

are i

n cu

bic u

nits

.1

21

2

ln5

3

4

5

6 9 10 11

12

13

14

15k

= 1

16 17 18 19a

Two

poss

ible

solu

tions

: sol

ving

, a

= 4

.953

31;

solv

ing

, the

n a

= –0

.953

31b

20 2164

Revision

Set

A1

–84

2a

b i ]

–1,

[ii

c

384

0

4a

i 0ii

2b

–2

x

2c x

0

26

2–

8 3---

1 4---2

ln

9 2---

14 3------ 7 6---

9 2---

15 4------

45 4------

22 3------

e1–

e2

–+

2y

3a

xa3

–=

1 15------ a

5

1e

1––

e1–

1e

e1–

–1

––

e1–

–

xt3

3t

10t

0

+

+=

x4

t3

tco

s+

1t

0

–

sin

=x

t24

e

1 2---t

––

2t

4t

0

+

+=

xt3

t2t

0

–

=10

08 27------

x2 3---

4t

+

32/

–2

t8

++

=

125 6----

-----

125

49--------- 6---

2---1

–

st

160 ----

-----1

16------ t

cos

–t

0

=

v4

kk t2----

t0

–+

=

2---e1

0e2

–

2 2--- 8 3---

23

ln–

2---5

51

sin

–

251

30---------

242 5----

-----

4--- 88 5------

3

3 4------

42

a2

k 2---

=

8 15------

a

1a2

+----

--------

---3

a2

2+

1a2

+----

--------

------

a3

6a2

–36

a–

204

+0

=

a36

a2

–36

a–

28–

0=

a10

0 ---------

=

28 15------

y

x

y =

–1

f1–

x

x1

+

ln

=

y

x

y =

–1

x =

–1

f

f1–

SL

Mat

hs 4

e re

prin

t.boo

k P

age

867

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

868

MATH


ANSW

ERS

5a (1, –2), (–1, 4) and (3, 0)

b

6a 2

b S = [0, [, range = [1,

[c f –1: [1,

[ , f –1(x) = (lnx) 2

8a i 512

ii 2b i

ii 9

a i –1 or 6ii

b i ii 0.2

iii 0

10a i 2 or 6

ii b i 0 < x < 1

ii iii

iv

11a

, b

12a i

ii b

c i ,

ii [–1,1]\{0}

130.5

15a k = 0 or 16

b c 0 < x < 3

16a 0 < x < 5

b 70c –2,

,117

a 9b –4

18±3

19a

b

20b

21b ii

, 22

a b 59136

23

24a ii {±1} b i

ii 25

26

a b

c

27a

b ]–,4]

c ]–,4[

28b

sq units

y

x

g(x)

f(x)

(–1, 4)

(3, 0)

(1, –2)

1

y

x

(2, 2)

(4, 6)

y

x

(1, 0) (2, 4)

–2y

x

(0, 2)

(4, 4)

(–2, 0)

7 a b

y

x

(1, –2)

(5, 4)

(3, 0)

c d

3x

2h3

xh2

h3

++

3x

23

xhh

2+

+3

e1

–-----------

\3

13 ---e

24

–

4

0.72

elog

e0.8

1e

0.8+

------------------0.69

gf

x

2

x1

x–

-----------–

=x

\1

P

24

x6

ln3

ln --------=

157 ------1

3+

fg

x

1x2

-----1

–=

gf

x

1

x1

–-----------

=

(0, 2) y

x–2 –1 0 1 2 3

–1–2 32

y

x1

–3

–3

–2 –1 0 1 2 3 4–1–2 432

y

x1

–3(–2, 0)

(3, 0)

(3, 1)(–2, 1)

14 a b c

2x

1–

3x

2+

x3

+

12 ---

–

y2

x–

=x

y–

xy

+-----------

x49 ---

y–

19 ---=

=

p5

35

p+

=p

5–5

p8

–=

29 ---

a35 ---

b–

64825---------

n–

10=

==

y6

x3

–

=

x9

y6

==

1792x

5

52 ---32 ---

–32 ---

12 ---–

172 ------

58 ---

2c 4c

d0

c, dy

x

3c, d

2c 3c 4c

d0

2 c, dy

x

4c, d

29 a b

SL

Mat

hs 4

e re

prin

t.boo

k P

age

868

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

869 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

35a

150

cmb

138

cmc 9

4 hr

sd

[0, 9

4]e

f Use

gra

phic

s cal

cula

tor.

g 17

.3 h

rs36

a b

]0, 2

]c N

o (x

= 0

)

3778

38a

0b

c , i

.e. d

oes n

ot ex

ist

39

40a

b

41b

, ran

ge =

]–

, 4]

42a

Use

gra

phic

s cal

cula

tor.

b c U

se g

raph

ics c

alcu

lato

r.43

–10

44a

exist

s; d

oesn

’t ex

ist.

b x

< –2

or x

> 2

45a

b d

oes n

ot ex

ist;

exist

s.c

46

a t =

2 o

r 3b

t = 3

c 47

a i 5

0ii

c d

i 50

ii 33

4.5

f Inc

reas

ing

at a

decr

easin

g ra

teg

~ 46

0 w

asps

h ii

t = 0

and

Re

vision

Set

B1

a 18

9b

99c –

96d

362

b –6

53

b 23

9 km

c 264

°d

153

kme 1

075

30 a

i &

ii

b i

ii $

200

iii $

4

iv

t = 0

42, 1

57

0

1

y

x

4

05

fx

x22

x–

2+

= y1 fx

--------

-=

1

y

t

200

1,40

0

31 a

b 3

y

xa

4

2

y = g

x

y =

fx

32 g

1–x

e2x

ex

+

=y

x

y =

e

33 a

i ]0

, [

ii b

c

d

a eb------

–

1a

b elo

g

xb

1 x---1

–=

y

x1 b

34 a

a =

–36

, b =

900

b

c 20

d 10

00

e

t >

195

f

t1

235

513

104

5270

295

723

9977

8B

t

y

t

100

1000

5y =

At

y =

Bt

h1–

x

12.5

t–

0.13

--------

--------

------

=

12

y

x

2–

r fd g

63 8------ x

5–

g1–

x

1–

x2

–x

2

+

=

3

3

y

x

y =

x

yg

x

=

yg

1–x

=

hx

4x

x0

–=

f1–

x

1x

–

e

x1

log

–=

r gd f

fog

r fd g

go

f

f1–

x

2x

–

2x

2

=

r gd f

1–f

1–o

g

r f1–

d gg

of

1–

F

x

x2

x2

–=

x1

+

y4

z

–

=

==

50e

135.

9

500

y

xy =

Q(t

)

y =

P(t

)

t10

9 elo

g=

SL

Mat

hs 4

e re

prin

t.boo

k P

age

869

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

870

MATH


ANSW

ERS

4a i A

: $49000; B: $52400; C: $19200ii A

: $502400; B: $506100; C: $379400 b 46%

c i 14 months

ii C never reaches its target5

a r = 05b 625 cm

6

b or

7b

8

a 289

a b

10a M

ax. value is for

or , w

here k is an integer;

min. value is

for , w

here k is an integerb

11a

b c

, 420

12

14a

b i ii

15

a ~342b 20 term

sc 0 < x < 2

d {1, 3, 8, 18, . . . }e

f $4131.4516

a b 4

17a 120°

b cm

2

18a i

mii

mb ~1.15 m

c

19a

b

20a 8 cm

b

213

22a

b

23a $77156.10

b

24a

b c 3

25b i BP = 660 m

, PQ = 688 m

26216°

27b 906 m

28

a b 0.08004 m

2c $493.71

29a

b range = [3, 3.5]c i 3

ii 2

30a i

ii iii

b Am

p = 5, period = 50 weeks

d $27.07 e during 7th & 46th w

eeks

31a $49000, $47900, $46690

b $34062.58c 18.8 years

d ~$24856432

a ii 26 cardsb 26, 40, 57, 77

c a = 3, b = –d 155 cards

e 33

a ~2.77 mb i 3.0 m

ii 2.0 mc 4.15 pm

d Use graphics calculator.

e 34

1.262 ha

35

36a

b c

371623 m

38

a 19.5°Cb

d Use graphics calculator.

e 8 am to m

idnight39

1939 m40

a ii b 2000, 2200, 2420, 2662, 2988.2

c 52 hrsd 176995

41a

b

c ii r =

iii

d i

ii

e Geom

etric

Revision Set C1

a b –8

c

2a

b

30

2634135

76 ------116---------

2 ---

3 ---23 ------

43 ------53 ------

02 ---

2

172 ------x

2 ---2

k+

=x

32 ------2

k+

=

175 ------x

k=

3 ---53 ------

un

746

n–

=n

16 ---74

p–

=112 ------

74p

–

68

p+

244

33

–

39

-------------------------------

6010928

25032300

2cosec

3 ---23 ------

un

233

n–

=

12 ---–

143

0.33

0.23

7313

3 ---43 ------

x3 ---

x43 ------

284

12 ------712------

1312

---------1912

---------

3 ---1

u1

3u

3

–3

3–

==

fx

32

x

cos

=76 ------

3840

tan

15

+2----------------

–=

W4

19.38P

4

14.82=

=W

20

10.95

P20

27.02=

=W

35

13.45

P35

23.25=

=

10 20 30 40 50

30252015105

y = W

(t)

y= P

(t)

y

t

c

tnn2 ---

3n

1+

=

216 ---

t6

13 ---

4 ---34 ------

x23 ------

23 ------

–=

1–1

–

y

x

23 ------x

23 ------

–

Dt

1–

212 ------t

cos+

=

N0

2000

10=

=

4

–

cm

24

–

2-----------------

cm2

12 ---A

n4

–

12 ---

n1

–n

1

2

==

3116 ------4

–

cm2

24

–

cm2

7i

–6

jk

++

a13

-------i

jk

++

=

x72 ---

y

+

z92 ---

5+

==

=x

3.5–1

----------------y1 ---

z4.5

–5---------------

==

SL

Mat

hs 4

e re

prin

t.boo

k P

age

870

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

871 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

4t =

2, (

16, –

8, 4

) 5

a b

100°

c 6

a b

7a

i 90°

ii

b i

ii iii

iv

8a

27°

b

9a

i ii

iii

b c i

no

ii lin

es ar

e ske

w

10

11a

b

12 o

r

13a

5b

14,

; 15

Yes

16a

i (1,

–1,

2);

ii (3

, 6, 2

); b

lines

do

not m

eet

17 18 19 o

r 2

20a

b (u

nits

in m

etre

s)c T

hey

do n

ot co

llide

.

21a

b c ~

129

.31

km

22a

b

c 0

.316

923

0.02

28

24a

0.12

b 0.

6087

25a

0.89

b c

26a

0.46

b

27a

3326

400

b i

ii

28a

0.97

72b

0.34

13

29a

0.93

6b

5

30a

792

b 35

31

a 15

1200

b 0.

1512

32

0.28

5233 34

a 0.

10b

0.40

c v

alue

s are

: 0, 0

.40)

, 1, 0

.50)

, 2, 0

.10)

d

35a

0.86

64b

0.72

10c 0

.903

4d

9.88

55 <

Y <

10.

2145

e 79.

3350

36a

315

b 17

280

37

38a

b

39a

, i.e.

geo

met

ricb

i 0.0

670

ii 0.

4019

iii

40a

b

b v

alue

s are

: ,

, ,

,

c EX

= , v

arX

= d

0.00

064

41a

0.30

85b

0.00

91c 0

.158

742

100

43

a b

c 44

b v

alue

s are

: 1,0

.4, 2

,0.3

, 3,0

.2,4

,0.1

c i 2

ii 5

iii 3

45

a 0.

8186

b 0.

1585

46

a v

alue

s are

: ,

, ,

b ii

. 0.0

064

iii. 0

.770

5

47

48a

b i 0

.308

5ii

0.17

47

49a

i 0.8

ii 0.

25b

i 0.4

ii

50a

i ii

iii

iv

v

51 52

53

54a

v

alue

s are

: ;

b c

55a

0.4

b 0.

096

c 0.2

25d

0.63

5

56a

3i

j–

2k

–4

i3

j–

3k

–

r Bm

in2

2=

t5

b2 5---

==

7 2---26

unit

2s

3p

+s

2p

+1 2---

s2

p+

1 2---–

s2

p+

1 2---17

unit

2

x y

2 3–

3 7

+

=x

23

y+

3–

7

+

==

x2

– 3----

-------

y3

+ 7----

--------

=

i–

11j

+283

5

a3 2---

=b

3 2---c

1 3---=

=

4 77----

------

5 4---i

–j

3 2---k

++

4 77----

------

–5 4---

i–

j3 2---

k+

+

5 3---5

OA

2i

2j

–k

+=

OB

4i

3k

–=

703

2

1 6----

---1 6

-------

2 6----

---

–

3 7---

6 7---2 7---

x1

– 3----

-------

y2

– 2----

-------

z3

+ 1----

-------

==

2 5---23 5----

--

2 3---

r A0

8000

0

t

3 2–

+=

2160

065

600

r4 2

t2 3

+=

LP

21– 11–

t2 3

+=

1 4---3 8---

21 40------

40 89------

9 23------

2 11------

2 77------

128

850

---------

0.15

06

xP

Xx

=

EX

0.70

var

X

0.

41=

=

193

512

--------- 2 3---

1 2---

PX

x=

1 6---5 6---

x

x

01

==

1 6---

13 44------

9 44------

xP

Xx

=

19 25------

3

7 25------

5

5 25------

10

3 25------

20

1 25------

105

25---------

4.2

11

400

625

--------

-------

18.2

4

1 2---1 7---

2 7---

xP

Xx

=

xP

Xx

=

03 16------

1

7 16------

2

5 16------

3

1 16------

0.

9586

0.

0252

==

10 21------

EX

0.8

var

X

14 25----

--=

=

1 8---47 72----

--1 8---

47 72------

9 47------

189

8192

--------

----

43 60------

0.71

67

117

145

---------

0.80

69 x

PX

x=

0

1 6---

1

1 3---

2

1 2---

E

X

4 3---va

rX

5 9---

==

2 3---5 24------

3 5---

SL

Mat

hs 4

e re

prin

t.boo

k P

age

871

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

872

MATH


ANSW

ERS

57b

values are: ,

, c

d

58a i

i iii

iv v

b

59

60a 0.1359

b 137.22c 137

d a = 141.21

61a

b c not independent

62a

b

63a 0.081

b

64a 0.0169

b i 0.9342ii 127

iii 0.00865

a 0.1587b 0.7745

c $0.23 66

a i

ii b i 11.44ii 2.6695

c i ,

, med = 12,

,

ii med = 12, m

ode = 12iii 4

67a i 0.24

ii 0.36b

c Q > 29.17

68a use graphics calculator

b i ,

ii A:

, B: c i Class A

: ,

, med = 36,

, C

lass B: ,

, med = 37,

,

ii Class A

: med = 36; m

ultimodal – 34, 35, 39, 43, 48

Class B: m

ed = 37; multim

odal – 27, 34, 38, 42, 49iii C

lass A: IQ

R = 12, Class B: IQR = 13

d Results from both classes are very close, how

ever, Class B does slightly better as it

has a larger median as w

ell as the larger maxim

um value.

Revision Set D1

a b

2

3a

b c 4 sq. units

4a 19.8°C

b 1.6°C per minute

c 17.3 min

5a

b

610 m

7

a b

or

8a i 0

ii 2b

c x 0d

9a

b –2

10a 74

b 0.69 11

1.455 12

a Absolute maxim

um at

; local min at 0, 0; x-intercept at ±1, 0

b Local min at

; asymptotes at

, y = 0.

13a

b

14

15720

xP

Xx

=

0425 ------

1

1225 ------

2925 ------

E

X

1.2va

rX

0.48

==

37 ---

815 ------715 ------

15 ---45 ---

47 ---x

pq

–

100

q+

100--------------------------------------

23 ---

13 ---

23 ---29 ---

b6

a+

0b

13 ---

413 ------

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

654321

frequency

x

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

252015105

cum. freq

x

xm

in4

=xm

ax

16=

Q3

13.5=

Q1

9.5=

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16x

1720.96

Q+

xA37.35

=xB

37.31=

sn8.801

=sn

9.025=

xm

in23

=xm

ax

56=

Q3

43=

Q1

31=

xm

in22

=xm

ax

57=

Q3

44=

Q1

31=

0 10 20 30 40 50 60

Class A

Class Bx

x2

4+

-------------------2

2x

22

x1

–

2

xsin

–cos

3013 ---

4elog

+

3 ---43 ------

x3 ---

x43 ------

x1

0

0

–

x

0

2

–

4x

x2

1+

2

----------------------4

2x

2x

cossin

–2

4x

sin–

x2

2–

x

4x

2–

------------------2

x2

––

2h

+

1h

+

2--------------------

h0

–

ms

1–

12-------

1

12-------

1

x

1

=

62

xsin22

xcos

x3

+

2x

3+

3

2/----------------------------

12 ---e

2x

4x

–e

2x

–+

c+

m3

SL

Mat

hs 4

e re

prin

t.boo

k P

age

872

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

873 MATH

EMATI

CS –

Sta

ndar

d Le

vel

ANSW

ERS

16a

b 98

°

17b

i 2ii

72

18

a A

rea =

, V

olum

e =

b

19a

(–1,

4),

(1, –

2), (

3, 0

)b

use g

raph

ics c

alcu

lato

rc

sq. u

nits

20 21a

b ra

dius

= 5

cm, h

eigh

t = 1

2.7

cm

22a

b

23a

b c

24

a b

25a

b ~

38.3

mill

ion

c i d

ecre

asin

gii

0.1

mill

ion/

year

d 50

yea

rs ti

me,

i.e. 2

030;

42.

2 m

illio

n

2676

222

27

a b

use g

raph

ics c

alcu

lato

rc

sq. u

nits

28a

b

29a

b 12

30

a b

i y =

xii

iii

c i

ii cu

bic u

nits

31

a b

32

a i 2

83 se

cii

250

sec

c 244

sec

33a

b sq

. uni

tsc i

At (

0,1)

: y =

–2x

+ 1

At

: ii

iii

sq. u

nits

d

ii cu

bic u

nits

34b

i 0 <

x <

0.5

ii x

= 0.

5iii

x <

0 o

r x >

0.5

c

d e i

y =

–x +

1ii

y = x

– 1

f i

sq. u

nits

ii sq

. uni

ts

35a

4.20

b i

ii –0

.40

36

cubi

c uni

ts

37a

= –1

, b =

6, c

= –

9

38a

b i

ii

39A

= 0,

B =

0.5

40

a sq

. uni

tsb

cubi

c uni

ts

41a

b c

d

e

42b

c cu

bic u

nits

43a

; c 1

, e; y

= ex

d sq

. uni

ts

(3, 3

) (4,0

)

(0,0

)

y

x

cm3

A8 15------ h

32/

=V

0.48

h32/

=5 14

4----

----- m

/min

16 3------

33 e

log

–

3 –1

–2

–1

x

y

h10

00

r2

--------

----=

3x

13

x2–

--------

--------

------

–ex

1ex

+

2----

--------

--------

--

3x2

h3

xh2

h3+

+3

x23

xhh2

++

3x2

23 4---

3–

3 elo

g

p't

0.8

10.

02t

–

e0.

02t

–= cm

3

A1

5–

B

13

C4

0

124

4 elo

g–

1x

xx

sin

+co

s+

1x

cos

+

2----

--------

--------

--------

--------

-----

x

x21

+----

--------

--

126

hh2

h0

++

A2

2e

1–

d dx

------

xex

k/–

1x 2---

–

ex

2/–

=4

22

a+

e

a2/

––

1 2---a

2–

2x

x2–

e

x–

210

e2–

–

x elo

g2

2 e1

–lo

g

A1 2---

22

12

ln–

ln

1 2---

e25

–

1e2

4–

y

2e2

4–

x

e2–

=y

x(m

, n)

(0, 1

)

1 2---e2

5–

12------

3e4

24e2

–37

+

1 2---1

2ln

–

1 2---1

2ln

–

1 2---1

x

y 3 8---1 2---

2ln

–1 8---

1 2---2

ln+

1 2---1 4---

tco

s

3 e

log 4

33

--------

--6

3x

3x

cos

sin

–x 2---

1 12------

6x

c+

sin

+

7 12------

7 15------

V

r2h

4 3---

r3+

=P

2

krh

6

kr2

+=

P2

kV r----

------

10 3

--------

- kr2

+=

0r

3V 4

-------

1

3/

r3

V10

--------

-

13/

= a 4---a 2---

336----

---a3

2x

x ex

+lo

g2

2 e3 4---

–lo

g1 2---

e1

–

SL

Mat

hs 4

e re

prin

t.boo

k P

age

873

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

874

MATH


ANSW

ERS

44

45a a = 2;

, b

cubic units

46a i

ii [0, [

b 0.5c {0, 4}

d i ii {x | x > 4

e i 4 sq. unitsii

cubic units

47a

b

48a

b

49a

b c

sq. units

50a

sq. unitsb

cubic units

51a [0, 5]

b use graphics calculatorc 0.625

d

52c M

inimum

, ; M

aximum

53a

b 30 secondsc 116

metres

54a

b 1 55

a b

56a use graphics calculator

b d 1.12 sq. units

x16 ---

x2

4 ---x3

3 ---=

==

f1–

x2

x+

=x

0

1363 ---------

x

y1

4

g(x)

f(x)

2 ---e

43

+

62

xcos 22x

sin–

12

x2

–1x

2–

------------------

xx

xcos

+sin

xx

xcos

+sin

yex

–e

e1–

++

=

x

y

normal

curve12 ---e

e2–

+

43 ---6415 ------

a12 ---

15 ---t0

t5

–

=

3a

253 ---

13/

3a

294 ---

13/

t

v

20 40

20

23 ---

ex–

xx

sin+

cos

–

1t 2 ----–

1+

t2

t12 ---t 2

c+

++

ln

Ax

2x

x0

x2 ---

cos

=

SL

Mat

hs 4

e re

prin

t.boo

k P

age

874

Wed

nesd

ay, M

ay 2

2, 2

013

10:

37 P

M

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