Mathematics Standard Level 4eultimate2.shoppepro.com/~ibidcoma/samples/Maths... · MATHEMATICS –...
Transcript of 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
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ay 2
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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
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013
10:
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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
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age
809
Wed
nesd
ay, M
ay 2
2, 2
013
10:
37 P
M
810
MATH
EMATICS – Standard Level
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
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810
Wed
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ay 2
2, 2
013
10:
37 P
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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
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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
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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
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n)b
10 s
c 500
md
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7 s
e 750
m14
a $7
2500
b N
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oss o
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c 250
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i $70
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c Fix
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, ...)
d Se
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620
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6000
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1
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e gra
ph in
par
t a
Rx
xpx
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x–
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100
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==
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–
=
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x50
x–
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00 3----
--------
m2
x25
y20
0 3---------
==
100 3----
-----
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1–
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0] b
y [0
, 1]
[0, 2
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y [0
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usag
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00 M
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Cos
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0x
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0.65
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9516
0x
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2.
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500
3.
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0x
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80
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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
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k P
age
811
Wed
nesd
ay, M
ay 2
2, 2
013
10:
37 P
M
812
MATH
EMATICS – Standard Level
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
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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
EMATICS – Standard Level
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
EMATICS – Standard Level
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
EMATICS – Standard Level
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
EMATICS – Standard Level
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
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821
Wed
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ay 2
2, 2
013
10:
37 P
M
822
MATH
EMATICS – Standard Level
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
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822
Wed
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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
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823
Wed
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ay 2
2, 2
013
10:
37 P
M
824
MATH
EMATICS – Standard Level
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
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824
Wed
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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
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825
Wed
nesd
ay, M
ay 2
2, 2
013
10:
37 P
M
826
MATH
EMATICS – Standard Level
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
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prin
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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
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k P
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827
Wed
nesd
ay, M
ay 2
2, 2
013
10:
37 P
M
828
MATH
EMATICS – Standard Level
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
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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
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k P
age
829
Wed
nesd
ay, M
ay 2
2, 2
013
10:
37 P
M
830
MATH
EMATICS – Standard Level
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
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830
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ay 2
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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
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hs 4
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831
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10:
37 P
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832
MATH
EMATICS – Standard Level
ANSW
ERS
Chapter 7Exercise 7.1.1
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
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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
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013
10:
37 P
M
834
MATH
EMATICS – Standard Level
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
Chapter 8Exercise 8.1.11
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
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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
EMATICS – Standard Level
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
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836
Wed
nesd
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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
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837
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838
MATH
EMATICS – Standard Level
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
Chapter 10Exercise 10.11
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
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838
Wed
nesd
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2, 2
013
10:
37 P
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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
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839
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10:
37 P
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840
MATH
EMATICS – Standard Level
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
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t.boo
k P
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840
Wed
nesd
ay, M
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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
EMATICS – Standard Level
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
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843
Wed
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ay, M
ay 2
2, 2
013
10:
37 P
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844
MATH
EMATICS – Standard Level
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
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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
EMATICS – Standard Level
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
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hs 4
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846
Wed
nesd
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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
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ay 2
2, 2
013
10:
37 P
M
848
MATH
EMATICS – Standard Level
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
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848
Wed
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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
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prin
t.boo
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849
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nesd
ay, M
ay 2
2, 2
013
10:
37 P
M
850
MATH
EMATICS – Standard Level
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
Chapter 14Exercise 14.11
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
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hs 4
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850
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ay, M
ay 2
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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
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013
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852
MATH
EMATICS – Standard Level
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
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hs 4
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852
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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
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ay 2
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013
10:
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M
854
MATH
EMATICS – Standard Level
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
EMATICS – Standard Level
ANSW
ERS
6a
b i 5 ms –1
ii 4 ms –1
c m
s –1d
sec
Chapter 19Exercise 19.11
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
EMATICS – Standard Level
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]
Chapter 20Exercise 20.11
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
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age
859
Wed
nesd
ay, M
ay 2
2, 2
013
10:
37 P
M
860
MATH
EMATICS – Standard Level
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
EMATICS – Standard Level
ANSW
ERS
6a i (0, 1), (2, 0)
ii y = –1, x = –2iii
iv d = \{–2}
b c
7a y =
8, x = 3
b
Chapter 21Exercise 21.11
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
EMATICS – Standard Level
ANSW
ERS
Chapter 22Exercise 22.1
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
EMATICS – Standard Level
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
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867
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ay 2
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013
10:
37 P
M
868
MATH
EMATICS – Standard Level
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
EMATICS – Standard Level
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
EMATICS – Standard Level
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
EMATICS – Standard Level
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