mathematics – higher level (core)ultimate2.shoppepro.com/~ibidcoma/samples/maths hl...mathematics...
TRANSCRIPT
995
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
ANSW
ERS
Exer
cise
2.1
2a
[–2,
7]b
]9,∞
[c ]
0,5]
d ]–
∞,0
]e ]
–4,8
[f ]
–∞,–
1[ ∪
]2,∞
[3
a b
c 4
a 4
b 4
+ c 6
d 31
+ 1
25
a 2
– b
+ 2
c 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 ab
f
g 0
h
i a +
b
5a
–4, 4
b c –
6, 1
8
d e
f g
h i –
3, 0
j k
l
6a
b c
d 0
e f
7a
, 0
b ±a
c Ød
0, 4
a
8a
b x
≤ 1
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
6p
< 3
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
--------
--------
--------
--------
-----
-50
5–4
–20
24
–5–2
02
5–4
04
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
+(
)
ab
ab
–----
--------
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----
--b
0≥
,,
1 3---2 3---
3 2---3
1 3--- ,–
1 4---–
2a 3------
–
3 2--- ± 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≥
∪x
3 4---<
x9 4---
>∪
6x
14<
<–
x28–
<x
44>
∪x
5 12------
–<
x1 12------
>∪
x4–
≤x
16≥
∪
HL
Mat
hs 4
e.bo
ok P
age
995
Tue
sday
, May
15,
201
2 8
:54
AM
996
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
7 a
b
c
8a
b c
9a
b
Exer
cise
2.3
.1
2a
2b
3c
3a
y = 2
x –
1b
y = 3
x +
9c y
= –
x –
14
a b
c d
–
5y =
2x
6y =
–x
+ 1
7
82
9a
b c
d
Exer
cise
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
–3b
–5c –
1.5
4a
m =
2, a
= 8
b m
= 1
0, a
= 2
4c
m =
–6,
a =
9.
5a
b c
d e
f
Exer
cise
2.3
.31
a b
c
d e
f g
h Ex
ercise
2.4
.1
1a
–5b
4, 6
c –3,
0d
1, 3
e –6,
3f
g 2
h –3
, 6i –
6, 1
j
2a
–1b
–7, 5
c d
–2, 1
e –3,
1f 4
, 5
3a
b
c d
e f
4a
b c
d e
f –
4, 2
g h
i j n
o re
al so
lutio
nsk
2 3---–
x2
<<
3–x
1≤
≤0
x2
<<
a1
a+
--------
----x
a1
a–
--------
----<
<1– 1
a+
--------
----x
11
a–
--------
----<
<
∞a
2– a
1+
--------
----a
2
a1
–----
--------
∞
,∪
,–
4 3---–x
4 3---<
<3 2---
–x
3 4---<
<
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
x
y 3
–3/4
x
y3
3
x
y 2
4
x
y
6-2
x
y
2
5
x
y
1
-3
t
p
5-2
/3x
y
-1 -4/5
t
q 2-4
x
y
1
1/2
fg
hi
j
kl
m
n
o
5 3---
1 2---–
1 3---3 2---
4 5---
yx
2+ 2
--------
----=
y5 2---
x=
y3 2---–
x3
+=
y5 6---
x1 2---
–=
y2
x–
1+
=
10a
bc
d
x
f(x)
b a---
–bx
f(x) b– a2
------
b
Not
e si
gns!
x
f(x) –a
a+1
x
f(x)
2a
2a2
–
x13 11----
--=
y,17 11----
--=
x9 14------
=y,
3 14------
=
x4 17------
=y,
22 17------
–=
x16 7----
--–
=y,
78 7------
=x
5 42------
=y,
3 28------
–=
x1
=y,
ab
–=
x1–
=y,
ab
+=
x1 a---
=y,
0=
xb
=y,
0=
xa
b–
ab
+----
--------
=y,
ab
–a
b+
--------
----=
xa
=y,
ba2
–=
x4
=y,
5–z,
1=
=x
0=
y,4
z,2–
==
x10
=y,
7–z,
2=
=x
1=
y,2
z,2–
==
∅x
2t
1–
=y,
tz,
t=
=
x2
=y,
1–z,
0=
=∅
25 3--- ,
–0
3 2--- ,
2 5---–
3,
1–
6±
35
±1
5±
1–
33± 8
--------
--------
--------
973
± 4----
--------
-------
185
± 6----
--------
-------
337
± 2----
--------
-------
533
± 2----
--------
-------
333
± 2----
--------
-------
757
± 2----
--------
-------
7–
65± 2
--------
--------
--------
1–
22
±5
–53
± 2----
--------
--------
----3
37± 2
--------
--------
---4
7±
HL
Mat
hs 4
e.bo
ok P
age
996
Tue
sday
, May
15,
201
2 8
:54
AM
997
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
l no
real
solu
tions
m
n o
p
5a
–2
< p
< 2
b p
= ±2
c p <
–2
or p
> 2
6a
m =
1b
m <
1c m
> 1
7a
b c
8a
b c
10 4
Exer
cise
2.4
.21
Gra
phs a
re sh
own
usin
g th
e ZO
OM
4 vi
ewin
g w
indo
w:
2 3G
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),
(1.5
4, 0
), (0
, –4)
4a
x =
1b
(1, –
1)c
i ii
(0, 1
)
213
± 2----
--------
-------
32
11±
5----
--------
--------
--6
31± 5
--------
--------
---6
29± 7
--------
--------
---
m2
2±
=]
∞2–
2[,
–]2
2∞
[,
∪]
22
22[
,–
k6
2±
=]
∞6–
2[,
–]6
2∞
[,
∪]
62
62[
,–
ab
c
de
f
gh
i
jk
l
ab
c
de
f
gh
i
jk
l
22
± 2----
--------
----0,
1
1
HL
Mat
hs 4
e.bo
ok P
age
997
Tue
sday
, May
15,
201
2 8
:54
AM
998
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
5a
x =
1b
(1, 9
)c
i
ii (0
, 7)
6a
b c
7a
b c
8a
b
c
9a
b
c d
10a
b
c d
Exer
cise
2.4
.31
a ]–
∞,–
2[∪
]1,∞
[b
[–3,
2]c ]
–∞,0
]∪[4
,∞[
d ]
,3[
e ]–∞
,–1.
5]∪
[–1,
∞[
f ]0.
75,2
.5[
2a
]–∞
,–2[
∪]–
1,∞
[b
]–2,
3[c ]
–∞,–
0.5]
∪[3
,∞[
d [–
2,2]
e ][
f ]–∞
,–2]
∪[3
,∞[
g [
]h
[–2.
5,1]
i ]–∞
,–3[
∪]0
.5,∞
[j ]
1,3[
k ]–
1,0.
5[l Ø
m Ø
n [–
1.5,
5]o
]–∞
,–2[
∪]
,∞[
3a
–1 <
k <
0b
c n ≤
–0.
54
a i ]
–∞,–
1[∪
]2,∞
[ii
[–1,
2]b
i ]–∞
,2[∪
]3,∞
[ii
[2,3
]c
i ]1,
3[
ii ]–
∞,1
]∪[3
,∞[
d i ]
–,1
[ ii
]–∞
,–]∪
[1,∞
[ e
i ]–∞
,–2[
∪]2
,∞[
ii [–
2,2]
f i ]
[ii
]–∞
,]∪
[,∞
[
5]0
,1[
6[–
2,0.
5]
7a
i ][∪
][
ii ]
\{2}
iii ]
b i [
]ii
all r
eal v
alue
s 8
a {x
: x <
–3}
∪{x
: x >
2}
b {x
: –1
< x
< 4}
c i {
x: x
< 0
.5}
ii {x
: –2
< x
< 0}
9
a i ]
0,1[
(k
= 1)
; ]–
1,0[
(k
= –1
)ii
Øb
k >
1.25
Ex
ercise
2.4
.4
1a
(–2,
–3)
(2, 5
)b
(–2,
–1)
(1, 2
)c
d
e f
g h
no re
al so
lutio
ns
i j (
–2, –
3), (
2, 1
)
k no
real
solu
tions
2a
(1, 4
), (–
7, 8
4)b
c (0,
2),
(3, 2
3)d
e Ø
f (2,
8)
g Ø
h
3a
b c
4
5
1.75
6
7
80.
5
9
13a
i (1,
3),
ii (–
2, 1
2),
c i A
(1,3
), B(
–2, 2
)ii
4 sq
. uni
ts
Exer
cise
3.1
1a
b c
d
e f
2
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---–
x22
x–
40 3------
+=
1 3---
1–
21– 2
--------
--------
-------
1–
21+ 2
--------
--------
--------
,
15
– 2----
--------
----1
5+ 2
--------
--------
,
1 3---
22
–k
22
<<
2 3---2 3---
23
–2
3+
,2
3–
23
+
x
y
1
1
26
–2
2–
,2
2+
26
+,
21
2–
()
21
2+
()
,2
13
–(
)2
13
+(
),
513
–2
--------
--------
---1
13+
2----
--------
-------
,
1 3---–
2–,
2
5,(
),
3 2---–15 4----
--–,
10,
()
,
9 2---–19 4----
--–,
12–,
()
,3
73+
4----
--------
-------
3–
73– 8
--------
--------
-------
,
3
73–
4----
--------
-------
3–
73+ 8
--------
--------
--------
,
,
113
–2
--------
--------
---1
13–
,
1
13+
2----
--------
-------
113
+,
,
117
–2
--------
--------
---5
317
–2
--------
--------
------
,
1
17+
2----
--------
-------
53
17+
2----
--------
--------
--,
, 4 3---56 9----
--–,
3 4---7 4---
–,
,
aa2 –,
–()
a 2---a2 2----
-,
,
1 2---23 4----
--,
26
±m
26
m2
6>
,–
<2
6m
26
<<
–
80 23 12------
–
∞3–,
–()
1∞,
()
∪
ca m----
=
14 3------
–19
6 3---------
,
7 3---
49 3------
,
2x3
–6x
2x
7–
++
x4x3
–12
x22
x9
–+
+6
x57
x3–
9x2
2x
6–
++
2x7
2x6
–x5
–8x
45
x3–
2x2
–7
x3
–+
+4x
64
x4–
12x3
x26x
–9
++
+
9x3
12x2
–18
x–
13+
3x
12
x1
–----
-------
++
HL
Mat
hs 4
e.bo
ok P
age
998
Tue
sday
, May
15,
201
2 8
:54
AM
999
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
3 4
5
6
7
81
9
–1
10–1
2
115
Ex
ercise
3.2
.11
a 55
b 32
.7%
c 9%
2a
diag
ram
b 26
%c 8
03
a C
ontin
uous
Exer
cise
3.2
.21
(x –
1)(
2x –
3) –
2
2(x
– 3
)(3x
2 + 1
0x +
29)
+ 9
0
3(x
+ 3
)(2x
3 – 7
x2 + 1
9x –
57)
+ 1
744
(2x
– 1)
(x2 –
2x
+ 4)
+ 1
5
(x +
2)(
2x3 –
4x2 +
11x
– 2
3) +
46
6(4
– x
)(x3 +
4x2 +
18x
+ 7
2) –
283
Exer
cise
3.3
1a
–6b
15c 8
d 1.
25e
26
3
a 7
b 70
c –21
x +
28
4–6
5
a 11
b 3
Ex
ercise
3.4
1a
b
c d
e f
g
h i
j
2
3
40,
(x +
4)(
x +
1)(x
– 3
) 5
(2x
+ 1)
(x +
2)2
6(x
– 5
)(x2 +
x +
2)
7(x
– 1
)2 (2x
– 1)
(3x
+ 2)
8
a b
1, –
2,
c
9a
= –1
, b =
–2
11a
= –2
, b =
113
a =
–9, b
= 2
4, (6
x2 + 9
x –
2)14
x3 – 2
x2 + 8
x +
2 15
3x3 –
5x2 +
6x
+ 4
16
17–8
18
19 20
a b
21m
= 3
, n =
–4,
k =
–12
; 22
23
26 Ex
ercise
3.5
.11
a –3
, –1,
2b
, 1, 2
c –2,
–1,
3d
, , 4
e
f –1,
3g
–4, 1
h –2
, i
, j
,
2, 1
, 5
3,
, 24
–1, 2
, 35
–4, 1
6
a =
0b
= 0
c =
0
7a
, 1b
–1, 1
c –4,
–1,
2d
1,
, 3
2x2
3x
–11
82
x1
–----
--------
---–
+
x2x
–5
–19
6x
–
x22
x–
3+
--------
--------
--------
---+
2x2
7x
–7
17 x1
+----
--------
–+
x21
–3
x–
x23
+----
--------
--+
3x
1–
()
16 3------
3x
–2
x2–
52 3------
+
155
27---------
–
x3
–(
)x
2–
()
x5
+(
)x
2–
()
x23
x5
++
()
x2
–(
)x
1–
()
x2
+(
)x
2–
()
x2
+(
)3
x1
+(
)x
3–
()
x3
+(
)2
x1
–(
)x
2–
()
x1
–(
)x2
3x
4+
+(
)x
2–
()2
x3
+(
)x
2–
()2
5x
4–
()
x4
+(
)–
2x
5–
()
5x
2+
()
x1
+(
)–
25
x1
–(
)6
x34
x2–
2x–
3+
47 8------
–
x1
–(
)–
x2
+(
)3
x1
–(
)21 3---
x
y
1–2
1 3---
ac
+b
d+
=
a1
b,3
c,–
3d,
1–=
==
=x
2–
()
2x
1+
()
3x2
+(
)
m18 5----
--n,
39 5------
k,78 5----
--–
==
=x
2–
()
x3
+(
),
x33x
24
x–
12–
+x
2–
()
x2
+(
)x
3+
()
=k
2n,
–3
==
a3
b,6
a;–
3b,
–6
==
==
α2
αβ
β2+
+(
)xα
βα
β+
()
–
1 2---–
1 3---–
3 2---1 2---
210
±1 2---
–2
–2
±1 3---
6±
1 2--- 3 2---–
1 2--- x2
+(
)x
3–
()
x4
+(
)x
1+
()
x0.
5–
()
x2
–(
)8
x316
x2–
2x–
4+
7–
337
± 12----
--------
--------
-------
3 2---
HL
Mat
hs 4
e.bo
ok P
age
999
Tue
sday
, May
15,
201
2 8
:54
AM
1000
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
8a
–1.7
5, 0
.432
, 1.3
2b
3.77
c 0.3
09d
–1.6
8, –
0.42
1, 0
.421
, 1.6
89
–3,
, 210
No
othe
r sol
utio
ns11
m =
1, n
= –
6; x
= 6
, –1,
212
1, 4
, 7Ex
ercise
3.5
.21
a ]–
1,1[
∪]2
,∞[
b ]–
∞,–
2]∪
[2,3
]c [
–3,–
2]∪
[2,∞
[d
]0,∞
[ \{1
}e {
–2}∪
[–0.
5,∞
[f ]
–∞,–
4[∪
]–2,
2[g
]–∞
,1[∪
]2,∞
[\{–
1}h
]–∞
,2]∪
{3}
2a ]
–3,–
1[∪
]2,∞
[b
]–∞
,]∪
[1,2
]c [
–2,–
1]∪
[3,∞
[d
],
[∪]4
,∞[
e [,∞
[
f ]3,
∞[
g ]–∞
,–4[
h [–
2,]∪
[,∞
[i ]
–∞,
]∪
j ]–∞
,–[∪
],
[k
]–∞
,1]
l ]–∞
,[∪
],∞
[
Exer
cise
3.6
1 2---–
1 2---–
1 3---–
3 2---1 2---
210
–2
10+
2–
2–
2–
21 2---
–]
,+
[
61 3---
61
13–
2----
--------
-------
113
+2
--------
--------
---
1a
b
c
x
y
2–2
x3
–21
x
y
2–1
2 0.5
g
h
i
x
y
2
–6
3–1
/3
x
y
2–0
.52 1/
3
x
y
4
x
y
0.5
–0.5
x
y
3
d
e
f
x
y 3
–33
1
j
k
l
x
y
2–2
x
y
1.5
–1
x
y
2
m
n
o
x
y
2–1
3
x
y
2–2
x
y
3–3
p
q
r
x
y
1–3
x
y
4–2
x
y
1–3
–1
s
t
x
y
2–2
x
y
2
0.5
1
2a
bc
x
y
14
–1
x
y
24
x
y
–30.
5
de
f
x
y
4–2
x
y
2–2
1
4
–1
x
y
2
HL
Mat
hs 4
e.bo
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age
1000
Tue
sday
, May
15,
201
2 8
:54
AM
1001
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
4a
b c
d e
f
5a
b c
Exer
cise
3.7
1a
Sum
= –
2 P
rodu
ct =
4b
Sum
= 3
Pro
duct
= –
7c S
um =
4 P
rodu
ct =
–3
d Su
m =
P
rodu
ct =
e S
um =
P
rodu
ct =
f Sum
=
Pro
duct
=
g Su
m =
P
rodu
ct =
h Su
m =
P
rodu
ct =
i Sum
=
Pro
duct
=
C
onsid
er th
e pos
sibili
ty o
f a ze
ro d
enom
inat
or!
2a
3b
–1c
d e
f g
h 4
i 7
3Th
e cub
ic ca
se:
giv
es
. The
fact
oriz
ed
vers
ion
is:
. The
onl
y sim
ple c
onclu
sion
is th
at th
e pro
duct
of
the r
oots
is
.4
This
is re
late
d to
the c
onju
gate
root
theo
rem
. The
coef
ficie
nts m
ust b
e rea
l.Ex
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 b
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%
gh
i
x
y
2–1
x
y
3–1
0.5
x
y
2–1
0.5
j
x
y
1–2
3a
iii
b i
ii
x
y
–bb
x
y
11
b2+
,(
)
x
y
b2
x
y
–b2
y1 15------
x3
+(
)x
1–
()
x5
–(
)–
=y
1 8---x
2–
()2
x4
+(
)=
y3 2---
x2x
3–
()
–=
y1 30------
x2
+(
)275
29x
–(
)=
y1 6---
x2
+(
)4
3x–
()
x3
–(
)=
yx3
–x2
–2x
8+
+=
y1 2---
x2
+(
)x
2–
()3
=y
1 35------ x
2x
3–
()
x5
–(
)=
y1 6---
x2
+(
)2x
1–
()
x3
–(
)–
=
6a
bc
x
y
–bc
bx
y
bx
y
bx
y
b
or
7a
b
x
y
aa+
1 a0,
()
a1
1,+
()
,{
}
x: x
a1
+>
{}
7 5---3 5---
5– 2------
3– 2------
4 9---2– 9------
7 3---4 3---
8– 3------
13– 5---------
3 4---1– 8------
1 2---1– 2------
5– 3------
9 5---3 2---
ax3
bx2
cxd
++
+0
=x3
b a---x2
c a-- xd a---
++
+0
=
xα
–(
)x
β–
()
xγ
–(
)0
=
αβγ
d– a------
=
b22b
cc2
++
a33
a2g
3ag2
g3+
++
13y
3y2
y3+
++
1632
x24
x28
x3x4
++
++
824
x24
x28x
3+
++
8x3
48x2
–96
x64
–+
1632 7----
-- x24 49----
-- x2
8 343
---------
++
+x3
124
01----
--------
x4+
8x3
60x2
–15
0x12
5–
+
27x3
108x
2–
144x
64–
+27
x324
3x2
–72
9x
729
–+
8x3
72x2
216x
216
++
+b3
9b2
d27
bd2
27d3
++
+
81x4
216x
3y
216x
2y2
96xy
316
y4+
++
+
x515
x4y
90x3
y227
0x2
y340
5xy
424
3y5
++
++
+12
5
p3----
-----15
0 p---------
60p
8p3
++
+
16 x4------
32 x------
–24
x28x
5–
x8+
+q5
10q4 p3
--------
----40
q3 p6----
--------
80q2 p9
--------
----80
q
p12----
-----
32 p15--------
++
++
+
x33
x3 x---
1 x3-----
++
+
160
x321
x5y2
448x
3–
810
x4–
216p
4
2041
2p2
q5–
2268
0p–
1400
000
– 64x6
960x
560
00x4
2000
0x3
3750
0x2
3750
0x
1562
5+
++
++
+
HL
Mat
hs 4
e.bo
ok P
age
1001
Tue
sday
, May
15,
201
2 8
:54
AM
1002
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
519
6–
7 8–
9–2
010 11 12 13
a 0
b –5
914 15 Ex
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 o
ther
-w
ise st
ated
.1
a 3,
5b
i 2(x
+a) +
3ii
2ac 3
2a
0,
b c
3a
, b
c no
solu
tion
b i
ii {3
, –2}
6b,
c, d
, e8
a, d,
e, f
9a
Win
dow
[–2
,2],
[–1
,1]
b [0
, 1[
10a
{y: y
> 1
} ∪ {y
: y ≤
–1.
25}
b 10
11b
112
a onl
y –
it is
the o
nly
one w
ith id
entic
al ru
les a
nd d
omai
ns13
a [–
3,∞
[b
[–3,
0]c [
3,∞
[d
[1.5
,3[ ∪
]3,∞
[14
a i
ii b
i ii
63 8------
231
16--------- 13
027---------
a3±
=n
5=
n9
=
a3
n,8
==
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---–
010 11----
--,
1 2---–
x2x
–3 2---
+1 2---–
x2x
3 2---+
+2
±
4a
x =
0, 1
b
Win
dow
[–2,
2], [
–1,1
] Ra
nge:
[–12
, 4]
5a
iii
22
22
–,{
}
px(
)8
216
x2–
+0
x4
<<
,=
Ax(
)x
16x2
–0
x4
<<
,=
x
y
16 8
r =
]8,
16[
4x
y 8r
= ]
0,8]
4
HL
Mat
hs 4
e.bo
ok P
age
1002
Tue
sday
, May
15,
201
2 8
:54
AM
1003
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
Exer
cise
5.3
.1
Exer
cise
5.3
.2
x
y
3
–1
1
x
y 2
x
y
(4,2
)
6
x
y 2
1 a
b
c
d
x
y
2
–2
2
2 a
b
x
y
1–1
x
y 1
–1
x
y
1
x
y
1
x
y 2
–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
–2 ax
y
2
a =
4
4
x
y 3
a =
1
6 a
b
c
d
x
ya
= 4
a(1
,5)
x
y
2
a =
4
4
x
y 3
a =
1
x
y 1a
2=
a2
–=
7 a
b
x
ya –a
x
y
4a
= 4
1 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
HL
Mat
hs 4
e.bo
ok P
age
1003
Tue
sday
, May
15,
201
2 8
:54
AM
1004
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
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
d
e
f
x
y 7x
y
2
–8
x
y
4
–4
x
y
1
–1
x
y 2
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
–12
x
y
1–1
–1
HL
Mat
hs 4
e.bo
ok P
age
1004
Tue
sday
, May
15,
201
2 8
:54
AM
1005
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
Exer
cise
5.3
.3
3‘b’
has
a di
latio
n ef
fect
on
f (x)
= a
x (alo
ng th
e y ax
is).
5a
[1,1
6]b
[3,2
7]c
[0.2
5,16
]d
[0.5
,4]
e [0
.125
,0.2
5]f [
0.1,
10]
8a
]2, 2
+ e–1
[b
[–1,
1[
c [1
– e,
1 +
e–1]
9a
b 2
7 a
x
y
2–2
b i Ø
ii
[–2,
2]iii
{±4}
8 a
x
y
3
1–3
x: x
3–≤
{}
x: x
1≥
{}
∪
1 a
b
c
d
x
y 1]0
,∞[
(1, 4
)
x
y 1]0
,∞[
(1, 3
)
x
y 1]0
,∞[
(1, 5
)
x
y 1]0
,∞[
(1, 2
.5)
x
y 1]0
,∞[
(1, 1
.8)
x
y 1]0
,∞[
(–1,
2)
x
y 1]0
,∞[
(–1,
3)
x
y 1]0
,∞[
(1, 3
.2)
e
f
g
h
i
j
k
l
x
y 1]0
,∞[
(–1,
5)
x
y 1]0
,∞[
(–1,
4/3
)
x
y 1]0
,∞[
(–1,
8/5
)
x
y 1]0
,∞[
(–1,
10/
7)
2 a
b
x
y
–1
2
x
y
0.5
1.5
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
HL
Mat
hs 4
e.bo
ok P
age
1005
Tue
sday
, May
15,
201
2 8
:54
AM
1006
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
EXER
CISE
5.3
.4
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
15 a
b
x
y
4 23
x
y
a
16 a
b
c
x
y 1]0
,1]
d
e
f
x
y 1[1
,∞[
x
y 1
]0,∞
[
x
y
12
[1,2
[x
–2
]–∞
,–2[
∪ ]
0,∞
[
y
x
y[0
,∞[
a >
1
a <
1]0
,∞[
a =
1{1
}
1 a
b
c
d
x
y
23 ]2
, ∞[
x
y
–2
]–3,
∞[
–3
x
y
]0,∞
[
x
y27
HL
Mat
hs 4
e.bo
ok P
age
1006
Tue
sday
, May
15,
201
2 8
:54
AM
1007
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
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,∞
[
e51 ]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,∞
[
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
HL
Mat
hs 4
e.bo
ok P
age
1007
Tue
sday
, May
15,
201
2 8
:54
AM
1008
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
EXER
CISE
5.3
.5
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,∞
[
e
f
g
h
y
1/2
–2x
x
y
2–1
y
x
3
–1/3
x
2
1.5
y
x
y
34/
3
x
y
1–3
/2x
y
3–1
–1.5
0.5
x
y
–2
–3
1 a
b
c
d
2 a
b
c
d
x
y 1
x
y
–4
–1
x
y
2–2
4
x
y 2
1
–2
HL
Mat
hs 4
e.bo
ok P
age
1008
Tue
sday
, May
15,
201
2 8
:54
AM
1009
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
EXER
CISE
5.4
.1
1a
i [0
,∞[
ii
[1,∞
[
iii
, b
i
ii
iii
2a
i ]–
1,∞
[ i
i ]–
0.25
,∞[
iii
, [–4
,4]
b i
ii
iii
3a
, b
]–∞
, ∞[,
]–∞
, ∞[
ii a
, b
[1, ∞
[, [1
, ∞[
iii a
,
b [0
, ∞[,
[–2,
∞[
iv a
,
b ,
v a
, b
[0, ∞
[, [0
, ∞[
vi a
, g
of(x
) doe
s not
exist
.b
]–1,
∞[
vii a
,
b ]0
, ∞[,
]0, ∞
[vi
ii a
, b
[–4,
∞[,
[0, ∞
[
ix a
,
b [–
2, ∞
[, [0
, ∞[
x a
fog(
x) d
oes n
ot ex
ist,
b [0
, ∞[
xi a
,
b [0
,1[,
[0, ∞
[
xii a
,
b [1
, ∞[,
xiii
a ,
b [1
, ∞[,
]0, ∞
[
xiv
a fo
g(x)
doe
s not
exist
, b
xv a
fog(
x) d
oes n
ot ex
ist,
b ]1
, ∞[
xvi a
,
b [1
, ∞[,
]0, ∞
[4
a b
c
5
6a
, ]–∞
,–1]
∪ [3
,∞[
b go
f(x) d
oes n
ot ex
ist.
c , ]
–∞,–
2.5]
∪ [2
.5,∞
[
7a
9b
39
a b
3
a
2b,
1=
=
x
y 3
–1–1
.5
4 a
b
c
x
y
1
i ii
–1
d
e
f
x
y
2–2
2
4–4
x
y
1
–1
2
x
y
2–2
x
y
1/2
–1/2
y 2
fg:
0
∞
whe
re
[,
[+
fg
+(
)x(
)x2
x+
=
fg
: ]0
∞
whe
re
[,
+f
g+
()
x()
1 x---x(
)ln
+=
fg:
3–
2–]
23,
[]
whe
re
∪,
[+
fg
+(
)x(
)9
x2–
x24
–+
=5
10,
[]
fg:
0∞
whe
re
[,
[fg(
)x(
)x2
xx5
2/=
=
fg:
]0∞
whe
re
[,
fg()
x()
x()
lnx
--------
-----=
fg:
3–2–
]2
3,[
]
w
here
∪
,[
fg()
x()
9x2
–(
)x2
4–
()
=
fg–
: ]
∞–∞
w
here
[
,f
g–(
)x(
)2
ex1
–=
fg–
: ]
1–∞
w
here
[
,f
g–(
)x(
)x
1+
()
x1
+–
=
fg–
: ]
∞–∞
[
whe
re
,f
g–(
)x(
)x
2–
x2
+–
=
f/g
: \
0{}
whe
re
,f/
g(
)x(
)ex
1ex
–----
--------
--=
f/g
: ]
1–∞
whe
re
[,
f/g
()
x()
x1
+=
f/g
: \
{–2}
w
here
f/g
()
x()
x2
–x
2+
--------
----=
fog
x()
x31
+=
gof
x()
x1
+(
)3=
fog
x()
x1
+=
x0
≥,
go
fx(
)x2
1+
=
fog
x()
x2=
gof
x()
x2
+(
)22
–=
fog
x()
xx
0≠
,=
gof
x()
xx
0≠
,=
\{0}
\{0}
fog
x()
xx
0≥
,=
gof
x()
x=
fog
x()
1 x2-----
1–
=x
0≠
,
fog
x()
x2x
0≠
,=
gof
x()
x2x
0≠
,=
fog
x()
x4
–=
go
fx(
)x
4–
=
fog
x()
x2
+3
2–
=g
ofx(
)x3
=
gof
x()
4x
–(
)x,
4≤
=
fog
x()
x2
x21
+----
--------
--=
gof
x()
xx
1+
--------
----
2x
1–≠
,=
fog
x()
x2x
1+
+=
gof
x()
x2x
1+
+=
0.75
∞[
,[
fog
x()
2x2
=g
of
x()
22x
= gof
x()
1x
1+
--------
----1
–x
1–≠
,=
\{–1
}
go
fx(
)4
x1
–----
-------
1+
=
fog
x()
4x
=x
0≥
,g
of
x()
40.5
x=
fog
x()
2x
3+
=x
∈,
gof
x()
2x
2+
=x
∈,
fof
x()
4x
3+
=x
∈,
gx(
)x2
1x
∈,
+=
fog
x()
1 x---x
1x
∈\
0{}
,+
+=
gog
x()
x1 x---
x
x21
+----
--------
--x
0≠
,+
+=
x1±
=x
13–,
=
HL
Mat
hs 4
e.bo
ok P
age
1009
Tue
sday
, May
15,
201
2 8
:54
AM
1010
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
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 17
a D
om f
= ]0
,∞[,
ran
f = ]e
,∞[,
Dom
g =
]0,∞
[, ra
n g =
b
fog d
oes n
ot ex
ist:
go
f ex
ists a
s
c 18
; ran
ge =
[0, ∞
[
23a
EXER
CISE
5.4
.2
1a
b c
d
e f
g h
1 x---x–
2x
1+
--------
-------
ho
fx(
)x
1–
()2
4+
x2
≥,
5x
–x
2<
,
=
(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
,∞[
==
go
f: ]
0,∞
[
, w
here
go
fx(
)x
1+
()
2ln
+=
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,
∞[
, whe
re g
of
x()
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
1 2---x
1–
()
x∈
,x
3x
∈,
3x
3+
()
x∈
,5 2---
x2
–(
)x
∈,
x21
–x
0>
,x
1–
()2
x1
≥,
1 x---1
–x
0>
,1
x1
+(
)2----
--------
-------
x1–
>,
1
01 2---
–,(
)
x
y
x
y9
–3x
y
2
–5y
x
1–1y
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–3 0
3–,
()
x
y
HL
Mat
hs 4
e.bo
ok P
age
1010
Tue
sday
, May
15,
201
2 8
:54
AM
1011
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
6a
b
c d
e f
8a
b
c d
e f
12[2
, ∞[
13\{
1.5}
4 x
± 1x2
–----
--------
------
1x
1<
<–,
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–
x()
x1
–(
)3
x1
>,
log
=f
1–x(
)x
5+
()
2x
5–>
,lo
g=
f1–
x()
1 2---x
1–
3lo
g(
)x
0>
,=
g1–
x()
13
x–
()
10x
3<
,lo
g+
=
h1–
x()
12 x---
+
3
x
\[–2
,0]
∈,
log
=g
1–x(
)1
x1
+----
--------
2x
1–>
,lo
g=
7 a
b
c
x
y
21
1
inve
rse
inve
rse x
y –4
–5
–4–5
x
y
3
inve
rse
inve
rse
x
y
inve
rse
d
e
f
x
y
(1, 1
)
(–2,
–2)
inve
rse
x
y –1
f1–
x()
2x1
x∈
,–
=f
1–x(
)1 2---
10x
⋅x
∈,
=
f1–
x()
21x
–,x
∈=
f1–
x()
3x1
+1
+x
∈,
=
f1–
x()
5x2/
5+
x∈
,=
f1–
x()
110
32
x–
()
–x
∈,
=
9 f
1–x(
)1
–x
1+
+x
1–>
,=
x
y
(–1,
–1)
dom
= [
–1, ∞
[, r
an =
[–1
, ∞[
10 a
b
c f
1–x(
)a
x–
=f
1–x(
)2
xa
–----
-------
a+
=f
1–x(
)a2
x2–
=
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(
)x
x24
+–
2----
--------
--------
-------
=
15f
1–x(
)a
x21
+(
)x
0≥
,=
x
y
a
a
x: f
x()
f1–
x()
={
}∅
=
HL
Mat
hs 4
e.bo
ok P
age
1011
Tue
sday
, May
15,
201
2 8
:54
AM
1012
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
18go
f exi
sts as
. I
t is o
ne:o
ne so
the i
nver
se ex
ists:
20a
i ii
b
i
ii
c i &
ii n
eith
er ex
ist
d
Adju
stin
g do
mai
ns so
that
the f
unct
ions
in p
art c
exist
, we h
ave:
and
e
Yes
as
rule
s of
com
posi
tion
OK
.21
a 1
b 0
.206
22a
b fog
exist
s but
is n
ot o
ne:o
ne
c i B
= [l
n2, ∞
[ii
iii
EXER
CISE
5.5
1a
even
b ev
enc n
eith
erd
neith
ere e
ven
f odd
g od
dh
even
i odd
3N
ot if
0 is
exclu
ded
from
the d
omai
n.6
16 a
b
f1–
x()
2x
1+
()
–,
x1–
<
x3
,–
x1–
≥
= x
y
1
1f
1–x(
)x(
)ln
1–
,0
x<
e≤
xe,
–x
e>
= 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–:
, f1–
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
iv {–
1, 0
, 1}
tom
x()
ex
x0
≥,
=m
otx(
)ex
x∈
,=
tom
()
1–x(
)x(
)ln(
)2x
1>
,=
mo
t(
)1–
x()
x2ln
x0
>,
=
t1–
om1–
x()
mo
t(
)1–
x()
=m
1–o
t1–
x()
tom
()
1–x(
)=
x
y
0.25
x
y
x
y
1(tom
)–1
(mot
)–1
x
y
–2
f
g
fog
()
1–: [
0,∞
[
w
here
, fo
g(
)1–
x()
x2
+(
)ln
=
x
y
ln2
ln2
fx(
)0
=x
ℜ∈
,
HL
Mat
hs 4
e.bo
ok P
age
1012
Tue
sday
, May
15,
201
2 8
:54
AM
1013
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
EXER
CISE
6.1
1a
b c
d
e f
g h
i j
k l
yx
4–
()2
=y
x2
+(
)2=
yx2
5+
=x
2–
()2
y+
2=
x2y
+4
=x2
y+
0=
y8
x4
–----
-------
x4
≠,
=y
8 x---1
–x
0≠
,=
x1
+(
)2y2
+4
=y2
9x
3–
--------
---x
3≠
,=
y3
+(
)29 x---
x0
≠,
=x
y2+
8=
–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
a i
b i
–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 ii
i
b ii
i
–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
a iv
b iv
–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
a ii
b ii
2 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
6 F
irst
fun
ctio
n in
bla
ck, s
econ
d fu
ncti
on in
blu
e
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
HL
Mat
hs 4
e.bo
ok P
age
1013
Tue
sday
, May
15,
201
2 8
:54
AM
1014
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
8a
b c
d e
f g
h i
j
k l
m
n o
9a
b c
d e
j
x
y 2
–4
4
7 N
ote:
coor
dina
tes w
ere a
sked
for.
We h
ave l
abel
led
mos
t of t
hese
with
sing
le
num
bers
.
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
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(
)f
x1
–(
)1
+=
gx(
)f
x2
+(
)4
–=
gx(
)f
x2
–(
)=
gx(
)f
x1
–(
)1
+=
gx(
)f
x1
–(
)3
+=
10 a
i
ii y
x
yx
y
x
y
x
(–0.
5, 0
)
(2, –
4)
(3, 3
)
(–1,
1)
–2
–1
0
–3y
x–3
–5
y
x
4
2
y
x
–2
–1
0
1
2
3
1
–1
y
x
2
3
b i
i
i
(–2,
–2)
i
ii
iv
i
ii
iv
c i
ii
y
x
–4
y
x
–6
–2
HL
Mat
hs 4
e.bo
ok P
age
1014
Tue
sday
, May
15,
201
2 8
:54
AM
1015
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
11 EXER
CISE
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
4 3 2 1
y
x
y
x–5
–4
–3
–2
–1
1
2
3
4
5
4 3 2 1
–1 –2
3a i
b i
–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
–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
–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
ii
i
i
ii
i
i
ii
iv
iv
HL
Mat
hs 4
e.bo
ok P
age
1015
Tue
sday
, May
15,
201
2 8
:54
AM
1016
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
a b
c d
e f
i
ii
2
–2
9 2---–
y
x–3
0
3
1
–1y
x
y
x
(0, 3
)
(3, 0
)
–2
2
(–3,
6)
y
x
iii
iv
yf
x()
=
yf
x()
=
yf
2 3---x
=
9 2---9 2---
4 a
i
ii
y
x–3
0
3
4
–4y
x
y
x
(0, 1
2)
(3, 0
)
–8
8
(–2,
24)
y
x
iii
i
vy
fx(
)=
yf
x()
=
9 2---
y4
fx(
)=
–3
0
3
b8
–8
fx(
)x
=y
f2
x(
)1
+=
fx(
)x2
=y
1 2---f
x2
–(
)3
–=
fx(
)1 x---
=y
1 2---f
x1 2---
–
=
fx(
)x3
=y
27f
x2 3---
–
=
fx(
)x4
=y
128
fx
1 2---–
2–
=f
x()
x=
y2f
x()
2+
=
7f
x()
x2
+(
)2
if
x0
≥4
x
if x
0<
–
=
i
i
i
i
ii
x
y 4
hx(
)x2
3–
if
x2
≥3
x
if x
2<
–
=
x
y
(2, 1
)3
hx(
)2
x2
if x
2≥
122
x
if x
2<
–
=
x
y
(2, 8
)
iv
v
vi
kx(
)4
x2
if x
1≥
62
x
if x
1<
–
=k
x()
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
)
8
x
y
x
y
x
y
(4, 0
)
(2, 2
)
(2, –
2)
(2, 1
)
(4, 0
)(8
, 0)
(4, 2
)y
1 2---f
x()
=y
f1 2---
x
=
yf
x()
=a
b
x
y
x
y
x
y
x
y(1, a
)(1
/a, 1
)
(–b,
0)
(1, 1
)b
b
(1, –
1)
1 a2------
1 a--- ,
1 a2------
1 a--- ,
a <
0
a >
0
9 a
b
c
d
HL
Mat
hs 4
e.bo
ok P
age
1016
Tue
sday
, May
15,
201
2 8
:54
AM
1017
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
10 EXER
CISE
6.3
2
a b
c d
e
3a
bc
d
x
y
x
y
b–
a----
------
0,
b a-------
0,
b a---
0,
b a---
–0,
a b---–
a–
ab
1 a
i
ii
–3
0
3
2
–2
3 2---–
3 2---
y
x–3
0
3
2
–2
3 2---–
3 2---
y
x
b
i
ii
–2
–1
0
1
2
3
1
–1y
x–3
–2
–1
0
1
2
3
1
–1y
x
c i
iiy
x
(0, 3
)
(–2,
0)
–2
y
x(0
, –3)
2
–2
–1
0
1
2
3
2y
x
1
(–1,
2)
(1, –
1)
3
y
x–2
–11
d i
ii
2
(2,6
)y
x
e i
ii
f i
ii
–3y2
1–1
–2
–1
0
1
2
3
2y
x1
(1, –
2)
(–1,
1)
–2
(–2,
–6)
y
x
yf–
x()
=y
fx–(
)=
yf
x1
+(
)=
yf
2x()
=y
2fx(
)=
x
y9
3–
0,(
)
30,
()
x
y4
22
0,(
)
22
–0,
()
x
y
21
x
y(–
2, 3
)
–1
–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
HL
Mat
hs 4
e.bo
ok P
age
1017
Tue
sday
, May
15,
201
2 8
:54
AM
1018
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
ef
gh
ij
kl
mn
op
qr
4a
b
cd
ef
gh
ij
5a
i ii
b i
ii
c i
ii
6a
b c
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
------
–
x
y
a
a2x
ya
–a2
x
y(3
, 2)
x
y
(–3,
2)
x
y 2(2
, 2)
(1, 3
)
HL
Mat
hs 4
e.bo
ok P
age
1018
Tue
sday
, May
15,
201
2 8
:54
AM
1019
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
d e
f
EXER
CISE
6.4
1a
b c
d e
f
2a
b c
d e
f
g h
i
j k
l
3a
b
4 EXER
CISE
6.5
1a
i ii
b i
ii
c i
ii
5
(–6,
2)
(–2,
2)
(–4,
3)
x
y
x
y 5(1
, 2)
x
y 1
(1, –
2)
x
y
3–3
0.5 –0
.5x
y
–2–1
1–1
12
3
x
y
2–0
.5
1 3---
x
y
1/2
x
y
3–2
–1–1
1
x
y
2
1 3---
1
2 3---
y
4
–2
x21
x
y
–2–4
–4
x
y 2
1/2
x
y
–1
x
y
1x
y
1–1
x
y
21
3x
y
1
10.
52
x
y 1
–1
x
y 2(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)
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
HL
Mat
hs 4
e.bo
ok P
age
1019
Tue
sday
, May
15,
201
2 8
:54
AM
1020
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
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
EXER
CISE
7.1
.1
1a
b c
d e
f g
h i
2a
b c)
d
e f
3a
b c
d 9
e
f g
h i
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
2x
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
27y1
5
8x3
--------
------
91
216
a6----
--------
---2
n2
+8
x11
27y2
--------
----3
x2y2
8----
--------
---
3n1
+3
+4n
1+
4–
24n
1+
4–
()
1b6 –
16b4
--------
----
642 3---
x
22y
1+
1 b2
x----
----y 2---
6
9 2---
n
2+
z2 xy-----
37n
2–
5n
1+
26n
1+
213
n–
x24
nn
2–
+x3
n2n
1+
+27
HL
Mat
hs 4
e.bo
ok P
age
1020
Tue
sday
, May
15,
201
2 8
:54
AM
1021
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
4
5a
b c
d e
f
6a
b c
d e
f
7a
b 1
c d
e f
g h
8a
b c
d e
f
EXER
CISE
7.1
.2
1a
2b
–2c
d 5
e 6f –
2.5
g 2
h 1.
25i
2a
–6b
–c –
3d
1.5
e 0.2
5f 0
.25
g h
i –1.
25
EXER
CISE
7.1
.3
1a
3.5
b 3.
5c –
3d
1.5
e 3.5
f 1.5
g 1.
8h
i 0
2a
–0.7
5b
–1,4
c 0,1
d 3,
4e –
1,4
f 0,2
3a
–1,1
,2b
–3,1
,3,4
c ,
, 2d
–1,1
,2e 3
,7,
,
EXER
CISE
7.1
.41
a i 5
.32
ii 9.
99iii
2.5
8b
i 2.2
6ii
3.99
iii 5
.66
c i 3
.32
ii –4
.32
iii –
6.32
d i –
1.43
ii 1.
68iii
–2.
862
a 0
b 0.
54c –
0.21
d–0.
75, 0
e 1.1
3f 0
, 0.1
6EX
ERCI
SE 7
.1.5
1a
2b
–1c 0
.5d
0.5
2a
1b
0.6
c 0
3a
0b
4a
–1,2
b –2
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HL
Mat
hs 4
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ok P
age
1021
Tue
sday
, May
15,
201
2 8
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AM
1022
MATH
EMATI
CS –
Highe
r Le
vel (C
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ANSW
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11a
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Exer
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7.3
1a
2b
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h 0
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k 0.
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2 2
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3a
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045
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c d
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5a
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real
soln
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tion
k l
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1,4
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1000
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HL
Mat
hs 4
e.bo
ok P
age
1022
Tue
sday
, May
15,
201
2 8
:54
AM
1023
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
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h
i j Ø
k l
15
a b
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219
e –1.
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21.5
j 2
Exer
cise
7.5
1a
10b
30c 4
02
a 31
.64
kgb
1.65
c
3a
4.75
b
4a
[0,1
[b
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2ii
1.11
iii 0
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yrs
c As c
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ease
s, re
liabi
lity
redu
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d e
5a
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b c
16.8
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Exer
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8.1
.11
i b 4
c ii
b –3
c i
ii b
–5 c
iv
b 0
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v b
2c
vi b
–2
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39,
174
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57
67
7–5
80
9a
41 b
31s
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11 a
i 2
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b i 4
ii 11
12
13
14a
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0Ex
ercise
8.1
.21
a 14
5b
300
c –17
0
2a
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b 69
0c 7
0.4
3
a –1
05b
507
c 224
4
a 12
6b
3900
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7 Ex
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8.1
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123
2
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0.5,
2, 4
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, 9.5
, 12
3
3.25
4
a =
3 d
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5
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a 50
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390
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45
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cise
8.2
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1a
b
c d
e f
1 3---e2
2.46
30=
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90.0
171
±=
e24
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3891
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320
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02
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410
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2.4
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e
LL
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m
L
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6
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1
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hs 4
e.bo
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age
1023
Tue
sday
, May
15,
201
2 8
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AM
1024
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
2a
b
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b 15
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4a
b c n
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d e
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11; 3
5429
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tonn
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Exer
cise
8.2
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Term
9 A
P =
180,
GP
= 25
6. S
um to
11
term
s AP
= 16
50, G
P =
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8.2
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NB:
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c
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9
10
11
Exer
cise
8.2
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3, –
0.2
2 3 4a
b c
599
006
3275
73
8
96
10
–
11a
12 b
26
12
9, 1
2
12±5
± 2----
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5 6---
n
1–
×=
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127
128
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HL
Mat
hs 4
e.bo
ok P
age
1024
Tue
sday
, May
15,
201
2 8
:54
AM
1025
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
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13±2
14
(5, 5
, 5),
(5, –
10, 2
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Exer
cise
9.1
1a
cmb
cmc c
mA
BC
13.
84.
11.
667
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°23
°2
81.5
98.3
55.0
56°
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34°
332
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1.61
30.7
30.7
3°90
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2.3
2.74
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33°
648
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39°
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51°
744
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2.93
13.0
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77°
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71.8
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086
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82.4
88.9
33.3
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22°
2a
b c 4
d e
f
4a
b
Exer
cise
9.2
1a
i 030
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330°
Tiii
195
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200
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i N25
°Eii
Siii
S40
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3
18.9
4m
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° 18'
5m
/s
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3’W
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23 k
m
719
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m
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m9
72.2
5 m
10
25.3
9 km
11
15.7
6 m
12
a 3.
01 k
m N
, 3.9
9 km
Eb
2.87
km
E 0
.88
km S
c 6.8
6 km
E 2
.13
km N
d 7.
19 k
m 2
53°T
1352
4m
Exer
cise
9.3
1a
39°4
8'b
64°4
6'
2a
12.8
1 cm
b 61
.35
cmc 7
7°57
'd
60.
83 cm
e 80°
32'
3
a21°
48'
b 42
°2'
c 26°
34'
4a
2274
b 12
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525
1.29
m
6a
103.
5 m
b 35
.26°
c 39.
23°
7b
53.4
3c 1
55.1
6 m
d 14
5.68
m
8b
48.5
4 m
9a
b c
d
1082
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m11
a 40
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mb
49.4
6 m
12
a 10
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cmb
75° 5
8'c 9
3° 2
2'
13a
1.44
mb
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3'c 6
2° 1
1'
Exer
cise
9.4
1a
1999
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2b
756.
8cm
2c 3
854.
8cm
2d
2704
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2e 5
38.0
cm2
f 417
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549.
4cm
2h
14.2
cm2
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81.5
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8.8
cm2
23
51
3+
()
21
3+
()
4 3---3
3+
()
106
5–
251
3+
()
403
3----
--------
-
26 9------
bc
–(
)2h2
+h a---
1–ta
nh
bc
–----
-------
1–ta
n
2b
c+
()
h2a2
+2
ab
c–
()2
h2+
+
HL
Mat
hs 4
e.bo
ok P
age
1025
Tue
sday
, May
15,
201
2 8
:54
AM
1026
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
l 387
.2cm
2m
139
.0cm
2n
853.
7cm
2o
314.
6cm
2
269
345
m2
310
0π –
cm
2
417
.34
cm
5a
36.7
7sq
units
b 14
.70
sq u
nits
c 62.
53 sq
uni
ts
652
.16
cm2
77°
2'
8
9A
rea o
f =
101
.78
cm2 ,
Are
a of
= 6
1.38
cm2
Exer
cise
9.5
.1a
cmb
cmc c
mA
BC
113
.337
.148
.210
°29
°14
1°2
2.7
1.2
2.8
74°
25°
81°
311
.00.
711
.360
°3°
117°
431
.939
.151
.738
°49
°93
°5
18.5
11.4
19.5
68°
35°
77°
614
.615
.05.
375
°84
°21
°7
26.0
7.3
26.4
79°
16°
85°
821
.610
.128
.539
°17
°12
4°9
0.8
0.2
0.8
82°
16°
82°
1027
.77.
433
.336
°9°
135°
1116
.420
.714
.552
°84
°44
°12
21.4
45.6
64.3
11°
24°
145°
1330
.927
.722
.675
°60
°45
°14
29.3
45.6
59.1
29°
49°
102°
159.
79.
87.
965
°67
°48
°16
21.5
36.6
54.2
16°
28°
136°
1714
.829
.327
.230
°83
°67
°18
10.5
0.7
10.9
52°
3°12
5°19
11.2
6.9
17.0
25°
15°
140°
2025
.818
.540
.130
°21
°12
9°Ex
ercise
9.5
.2a
bc
A°
B°C°
c*B*
°C*
°1
7.40
18.1
021
.06
20.0
056
.78
103.
2212
.95
123.
2236
.78
213
.30
19.5
031
.36
14.0
020
.77
145.
236.
4915
9.23
6.77
313
.50
17.0
025
.90
28.0
036
.24
115.
764.
1214
3.76
8.24
410
.20
17.0
025
.62
15.0
025
.55
139.
457.
2215
4.45
10.5
55
7.40
15.2
019
.55
20.0
044
.63
115.
379.
0213
5.37
24.6
36
10.7
014
.10
21.4
126
.00
35.2
911
8.71
3.94
144.
719.
297
11.5
012
.60
22.9
417
.00
18.6
814
4.32
1.16
161.
321.
688
8.30
13.7
018
.67
24.0
042
.17
113.
836.
3613
7.83
18.1
7
913
.70
17.8
030
.28
14.0
018
.32
147.
684.
2716
1.68
4.32
1013
.40
17.8
026
.19
28.0
038
.58
113.
425.
2414
1.42
10.5
811
12.1
016
.80
25.6
323
.00
32.8
512
4.15
5.30
147.
159.
8512
12.0
014
.50
24.3
521
.00
25.6
613
3.34
2.72
154.
344.
6613
12.1
019
.20
29.3
416
.00
25.9
413
8.06
7.57
154.
069.
9414
7.20
13.1
019
.01
15.0
028
.09
136.
916.
3015
1.91
13.0
915
12.2
017
.70
23.7
330
.00
46.5
010
3.50
6.93
133.
5016
.50
169.
2020
.90
27.9
714
.00
33.3
413
2.66
12.5
914
6.66
19.3
417
10.5
013
.30
21.9
620
.00
25.6
713
4.33
3.03
154.
335.
6718
9.20
19.2
026
.29
15.0
032
.69
132.
3110
.80
147.
3117
.69
197.
2013
.30
18.3
319
.00
36.9
712
4.03
6.82
143.
0317
.97
2013
.50
20.4
025
.96
31.0
051
.10
97.9
09.
0112
8.90
20.1
021
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
.4a
cmb
cmc c
mA
BC
113
.59.
816
.754
°36
°90
°2
8.9
10.8
15.2
35°
44°
101°
322
.825
.612
.863
°87
°30
°4
21.1
4.4
21.0
85°
12°
83°
515
.910
.615
.174
°40
°66
°6
8.8
13.6
20.3
20°
32°
128°
79.
29.
513
.244
°46
°90
°8
23.4
62.5
58.4
22°
89°
69°
910
.59.
615
.741
°37
°10
2°10
21.7
36.0
36.2
35°
72°
73°
117.
63.
49.
449
°20
°11
1°12
7.2
15.2
14.3
28°
83°
69°
139.
112
.515
.835
°52
°93
°14
14.9
11.2
16.2
63°
42°
75°
152.
00.
72.
538
°13
°12
9°16
7.6
3.7
9.0
56°
24°
100°
1718
.59.
824
.145
°22
°11
3°18
20.7
16.3
13.6
87°
52°
41°
691
ba
θta
n×
+(
)22
θta
n----
--------
--------
--------
--------
-
ΔA
CD
ΔA
BC
HL
Mat
hs 4
e.bo
ok P
age
1026
Tue
sday
, May
15,
201
2 8
:54
AM
1027
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
1914
.622
.429
.928
°46
°10
6°20
7.0
6.6
9.9
45°
42°
93°
2121
.820
.823
.858
°54
°68
°22
1.1
1.7
1.3
41°
89°
50°
231.
21.
20.
485
°76
°19
°24
23.7
27.2
29.7
49°
60°
71°
253.
44.
65.
240
°60
°80
°Ex
ercise
9.5
.51
a 10
.14
kmb
121°
T 2
7° 3
3' 3
4.12
cm
457
.32
m
531
5.5
m
612
4.3
kmb
W28
° 47'
S
Exer
cise
9.5
.61
39.6
0 m
52.
84 m
2
30.2
m
354
°,42°
, 84°
4
37°
502
8°T.
6
108.
1 cm
7
a 13
5°b
136
cm8
41°,
56°,
83°
9a
158°
left
b 43
.22
km
1026
4 m
11
53.3
3 cm
12
186
m
1350
.12
cm14
5.17
cm
15a
5950
mb
1334
1 m
c 160
°d
243°
17
a 20
.70°
b 2.
578
mc
1.99
4m
3
18a
4243
m2
b 86
mc 1
01m
Ex
ercise
9.6
15.
36 cm
2
12.3
m
324
m
440
.3 m
, 48.
2°
516
.5 m
in, 8
.9°
6~1
0:49
am
7a
i ii
b o
r c
Exer
cise
9.7
1a
, b
,
c ,
d ,
e ,
f ,
g ,
h ,
i ,
j ,
k ,
l ,
m
, n
,
o ,
2, 3
6°
30.
0942
m3
4
579
cm6
5.25
cm2
7a
31.8
3 m
b 40
6.28
mc 1
1°
8
9
10a
b
i 37.
09 cm
ii 88
.57
cmc 3
70.9
2 cm
2 11
26.5
7
1219
3.5
cm
13a
105.
22 cm
b 11
8.83
cm
14a
9 cm
b 12
cmc
15b
c 0.4
9
1614
39.1
6 cm
2
dφ
sin φ
θ–
()
sin
--------
--------
---------
dθ
sin φ
θ–
()
sin
--------
--------
---------
dφ
αta
nsi
nφ
θ–
()
sin
--------
--------
--------
--d
θβ
tan
sin φ
θ–
()
sin
--------
--------
--------
--d
φθ
cos
sin
φθ
–(
)si
n----
--------
--------
-----1
–
169
π15
0----
--------
cm2
5.2
13π
15--------
-+
cm52
9π
32--------
----cm
223
23π 8----
-----
+cm
242
πcm
288
11πc
m+
1156
π75
--------
-------
m2
13.6
68π
15--------
-+
m
96π
625
--------
- cm
21.
2812
π25--------
-+
cm36
1π
15--------
----cm
215
.219
π 3--------
- cm
+
5248
.8πm
264
832
.4πc
m+
1294
3π30
0----
--------
------ c
m2
17.2
301
π30--------
----cm
+
1922
π75
--------
-------
cm2
12.4
124
π15--------
----+
cm15
884
π3
--------
--------
-- cm
215
241
8π
3----
--------
+cm
12πc
m2
242π
cm+
98π 3----
----- c
m2
2814
π 3--------
-+
cm
196
π75--------
----cm
25.
628
π15--------
- cm
+11
532
π25
--------
--------
-- cm
249
.618
6π 5----
--------
+cm
3π 50------ c
m2
2.4
π 10------ c
m+
0.63
c
1.64
c
1.11
c
0.75
c
1.85
c cm2
36°5
2′
y1
αta
n----
--------
=
y5.
5π
–α
–=
5.5
0.49
α
y
HL
Mat
hs 4
e.bo
ok P
age
1027
Tue
sday
, May
15,
201
2 8
:54
AM
1028
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
Exer
cise
10.
11
a 12
0°b
108°
c 216
°d
50°
2a
b
c
d
3
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
1d
1e
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 27
ii
15a
b
16a
b i
ii
17a
b
18
πc3
π 2------c
7π 9------c
16π 9----
-----c
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
–----
--------
------
–
θsi
nθ
cot
θco
tθ
tan
π 3---2
π 3------
,π 3---
5π 3------
,π 3---
4π 3------
,5
π 6------
7π 6------
,5π 6----
--11
π 6--------
-,
7π 6------
11π 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-------
–
π 3---2
π 3------
4π 3------
5π 3------
,,
,π 2---
7π 6------
11π 6----
-----
,,
0π 6---
5π 6------
π2π
,,
,,
π 2---3
π 2------
,
2a
a21
+----
--------
---a2
1–
a21
+----
--------
---
1x2
1–
–x
--------
--------
--------
---1
x21
–+
x----
--------
--------
--------
2 x2-----
1–
5 2---9 8---
2±π 6---
2kπ
k∈
,+
7π 6------
2kπ
k∈
,+
1 54-----
1 3---
12k
+1
k–
()
12k
+
1a
–
2a
--------
----2
2aa2
–+
2aa2
––
1a
–----
--------
--------
-----
2 3---0
22
3----
------
±,
0π 3---
2π 3------
π,
,,
HL
Mat
hs 4
e.bo
ok P
age
1028
Tue
sday
, May
15,
201
2 8
:54
AM
1029
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
Exer
cise
10.
2.2
1a
b c
d e
f
2a
b c
d
e f
g h
i
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 –1
1
18 Exer
cise
10.
3
1a
4πb
c 3π
d 4π
e 2f
2a
5b
3c 5
d 0.
5 3
a b
c d
e f
g h
i j
αφ
αφ
sin
cos
+co
ssi
n3
α2
β3
α2β
sin
sin
–co
sco
s2
xy
2x
ysi
nco
s–
cos
sin
φ2
αφ
2αsi
nsi
n+
cos
cos
2θ
αta
n–
tan
12θ
αta
nta
n+
--------
--------
--------
--------
----φ
3ωta
n–
tan
1φ
3ω
tan
tan
+----
--------
--------
--------
--------
2α3β
–(
)si
n2
α5β
+(
)co
sx
2y
+(
)si
nx
3y
–(
)co
s
2α
β–
()
tan
xta
nπ 4---
φ–
tan
π 4---α
β+
+
si
n2x
sin
56 65------
–33 65----
--16 63----
--–
16 65------
63 65------
56 33------
511
18----
--------
-–
7 18------
–5
11 7----
--------
-35
1116
2----
--------
----
3 5---–4 5---–
3 4---24 7----
--
13
+ 22
--------
--------
13
+ 22
--------
--------
13
+ 22
--------
--------
–3
2–
2ab
a2b2
+----
--------
------
a2b2
+ 2ab
--------
--------
--a4
6a2
b2–
b4+
a2b2
+(
)2----
--------
--------
--------
--------
----2
ab
b2a2
–----
--------
-----
21
–
0π 3---
π5π 3----
--2
π,
,,
,π 6---
5π 6------
3π 2------
,,
0π
2π
απ
α2
πα
α,–
,±
,,
,,
1 2----
---
1–
tan
=
Ra2
b2+
αta
n,
b a---=
=
Ra2
b2+
αta
n,
b a---=
=
23
–
2π 3------
π 2---
2π2,
6π
3,π
ππ
4,π
3,6
π2π 3----
--1 4--- ,
3π
8π 3------
2 3--- ,
4 a
b
c
d
x
y
2 π–33
x
y 1 –1
π–π
x
y 2 –2
3πx
y0.
5
–0.5
π
e
f
g
h
x
y
π2π
x
y
–πx
y
–π/3
π/3
1/3
–1/3
x
y
2
3
–3
– ππ
5 a
b
c
d
x
y 3
2π
x
y
–2
π–π
–1
x
y3 π
–4x
y 2
π
1.5
2.5
e
f
g
h
x
y
πx
y
–πx
y2/
3
– π/3
π/3
x
y
–5
π–π
1
–1
6 a
b
c
d
x
y
2π
–3
3
x
y 1 –1
π–π
y
x
–2
xπ
–1/2
HL
Mat
hs 4
e.bo
ok P
age
1029
Tue
sday
, May
15,
201
2 8
:54
AM
1030
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
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
e
f
g
h
x
y
2πx
y
– πx
y1/
3
–π/3
π/3
x
y 3
–3
π–π
π–4 7 a
b
c
d
x
y
2π–2
π
1
–1
x
y –1
–22 π
–2π
1
x
y
2π–2
πx
y 2
2π–2
π
–2
x
y
–22 π
–2π
3 –1
x
y2
–2
2π–2
π–2
x
y
–22π
–2π
x
y
–22 π
–2π
3
–3
i
j
k
l
x
y1
–2π
–3
2πx
y
2π–2
πx
y
2π–2
π4 –4
x
y 1
2 π–2
π
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 πx
yse
cx 2πx
y
π
cotx
π 4---π 2---
ππ 3---
π 4---π 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
HL
Mat
hs 4
e.bo
ok P
age
1030
Tue
sday
, May
15,
201
2 8
:54
AM
1031
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
Exer
cise
10.
5
1a
b c
d
e f
2a
b c
d π
e f
3a
b c
d e
f 3
4a
90°,
330°
b 18
0°,2
40°
c 90°
,270
°d
65°,3
35°
e f 0
, π, 2
πg
h
5a
60°,3
00°
b c
d 23
°35’,
156°
25'
e f
g h
3.35
59c , 5
.210
5ci
j k
l 68°
12',2
48°1
2'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°,1
99°2
8',34
0°32
'b
(199
°28',
340°
32')
24
Exer
cise
10.
6
1a
5, 2
4, 1
1, 1
9b
c 23.
6°
2a
3, 4
.2, 2
, 7b
x
y
1–1
π/2
Cos
–1 Sin
–1
12 a
b
c
i
ii
π 2---π 2---
x
y
1–1
π/2
13
π 4---1
n1
+----
--------
1–ta
n– π 4---
3π 4------
,7
π 6------
11π 6----
-----
,π 3---
2π 3------
,π 18------
5π 18------
13π
18--------
-17
π18--------
-25
π18--------
-29
π18--------
-,
,,
,,
π 3---5
π 3------
,5 4---
7 4---13 4----
--15 4----
--21 4----
--23 4----
--,
,,
,,
π 4---7π 4----
--,
2π 3------
4π 3------
,π 6---
11π 6----
-----
,π 6---
5π 6------
7π 6------
11π 6----
-----
,,
,3 2---
5 2---11 2----
--,
,
π 6---7π 6----
--,
3π 4------
7π 4------
,π 3---
4π 3------
,4
21–ta
nπ 3---
5π 6------
4π 3------
11π 6----
-----
,,
,
π 12------
5π 12------
13π
12--------
-17
π12--------
-,
,,
π 3---2
π 3------
4π 3------
5π 3------
,,
,3π 8----
--7
π 8------
11π 8----
-----
15π 8----
-----
,,
,
4π 3------
5π 3------
,π 6---
7π 6------
,π 3---
2π 3------
4π 3------
5π 3------
,,
,2
π 3------
5π 3------
,
5π 6------
9π 6------
,π 3---
4π 3------
,π 3---
2π 3------
4π 3------
5π 3------
,,
,π 6---
2π 3------
7π 6------
5π 3------
,,
,
π 3---5
π 3------
,π 4---
3π 4------
5π 4------
7π 4------
,,
,
3π 4------
–π 4--- ,
π 3--- ±7π 8----
--–
3π 8------
–π 8---
5π 8------
,,
,π 2---
–π 2--- ±
π 8---7
π 8------
9π 8------
15π 8----
-----
,,
,π 2---
3π 2------
,π 2---
3π 2------
,
3π 4------
7π 4------
2 3---
1–ta
nπ
2 3---
1–ta
n+
,,
,π 3---
2π 3------
3π 4------
4π 3------
5π 3------
7π 4------
,,
,,
,
π 12------
5π 12------
7π 12------
11π
12--------
-13
π12--------
-17
π12--------
-19
π12--------
-23
π12--------
-,
,,
,,
,,
2π 3------
4π 3------
,
π 3---5
π 3------
,π
cos
1 4---
1–±
3π 4------
7π 4------
3()
1–ta
nπ
3()
1–ta
n+
,,
,π 6---
7π 6------
π 2---3
π 2------
,,
,
3 2---
1–ta
nπ
2()
1–ta
n–
π3 2---
1–ta
n+
2π2(
)1–
tan
–,
,,
2x
π 6---+
sin
02π 3----
--2
π,
,
2x
π 3---–
sin
π 6---3
π 2------
,
π 3---2π 3----
--,
π 6---5
π 6------
,
13
π 6--------
-17
π 6--------
-,
∪
π1 3
-------
1–si
n+
2π
1 3----
---
1–
sin
–,
3π
1 3----
---
1–
sin
+4
π1 3
-------
1–si
n–
,
∪
0π 4---
,5
π 4------
2
π,
∪0
π 6---
π 2---5
π 6------
,
3
π 2------
2π
,
∪∪
,
xx
kπα
1–()k
+k
∈
,=
{}
x2
kπα
+x
2k
1+
()π
α–
≤≤
k
∈,
{}
xx
2k
1+
()π 5---
=
x
x2
kπ=
{}
k∈
,∪
xx
2kπ 5--------
-π 10------
+=
xx
2kπ
π 2---–
=
k
∈,
∪
0π 3---
5π 3------
2π
,,
,2
2 2-------
,
2π 9---
cos
25
π 9------
cos
27
π 9------
cos
,,
π 4--- ±2
π 3------
3π 4------
±,±,
xy,
()
x2
kππ 2---
+=
y2
kπ=
,
x
y,(
)x
2kπ
π 2---–
=y
2kπ
π+
=,
k∈
,∪
T5
πt 12------
3–
19+
sin
=
L3
πt 2.1
-------
3–
7+
sin
=
HL
Mat
hs 4
e.bo
ok P
age
1031
Tue
sday
, May
15,
201
2 8
:54
AM
1032
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
3a
5, 1
1, 0
, 7b
4a
1, 1
1, 1
, 12
b
5a
2.6,
7, 2
, 6b
6a
0.6,
3.5
, 0, 1
1b
7a
0.8,
4.6
, 2.7
, 11
b
8a
3000
b 10
00, 5
000
c
9a
6.5
m, 7
.5 m
b 1.
58 se
c, 3.
42 se
c 10
a 75
0, 1
850
b 3.
44c m
id-A
pril
to en
d of
Aug
ust
11a
1500
0b
12 m
onth
sc
d 4
mon
ths
12a
π, –
2, 2
b m
c m
13a b
c 3d
38.4
%
14a
b i 7
,11,
19,2
3ii
c 14
.9 m
Exer
cise
11.
1
1a
i 2ii
–3iii
6iv
0v
vi
b i 2
ii iii
–5
iv
v vi
–1
c i
ii iii
iv
v
vi
2a
b c
d e
f
3a
b c
d e
f
4a
b c
d
e 1
f
5a
b c
d
e f
6a
b c
7a
b c
8a
i ii
b i –
1ii
iii –
1iv
–1
9
12a
b
13
a b
14
a b
–4c
15
a b
161
17 18
19a
b
21
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
V5
2πt 11--------
sin
7+
=
P2
π 11------
t1
–(
)si
n12
+=
S2.
62π 7----
--t
2–
()
sin
6+
= P0.
64
πt 7--------
sin
11+
= D0.
8π 2.3
-------
t2.
7–
()
sin
11+
=
4 9---
t
R 915
12
d 4
mon
ths
1 3---4 3---
t
d 915
12
07,
[]
1119,
[]
2324,
[]
∪∪
3 2---1 3---
22 5---
–1 2---
22
i–
3–
2i
–6
5i+
2 5---i
3 2---1 2---
i–
1 3---i
+
7i
+1
3i–
158i
–1
–8
i–
1011
i+
2–
3i+
1–
3i
+5
i–
4–
3i
+6
i4
–7i
+2
–3i
+1 2---
1i
+(
)1 2---
5i
+(
)–
1–
2i
–1 2---
i1 13------
5i
+(
)–
148i
+2
–2i
–2
2–
i–
1 5---2
i+
()
2i
–1 5---
13
i+
()
1 2---1 2---
32
+(
)3
2+
x4
y,1 2---–
==
x5
y,–
12=
=x
0y,
5=
=
1i
1i
1i
,,
–,–
,,
i1
i1
i–,
,,
–,–
i–
x12
029---------
y,–
39 29------
==
x0
or
y0
or
both
==
x2y2
–1
=3
i–
2i
–4
ii–
x13
y,4
==
x4
y,4 3---
==
1 3---1
22
i+
()
– uv,
()
1 2---2
2+
()
1 2---2
,
1 2---
22
–(
)1 2---
2–,
,=
7 2---–
1 5---–
21
i+
()
2----
--------
--------
---±
HL
Mat
hs 4
e.bo
ok P
age
1032
Tue
sday
, May
15,
201
2 8
:54
AM
1033
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
22a
b c
d e
24a
b 32
5c
25
26
a b
27a
b c
d
28a
d
Exer
cise
11.
21
The p
oint
s to
plot
are:
(2,1
), (0
,–6)
, (4,
–3),
(2,–
2), (
–3,3
),(–3
,4),(
0.–0
.5),(
1,–1
). 2
a i
ii iii
iv
; A
ntic
lock
wise
rota
tion
of 9
0°.
b i R
efle
ctio
n ab
out t
he R
e(z)
axis.
ii Re
sults
will
alw
ays b
e a re
al n
umbe
r, so
the
poin
t will
alw
ays l
ie o
n Re
(z) a
xis.
iii P
oint
will
alw
ays l
ie o
n th
e ax
is.
3a
b c
d
e f
g h
4a
2;
b 2;
c
; d
3;
e 1;
f 1;
g 6;
0h
;
5a
; ;
b i
or
ii 0
or π
6a
b
7
Tria
ngle
pro
pert
y; th
e sum
of t
he le
ngth
s of t
wo
sides
of a
tria
ngle
is la
rger
than
the
third
side
. 8
0 9
a 15
b 5
c 10
12b
14a
5; –
53.1
3°b
; –45
°c 1
; –90
°
15, π
16a
1;
b 1;
c 1
; –
17a
(for
Prin
cipa
l arg
umen
t) ot
herw
ise,
, whe
re k
is an
inte
ger.
b (f
or P
rinci
pal a
rgum
ent)
othe
rwise
, , k
is an
inte
ger.
c (f
or P
rinci
pal a
rgum
ent)
othe
rwise
, , k
is an
inte
ger.
18a
i ii
b
19a
i ii
b
Exer
cise
11.
3
1a
b c
2a
b c
d e
f g
h i
3a
2b
c d
e f
4a
b 1
c 0
5a
b c
7a
b 2
c d
e f
8a
b –4
c d
e f
9a
b c
d
e f
10a
b c
d e
f
11a
b c
d 25
6e
f
12b
i –1
ii –1
iii
13
a b
c
θα
+(
)i
θα
+(
)si
n+
cos
θα
–(
)i
θα
–(
)si
n+
cos
r 1r 2
θα
+(
)i
θα
+(
)si
n+
cos
()
x22
xθ(
)co
s–
1+
x22
xα(
)si
n1
++
3i
+x2
y2+
()2
z4
b,4–
==
θ()
iθ(
)si
n+
cos
4θ
()
i4
θ(
)si
n+
cos
α2
–0
0β2 –
α4
0
0β4
i α---–0
0i β---
α4
n0
0β4
n
θ()
sin
–i
θ()
cos
+θ(
)i–
θ()
sin
cos
i1
–1
i+
()
–1
i–
1i
+
Imz(
)
3–
4i+
1 2---1
i–
()
1–
3i
+1
3i+
1–
3i–
23i
–1 2---
3i
+(
)1 5---
3i
–(
)
π 3---π 3---
–3
arct
an2
()
π 2---
2π 3------
–π 4---
5 4---ar
ctan
4 3---
–
a2b2
+a2
b2+
a2b2
+π 2---
π 2---–
2x2
18+
3±
1i
+
2π 4---
θπ 2---
θ–
θ
αse
cα
π–
,α
kπ+
αse
cα
π 2---+
,α
π 2---+
kπ+
2θ 2---
cos
θ 2--- ,θ 2---
kπ+
3 2-------
i 2---+
π 6---5
π 6------
–
2θ 2---
sin
θπ
– 2----
--------
Im(z
) θπ
– 2----
--------
2θ 2---
sin
Re(
z)O
2ci
sπ 4---
2ci
s3π 4----
--
2
cis
3π 4------
–
22
cis
π 4---
2ci
sπ 6---
42
cis
π 4---–
5ci
s53
°7'
()
5ci
s15
3°2
6'(
)
13ci
s12
3°41
'–(
)2
cis
5π 6------
cis
π 3---–
10
cis
18–°2
6'(
)
i3
32
--------
--3 2---
i+
1i
–5
i–
4–
43i
+1 6---
26
i+
()
5 3---
13
i–
1i
–1
3–
()
13
+(
)i+
22
2π 4---
2π 3------
11π
12--------
-
41
i+
()
–16
–16
i+
8–
83i
–16
3–
16i
–11
7–
44i
–1 8---
1–
i+
()
1 4---–
1 32------
1i
+(
)–
1 32------
1–
3i
+(
)
1 64------
3–
i+
()
115
625
--------
-------
117
–44
i+
()
8i
–81 2----
--1
–3
i+
()
1 2---i
1 125
---------
i–
1 16------
13
i+
()
–2 81------
–1
3i
+(
)
128
1i
–(
)4
34i
–32
i–
1175
362
5----
--------
---10
296
625
--------
-------
i–
2i–
i
i–6
21
i+
()
22
––
22
+i
+
HL
Mat
hs 4
e.bo
ok P
age
1033
Tue
sday
, May
15,
201
2 8
:54
AM
1034
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
14a
;
b i
ii
c 323
[or
] 25
a co
sec
b
Exer
cise
11.
4.1
1a
b c
d e
f g
h i
2a
b c
d e
f g
h i
3a
b c
4
a b
c d
e f
Exer
cise
11.
4.2
1a
b c
2a
b c
d
e f
3a
b c
d e
f
4
5
6
7
8a
b
9
10
11
12
13
14a
b
c d
15a
b
c d
16
,
172
± i,
Exer
cise
11.
4.3
1a
, 3b
c
d e
f ±2,
,
2;
3a
b c
4a
b
c
5a b
c
d e
,
f
2 2-------
1i
+(
)1 2---
13
i+
()
2 4-------
13
–(
)1
3+
()i
+(
)2 4-------
13
+(
)2 4-------
13
–(
)
cis
θ–()
()3
2θ
2α
cos
+co
s(
)θ
α–
()
iθ
α–
()
sin
–co
s(
)2
αθ
–(
)co
s
θθ
π 2---–
x3
–i
+(
)x
3–
i–
()
x2
3i
++
()
x2
3i–
+(
)x
1–
i+
()
x1
–i
–(
)
z2
i+
+(
)z
2i
–+
()
z3
7i
+(
)2
--------
--------
-------
–
z
37
–i
+(
)2
--------
--------
--------
--–
z5
5i
++
()
z5
5i
–+
()
4w
14
i– 2
--------
------
+
w
14
i+ 2
--------
------
+
3w
1–
i+
()
w1
–i
–(
)2
w2
–6
i2---------
–
w
2–
6i
2---------
+
–
2–
2i±
111
i±
2----
--------
--------
-3
3i
± 6----
--------
------
5–
7i± 4
--------
--------
-------
5–
2i
±
4i
±6
–2
i±
3–
i±
5 3---i
±
2±i±,
3±i±,
3±2
i±,
z5
i–
()
z5
i+
()
z7
i–
()
z7
i+
()
z2
i+
+(
)z
2i
–+
()
z3
2i
++
()
z3
2i
–+
()
z2
i–
()
z2
i+
()
z2
–(
)z
2+
()
z2
i–
()
z2
i+
()
z3
–(
)z
3+
()
z2
+(
)z
i+
()
zi
–(
)z
9–
()
zi
+(
)z
i–
()
z2
–(
)z
2i
+(
)z
2i
–(
)w
15
i–
+(
)w
15
i+
+(
)w
2–
()
z1
–(
)z
2–
i+
()
z2
–i
–(
)z
1–
()
z1
i+
+(
)z
1i
–+
()
x2
+(
)x
2–
()
xi
+(
)x
i–
()
w2
+(
)w
1–
i+
()
w1
–i
–(
)z
5+
()
z5
–(
)z
5i
+(
)z
5i–
()
13
4i±
,2
32
i±
,2
3,–
1i
±,
1 2---1
–i
±,
5 3---–3 2---
12
i±
,,
1–3
–i
±,
1 3---1–
3i± 2
--------
--------
---, 1 2---–
12i
±,
z3
–(
)z
2–
3i+
()
z2
–3
i–
()
12i±
1–
11i
± 2----
--------
--------
------
,
2z
1–
()
zi
+(
)z
i–
()
z3
+(
)z
3–
()
z2i
+(
)z
2i
–(
)2
i1–,
±
23
i13 4----
--–,
±
2i
1 2--- ,± z
2–
()
z4
–i
+(
)z
4–
i–
()
z2
–(
)z
1–
i+
()
z1
–i
–(
)
21
1–
3i
± 2----
--------
--------
---,
,–
13
i±
,1
21
3i± 2
--------
--------
--,
,1
–3
i±
,3
±i±,
5±
i±,
x37x
217
x15
–+
–0
=x4
5x
–3
10x2
10–x
4+
+0
=
x35
x210
x12
–+
–0
=x4
2x
+3
2x2
2x
–21
++
0=
1–
i3
–1
5±
1 2---3
–5
±(
)
3 2---–
33
2----
------
±i
33
2----
------
±3 2---
i3
i–,
+2
i3
±i
–,
2–
2i2
–2i
22
i2
2i
+,
–,
+,
–3 2---
22
–2
2+
i–
()
3 2---2
2–
22
+i
–(
)3 2---
22
+2
2–
i+
()
,–,
3 2---–2
2+
22
–i
+(
)3
i±
3–
i±
1i
1i
±–,
±z
1–
i–
()
z1
–i
+(
)z
1i
–+
()
z1
i+
+(
)1 2
-------
1i
+(
)±
2i
2–
i–
,+
2 2-------
13
i+
()
±
26
cis
π 12------
–
26
cis
7π 12------
26
cis
9π 12------
–
,,
23
cis
2π 9------
23
cis
8π 9------
23
cis
4π 9------
–
,,
cis
π 6---
cis
5π 6------
cis
π 2---–
,
,
28
cis
π 16------
28
cis
9π 16------
28
cis
15π
16--------
-–
2
8ci
s7
π 16------
–
,,
,
cis
π 8---
cis
5π 8------
cis
9π 8------
cis
13π 8
--------
-
,
,,
i3 2-------
1 2---i
–±,
2ci
sπ 12------
–
2ci
s5
π 12------
2ci
s11
π12--------
-
2
cis
7π 12------
–
,,
,2
31
i+
±()
4i–
1 2---3
1+
()
31
–(
)i+
()
±
HL
Mat
hs 4
e.bo
ok P
age
1034
Tue
sday
, May
15,
201
2 8
:54
AM
1035
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
6a
1,
7
Exer
cise
12.
4.1
1a
b c
d e
f g
h
2a
b c
d e
Exer
cise
12.
4.2
1a
b c
d
e f
2
3
4
5
7
8
9
12
Exer
cise
13.
11
a i 1
4 50
0ii
2 00
0b
305
(304
.5)
2Sa
mpl
e siz
e is l
arge
but
may
be b
iass
ed b
y fa
ctor
s suc
h as
the l
ocat
ion
of th
e cat
ch.
Popu
latio
n es
timat
e is 5
000.
3
a i 1
500
ii 12
0b
100
c 100
0 4
a, c n
umer
ical
; b, d
, e ca
tego
rical
5
a, d
disc
rete
; b, c
, e co
ntin
uous
Exer
cise
13.
21 2 3
Set A
Mod
e = 2
9.1
Mea
n =
27.2
Med
ian
= 27
.85
Set B
Mod
e = 9
Mea
n =
26.6
Med
ian
= 9.
Set
B is
muc
h m
ore s
prea
d ou
t tha
n se
t A
and
alth
ough
the t
wo
sets
have
a sim
ilar m
ean,
they
hav
e ver
y di
ffere
nt m
ode a
nd
med
ian.
Exer
cise
13.
31
Mod
e = 2
36–2
38 g
; Mea
n =
234
g; M
edia
n =
235
g2
Mod
e = 1
.8–1
.9 g
; Mea
n =
1.69
g; M
edia
n =
1.80
g
3Se
t A M
ode =
29.
1; M
ean
= 27
.2;
Med
ian
= 27
.85
Set B
Mod
e = 9
; M
ean
= 26
.6;
Med
ian
= 9.
4a
$275
22b
$210
25c M
edia
n5
a $2
3330
0b
$169
000
c Med
ian
6
a 14
.375
b 14
.354
7b
A: 4
9.56
hr,
B: 5
6.21
hr
c Typ
e B8
12.2
2 ca
rds
9a
= 16
, b =
310
b 60
10Ex
ercise
13.
41
a Sa
mpl
e A M
ean
= 1.
99kg
; Sam
ple B
Mea
n =
2.00
kgb
Sam
ple A
Sam
ple s
td =
0.0
552
kg; S
ampl
e B S
ampl
e std
= 0
.187
7kg
c Sa
mpl
e A P
opul
atio
n std
= 0
.054
7kg
; Sam
ple B
Pop
ulat
ion
std =
0.1
858
kg
2a
16.4
b 6.
83
3M
ean
= 49
.97;
Std
= 1
.365
4a
$84.
67b
$148
5
a 2.
35b
1.25
6
a $2
32b
$83
7c 4
08
a i 2
0.17
ii 7.
29b
31c 2
0.76
9a
20b
x +
1
1 2---–
3 2-------
i±
21
3i
±,
–
2n
2+
5n
2–
1 n---3n
n21
+n
n1
+----
--------
n1
+ n----
--------
23n
–
n22n
2+
+2n
2n
2–
+n3
2n
–20
n2n4
–+
4----
--------
--------
--------
--1
n6n
2n3
–+
+
n2
n23
n7
++
()
6----
--------
--------
--------
--------
---n2
n1
+(
)24
--------
--------
--------
--1 4---
11 5n-----
–
n2
2n2
1–
()
n2
n29
n7
++
()
6----
--------
--------
--------
--------
---n
2n
1+
--------
-------
nn
1+
()
2----
--------
--------
180
n2
–(
)n
--------
--------
--------
--
n2n
2+
+2
--------
--------
--------
nn
1+
()
2n
1+
()
6----
--------
--------
--------
--------
-----
2n2
2n
–1
+
n46n
3–
23n2
18n
–24
++
24----
--------
--------
--------
--------
--------
--------
--------
-------
15n
n1
+(
)2
--------
--------
----n
n2
+(
)2
--------
--------
----3
nn
1+
()
2----
--------
--------
---,
,,
2n
1+
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
220
2468
230
240
1.2
2468
1.5
2.0
HL
Mat
hs 4
e.bo
ok P
age
1035
Tue
sday
, May
15,
201
2 8
:54
AM
1036
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
10 Exer
cise
13.
51
a M
ed =
5, Q
1 =
2, Q
3 =
7, IQ
R =
5b
Med
= 3
.3, Q
1 =
2.8,
Q3
= 5.
1, IQ
R =
2.3
c M
ed =
163
.5, Q
1 =
143,
Q3
=182
, IQ
R =
39d
Med
= 1
.055
, Q1
= 0.
46, Q
3 =
1.67
, IQ
R =
1.21
e
Med
= 5
143.
5, Q
1 =
2046
, Q3
= 62
52, I
QR
= 42
062
a M
ed =
3, Q
1 =
2, Q
3 =
4, IQ
R =
2b
Med
= 1
3, Q
1 =
12, Q
3 =
13, I
QR
= 1
c M
ed =
2, Q
1 =
2, Q
3 =
2.5,
IQR
= 0.
5d
Med
= 4
0, Q
1 =
30, Q
3 =
50, I
QR
= 20
e M
ed =
20,
Q1
= 15
, Q3
= 22
.5, I
QR
= 7.
5 3 Ex
ercise
13.
61
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,
1614
.18
Exer
cise
14.
11
15
2a
25b
625
3a
24b
256
4a
24b
48
515
6
270
712
0 8
336
960
10
a 36
2880
b 80
640
c 172
8 11
20
12a
10!
b 2×
8!c
i 2×9
!ii
8×9!
13
3465
0 14
4200
15
4 Ex
ercise
14.
21
792
2a
1140
b 17
1 3
1050
4
70
526
88
6a
210
b 42
0 7
2400
0 8
8 9
155
105
Exer
cise
14.
31
a 12
0b
325
yx
y+
--------
----
1020
30
102030
x
40
40
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=
HL
Mat
hs 4
e.bo
ok P
age
1036
Tue
sday
, May
15,
201
2 8
:54
AM
1037
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
250
40
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
Ex
ercise
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
11a
0.13
99b
i 0.8
797
ii 0.
6 12
b 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.
90
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------
HL
Mat
hs 4
e.bo
ok P
age
1037
Tue
sday
, May
15,
201
2 8
:54
AM
1038
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
3a
b
c d
4a
0.5
b 0.
30c 0
.25
5a
b c
6 7a
b c
8 9a
0.88
b 0.
42c 0
.6d
0.28
10
a 0.
33b
0.49
c 0.8
2 d
0.55
111
a 0.
22b
0.98
5c 0
.862
9 12
a 0.
44b
0.73
314
a 0.
512
b 0.
128
c 0.8
571
15a
0.26
25b
0.75
c 0.4
875
d 0.
7123
16
a 0.
027
b 0.
441
c 0.4
53Ex
ercise
15.
41
a 0.
042
b 0.
7143
2a
0.46
67b
0.38
683
a b
4
5bi
ii
0.2
6ai
ii
b
7 8a
0.07
b 0.
3429
c 0.3
0d
0.02
829
a 0
.800
8b
0.97
67c 0
.000
3 10
a 0.
0464
b 0.
5819
c 0.9
969
11 12
13a
0.8
b 0.
005
14 15M
1
16a
0.57
b
Exer
cise
15.
5
1a
b c
2a
b c
d
3a
b c
4
5a
b
6a
b c
7 8a
b
9a
b
10a
b c
11a
b
4/9
5/9
3/5
2/5
3/5
2/5
R R_
R_ R_R R
8 45------
22 45------
6 11------
2H
2T F
H T01
1/2
H T
0 1
1/2
H T
1 2---2 3---
1 3---
5/10 3/
10
2/10
3/9
4/9 2/
9
5/9 2/
92/
9
5/9
3/9
1/9
Y B G Y B G Y B G
31 45------
2 9---
2 3---
5 7---9 13------
5 9---
1 40------
2Nm
–2
N----
--------
------
2N
m–
()
2N
m–
--------
--------
------
m
mN
m–
()2
n+
--------
--------
--------
--------
----
9 19------ 1 31------ 10 31------
10 21------
18 57------
5 126
---------
5 18------
1 126
---------
1 5---1 10------
2 5---3 5---
72 5525
--------
----1
5525
--------
----1
1201
--------
----
2 5---
63 143
---------
133
143
---------
5 12------
5 33------
5 6---
3 11------ 4 13----
--9 13------
67 91------
22 91------
1 4---1 28------
5 14------
5 28------
1 28------
HL
Mat
hs 4
e.bo
ok P
age
1038
Tue
sday
, May
15,
201
2 8
:54
AM
1039
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
12
13a
b
14a
b
15a
b 0.
6
16 Exer
cise
16.
11
0.3
2
a 0.
1bi
0.2
ii 0.
7
3a
b c
4a
{2, 3
, 4, 5
, 6, 7
, 8, 9
, 10,
11,
12}
b
5a
b c
6a
bi
ii
c
7
a i 0
.904
8ii
0.09
048
b 0.
0002
80.
3712
9
a b
i ii
10
11 a
b
12a
0.81
b 0.
2439
x2
34
56
78
910
1112
p(x)
6 13------ 1 6---
1 4---
1 210
---------
7 9---
719
38----
--------
11 21------
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
n1
23
4P(
N =
n)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---
2x
35p(y)
32468101214
14
p0(
)11 30----
--p
1–()
,1 2---
p3(
),
2 15------
==
=11 30----
--13 15----
--
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------
HL
Mat
hs 4
e.bo
ok P
age
1039
Tue
sday
, May
15,
201
2 8
:54
AM
1040
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
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
17
a b
18a
E(X)
= 4
, Var
(X) =
20
EXER
CISE
16.
31
a 0.
2322
b 0.
1737
c 0.5
941
2a
0.32
92b
0.86
83c 0
.209
9d
0.13
17
3a
0.15
26b
0.48
12c 0
.567
8
4a
0.77
38b
c 0.9
988
d
5a
0.27
87b0
.405
9 6
a 0.
2610
b 0.
9923
7
a 0.
2786
b 0.
7064
c 0.1
061
8a
0.13
18b
0.84
84c 0
.054
d 0.
326
9a
0.23
8b
0.65
31c 0
.002
7d
0.72
6e 1
2.86
10
a 0.
003
b 0.
2734
c 0.6
367
d 0.
648
11a
0.31
25b
0.01
56c 0
.343
8d
312
a 0.
2785
b 0.
3417
c 120
13
a 0.
0331
b 0.
565
14a
0.43
05b
0.61
c $72
0d
0.20
59
15a
i 1.4
ii 1
iii 1
.058
iv 0
.079
5v
0.00
47b
i 3.0
4ii
3iii
1.3
73iv
0.2
670
v 0
.139
016
38.2
3 19
a i 0
.107
4ii
iii 0
.375
8b
at le
ast 6
20a
b c
d
21a
20b
3.46
41
22a
102.
6b
0.00
0254
23
a i 6
ii 2.
4b
i 6ii
3.6
24
0.17
97
251.
6, 1
.472
26
a 0.
1841
b $1
1.93
27
a $8
b $1
60
28a
0.07
02c
29b
30b
0.80
35c 3
9.3
31a
, 0 <
p <
1
EXER
CISE
16.
41
a 0.
3263
b 0.
0932
2
a 0.
0015
b 0
c 0.9
714
3a
1.2
b 0.
56
4a
3.48
48b
c 0.7
071
5a
b
31 60------
μ2 3---
σ2
,0.
3556
==
np
31 2---
×1.
5=
=
1 25------
p0(
)35 12
0----
-----=
p1(
),
63 120
---------
p2(
),
21 120
---------
p3(
),
1 120
---------
==
=
3 3-------
n0
12
P(N
= n
)
28 45------
16 45------
1 45------
a2 3---
=0
b1
≤≤
,E
X()
b1
+ 3----
--------
=V
arX(
),
1 9---2
7b
b2–
+(
)=
3.12
510
7–×
310
5–×
7.9
104–
×4 3---
10 9------
1 6---5 288
---------
0.6
1
p
1
1
p
np
np
1p
–(
)n1
––
11
p–
()n
–np
1p
–(
)n1
––
--------
--------
--------
--------
--------
--------
--------
--------
-----
10 3------
113
65----
--------
22 91------
HL
Mat
hs 4
e.bo
ok P
age
1040
Tue
sday
, May
15,
201
2 8
:54
AM
1041
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
6 7a
b
8a
b 3
9a
b
10a
b c
11a
b
12
13a
hype
rgeo
met
ricb
c
14
15a
b
16a
b 0.
9999
17
180.
8 19
a b
i 0.3
365
ii 0.
0106
6c 0
.968
2
20a
b c
, 10
days
bef
oreh
and
(pla
ce o
rder
on
11 Ju
ly)
22a
rem
aind
er ~
0
b , ~
90%
EXER
CISE
16.
5
1a
b i 0
.135
3ii
0.27
07iii
0.5
940
iv 0
.455
72
a 0.
0383
b 0.
1954
3
a 0.
2052
b 0.
9179
4
a 0.
2623
b 0.
8454
5
a 0.
0265
b 0.
0007
6
a 0.
1889
b 0.
7127
7
a 0.
7981
b 0.
2019
c 0.1
835
8a
0.26
61b
0.52
21
90.
1912
10
a 0.
3504
b 0.
6817
11
a 0.
0012
7b
0.05
00
12a
0.18
04b
0.01
66c 0
.323
3 13
a 0.
8131
; 0.5
511
No
1414
. 0.4
781
15a
0.36
79b
0.26
42c 0
.213
5 16
a i p
ii iii
c 0
.478
5EX
ERCI
SE 1
7.1
1a
0.69
15b
0.96
71c 0
.947
4d
0.99
65e 0
.975
6f 0
.005
4g
0.02
87h
0.05
94i 0
.007
3j 0
.828
9k
0.64
43l 0
.082
3
2a
0.03
60
b 0.
3759
c 0.0
623
d 0.
0564
e 0.0
111
f 0.2
902
g 0.
7614
h 0.
0343
i 0.6
014
j 0.1
450
k 0.
9206
l 0.2
668
m 0
.702
0n
0.91
32o
0.52
03p
0.81
60q
0.93
88r 0
.725
8 EX
ERCI
SE 1
7.2
1a
0.02
28b
0.93
32c 0
.308
5d
0.88
49e 0
.066
8f 0
.977
2
2a
0.97
72b
0.06
68c 0
.691
5d
0.11
51e 0
.933
2f 0
.022
8
3a
0.34
13b
0.13
59c 0
.048
9
4a
0.68
27b
0.13
59c 0
.393
4 5
a 0.
8413
b 0.
4332
c 0.7
734
6a
0.11
51b
0.10
39c 0
.158
7 7
a 0.
1587
b 0.
6827
c 0.1
359
8a
0.19
08b
0.47
54c 1
6.88
9
a 0.
1434
b 0.
6595
10
a 0.
2425
b 0.
8413
c 0.5
050
11a
–1.2
816
b 0.
2533
12
a 58
.224
3b
41.7
757
c 59.
80
1339
.11
149.
1660
15
42%
16
0.70
21
17a
0.29
03b
0.45
83c 0
.251
418
23%
19
0.5
1921
1938
--------
----
27 91------
87 91------
x1
23
45
P(X
= x)
0.02
380.
2381
0.47
620.
2381
0.02
38
9 22------
1 22------
46 255
---------
184
595
---------
32 357
---------
7 22------
37 44------
49 60------
x0
12
34
P(X
= x)
330
1365
--------
----66
013
65----
--------
330
1365
--------
----44 13
65----
--------
113
65----
--------
66 91------
1 14------ 1 8---
47 72------
44 45------
11 21------ 48 55----
--
28 45------
16 45------
1 45------
x1
23
45
P(X
= x)
0.59
80.
315
0.07
50.
010
0.00
1
x0
12
34
56
P(Ac
cept
)1
4 5---28 45----
--7 15------
1 3---2 9---
2 1---
PX
x=
()
e2–
2x
x!----
--------
--x,
01
2…
,,
,=
=
pp
ln–
1p
–p
pln
+
HL
Mat
hs 4
e.bo
ok P
age
1041
Tue
sday
, May
15,
201
2 8
:54
AM
1042
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
2011
%
215%
22
14%
23
1.8
2425
2 25
0.15
17
260.
3821
27
0.22
28
322
29
0.15
45
307
3187
32
a i 0
.006
2ii
0.04
78iii
0.9
460
b 0.
0585
33a
$5.1
1b
$7.3
9
34a
0.00
62b
i 0.7
887
ii 0.
0324
c $14
72
35a
μ =
66.8
6, σ
= 1
0.25
b $0
.38S
36
a μ
= 37
.2, σ
= 2
8.2
b 20
(19.
9)
37a
i 0.3
446
ii 0.
2347
b i 0
.333
9ii
0.38
52c 0
.999
5Ex
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
.49º
C/se
c 7
a 12
7π cm
3 /cm
b i 1
9.66
67π
cm3 /c
mii
1.99
67π
cm3 /c
miii
0.2
000π
cm3 /c
m
81.
115
9
a –7
.5ºC
/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
ii
2 iii
1.2
6a
b i 2
0 cm
2ii
17.4
1 cm
2iii
2.5
9
3 4---3a 4
b----
--1–
15 8------
–
t
h1
a
b
t
h
2 a
b
c
t
h
t
h
d
e
f
t
h
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
ms
1–m
s1–
ms
1–
d Fi
nd (l
imit)
as
e 4
t – 3
h0
→
20s(
t)
t
iv
–1.
29 cm
2 /day
c cm
2 /day
d i –
1.38
63 cm
2 /day
ii –1
.293
5 m
2 /day
201
20.
1h
––
()
HL
Mat
hs 4
e.bo
ok P
age
1042
Tue
sday
, May
15,
201
2 8
:54
AM
1043
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
Exer
cise
18.
4
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
5
a 1
ms–1
b (2
– a
) ms–1
6a
b i 5
ms–1
i
i 4 m
s–1c
ms–1
d se
c
Exer
cise
19.
11
a 5x
4b
9x8
c 25x
24d
27x2
e –28
x6f 2
x7g
2xh
20x3 +
2i –
15x4 +
18x
2 –1
j k
9x2 –
12x
l
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
Exer
cise
19.
2.1
1;
2;
;
3–1
2
4a
3b
c 12
d 4
e 4f
g h
5
6a
2x –
12
b –1
8c (
8, –
32)
7a
–3x2 +
3b
0c
8a
,(0, 0
)b
9 10a
–2, 6
, 3b
–2
11a
= 1
b =
–8
12
13a
b
14–5
6Ex
ercise
19.
2.2
1a
bc
2
1 9---–
17 16------
4.9
h22
h+
()
x2–
–2
x1
+(
)2–
–0.
5x1
2/–
4
8 3---25
627---------
,
x(
t)
t
ms
1–8
t3
t2–
8 3---
4 3---–x3
10+
32 5---
x4
x3+
+
3 x4-----
–3 2---
x5 2---
x31
3x2
3----
--------
--2 x
------
9x
1 x----
--3 x2-----
+3 2---
x1
2x3
--------
-----
–
10 3x
3----
-------
9–
51
2x
--------
--8 5x3
---------
––
4 x----
--15 x6----
--1 2---
+–
1
2x3
--------
-----–
1 x----
--–
x2+
3 2---x
1 x----
--+
4x3
3x2
1–
+3
x21
+1 x2-----
1 x3----
-----
1 2---1
4x3
--------
-----–
2x
8 x3-----
–2
x2 x2-----
4 x5-----
––
1 2---3 x---
1
6x3
--------
-----+
2x
12 5------
x5
2
5x3
5----
--------
--+
–
3
2x
--------
--1 x---
1+
1 x----
--x
–
2–
mP
Q4
h+
=m
PQ
h0
→lim
4=
P1
1,(
)Q
1h
22
h+
--------
----,
+
,
mP
Q1
2h
+----
--------
–=
mP
Qh
0→lim
1 2---–
=
1 4---–7 6---
1 12------
–53 16----
--
8 3---±
22
±,±(
)
2 2-------
±1 16------
–,
x:
1–
2----
---x
0<
<
x:
x1 2
-------
>
∪
x1 3---
=1–,
f'
ab
+(
)2
ab
+(
)2
a2
b+
==
4a2
2a
a0
≥,
–4
1 a---a
0>
,–
03 4--- ,
y'
x
y'
x4
–1
1y'
x5
y'
x4
–1
y'
x
y'
x5
y'
x
y'
x
2
–1
1
2
3
5 –5
a
b
y'
xa
b
c d
gh
i
de
f
y
x
1
1
HL
Mat
hs 4
e.bo
ok P
age
1043
Tue
sday
, May
15,
201
2 8
:54
AM
1044
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
3 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
11
0
120
y
x1
2
4 2 48t3
1 2t
---------
–2n
2 n2-----
–4 n5-----
–3 2---
r5
6r
6----
------
1 r----
--–
+2θ
9 2---θ
–3
1
2θ
--------
--–
+
403L
2–
100
v3---------
–1
–6l
25
+2
π8
h+
4n3
1
3n2
3----
--------
--π
+–
8 3t3
--------
2πr
20 r2------
–5 2---
s32/
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
–2
x–
++
4 x5-----
–6x
58
x32
x+
+
2
x1
–(
)2----
--------
-------
–1
x1
+(
)2----
--------
-------
1x2
–2
x–
x21
+(
)2----
--------
--------
------
x43
x22
x+
+(
)–
x31
–(
)2----
--------
--------
--------
--------
-----
2x2
2x
+
2x1
+(
)2----
--------
--------
---1
12x
–(
)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
ln()2
--------
---------
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
–+
44x
1+
--------
-------
1–
1 2---x 2---
cos
22x
sin
–7
7x2
–(
)co
s1
2x
--------
--1 x---
–1 x---
66
xsi
n+
2x
x2co
s2
xx
cos
sin
+2
sec2
2θ
θco
s
sin2
θ----
--------
-–
1
2x
--------
--x
cos
1 x2-----
1 x---
sin
3θ
cos2
θ⋅
sin
–ex
ex ()
cos
1 x---se
c2x e
log
()
2si
nx
–
2xco
s----
--------
-------
θco
s–
θsi
n(
)si
n⋅
4θ
sin
sec2
θ⋅
55
xco
s–
csc2
5x
()
⋅6
csc2
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+
θ()e
θsi
nco
s
22
θ(
)e2
θco
s–
sin
2x
2ex–
ex–
1+
()2
--------
--------
--------
-3
exe
x–+
()
exe
x––
()2
ex2
+2
x–
9+
()e
x2–
9x
2–
+
2x
x21
+----
--------
---θ
cos
1+
θsi
nθ
+----
--------
--------
-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 θ----
---θ
θco
s⋅
sin
–
3x2
4x4
–(
)e2
x23
+–
xln
1+
()
–x
xln
()
sin
1x
xln
--------
---
2x
4–
()
x2 ()
sin
⋅2
xx2 (
)x2
4x
–(
)co
s⋅
–
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
2xx
cos
+(
)–
2xx
sin
()
sin
⋅e5
x2
+9
20x
–(
)1
4x–
()2
--------
--------
--------
--------
--------
cos2
θsi
n2θ
θsi
n(
)ln
+ sin
θcos
2θ
--------
--------
--------
--------
--------
--------
--------
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
–----
--------
--------
+
HL
Mat
hs 4
e.bo
ok P
age
1044
Tue
sday
, May
15,
201
2 8
:54
AM
1045
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
131
14 –
2e
15a
b c
16b
i 2xs
inxc
osx
+ x2 co
s2 x - x
2 sin2 x
ii
17a
i ii
b i
ii
18
19 20 21a
b c
d
e f
22a
b c
d e
f g
h i
23a
b c
d e
f
g h
i 0
Exer
cise
19.
4
1a
b c
d e
f g
h i
j k
l
2a
b c
d
e f
g h
i
j k
l
m
n o
3a
b c
d e
f g
h
i
4
6b
7a
and
; =
b ;
,
c ;
= [–
1, 1
]
d an
d ;
= ]–
∞, ∞
[
e ;
f an
d
or
; =
[–1,
1]
cos2
xsi
n2
x–
π 180
---------
x°co
sπ 18
0----
-----x°
sin
–
ex3
–2
2x
xco
sln
cos
3x2
–2
xx
cos
lnsi
n2
sin
xx
tan
–(
)
3 x---–x
ln()2
3x2
1x3
–----
--------
--–
2e2
x–
e2
x–
()
cos
⋅–
2x
x2co
s–
ex2
sin
–⋅
1 5---–k
xa
bm
bn
a+
mn
+----
--------
--------
,,
= θ:n
θmta
nθn
tan
⋅m
θmn
–=
{}
44
x(
)cs
c–
22x(
)se
c2x(
)ta
n3
3x()
cot
3x()
csc
33x(
)si
n–
π 4---x
–
2
csc
2–2
x(
)se
c2
x(
)ta
n
2xx2
()
sec
x2(
)ta
nse
c2x
xta
n3
cot2
xcsc
2x
–x
xx
sin
+co
s
2xc
sc2
xco
t–
4x3
4x
()
csc
4x4
4x
()
cot
4x
()
csc
–
2xs
ec2
2x
()
cot
csc2
x2
x(
)ta
n–
xx
xsi
n–
tan
sec
2x
xse
c+
cos
--------
--------
--------
--------
-------
ex
sec
xx
tan
sec
exex (
)se
cex (
)ta
nex
x()
sec
exx(
)se
cx(
)ta
n+
csc2
xlo
g(
)–
x----
--------
--------
--------
--5
5x
()
csc
–5x(
)se
cx(
)co
t x----
--------
---x(
)2
csc
xlo
g–
xx
sin
()
cot
cos
–x
sin
()
csc
xcs
c(
)co
s–
xx
csc
cot
2
4x2
1+
--------
--------
--1
9x2
–----
--------
------
2–
14x
2–
--------
--------
------
4
116
x2–
--------
--------
---------
2
x24
+----
--------
---
1
2x
x2–
--------
--------
-----
1–
16x2
–----
--------
--------
-1
4x
1+
()2
–----
--------
--------
--------
-----1
4x
–(
)21
+----
--------
--------
--------
1–
4xx2
–----
--------
--------
-6
4x2
9+
--------
--------
--1–
x2–
x2
++
--------
--------
--------
--------
2x
x41
+----
--------
---1
2x
x2–
--------
--------
-----
1
2x3
x2–
--------
--------
--------
xsi
n– 1
cos2
x–
--------
--------
--------
---1
if
x0
>si
n– 1
if
x0
<si
n
=
1
2xx
1–
--------
--------
------
1
1x2
–S
inx1–
--------
--------
--------
--------
---ex
1e2
x+
--------
--------
-1
e2x
1–
--------
--------
-----
earcs
inx
1x2
–----
--------
------
4–
4x2
1+
()
tan
1–2
x(
)[
]2----
--------
--------
--------
--------
--------
--------
---1–
1x2
–si
n1–
x()
()3
2/----
--------
--------
--------
--------
--------
---------
2
1x2
–co
s1–
x()
()3
--------
--------
--------
--------
--------
--------
4x– 14
x2–
--------
--------
------
4x
– 14
x2–
--------
--------
------
1–
x21
x2–
--------
--------
--------
Tan
1–x
x
1x2
+----
--------
---+
x1
x2–
sin
1–x
–
x21
x2–
--------
--------
--------
--------
--------
---x
1x2
–co
s1–
x+ co
s1–
x(
)21
x2–
--------
--------
--------
--------
--------
-----
2x2
–ta
n1–x
x2
tan
1–x
–+
x3x2
1+
()
--------
--------
--------
--------
--------
--------
--------
-------
2x2
x1
x4–
sin
1–x2
()
+lo
g
x1
x4–
--------
--------
--------
--------
--------
--------
--------
--------
-----
1x
–co
s1–
x–
x–
2x3
2/1
x–
--------
--------
--------
--------
--------
--------
------
exta
n1–
ex ()
e2x
1e2
x+
--------
--------
-+
2xta
n1–
x 2---
2+
1x
4x2
–----
--------
------ S
in1–
x 2---
–
0k,
π 2---=
kπ 2---
=
f'
x()
π–
xx2
π2–
--------
--------
---------
xπ
>,
=π
xx2
π2–
--------
--------
---------
xπ–
<,
dom
f()
]∞
π–[
]π∞
[,
∪,
–
f'
x()
1
x2
x1
–----
--------
--------
--x
1 2--->
,=
dom
f'
()
1 2---
∞
,=
dom
f()
1 2---∞
,
=
f'
x()
1
1x2
–----
--------
------ C
os1–
x 2---
1
4x2
–----
--------
------ S
in1–
x()
1x
1<
<–,
–=
dom
f()
f'
x()
2
x21
+----
--------
---–
x0
>,
=2
x21
+----
--------
---x
0<
,do
mf()
f'
x()
a
1a2
x2–
--------
--------
---------
x1 a---
<,
=do
mf()
1 a---–
1 a--- ,=
f'
x()
2
1x2
–----
--------
------
1 2----
---x
1 2----
---<
<–,
=f
'x(
)2– 1
x2–
--------
--------
--1
x1 2
-------
–<
<–,
=
1 2----
---x
1<
<do
mf()
HL
Mat
hs 4
e.bo
ok P
age
1045
Tue
sday
, May
15,
201
2 8
:54
AM
1046
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
g
and
; =
]–∞
, ∞[
h
and
; =
]–∞
, ∞[
8a
b
0c
d
e f
9a
b
Exer
cise
19.
5
1a
b c
d 3
e 7f 2
g h
3i –
5
2a
x +
b
c
d e
f
3a
b
c d
e
f g
h i
j k
l
4
5
6
7
81.
25
9a
b c
d
10a
b c
d e
f
g h
i
11a
b c
d
e f
g
h i
12a
b c
d
Exer
cise
19.
61
a b
c d
e f
g h
i j
k l
m
n o
p q
r s
t
2a
b c
d e
f g
h i
3
f'
x()
2
x21
+----
--------
---x
0>
,=
2–
x21
+----
--------
---x
0<
,do
mf()
f'
x()
2
x21
+----
--------
---x
1<
,=
2–
x21
+----
--------
---x
1>
,do
mf()
nxn
1–
1x2
n+
--------
--------
--n
1x2
+----
--------
---ar
ctan
x(
)n1
–+
21
x2–
1
2a
x–
()
xb
–(
)----
--------
--------
--------
--------
----
1
21
x2+
()
--------
--------
-------
1
x21
+----
--------
---–
x0
≥1–
x1
+(
)x
--------
--------
--------
x0
>,
4ln(
)4x
3ln(
)3x
8ln(
)8x
5ln(
)5x
6ln(
)6x
10ln(
)10x
6ln(
)6x
2–
2ln(
)23
x1
+7
ln()7
3x
–
3ln(
)3x
3x4
2x
()2
x2
2()
2x
()2
xsi
nln
+co
s5(
)5x e
x–5x e
x––
ln
28
x–×
8()8
x–x2
ln–
14x
+(
)x
2+
()
4()4
xln
– 14x
+(
)2----
--------
--------
--------
--------
--------
--------
--------
--x
sin
5x5(
)5x
xco
sln
+–
1 5ln(
)x----
--------
----1 10
ln()x
--------
--------
---1 4
ln()x
--------
--------
19
ln()
x1
+(
)----
--------
--------
--------
--2
x
2ln(
)x2
1+
()
--------
--------
--------
--------
-
12
5ln(
)x
5–
()
--------
--------
--------
--------
-x 2
log
1 2ln----
----+
3ln(
)3x
x 33x 3
ln()x
--------
--------
+lo
ga
ln()a
xx a
ax aln(
)x----
--------
----+
log
aln(
)2xa
xx
ax–
alo
g
aln(
)xx a
log
()2
--------
--------
--------
--------
--------
--------
-10(
)ln(
)x
1+
()
1–
10lo
g
10()
lnx
1+
()
10lo
g(
)2----
--------
--------
--------
--------
--------
--------
---------
2()
ln()2
x2
–2
log
2()
lnx 2
log
()2
--------
--------
--------
--------
--------
----
1 2ln----
----
02 2
ln--------
–, 13
ln– 3
--------
--------
-
π2π–
33
ππ
ln 2----
--------
--------
+
2010
10ln
+4
ln()
1()
cos
1 2---10
10 10ln----
-------
–
454
x1
+×
5ln
3xx3
–1
3x2
–(
)3
ln2
102
x3
–(
)10
ln
9x
x–
1
2x
--------
--1
–
9
ln2
2x
()
1+
cos
22
xsi
nln
–4–
2x
cos
42
xsi
nln 2
xco
s----
--------
--------
--------
--------
--------
---
2x2x
cos
2ln
2x
sin
xco
s2
ln7–
1 x---2
x–
2x
2–+
()
7ln
22
xco
t 2ln
--------
--------
-x
x21
–(
)5
ln----
--------
--------
--------
1
2x
10–
()
x10
ln----
--------
--------
--------
--------
--------
--4–
2x
2se
c2
42
2x
tan
–(
)ln----
--------
--------
--------
--------
----
1
2x
x2–
x1–
sin
2ln
--------
--------
--------
--------
--------
--------
---1–
1x2
–(
)1
+(
)1
x–
()
3ln
1–ta
n----
--------
--------
--------
--------
--------
--------
--------
--------
-----
3x2
x33
–(
)3
ln----
--------
--------
--------
1–2
2x
–(
)2
ln----
--------
--------
--------
12
10ln
--------
------
–x 2---
2–
tan
xxx
1+
ln()
xx
sin
xx
xsi
n x----
------
+ln
cos
1x
ln–
()x
1 x---2
–2
x()
lnx
x1
–ln
20x3
481
2x
+(
)22 x3-----
2
1x
+(
)3----
--------
-------
26
x2
–(
)3----
--------
-------
42 x8------
241
2x–
()
1 x2-----
–2
x21
+(
)1
x2–
()2
--------
--------
-------
–16
4θ
sin
–2
xx
xsi
n–
cos
6x2
x6
xx
x3x
sin
–si
n+
cos
1 x---10
2x
3+
()3
--------
--------
-------
6xe
2x
12x2
e2x
4x3
e2x
++
84
x15
4x
cos
–si
n
ex----
--------
--------
--------
--------
--------
-
2x2
cos
4x2
x2si
n–
48x2
2x5
+(
)–
4x3
1–
()3
--------
--------
--------
--------
---10
x3
–(
)3----
--------
-------
2x–
x21
+(
)2----
--------
--------
--x
1x2
–(
)32/
--------
--------
--------
---x–
1x2
–(
)32/
--------
--------
--------
---2
x21
+(
)2----
--------
--------
--2
x1
–
4x
x2–
()3
2/----
--------
--------
--------
--
2x
3x2
–
4x3
x2–
()3
--------
--------
--------
--------
2ar
ctan
x()
x3----
--------
--------
------
1
x21
x2+
()
--------
--------
--------
-–
13
x2+
xx3
+(
)2----
--------
--------
--–
1
4x
x3–
--------
--------
-----
x32/
24
x2–
()3
2/----
--------
--------
--------
---1 4---
Sin
1–x 2---
x3
2/–
–+
exx
–2
e2x
2–
+(
)
e2x
1–
()
1e2
x–
--------
--------
--------
--------
--------
----
6x
5–
ln
x4----
--------
--------
n2x
lnn
xln
2n
–1
–+
xn2
+----
--------
--------
--------
--------
--------
--------
--,
HL
Mat
hs 4
e.bo
ok P
age
1046
Tue
sday
, May
15,
201
2 8
:54
AM
1047
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
4
5 7a
b
c
8a
b
9–1
10
[0,1
.076
8[ ∪
]3.6
436,
2π]
Exer
cise
19.
7.1
1a
b c
d e
f g
h i
j –1
k l
2(1
,5),
0
4,
5a
c d
e Hyp
erbo
la
6a
Dom
= R
an =
[–2,
2]b
c d
smal
l
e Dom
= R
an =
[–k,
k]f
7a
b
8a
b –1
9a
b
10a
unde
fined
b At
(0.8
042,
0.5
), gr
ad =
1.3
2; at
(0.0
646,
0.5
), gr
ad =
3.7
4 Ex
ercise
19.
7.2
1 a
b
c d
f g
h i
2b
–3c 2
d
3b
–1c 1
0d
4a
Dom
[0,6
] Ran
[–4,
2]b
,
5
Exer
cise
19.
8
1a
b c
d e
f
g h
l
2a
b
c d
3–3
40
5a
b c
6
f'
x()
1
x1
+(
)2----
--------
-------
–=
f''
x()
2
x1
+(
)3----
--------
-------
f(ii
i)x(
)6
x1
+(
)4----
--------
-------
–=
,=
f(iv
)x(
)24
x1
+(
)5----
--------
-------
=,
,
f(v)
x()
120
x1
+(
)6----
--------
-------
……
f(n)
x()
1–()n
n!
x1
+(
)n1
+----
--------
--------
--------
=,
,–
=
fx(
)x
1+
x1
–----
--------
n
f''
x()
4
nn
x+
()
x21
–(
)2----
--------
--------
---x
1+
x1
–----
--------
n
==
anea
x1–(
)n2n
n!
2x
1+
()n
1+
--------
--------
--------
-------
n2
k :
=
n2
k1
:–
=
yn(
)x(
)1–(
)ka2
kax
b+
()
k,si
n1
2…
,
,=
=
yn(
)x(
)1–(
)k1
+a2
k1
–a
xb
+(
)k,
cos
12
…,
,=
=
21
82
--------
--+
3π
+ 2----
--------
2x
–x y--
–1 x3
y----
----y
x1
+----
--------
–ye
x
1ex
+----
--------
--–
xsi
ny
–x
--------
--------
---x–
13x
4y
– x5----
--------
---------
yx
2+
cos
xsi
n----
--------
--------
----–
4x3
3y2
1+
--------
--------
--x
y+
1–
32
10.6
–2
--------
--------
--------
---80
426
5+
40----
--------
--------
--------
,
3
210
.6+
2----
--------
--------
-------
804
265
–40
--------
--------
--------
----,
yx
5x2
80–
±2
--------
--------
--------
--------
--=
dy
dx
------
2x
y+
2y
x–
--------
-------
=5
x5x
280
–± 2
5x2
80–
--------
--------
--------
--------
-----
x3 y3-----
–x3
16x4
–4 (
)3----
--------
--------
--------
--–
dy
dx
------
x2n
1–
y2n
1–
--------
--------
–=
ν– pγ
------
nm
1–
()x
m2
–
mn
1+
()y
n----
--------
--------
--------
--------
1 11------
yxy
x–
--------
------
1y2
+(
)y
1–·
1–ta
n(
)
1x
–y2
+----
--------
--------
--------
--------
--------
------
dy
dx
------
5 x2-----
–=
d2y
dx2
--------
,10 x3----
--=
dy
dx
------
2x3
18x2
+
x6
+(
)2----
--------
--------
-------
=d2
y
dx2
--------
,2
x336
x221
6x
++
x6
+(
)3----
--------
--------
--------
--------
--------
--=
dy
dx
------
x y--–
=d2
y
dx2
--------
,4 y3-----
–=
dy
dx
------
1
3e3
y----
-------
–=
d2y
dx2
--------
,1
3e6
y----
-------
–=
dy
dx
------
22
xco
s yco
s----
--------
------
=d2
y
dx2
--------
,4
2xsi
ny
cos
--------
---------
–4
2x2co
sy
tan
y2co
s----
--------
--------
--------
----+
=d
yd
x----
--y x--
–=
d2y
dx2
--------
,x
xy+ 2x
2----
--------
------
=
dy
dx
------
x3
y+
3x
y+
--------
-------
–=
d2y
dx2
--------
,8
x248
xy8y
2+
+
3x
y+
()3
--------
--------
--------
--------
--------
--=
dy
dx
------
y x--–
=d2
y
dx2
--------
,2y x2----
--=
dy
dx
------
3x
–y
1+
--------
----=
d2y
dx2
--------
9
y1
+(
)3----
--------
-------
–=
7 a---–
15 16------ x
72⁄
––
240
x6–
+24
t5–
–19
202
x1
–(
)5–
813
x(
)16
2x
()
sin
+co
s15 16----
--x
2+
()
72⁄
–24
x1
+(
)5----
--------
-------
81e
3x
–6 x4-----
–11
9e2
x3
x(
)12
0e2
x3
x(
)si
n+
cos
–24
x()
ln x5----
--------
-------
50 x5------
–
xx(
)3
x()
cos
–si
n4
2x()
cos
–
164x(
)si
n–
482x(
)4
642
x(
)16
+2
tan
+ta
n
2()
ln()3
2()
ln–
2 3()
ln--------
-----
3ln(
)23
3()
ln+
[]
642(
)ln(
)4
HL
Mat
hs 4
e.bo
ok P
age
1047
Tue
sday
, May
15,
201
2 8
:54
AM
1048
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
Exer
cise
20.
11
a y =
7x
– 10
b y =
–4x
+ 4
c 4y =
x +
5d
16y =
–x
+ 21
e 4y =
x +
1
f 4y =
x +
2g
y = 2
8x –
48
h y =
4
2a
7y =
–x
+ 30
b 4y
= x
– 1
c y =
–4x
+ 1
4d
y = 1
6x –
79
e 2y =
9 –
8x
f y =
–4x
+ 9
g 28
y = –
x +
226
h x
= 2
3
a y =
2ex
– e
b y =
ec y
= π
d y =
–x
e y =
xf
g
y = ex
h y =
2x
+ 1
4
a 2e
y = –
x +
2e2 +
1b
x =
1c x
= π
d y =
x –
2π
e y =
–x
+ π
f g
ey =
–x
h 2y
= –
x +
2
5A
: y =
28x
– 4
4, B
: y =
–28
x –
44, I
sosc
eles
.
62
sq. u
nits
, y =
2 x
= 1
7
4y =
3x
8
9
y = 4
x –
9
10y =
log e
4
11;
12A
: y =
–8x
+ 3
2, B
: y =
6x
+ 25
,
13y =
–x,
Tan
gent
s:
tan
gent
and
norm
al m
eet
at (0
.5, –
0.5)
14
a y =
3x
– 7
b
15m
= –
2, n
= 5
16a
b c
d
e
17a
y = 1
b At
(1, 2
) y =
2; A
t (–1
, –2)
y =
–2
18a
, b
,
c ,
19
Exer
cise
20.
21
ab
cd
2a
max
at (1
, 4)
b m
in at
c m
in at
(3, –
45) m
ax (–
3, 6
3)
d m
ax at
(0, 8
), m
in at
(4, -
24)
e max
at (1
, 8),
min
at (–
3, –
24)
f min
at
, max
at
g m
in at
(1, –
1)
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
)
ey2
e1
–(
)xe2
–2
e1
–+
=
2e
1–
()y
ex–
3e2
4e
–1
++
=
z0
a23
a4–
,(
)≡
by
a2b2
–x
=
8y
4π
2+
()x
π2–
=4
π2
+(
)y8
x–
4π
π2+
+=
1 2---28,
y1 2---
y,1 2---
–=
=1 2---–
1 2--- ,
1 2---
1 2---–,
,
Q2
1–,(
)≡
y4
x2
–=
37y
26x
70+
=16
yx
65+
=y
4 π---π2
4π
2–
()
--------
--------
----π2
4π
2–
()
--------
--------
----x
–+
=
5y
6x
1–
=
l 1:
3y
2x
–1
+=
l 2:2
y3
x5
+=
l 1:
2y
x=
l 2:y
2x
–5
+=
l 1:
6y
x16
+=
l 2:
y6
x–
15+
=
2 3---1,
(1, 2
)
(3, –
2)
y
x(2
, 0)
(0, –
4)y
x(4
, 0)
y
x3
y
x
(4, 4
)
9 2---–
81 4------
–,
113
+3
--------
--------
---70
2613
–27
--------
--------
--------
----,
113
–3
--------
--------
---70
2613
+27
--------
--------
--------
----,
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
HL
Mat
hs 4
e.bo
ok P
age
1048
Tue
sday
, May
15,
201
2 8
:54
AM
1049
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
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,
π, 2
π
c St.p
ts.
, In
f. pt
s. (0
, 1),
(π, –
eπ ), (2
π, e2π
)
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
)e p
t A: i
No
ii C
usp
(loca
l min
); pt
B: i
Yes
ii St
atio
nary
pt o
f inf
lect
; pt
C: i
Yes
ii S
tatio
nary
poi
nt (l
ocal
max
)f 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 p
oint
(lo
cal/g
loba
l min
); pt
C: i
No
ii Ta
ngen
t par
alle
l to
y–ax
is.13
a i A
ii B
iii C
b i C
ii B
iii A
14a
b
c
x
y1 36------
1 108
---------
,
1 16------
i
j
(1,–
1)
4x
y
2–
4–,(
)2
4–,(
)
x
y
–2
2
π 4---5
π 4------
,π 2---
3π 2------
,
π 2---e
π 2---–
,
3
π 2------
e
3π 2------
–
–,
π/4
π2π
x
y
x3
π 4------
7π 4------
,=
xπ 2---
3π 2------
,=
3π 4------
1 2----
---e3
π 4------
,
7
π 4------
1 2----
---–
e7π 4------
,
π 2---
eπ 2---
,
3
π 2------
e3π 2------
–,
π/2
π2π
x
y
π 4---5
π 4------
,
π 4---1 2
-------
eπ 4---
,
5π 4----
--1 2
-------
–e5
π 4------
,
¼/2
3¼/2
1
y
x
(1, e
–1)
(2, 2
e–2)
y
x
279
356
.16
≈
f'
x()
f''
x()
y
x–a
a
y
x
f'
x()
f''
x()
f'
x()
f''
x()
y
x
HL
Mat
hs 4
e.bo
ok P
age
1049
Tue
sday
, May
15,
201
2 8
:54
AM
1050
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
15 16
17 18 19 m
= –
0.5,
n =
1.5
20a
i ii
b i
ii
21a
= 2,
b =
–3,
c =
0
22St
atio
nary
poi
nts:
loca
l min
at (–
1, 0
) and
loca
l max
at
.In
flect
ion
pts a
re:
and
23Ab
solu
te m
in at
~
, loc
al m
ax at
~In
flect
ion
pts a
t ~ (–
0.43
84, –
1.44
89) a
nd (–
4.56
15, 0
.148
8)24
–27
are l
eft a
s que
stio
ns fo
r cla
ssro
om d
iscus
sion.
28 a
= 1
, b =
–12
, c =
45,
d =
–34
29b
b =
1c
d
30a
2.79
83, 6
.121
2, 9
.317
9b
Use
a gr
aphi
cs ca
lcul
ator
to v
erify
you
r ske
tch.
Exec
ise
20.3
1a
Loca
l min
. at
, loc
al m
ax at
b Lo
cal m
ax. a
t x =
0, l
ocal
min
. at x
= ±
1c L
ocal
max
. at x
= 0
.25
d Lo
cal m
ax. a
t x =
1e n
one
f Loc
al m
ax. a
t x =
0.5
, loc
al m
in. a
t x =
1, 0
g Lo
cal m
ax. a
t x =
1, l
ocal
min
. at x
= –
1h
none
2
a m
ax. =
120
, min
. =
b m
ax. =
224
, min
. = –
1
c max
. = 0
.5, m
in. =
0d
max
. = 1
, min
. = 0
.3 4
yx3
6x2
9x
4+
++
=
(–3,
4)
y
x
4
–1–4
fx(
)1 3---
x3x2
–3
x–
6–
=
fx(
)3
x520
x3–
=
0.05
x
y
1 20------
1 20------
1 2---20
ln–
,
x4
y
x4
3 2---x
4–
3x
10–
2x
4–
--------
--------
---
14
e1–
,(
)
12
64
2+
()e
12
+(
)–
,+
()
12
64
2–
()e
12
–(
)–
,–
()
3–
13+ 2
--------
--------
--------
2.17
33–,
3–
13– 2
--------
--------
-------
0.20
62,
y
x
(3, 2
0)
(5, 1
6)
–34
a1 2
-------
=f
x()
1 2----
---xe
x2–
=y
x
x4 3
-------
=x
4 3----
---–
=
128
33
--------
--–
(2π/
3,–3
/4)
(4π/
3,–3
/4)
(π,–
1/2)
1.5
(2π,
1.5
)
π 3---3
34
--------
--,
5π 3------
33
4----
------
–,
¼
HL
Mat
hs 4
e.bo
ok P
age
1050
Tue
sday
, May
15,
201
2 8
:54
AM
1051
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
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 G
loba
l min
. at (
2, 1
+ ln
2); I
nfl.
pt. a
t (4,
2 +
ln4)
d no
ne
10G
loba
l min
. at
11a
b i
; non
e
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).
12b
Loca
l min
. at
c
13a
Glo
bal m
in. a
t (1,
c –
1); c
≠ 1
b
14 15G
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
+,
2π 3------
5 4--- ,
4
π 3------
5 4--- ,
(π, 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
+,
π 6---3
,
f'
x()
x2
–(
)a1
–x
2+
()b
1–
ab
+(
)x2
ab
–(
)+
()
=f
x()
x2
–x
2+
--------
----=
fx(
)x
2–
()2
x2
+(
)=
2 3---25
627---------
,–
fx(
)x
2–
()2
x2
+(
)2=
c5
cc
1–+
,(
)
x
y
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
HL
Mat
hs 4
e.bo
ok P
age
1051
Tue
sday
, May
15,
201
2 8
:54
AM
1052
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
6a
i (0,
1),
(2, 0
)ii
y = –
1, x
= –
2iii
iv
d =
\
{–2}
b c
7a
y =
8, x
= 3
b
9A
sym
ptot
es: a
b
c d
10A
sym
ptot
es: a
b
c d
11A
sym
ptot
es: a
b
c d
12
a i (
0, 4
), (2
, 0)
ii 13
a
b
Exer
cise
21.
11
a i x
< 0
ii
x >
4iii
0 ≤
x ≤
4b
i –1
< x
< 2
ii x
< –1
, 2 <
x <
5iii
c
i –1
< x
< 1
ii x
< –1
iii x
≥ 1
d i
0 <
x <
1 ii
2 <
x <
3iii
x <
0, 1
≤ x
< 2
e i
ii
–2 <
x <
4iii
f i
–4
< x
< –1
, 2 <
x <
5
ii –1
< x
< 2
, 5 <
x <
8iii
Ex
ercise
21.
21
4.4
(4 d
eer p
er y
ear,
to n
eare
st in
tege
r)
2a
200
cm3
b 73
.5 cm
3 /day
5 a
b
x =
1
y =
2
y
x
x =
–1
y
x
1
x
y
y =
–1
x =
–2
f1–
: \
{-1}
w
here
,f
1–x(
)2
1x
–(
)1
x+
()
--------
--------
---=
x
y
y =
1
x =
–2
x
y
y =
8
Rang
e =
\{8}
8 d
om =
\{
0}, r
an =
\{
2}
yf
x()
=y
gx(
)=
2
–0.5
dom
=
\{–0
.5,0
}, r
an =
\{
0.5}
y2
xx,
0=
=y
1 2---x
x,0
==
yx–
x,0
==
yx
x,0
==
yx2
x,0
==
yx2
x,0
==
yx
x,0
==
yx3
x,0
==
yx
3x,
+0
==
yx
–2
x,+
0=
=y
2x
2x,
–0
==
yx
2x,
+0
==
y3
xx,
–1
==
0.5
(3, 5
)
(1, 1
)
y
xy =
x +
1
y
x
y =
x +
3
2
1
14 a
b
c
15
32
2,
()
y
xx
= 1
∅
∅∅
∅
HL
Mat
hs 4
e.bo
ok P
age
1052
Tue
sday
, May
15,
201
2 8
:54
AM
1053
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
3a
75b
No
4
a $2
07.6
6b
$ 40
.79
per y
ear
c $41
.54
per y
ear
5a
2.50
b 3.
33c 2
.50
6
a 12
30 <
x <
487
70 ap
prox
.b
i 0 <
x <
250
00ii
2500
0 <
x <
5000
0
766
667
to n
eare
st in
tege
r, 1
4469
92 to
nea
rest
inte
ger
8b
133.
33d
46.6
7e 0
< x
< 5
700
9
a b
22.2
2 22
item
s/do
llar
10a
b i
ii
11a
i 0 m
m/s
ii ~
90.6
9 m
m/s
b 0.
6 se
c 12
a 8.
53 cm
/sb
neve
rc n
ever
13–e
–1 m
s–2
Exer
cise
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
neve
r at r
est
c i 5
m fr
om O
in n
egat
ive d
irect
ion
ii 4
ms–1
d 40
me
3a
1 m
s–1b
neve
rc
d 20
ms–2
4a
; b
~ 14
1 se
cc o
nce
d us
e gra
phic
s cal
cula
tor
5a
3 m
in p
ositi
ve d
irect
ion
b i 5
mii
2 m
c 5 m
s–1
e osc
illat
ion
abou
t orig
in w
ith am
plitu
de 5
m an
d pe
riod
2π se
cond
s7
a 10
0 m
, in
nega
tive d
irect
ion
b 3
times
c i 8
0 m
s–1ii
–34
ms–2
d 14
81 m
8
a m
ax. =
5 u
nits
, min
. = –
1 un
itb
sc
i ii
9a
0318
m ab
ove
b i
ii c 0
322
md
10
a 0
< t <
05
or t
> 1
b t >
05
c t =
1 o
r 168
≤ t
≤ 5
11a
This
ques
tion
is be
st do
ne u
sing
a gra
phic
s cal
cula
tor:
b Fr
om th
e gra
ph th
e par
ticle
s pas
s eac
h ot
her t
hree
tim
esc 0
45 s;
285
s; 3
87 s
d i
ms–1
ii m
s–1
e Yes
, on
two
occa
ssio
ns.
12a
2m
in p
ositi
ve d
irect
ion
b i 2
sii
neve
rc 0
026
ms–2
13a
b 02
95
14a
b i –
3999
ms–1
ii –3
620
ms–2
c 227
sd
514
m
Exer
cise
21.
41
a cm
2 s–1b
cms–1
26
cm2 s–1
3a
cm2 s–1
(x =
side
leng
th)
b cm
s–1
4a
37.5
cm3 h–1
b 30
cm2 h–1
c 0.9
6 g–1
cm3 h–1
5
~ 0.
37 cm
s–1
6–0
.24
cm3 m
in–1
7a
0.03
5 m
s–1b
0.03
5 m
s–1
88π
cm3 m
in–1
985
4 km
h–1
10 112
rad
s–1
12a
V =
h2 +
8h
b m
min
–1c 0
.56
m2 m
in–1
13 m
min
–1
D'
x()
4000
0–
2x12
+(
)x2
12x
20+
+(
)2----
--------
--------
--------
--------
-----
5x
18≤
≤=
3000
x32
+(
)2----
--------
--------
--x
0≥
x∅
∈
v1
t1
–(
)2----
--------
------
t1
>,
–=
a2
t1
–(
)3----
--------
------
t1
>,
=v
2e2
te
2t
––
()
t0
≥,
=
a4
e2t
e2
t–
+(
)t
0≥
,=
v2 4
t2–
--------
--------
-0
t≤
2<
,=
a2
t
4t2
–(
)32/
--------
--------
--------
--0
t2
<≤
,=
vt
t1
+(
)10
ln----
--------
--------
--------
t1
+(
)10
t0
≥,
log
+=
a1 10
ln--------
---1
t1
+(
)2----
--------
------
1t
1+--------
---+
t0
≥,
=
va
2b
tet2
–t
0≥
,–
=a
2b
et2
–2
t21
–(
)t
0≥
,=
v2
ln()
2t
1+
3ln(
)–
×3t
×t
0≥
,=
v2
ln()2
2t1
+3
ln()2
–×
3t×
t0
≥,
=
s
t1
–5
t1 3---
or
t1
==
v6
t2–
12+
=a
12t
–=
π 2---a
122
tπ
–(
)co
s–
=a
4x
2–
()
–=
v3.
75e
0.25
t–
3–
=a
0.93
75e
0.25
t–
–=
a0.
25v
3+
()
–=
B
Av A
0.3
e0.3
t–
=v B
10e
t–1
t–
()
=
win
dow
: [0,
3π]
by
[0, 1
4]
win
dow
: [0,
6]
by [
–6, 6
]
4πr
4π
dA dt
-------
3 2---2
x–
=3 2---
2–
53 6------
4 15------
310
200
--------
-----
HL
Mat
hs 4
e.bo
ok P
age
1053
Tue
sday
, May
15,
201
2 8
:54
AM
1054
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
14 cm
3 s–1
150.
9 m
s–1
16–3
.92
ms–1
17a
b [0
, 200
]c
i 153
1 cm
3 s–1ii
15.9
0 cm
2 s–1
d
18~1
.24
ms–1
19
~0.0
696
ms–1
20
a b
~0.5
16 m
s–1
21a
0.09
5 cm
s–1b
0.67
47 cm
2 s–1
22a
i ii
b c 1
30 k
mh–1
d 14
.66
kmh–1
23
–0.7
7 m
s–1
240.
40 m
s–1
253.
2 m
s–1
260.
075
m m
in–1
271.
26° p
er se
c28
rad
per s
econ
d29
a 9%
per
seco
ndb
6% p
er se
cond
300.
064
31
8211
per
yea
r 32
4% p
er se
cond
33
–0.2
5 ra
d pe
r sec
ond
Exer
cise
21.
51
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
102
x30
0.15
t–
=
1130
97V
t
100
200
y11
920
t4
t2–
+= x
70t
=y
80t
=13
0t
525
64----
--------
0.00
2≈
r50
4π
+----
--------
7.00
≈=
504
π+
--------
----50
4π
+----
--------
×
θπ 6---
=
π 3---3
34
--------
--,
5π 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 π------
3
HL
Mat
hs 4
e.bo
ok P
age
1054
Tue
sday
, May
15,
201
2 8
:54
AM
1055
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
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
49a
b Fi
rst i
nteg
er g
reat
er th
an
50 m
51a
isosc
eles
tria
ngle
b iso
scle
s rig
ht-a
ngle
d tr
iang
le52
sq. u
nits
53 sq
. uni
ts54
a c =
– r
Exer
cise
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
h24
r2
r214
4–
--------
--------
---=
8πr
4
r214
4–
--------
--------
---12
2
1 3---
22 3
--------
-- π
4 3--- 10 3π
--------
-- r3
2=
h6
2=
θar
csin
5 6---
=
θta
nxl
x2k
lk
+(
)+
--------
--------
--------
------
=x
k2kl
+=
πr2h
2 3---πr
3+
3πr2
2πr
h+
a23/
b23/
+(
)32/
αβe
2ln
--------
------- α
ln β2
ln----
--------
1β
αln
--------
-----α
ln β2
ln----
--------
R =
S
S
Rα
ln β2
ln----
--------
295
145
×20
7≈
r21 2---
1 4---3
+
4k2
4k
1–
+8
kk
1+
()
--------
--------
--------
------
1 4---x4
c+
1 8---x8
c+
1 6---x6
c+
1 9---x9
c+
4 3---x3
c+
7 6---x6
c+
x9c
+1 8---
x4c
+
5x
c+
3x
c+
10x
c+
2 3---x
c+
4x
–c
+6x
–c
+
3 2---x
–c
+x
–c
+
x1 2---
x2c
+–
2x1 3---
+x3
c+
1 4---x4
9x–
c+
2 5---x
1 9---x3
c+
+
1 3---x3
2/1 x---
c+
+x5
2/4
x2c
++
1 3---x3
x2c
++
x3x2
–c
+x
1 3---x3
–c
+
1 3---x3
1 2---x2
6x–
c+
–1 4---
x42 3---
x3–
3 2---x2
–c
+1 4---
x3
–(
)4c
+
2 5---x5
1 2---+
x41 3---
x31 2---
x2c
++
+x
1 2---x2
2 3---x3
2/2 5---
x52/
–c
+–
+
2 7---x7
2/4 5---
x52/
2 3---x3
2/2x
–c
++
+
1 2---x2
3x
–c
+2
u25u
1 u---c
++
+1 x---
–2 x2-----
–4 3x
3----
----–
c+
1 2---x2
3xc
++
1 2---x2
4x–
c+
1 3---t3
2t1 t---
–c
++
4 7---x7
42
x5
x–
c+
+1 3---
x31 2---
x24 7---
x72/
–4 5---
x52/
–c
++
1 2z2
--------
–2 z---
2z2
zc
++
++
1 2---t4
tc
++
2 5---t5
2t3
c+
–1 3---
u32
u24
uc
++
+
1 8---2
x3
+(
)4c
+3
x24
+c
+
HL
Mat
hs 4
e.bo
ok P
age
1055
Tue
sday
, May
15,
201
2 8
:54
AM
1056
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
Exer
cise
22.
2
1a
b c
d e
f g
h
2 3$3
835.
034
9.5
5 cm
3
629
2
7 81,
–8
9 10
11a
b
12
13
14Vo
l ~ 4
3202
cm3
1511
0 cm
2
Exer
cise
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
n o
5a
b c
d
e f
g h
i j
k
l m
n
o
6a
b c
d e
f
g h
i
7a
b
c d
814
334
913
.19m
s–1 or
1.1
9ms–1
102.
66 cm
11
x2x
3+
+2
x1 3---
x31
+–
8 3---x3
1 2---x2
40 3------
––
1 2---x2
1 x---2
x3 2---
–+
+x
2+
()3
3 4---x4
31 4---
x4x
++
1 3---x3
1+
x4x3
–2x
3+
+
1 2---x2
1 x---5 2---
++
251 3----
-----π
5 7---x3
23 7------
+
Px(
)25
5x
–1 3---
x2+
=
N20
000
201
--------
-------
t2.0
150
0t
0≥
,+
= y2 5---
x2–
4x
+=
y1 6---
x35 4---
x22
x+
+=
y2
x3x2
x+
+(
)=
fx(
)3 10------ x
3–
49 10------ x
13 5------
–+
=
1 5---e5
xc
+1 3---
e3x
c+
1 2---e2
xc
+10
e0.1
xc
+1 4---
–e
4x
–c
+e
4x
––
c+
0.2
e0.
5x
–c
+–
2e1
x–
c+
–5
ex1
+c
+e2
2x
–c
+
3ex
3/c
+2
exc
+
4x e
cx
0>
,+
log
3x e
log
–c
+x
0>
,2 5---
x elo
gc
x0
>,
+
x1
+(
)e
log
cx
1–>
,+
1 2---x e
log
cx
0>
,+
x2
x elo
g1 x---
cx
0>
,+
––
1 2---x2
2x
–x e
log
cx
0>
,+
+3
x2
+(
)c
+ln
1 3---–3x(
)co
sc
+1 2---
2x
()
sin
c+
1 5---5
x(
)ta
nc
+x(
)co
sc
+
1 2---–
2x()
cos
1 2---x2
c+
+2x
31 4---
4x()
sin
–c
+1 5---
e5x
c+
4 3---–e
3x
–2
1 2---x
cos
–c
+3
x 3---
sin
1 3---3
x(
)co
sc
++
1 2---e2
x4
x elo
gx
–c
x0
>,
++
1 2---e2
x2e
xx
c+
++
5 4---4
x(
)co
sx
x elo
gc
x0
>,
+–
+1 3---
3x
()
tan
2x e
log
2ex
2/c
x0
>,
++
–
1 2---e2
x2
x–
1 2---e
2x
––
c+
1 2---e2
x3
+c
+1 2---
2xπ
+(
)co
s–
c+
xπ
–(
)si
nc
+
41 4---
xπ 2---
+
co
s–
c+
2ex
2+ ex
--------
------
c+
1 16------
4x
1–
()4
c+
1 21------
3x
5+
()7
c+
1 5---2
x–
()5
–c
+1 12------
2x
3+
()6
c+
1 27------
73x
–(
)9–
c+
1 5---1 2---
x2
–
10
c+
1 25------
5x
2+
()
5––
c+
1 4---9
4x
–(
)1–
c+
1 2---x
3+
()
2––
c+
x1
+(
)ln
cx
1–>
,+
2x
1+
()
lnc
x1 2---–
>,
+
23
2x
–(
)ln
–c
x3 2---
<,
+3
5x
–(
)ln
cx
5<
,+
3 2---3
6x–
()
ln–
cx
1 2---<
,+
5 3---3
x2
+(
)ln
cx
2 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---
4x
5+
()3
=f
x()
24
x3
–(
)2
+ln
=
fx(
)1 2---
2x
3+
()
1+
sin
=f
x()
2x
1 2---e
2x
–1
+1 2---
e+
+=
2ex
2/1 2---
2x
()
sin
–2
–
HL
Mat
hs 4
e.bo
ok P
age
1056
Tue
sday
, May
15,
201
2 8
:54
AM
1057
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
12a
, b
13a
0.25
ab
14b
666
g
15a
b 73
.23%
c ~25
.24
litre
s
16a
b 70
00c 1
.16
day
d 2
days
Exer
cise
22.
4
1a
b c
d –8
2a
b c –
2d
0e
f g
h i
j 0k
l
4a
e b
c 0d
e f
g h
i
6a
b 2l
n5c 4
+ 4
ln3
d e
f 2ln
2
g h
4ln2
– 2
i
8a
1b
c d
–2e
f 0g
0h
i 0j 2
9a
b c 0
d e
f 1 –
ln2
g
h i
10
11; 0
12a
b
c d
e
13a
; b
i 99
acci
dent
sii
14a
1612
subs
crib
ers
b 46
220
15b
~524
flie
sEx
ercise
22.
5
1a
4 sq
.uni
tsb
sq.u
nits
c 4
sq.u
nits
d 36
sq.u
nits
e sq
.uni
ts
2a
e sq.
units
b sq
.uni
tsc
sq.u
nits
d sq
.uni
ts
3a
sq.u
nits
b 2l
n5 sq
.uni
tsc 3
ln3
sq.u
nits
d 0.
5 sq
.uni
ts
4a
2 sq
.uni
tsb
sq.u
nits
c sq
.uni
tsd
sq. u
nits
e sq
.uni
ts5
12 sq
. uni
ts
7sq
.uni
ts.
8ln
2 +
1.5
sq.u
nits
. 9
2 sq
.uni
ts.
10 sq
. uni
ts
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
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
15 2------
38 3------
5 36------
35 24------
8 5---2
2–
1 20------
4 3---–
7 6---
5 6---20 3----
--20 3----
--2 3-------
–
2e
2–e
4––
()
2e
e1–
–(
)e2
4e
2––
+1 2---
ee5
–(
)
2e
3–
1 4---16
e14/
e4–
15–
()
1 2---e
1–e3
–(
)
32
ln17
17 4----
--------
3 2---3
ln
3 4---2
ln
33
2----
------
3 2-------
π2
32------
1–
3 2-------
1 2---–
31 5------
77
3----
------
3–
5 72------
32
33 2---
–76 15----
--
16 15------
2 3---e
1+
()3
2/1
e3
2/–
–(
)
21 5------
ln
2x
2x
2x
cos
+si
n 2m
n–
ma
b–
+3
n–
m2
ab
–(
)n
a2
e0.1
x0.
1xe
0.1
x+
10xe
0.1
x10
0e0
.1x
–c
+
N12
t10
te0.
1t
100
e0.1
t–
978
++
=
32 3------
1 6---
1 2---e4
2–
e2–
()
2e
e1–
2–
+(
)
2e2
2–
e–
()
5 4---
ln
π 2---3 8---
π22
2–
+2
43
43
1 3---–
37 12------
26
2–
()
8 3---
1 4---2
ln
9 2---
14 3------
HL
Mat
hs 4
e.bo
ok P
age
1057
Tue
sday
, May
15,
201
2 8
:54
AM
1058
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
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.
6
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.
7.1
1,
2
0.1
3k
= 8,
4
0.05
637
5b
0.00
676
a b
0.36
9
7b
0.13
5c 1
.8%
d 3
8a
0.75
b 0.
269
c 0.1
495
d 0.
575
10 Exer
cise
22.
7.2
1a
Both
0.5
b Va
rianc
e =
; SD
≈ 0
.288
7
2a
Mod
e = 1
; Mea
n =
0.75
; Med
ian
≈ 0.
7937
b Va
r. =
0.03
75; S
D ≈
0.1
936
3
a A
ll 5
gb
Var.
= 0.
2; S
D ≈
0.4
47c [
4.10
65, 5
.894
]4
a M
ode =
2, M
ean
= , M
edia
n ≈
1.41
4b
SD ≈
0.4
714
c Mea
n
5a
0b
≈ 3.
5 s
c 5 s
d 5
se 1
5 s
6A
ll π.
7M
ean
= 8.
5 cm
Var
. = 1
.25.
8a
b c U
se g
raph
ics c
alcu
lato
r.
9b
c mod
e = 1
.25
10a
b c 4
.62
days
d 4.
75 d
ays
11a
b i 1
.5ii
; 0iii
0.4
Exer
cise
22.
8A
ll va
lues
are i
n cu
bic u
nits
.1
21π
2
πln5
3
4
5
6
7 8 12 13
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--------- π 6---
π 2---1
–
st()
160 π----
-----1
π 16------ t
cos
–t
0≥
,=
v4
kk t2----
t0
>,
–+
=
k1 9---
=1 27------
5 16------
1 3ln----
----
2
1 12------
4 3---
804 3---
23.0
1≈
lnF
t()
1e
t80/
–t
0≥
,–
=
a3 7---
b,5 2---
==
k1
25
1–
()
--------
--------
--------
=5
3– 5
1–
--------
--------
----0.
4078
≈
1x
–(
)32/
3 2---x
1x
––
11 2---
2
3/0.
37≈
–
4 5---π π 2---e10
e2–
()
π2 π 2--- 109 3----
-----π
π8 3---
23
ln–
π 2---5
51
sin
–(
)
251
30---------
π
HL
Mat
hs 4
e.bo
ok P
age
1058
Tue
sday
, May
15,
201
2 8
:54
AM
1059
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
14a
40π
b
15a
b
16a
b
17
18k
= 1
19 20 21b
i ii
22a
Two
poss
ible
solu
tions
: sol
ving
, a
= 4
.953
31;
solv
ing
, the
n a
= –0
.953
31b
23 24a
b 64
πc
Exer
cise
23.
1.1
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
e f
g
h i
j k
l m
n
o
4a
b c
d
e f
5a
b c
d
e f
g h
i j
k l
6a
b c
d e
f g
h i
j k
0l
m
n o
p q
r s
t u
v w
x
242 5----
-----π
8 35------ π
π 4---
9 2---π
88 5------
3π
3π 4------
4π2
a2
kπ 2---
=
πa2
1a
2+
()
--------
--------
-------
8π 15------
a
1a2
+----
--------
---3
a22
+
1a2
+----
--------
------
a3
6a
2–
36a
–20
4+
0=
a36
a2–
36a
–28
–0
=a
100 π----
-----=
28 15------ π 14
72 15--------
----π
576 5----
-----π
2 3---5
x22
+(
)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
3x
1+
+(
)2----
--------
--------
--------
--------
--c
+4
x22
+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
+
ex2
1+
c+
6e
xc
+1 3---
e3
xta
nc
+e–
ax2
bx
+(
)–
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+
()
ln()2
4----
--------
--------
--------
--------
c+
x21
+(
)c
+co
s–
10x
c+
cos
–2
21 x---
+
c
+si
n–
2 3---x
cos
()3
2/–
c+
1 3---3
xco
s(
)c
+lo
g–
4 3---1
3x
tan
+(
)c
+lo
g4–
33x(
)1
+ta
n(
)----
--------
--------
--------
--------
c+
2x
ln()
c+
sin
1 6---1
2xco
s+
()3
2/–
c+
ex ()
c+
sin
ex3
–2
+(
)–
c+
1 2---x
sin
ln2
c+
xc
+se
c1 4---
12
ex+
()
ln[]2
c+
1 3---x3
3x
–
c
+ta
n
Tan
x 2---
c+
1–T
anx 3---
c+
1–T
anx 5
-------
c+
1–S
inx 5---
c+
1–
Sin
x 4---
c+
1–C
osx 3---
c+
1–
3T
anx
c+
1–5
Sin
xc
+1–
Sin
x 2---
c+
1–S
inx 3---
c+
1–
1 2---S
in2
xc
+1–
1 2---S
in2
x 3------
c+
1–1 5---
Sin
5x 2------
c+
1–T
an2x
1–
1 6---T
an2
x 3------
c+
1–1 12------ T
an4
x 3------
c+
1–5
15-------
Tan
5x
3----
------
c+
1–1 5
-------
Sin
5 3---x
c+
1–
5313
779
--------
--------
--2
2–
21
e+
+3
22
2+
()
ln4
Tan
π 2---
1–e
e1–
()
sin
–si
n
2 3---1
π 2---
3
2/co
s–
2 3---e
e1–
–2
ln7
73
--------
--3 5---
π 4---
π 2---T
an1–
2()
–1 3---
Tan
91–
1 2---S
in1–
2 3---
1 4---π
2Si
n1–
2 3---
–
1 3---T
an1–
3 2---
1 6---T
an1–
3 4---
Tan
1–3 16------
–
1 64------
2 3---2 3-------
–
1 3---3
π 4------
1 60------
–
HL
Mat
hs 4
e.bo
ok P
age
1059
Tue
sday
, May
15,
201
2 8
:54
AM
1060
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
Exer
cise
23.
1.2
1a
b c
d
e f
g h
i j
k l
2a
b
c d
e f
g h
i
3a
0b
c d
e
f g
h
i
4a
b c
d e 1
f
g 24
414
h i
5a
b c
d e
f
6a
b c
d e
f 3 +
2ln
4
7a
b c
d e
f g
h i
8a
9a
b i
ii c
10 11
12 13a
b c
d e
f g
Exer
cise
23.
2.1
1a
b c
d e
f g
h i
j
k
2a
b c
3a
b c
4a
b
c
5a
b c
d e
f
6 7
8a
b
c
Exer
cise
23.
2.2
1a
b c
d e
2 3---x2
1+
()3
2/c
+2 3---
x31
+(
)32/
c+
1 3---4
x4–
()1.
5–
c+
x31
+(
)c
+ln
1
183
x29
+(
)3----
--------
--------
--------
----–
c+
ex2
4+
()
c+
z24
z5
–+
()
c+
ln3 8---
2t2
–(
)43/
–c
+
ex
sin
c+
ex1
+[
]c
+ln
1 5---si
n5x
c+
2 5---x
1+
()5
2/2 3---
x1
+(
)32/
–c
+
1 10------
2x1
–(
)52/
1 6---2
x1
–(
)32/
c+
+2 3---
1x
–(
)32/
–4 5---
1x
–(
)52/
2 7---1
x–
()7
2/–
c+
+
2 5---x
1–
()5
2/4 3---
x1
–(
)32/
c+
+e
xta
nc
+1
2x2
–(
)c
+ln
–1
12x
2–
--------
--------
--c
+
1 2---x
ln()2
c+
1e
x–+
()
c+
ln–
xln(
)c
+ln
22
ln 3----
-------
77 54------
ln2
ln1 3---
2ln
1 4---76 15----
--16 15----
--
2 3---1
e+
()3
2/1
e3
2/–
–(
)
77
3----
------
8 3---–
3 8---π2
cos
1–
()
1042 5
--------
----4
ln5 4---
e5e
1––
()
32
–1 4---
3ln
1 4---2
2 3---3
–31 80----
--4
22
–2
ln2 3---
2 5---–
32 5---
326 3----
--4 3---
–56 15----
--2
–
Tan
x3
+(
)c
+1–
2 3----
---T
an1–
2x
1– 3
--------
-------
c+
Sin
x2
– 5----
-------
c+
1–
3Si
n1–
x1
+ 3----
--------
c+
22x
3– 29
--------
-------
c+
1–si
n1 2---
x2 3-----
c+
1–si
n1 2---
arcs
inx
()2
c+
1 3---–
arcc
osx
()3
c+
1 2---–
arcs
inx
()
2–c
+
A1
B,2–
==
Tan
k1–
π 6---π 4---
π 2---π,
2x
2x
1+
()
ln–
22
2ln
–,
3k2
π8
--------
----
πa2
4--------- π 3---
8Si
n1–
2 3---
π 4---1 2---
Sin
1–1(
)2
22
–π 2---
–π 4---
π2
Tan
1–1 3---
–
xx
xc
+co
s–
sin
4x 2---
cos
2x
x 2---c
+si
n+
24
x 2---si
n2
xx 2---
cos
–
c
+
ex–
x1
+(
)–
c+
5–e
4x
–x 4---
1 16------
+
c
+x
xx
–c
+ln
x2 2-----
xx2 4----
-–
c+
ln
1 25------
5x
5x
5xsi
n+
cos
()
–c
+12
xx 3---
cos
3x 3---
sin
–
c
+x
xx
c+
tan
+co
sln
2 3---x
xx
ln4 9---
xx
–c
+
2 15------
3x
2–
()
x1
+(
)32/
c+
2 15------
3x4
+(
)x
2–
()3
2/c
+2 15------
3x
1+
()
x2
+(
)32/
c+
xCos
x1–1
x2–
–c
+xT
anx1–
1 2---x2
1+
()
c+
ln–
xSin
x1–1
x2–
c+
+
1 2---x2
1 4---–
C
osx1–
1 4---x
1x2
––
c+
1 2---x2
1+
()T
anx1–
x 2---–
c+
1 4---2
x21
–(
)Sin
x1–1 4---
x1
x2–
c+
+
1 4---1 4---
e21
+(
)1 4---
e24
–(
)1 4---
4π
2π
–4
2–
32----
--------
--------
--------
--------
----1 2---
1 6---2
π 12------
1 6---–
+ln
1 2---2
21
+(
)ln
+[
]
x 2---x
ln()
cos
x 2---x
ln()
c+
sin
+x 2---–
xln(
)co
sx 2---
xln(
)c
+si
n+
1 15------
1x2
–(
)2
3x2
+(
)1
x2–
–c
+
exx2
2x
–2
+(
)c
+3
x 2---2
x2x
21
– 4----
--------
------
2x
sin
+co
s
c
+x4 4----
-2
xlo
gx4 16----
--–
c+
ex 5-----2
2x2x
sin
–co
s(
)–
c+
2x 9------
3x
cos
9x2
2–
27----
--------
------
3x
c+
sin
+
HL
Mat
hs 4
e.bo
ok P
age
1060
Tue
sday
, May
15,
201
2 8
:54
AM
1061
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
f g
h i
j
k l
m
n
o p
q
2a
b c
d
e f
Exer
cise
23.
3
1 2a
4.66
mb
5 m
c 12
.28
m3
4a
c i
sq. u
nits
ii sq
. uni
ts
5a
b cu
bic u
nits
6a
b 0.
5956
Exer
cise
24.
11
vect
or
2sc
alar
3
scal
ar
4ve
ctor
5
vect
or
6ve
ctor
7
scal
ar
8sc
alar
Ex
ercise
24.
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
AC
b A
Bc A
Dd
BAe 0
6
a Y
b N
c Yd
Ye N
e2
x– 4
--------
---2
x2x
sin
–co
s(
)–
c+
8x3
x 2---co
s6
x2x 2---
sin
–24
xx 2---
cos
48x 2---
sin
+–
–c
+
1 2---x
ln()2
c+
2x
2x
3x
()
x3
x(
)ln(
)2c
++
ln–
xco
s 2----
-------
–3x
cos 6
--------
------
–c
+
1
1a4
+----
--------
---a3
eax
x a---
aea
xx a---
sin
+co
s
c
+2
x3 7--------
4x2 35--------
32x
105
---------
128
105
---------
+–
+
x
2+
c+
x4 4-----
ax
lnx4 16----
--–
c+
2x 2---
1–x 2---
4x2
––
c+
sin
3 2---x
x29
–9
xx2
9–
+(
)ln
+(
)c
+1 2---
x24
+(
)c
+ln
x2T
anx 2---
c+
1––
π2 16------
1 4---–
π 8---1 2---
e2π
eπ2/
–(
)1
21 2---
2ln(
)2–
ln–
a
a2b2
+----
--------
------
e2a
πb
--------
--ea
π b------
+
e
2–
9ln 3----
----
π 3---
x
y
1–1
π
π/2
- π/2
Cos
–1
Sin–1
21
–2
2–
4 5---1
e4
π–
e3
π–
eπ–
++
–(
)9
π 20------
1e
8–π
–(
)
3
63
2π–
--------
--------
-------
a b c
4 a
b
c
d
a
b
ab
ab
a b
e
f
g
ab
ab
ab
d cu
bic u
nits
ππ 4---
1 2---–
HL
Mat
hs 4
e.bo
ok P
age
1061
Tue
sday
, May
15,
201
2 8
:54
AM
1062
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
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
24.
3
1a
b c
2a
b c
d
3a
0b
PSc A
Yd
6OC
4a
b c
7a
b c
8a
b
15 16m
=
Exer
cise
24.
41
a b
c
d 2
a b
c d
3a
b c
d
4
5, (
–2, 3
)
6a
b c
d
7a
b c
d
8A
= –4
, B =
–7
9
a (2
, –5)
b (–
4, 3
)c (
–6, –
5)
10D
epen
ds o
n ba
sis u
sed.
Her
e we u
sed:
Eas
t as i
, Nor
th j
and
vert
ical
ly u
p k
b c
Exer
cise
24.
51
a b
c d
3e
f g
h
2a
b c
d
e f
g h
3a
Dep
ends
on
the b
asis:
o
r b
4a
b
5
6 Exer
cise
24.
61
a 4
b –1
1.49
c 25
2
a 12
b 27
c –8
d –4
9f 4
g –2
1h
6i –
4j –
10
3a
79°
b 10
8°c 5
5°d
50°
e 74°
f 172
°g
80°
h 58
° 4
a –8
b 0.
55
a –6
b 2
c Not
pos
sible
d 5
e Not
pos
sible
f 06
a b
c d
Not
pos
sible
71
810
5.2°
9
10
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
°co
s–
()
105
411
0°
cos
–iv
v33
°41
′
7°3
7′
ca
–b
c–
1 2---b
a+
()
ba
–b
2a–
2b
3a–
1 2---b
2a
+(
)
1 2---b
a+
()
1 3---2
ba
+(
)1 4---
ab
2c
++
()
cb
–c
a+
ac
2b
–+
221
226
m13 23----
--n,
50 23------
==
4 3---
4i28
j4
k–
+12
i21
j15
k+
+2
i–
7j7
k–
+6i
–12
k–
3i4j
–2
k+
8i
–24
j13
k+
+18
i32
j–
k+
15i
–36
j12
k+
+
11 0 8
27– 1 22–
3– 6– 12
16 1– 14
5– 3
2– 3
8i
4j
–28
k–
19i
–7j
–16
k–
17i
–j
22k
++
40i
4j
20k
–+
20 1 25
12 2 16
4– 38– 32–
20– 22– 40–
D60
0i80
0j60
k+
A,–
1200
i–
300j
–60
k+
==
1800
i50
0j
–
105
230
5341
1417
1 2----
---i
j+
()
1 41----
------
4i
5j
+(
)1 5
-------
i–
2j
–(
)1 46
--------
--i
6j3
k–
+(
)
1 5----
---i
2k+
()
1 17----
------
2i2
j–
3k
–(
)1 3---
2 1 2
1
33
--------
--1– 5 1
3i
–4j
k+
+4
i–
3j–
k+
26
3i
j–
k+
()
1 4---3
ij
–2k
+(
)
11± 13 4
23
–2
34
–14
23
–
x16 7----
--–
=y,
44 7------
–=
1 11----
------
i–
j3
k+
+(
)±
HL
Mat
hs 4
e.bo
ok P
age
1062
Tue
sday
, May
15,
201
2 8
:54
AM
1063
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
12a
b e.g
.
14 if
o
r
15a
b
16a
b 13
1.8°
, 48.
2°, 7
0.5°
18a
b
20 25a
Use
as
a 1
km ea
stwar
d ve
ctor
and
as a
1 km
nor
thw
ard
vect
or.
b ,
and
c
d e
Exer
cise
24.
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
24.
7.2
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
λ16
i–
10j
–k
+(
)i
j3 7---
k+
+
ab
c–
⊥b
c≠
bc
=
3 5---4 5--- ,
2 2-------
1 2---1 2---
–,
,
2 3---–
2 3---1 3---
,,
1 3---1 3
-------
6
–32
6
θ
b i
ii
c 81.
87°
u1 10
--------
--3
ij
–(
)=
v1 5
-------
i2j
+(
)=
19 a
u
v
y
x
1 2---i
–j
2k+
+(
)
ij
WD
4i
8j
+=
WS
13i
j+
=D
S9
i7
j–
=1 80
--------
--4
i8
j+
()
d 80----
------
4i
8j
+(
)3
i6
j+
ri
2j
+=
r5
i–
11j
+=
r5i
4j–
=r
2i
5j
λ3
i4
j–
()
++
=r
3i
–4j
λi
–5j
+(
)+
+=
rj
λ7
i8
j+
()
+=
ri
6j
–λ
2i
3j
+(
)+
=r
1– 1–
λ2– 10
+=
ri
–j
–λ
2i
–10
j+
()
+=
r1 2
λ5 1
+=
ri
2j
λ5i
j+
()
++
=
r2
i3
jλ
2i
5j
+(
)+
+=
ri
5j
λ3
i–
4j–
()
++
=r
4i
3j
–λ
5–i
j+
()
+=
r9
i5
jλ
i3j
–(
)+
+=
r6
i6
j–
t4
i–
2j
–(
)+
=
ri
–3
jλ
4–i
8j
+(
)+
+=
ri
2j
μ1 2---
i1 3---
j–
++
=
x8
–2
μ+
=
y10
μ+
=
x7
3μ
–=
y4
2μ
–=
x5
2.5μ
+=
y3
0.5μ
+=
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
3ij
+(
)+
=r
5i
ti
j+
()
+=
r6–
it
2ij
+(
)+
=
6i
13j
+16 3----
-- i–
28 3------ j
–
r2
i7
jt
4i3
j+
()
++
=r
4i
2j–
λ5
i–
3j
+(
)+
=M
L||
ML
=
4x3
y+
11=
3– 13----
------
2 13----
------
,4 5---
3 5--- ,
rk 7---
19i
20j
+(
)= 92 11----
--31 11----
--,
r2i
j3
kt
i2
j–
3k
+(
)+
++
=r
2i3j
–k
–t
2i–
k+
()
+=
r2
i5k
ti
4j3
k+
+(
)+
+=
r3
i4
j–
7kt
4i
9j5k
–+
()
++
=r
4i
4j
4kt
7i7
k+
()
++
+=
x 3---y
2– 4
--------
---z
3– 5
--------
---=
=x
2+ 5
--------
----z
1+ 2–--------
---=
xy
z=
=
x5
7t
–=
y2
2t
+=
z6
4t
–=
r5 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–=
=
HL
Mat
hs 4
e.bo
ok P
age
1063
Tue
sday
, May
15,
201
2 8
:54
AM
1064
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
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
19
2064
°21
3 or
–2
22 (o
r any
mul
tiple
ther
eof)
23
Not
par
alle
l. D
o no
t int
erse
ct. L
ines
are s
kew.
Exer
cise
25.
1.1
1a
5b
c 0d
6e 0
3a
b 16
4
5
6
a b
Exer
cise
25.
1.2
1a
b c
d e
f 2 5
a i 0
ii 0
6 7a
b 8
a 90
°b
79.1
°12
They
mus
t be p
aral
lel.
Exer
cise
25.
1.3
1a
b
2a
b c
67.8
4°
3
4 5
6 sq
. uni
ts
7
–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
=
11 a
b
x1
t+
=
y4
t–
=
z2–
=
r1 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,
,(
)0
1 2---0
,,
x 4---y 9---
z 3---=
=
k7 2---–
=
12i
6j
7k
–+
43
a
b
45°
2
a
b
c
60°a b
ba
45°
45°
ab
×
ab
×a
b×
ba
×
ba
×
ba
×
aa
×0
=
aa
×0
=a
a×
0=
291
56°2
7′
35 2
35
3
12i
–4
k+
10i
2j–
2k
–18
i9j
–10
i2j
2k–
+6
i–
9j8
k+
+20
i13
j–
4k–
10–i
6j
2k
–+
6i
–2j
2k
–+
11----
--------
--------
--------
-----
λkλ
9i
3j
–9
k+
()
5423
41 2---
3i
–13
j–
29k
+(
)1 2---
1019
1 2---23
31
122
43°3
6′
293
1 2---35
HL
Mat
hs 4
e.bo
ok P
age
1064
Tue
sday
, May
15,
201
2 8
:54
AM
1065
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
9a
12
66 cu
bic u
nits
13b
k =
0.5
Exer
cise
25.
2.2
1a b c d
2a
b c
d
3a
i ii
b i
ii
4a
i ii
b i
ii c
d
Coe
ffici
ents
are t
he n
egat
ive o
f tho
se in
par
t b.
Exer
cise
25.
2.3
1a
b c
d 2
c and
d
3a
b c
4
29.5
°b
70°
c 90°
d 11
°5
a 83
°b
50°
c 49°
6a
b
7
8
9
10a
b c 4
9.8°
Exer
cise
25.
2.4
1a
; b
;
2
5a
3b
c d
6
7
9 Exer
cise
25.
3.1
1a
(7, 5
, –3)
2Li
nes t
hat i
nter
sect
are b
and
c; (7
, –4,
10)
; 46.
7°3
(5, –
2, –
3)4
(4, 0
, 6)
Exer
cise
25.
3.2
1a
i (7,
4, 2
)ii
36.3
°b
i (5,
2, –
5)ii
10.1
°c
i (6,
–5,
–7)
ii 4.
4°d
i (3,
–1,
1)
ii 29
.1°
2a
b (0
, 4, 1
)
3a P
lane
is p
aral
lel t
o th
e z-a
xis s
licin
g th
e x-y
pla
ne o
n th
e lin
e x +
y =
6.
b x
= 4
form
s a p
lane
. y =
2z i
s in
this
plan
e par
alle
l to
the y
-z p
lane
. (4,
2, 1
)4
13Ex
ercise
27.
3.3
1a
or
;
b o
r ;
c pla
nes p
aral
lel
d ;
3 Ex
ercise
27.
3.4
1e.g
. the
face
s of a
tria
ngul
ar p
rism
. 2
a o
r
3a
or
OA
αi
cos
αj
sin
+=
OB
βico
sβj
sin
+=
,
ri
kλ
3i2
jk
++
()
μ2
i–
j–
k+
()
++
+=
ri
–2
jk
λi
j–
2k
+(
)μ
i–
j–
k+
()
++
++
=r
4i
j5k
λ2
i2
jk
–+
()
μ2i
j–
3k
+(
)+
++
+=
r2
i3
j–
k–
λ3
i–
j2
k–
+(
)μ
i2
j–
1 2---k
+
+
+=
3x5y
–z
+4
=x
3y–
2z–
9–=
5x8y
–6z
–18–
=7x
y10
z–
+21
=
r
2 3 4
λ3– 1– 3–
μ2– 2 2
++
=x
3y
2z
–+
3=
r
3 1– 5
λ
2– 5 11–
μ1– 4 1–
++
=13
x3
yz
–+
31=
r
2 2– 3
λ
2 1– 1–
μ3 1 2
++
=r
λ2 1– 1–
μ3 1 2
+=
x7y
5z
–+
27–=
x7y
5z
–+
0=
i–
7j–
5k
+
2xy
–5
z+
7=
4x–
6y
8z
–+
34=
x–
3y2
z–
+0
=5x
2y
z+
+0
=
3x
–y
–2z
+3
=y
2=
2x
2y
z–
+3–
=
2x
y2
z+
+12
=8
x17
yz
–+
65=
x2
y–
3z+
2–=
3x2y
–5
z+
2–=
a24 13----
--=
b,18 13----
--=
r3i
2jk
t2i
5j
5k
++
()
++
+=
3i2
jk
++
2x
7y6z
–+
33=
r2 7 6–
•33
=x
2y
–0
=r
1 2– 0
•0
=
3xy
–z
–2
=5 3---
1120 21
--------
--
3x4
y5
z–
+4–
=2x
3y
3z
–+
5=
x5
y6
z–
+19–
=
3 2---5 2---
2,
,
x2
y–
9+
2z–
3–
==
x3
+ 2–--------
----y
6–
z=
=22
°12
′
2110
x– 9–
--------
--------
-----y
710
z– 7
--------
--------
-=
=7
x29
– 11–----
--------
------
7y
9+ 6–
--------
-------
z=
=70
°48
′
x4
z,2
–2y
–=
=65
°54
′73
°42
′
42x
–2
y4
– 5–----
--------
---z
==
x 1---y
8+ 5
--------
----z
4– 2–--------
---=
=
54x
–y
84z
– 7----
--------
--=
=x 1---
y5
– 4–--------
---z
6+ 7
--------
---=
=
HL
Mat
hs 4
e.bo
ok P
age
1065
Tue
sday
, May
15,
201
2 8
:54
AM
1066
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
4a
No
solu
tion
b U
niqu
e sol
utio
n (5
, 1, 4
)c U
niqu
e sol
utio
n (5
, 1, –
3)d
Inte
rsec
t on
plan
e
6N
one o
f the
se p
lane
s is p
aral
lel b
ut th
e lin
es o
f int
erse
ctio
n of
pai
rs o
f pla
nes a
re
skew
.
7k
= 2;
o
r
8a
b c
i 5iii
not
5
9b
c
10a
2, 3
b 3
c For
k =
2,
Revision
Exe
rcises
– S
et A
1–8
4
2a
b i ]
–1, ∞
[ii
c
384
0
4a
i 0ii
2b
–2
≤ x
≤ 2
c x ≥
0
5a
Abso
lute
max
. at
; loc
al m
in. a
t (0,
0);
x-in
terc
ept a
t (±1
, 0)
b Lo
cal m
in. a
t ; a
sym
ptot
es at
, y
= 0
.
6a
(1, –
2), (
–1, 4
) and
(3, 0
)b
7a
2b
S =
[0, ∞
[, ra
nge =
[1, ∞
[c f
–1: [
1, ∞
[ , f
–1(x
) = (l
nx)2
5x
19+ 8–
--------
--------
--5y
13– 1
--------
--------
--z
==
DE
(4, 2
, 0)FG(0
, 2, 3
)B
CH
Oz
y
x4
2
3b
(2
, 1, 1
.5)
c
d
e 59.
2°
x4t
=
y2t
=
z3
t=
x4s
=
y2s
=
z3
3s
–=
3x
6y4z
–+
12=
8 3---4 3---
1,
,
58
°52
′
5 a
r0 3.5
1.5
t
1 2.5
– 0.5
–
+=
r3 4– 0
λ2– 5 1
+=
2i–
2j–
4k
+r
t
1– 1– 2
=
ab
–c
ab
ca
–b
c+
+,
–+
,+
()
1 a---1 b---
1 c---1 a---
1 b---1 c---
–+
1 a---–
1 b---1 c---
++
,,
+–
x1
–4
y–
z=
=
y
x
y =
–1
f1–
x()
x1
+(
)ln
=
y
x
y =
–1
x =
–1
f
f1–
1 2----
---1,
±
1 2----
---1,
±
x1±
=
x =
–1
x
= 1
x
y
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
)
–2 y
x
(0, 2
)
(4, 4
)
(–2,
0)
8
a
b
y
x
(1, –
2)
(5, 4
)
(3, 0
)
c
d
HL
Mat
hs 4
e.bo
ok P
age
1066
Tue
sday
, May
15,
201
2 8
:54
AM
1067
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
9a
i 512
ii 2
b i
ii 10
a i –
1 or
6ii
b i
ii 0.
2iii
0
11a
i 2 o
r 6ii
b i 0
< x
< 1
ii iii
iv
12a
, b
13a
i ii
b c
i ,
ii [–
1,1]
\{0}
140.
515
a b
Cubi
c thr
ough
w
ith lo
cal m
ax.
at (–
1, 1
6) an
d lo
cal m
in. a
t (3,
–16
)c
i k <
16
ii k
= ±1
6iii
–16
< k
< 1
6
17a
k =
0 or
16
b c 0
< x
< 3
18
a 0
< x
< 5
b 70
c –2,
,1
19a
i ii
b i 9
ii –4
20a
±3b
c –2
< x
< 0
or x
> 2
21a
b c
22a
= 1,
b =
6
23b
24b
ii ,
c o
r
25a
b 59
136
c
d a
= 2
and
b =
1 or
a =
–1
and
b =
–8
26b
c 27
a –5
< k
< 3
b p
= 1,
q =
0c
i ii
28
29a
ii {±
1} b
i ii
30
31a
b c
32a
b ]–
∞,4
]c ]
–∞,4
[ 33
b sq
uni
ts
3x2
h3
xh2
h3
++
3x2
3xh
h2+
+3
e1
–----
-------
\3{
}
1 3---e2
4–
()
40.
72≈
elo
ge0
.8
1e0
.8+
--------
--------
--0.
69≈
gf
x()
()
2x
1x
–----
-------
–=
x
\1±{
}∈
P2
4,(
)≡
x6
ln3
ln--------
=15 7----
--1
3+
fg
x()
()
1 x2-----
1–
=g
fx(
)(
)1
x1
–----
-------
=
11
23
12
3–
,+
,1
23
0,–
()
10,
()
12
3+
0,(
),
,
(0, 2
)
y
x–2
–1
0
1
2
3
–1 –23 2
y
x1
–3
–3
–2 –
1 0
1
2
3
4
–1 –24 3 2
y
x1
–3(–
2, 0
)(3
, 0)
(3, 1
)(–
2, 1
)
16
a
b
c
2x
1–
()
3x
2+
()
x3
+(
)1 2---
–
Px(
)x
3+
()
x2
–(
)2
x1
–(
)=
x3
x1 2---
<<
–
x
x2
>{
}∪
2x3
2x2
–x
–1
42
x1
–----
--------
---–
+
1 3---x
5<
<y
2x
–=
xy
–x
y+
--------
---
x4 9---
y,–
1 9---=
=
p53
5p
+=
p5–
5p
8–
=1 2---
x2 3---
<<
x3 4---
>
2 9---P
x()
x3
+(
)x
12
++
()
x1
2–
+(
)=
x1
–(
)23
x–
()
xx
1<
{}
x1
x3
<<
{}
∪x
6–x
3–<
<{
}x
1–x
4<
<{
}∪
xx
1<
{}
a3 5---
b,–
648
25---------
n,–
10=
==
y6
x3
–(
)=
x9
y,6
==
1792
x5
5 2---3 2---
–,3 2---
1 2---–,
17 2------
5 8---
2c
4c
d 0
c, d
y
x
3c, d
2c
3c
4
c
d 0
2c, d
y
x
4c, d
34 a
b
35 a
i &
ii
b
i
ii
$200
iii $
4
i
v t =
042
, 157
0
1
y
x
4
05
fx(
)x2
2x
–2
+= y
1 fx(
)----
-----
=
1
y
t
200
1,40
0
36 a
b 3
y
xa
4
2
y =
gx
y =
fx
37 g
1–x(
)e2
xe
x∈
,+
=y
x
y =
e
HL
Mat
hs 4
e.bo
ok P
age
1067
Tue
sday
, May
15,
201
2 8
:54
AM
1068
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
40a
150
cmb
138
cmc 9
4 hr
sd
[0, 9
4]e
f Use
gra
phic
s cal
cula
tor.
g 17
.3 h
rs41
x =
–8, y
= 1
1, z
= –6
42
a b
]0, 2
]c N
o (x
= 0
) 43
. 78
44a
0b
c , i
.e. d
oes n
ot ex
ist
45
46a
b
47b
, ran
ge =
]–∞
, 4]
48a
Use
gra
phic
s cal
cula
tor.
b c
Use
gra
phic
s cal
cula
tor.
49–1
050
51a
exist
s; d
oesn
’t ex
ist.
b x
< –2
or x
> 2
52 53a
c S =
]–3,
2[
54a
b d
oes n
ot ex
ist;
exist
s.c
55
a t =
2 o
r 3b
t = 3
c 56
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
Revision
Exe
rcises
– S
et B
1a
189
b 99
c –96
d 36
2
b –6
53
b 23
9 km
c 264
°d
153
kme 1
075
4a
i A: $
4900
0; B
: $52
400;
C: $
1920
0ii
A: $
5024
00; B
: $50
6100
; C: $
3794
00
b 46
%c
i 14
mon
ths
ii C
neve
r rea
ches
its t
arge
t 5
a b
c
6a
r = 0
5b
625
cm
7b
or
c
8a
i ii
iii
9b
11a
28b
i ii
12
a o
r b
13a
b c
i ii
38 a
i ]0
, ∞[
ii b
c
d
a eb------
–
∞,1
ab e
log
,(
)x
b
1 x---1
–=
y
x1 b
39 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
x2
–r f
d g⊆
63 8------ x
5–
g1–
x()
1–
x2
–x
2≥
,+
=
3
3
y
x
y =
x
yg
x()
=
yg
1–x(
)=
hx(
)4
xx
0≥
,–
=
f1–
x()
1x
–(
)e
x1
<,
log
–=
x7
λy,
λz,
11λ
λ∈
,–
==
=
r gd f
fog
⊆
r fd g
go
f
⊆
xλ
y,2
λz,
–3
2λ
λ∈
,+
==
=
xλ
y,8
3λ
– 5----
--------
---z,
6λ
– 5----
--------
λ∈
,=
==
f1–
x()
2x
–(
)2x
2<
,=
r gd f
1–f
1–o
g
⊆
r f1–
d gg
of
1–
⊆F
x()
x2
x2
≤,
–=
x1
λ+
y,4
λz,
–λ
λ∈
,=
==
50e
135.
9≈
500
y
xy =
Q(t
)
y =
P(t
)
t10
9 elo
g=
1 2---26
1 2---5
i–
()
–1 2---
–
26°3
4′
135
°2
3–
1 2---1 3---
1k
1+
()
k1
–(
)!----
--------
--------
--------
-------
7π 6------
11π 6
--------
-π 2---
,, 13 36----
--17 2----
--
34
i+
3–
4i
–ci
s2
θ(
)
π 3---2
π 3------
4π 3------
5π 3------
,,
,0
π 2---π
2π
,,
,R
1α,
π 6---=
=π 2---
π 6--- ,
HL
Mat
hs 4
e.bo
ok P
age
1068
Tue
sday
, May
15,
201
2 8
:54
AM
1069
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
14a
Max
. val
ue is
fo
r o
r , w
here
k is
an in
tege
r;
min
. val
ue is
fo
r , w
here
k is
an in
tege
rb
15a
b c
, 420
17a
b
18
19a
b
22a
b i
ii
23a
b c
26
27a
~342
b 20
term
sc
0 <
x <
2d
{1, 3
, 8, 1
8, .
. . }
e f $
4131
.45
28a
b 4
29a
120°
b cm
2
30a
i m
ii m
b ~1
.15
mc
31a
b
32a
8 cm
b
33a
b i 6
4 +
0iii
iii 0
– 2
iiv
v
34a
i ii
b o
r
353
36a
b
37a
$771
56.1
0b
38a
b c
3
39b
i BP
= 66
0 m
, PQ
= 6
88 m
40
216°
41
b 90
6 m
42
a b
0.08
004
m2
c $49
3.71
43a
5b
i
ii c a
= –
8
44a
b ra
nge =
[3, 3
.5]
c i 3
ii 2
e dom
ain
=
45a
i ii
iii
b A
mp
= 5,
per
iod
= 50
wee
ks
d $2
7.07
e d
urin
g 7t
h &
46t
h w
eeks
46a
$490
00, $
4790
0, $
4669
0b
$340
62.5
8c 1
8.8
yrs
d ~$
2485
6447
a ii
26 ca
rds
b 26
, 40,
57,
77
c a =
3, b
= –
d 15
5 ca
rds
e 48
a ~2
.77
mb
i 3.0
mii
2.0
mc 4
.15
pmd
Use
gra
phic
s cal
cula
tor.
e 49
1.26
2 ha
50
51a
b c
5216
23 m
53
a 19
.5°C
b d
Use
gra
phic
s cal
cula
tor.
e 8 am
to m
idni
ght
5419
39 m
55
a ii
b 20
00, 2
200,
242
0, 2
662,
298
8.2
c 52
hrs
d 17
6995
56a
b
c ii
r =
iii
d i
ii
e G
eom
etric
17 2------
xπ 2---
2kπ
+=
x3
π 2------
2kπ
+=
17 5------
xkπ
=π 3---
5π 3------
,
u n74
6n
–=
n1 6---
74p
–(
)=
1 12------
74p
–(
)68
p+
()
Px(
)x2
2x
–2
+(
)x
3+
()
=P
x()
x1
–i
–(
)x
1–
i+
()
x3
–(
)=
244
33
–(
)39
--------
--------
--------
-------
π 2---1 2---
60°
109
°28
′25
0°3
2′
300
°,
,,
2co
secθ
π 3---2
π 3------
,
z2
2i
z2 ,–
8i
–=
=0.
96–
0.72
i+
z2
–3
z,±
1 2---1 2---
3i
±=
=
kπ 4---
=
u n23
3n
–=
1 2---–
143
0.3
30.
23
73°1
3′
π 3---4
π 3------
,x
π 3---x
4π 3------
<<
28°4
′5
π 6------
43
i–
()
1 64------
3i
+(
)12
8–
128
3i
–
1 32------
1 64------
1–
3i
+(
)2
i±
1i
±
π 12------
7π 12------
13π
12--------
-19
π12--------
-,
,,
π 3---1,
u 13
u 3,
–3
3–
==
fx(
)3
2x
()
cos
=7
π 6------
38°4
0′
4 27------
11π
12--------
-–
αta
n1
5+ 2
--------
--------
–=
3–
3,
[]
W4(
)19
.38
P4(
),
14.8
2=
=W
20()
10.9
5P
20()
,27
.02
==
W35(
)13
.45
P35(
),
23.2
5=
=
10
2
0
30
40
50
30 25 20 15 10 5
y =
W(t
)
y= P
(t)
y
t
c
t nn 2---
3n
1+
()
=
21 6---
t6
1 3---<
<
π 4---3
π 4------
,
x2
π 3------
2π 3------
,–
=1 –1
–π
π
y
x
2π 3------
x2
π 3------
<<
–
Dt()
1–
2π 12------ t
cos
+=
N0
2000
α,10
==
4π
–(
)cm
24
π–
()
2----
--------
-----cm
21 2---
An
4π
–(
)1 2---
n
1–
n,×
12
…,
==
31 16------
4π
–(
)cm
22
4π
–(
)cm
2
HL
Mat
hs 4
e.bo
ok P
age
1069
Tue
sday
, May
15,
201
2 8
:54
AM
1070
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
57a
h Ra
tio te
nds t
o 0.
5. T
his m
eans
that
for l
arge
grid
s the
re ar
e app
roxi
mat
ely tw
ice a
s m
any
smal
l tria
ngle
s as t
here
are n
odes
.
Revision
Exe
rcises
– S
et C
1a
b c 0
.316
9
20.
0228
3
a 0.
12b
0.60
87
40.
0527
5a
b E(
X) =
, v
ar(X
) =
6a
0.89
b c
7a
0.46
b
8a
3326
400
b i
ii
9a
0.97
72b
0.34
13
10a
2b
0.32
3311
a 0.
936
b 5
12a
X ~
Hg
(n =
4, D
= 4
, N =
8)
b E(
X) =
2, v
ar(X
) =
13a
792
b 35
14
a 15
1200
b 0.
1512
150.
2852
16
a 0.
0067
b Po
isson
dist
ribut
ion
with
par
amet
er
c 0.5
134
17
18a
0.10
b 0.
40c
val
ues a
re: (
0, 0
.40)
, (1,
0.5
0), (
2, 0
.10)
d
19a
0.86
64b
0.72
10c 0
.903
4d
9.88
55 <
Y <
10.
2145
e 79.
3350
20a
315
b 17
280
21 22
ab
23a
(i.e.
geo
met
ric)
b i 0
.067
0ii
0.40
19iii
24a
b
25b
val
ues a
re:
, ,
, ,
c E(X
) =
, var
(X) =
d
0.00
064
26
a 0.
3085
b 0.
0091
c 0.1
587
27
100
28a
b c
29
a 0.
5940
b ~
34 d
ays
30b
val
ues a
re: (
1, 0
.4),
(2, 0
.3),
(3, 0
.2),(
4, 0
.1)
c i 2
ii 5
iii 3
31
a 2,
1.3
5b
c Bin
omia
ld
e i 0
.38
ii 0.
390
32a
i ii
0.34
56b
4.87
%c
i 0.8
186
ii 0.
1585
33a
val
ues a
re:
, ,
,
b ii
0.00
64iii
0.7
705
c i 3
8ii
34
35
a b
i 0.
3085
ii 0.
1747
c i 0
.264
2ii
0.8
36a
i 0.8
ii 0.
25b
i 0.4
ii
37a
b c
d e
38a
i 0.1
353
ii 0.
2707
iii 0
.864
7b
~ 0
.023
1
39 40 41a
val
ues a
re:
; b
c
42a
0.09
93b
$2.0
3 pe
r met
re. I
f the
leng
ths a
re in
crea
sed,
the n
umbe
r of f
aults
per
le
ngth
wou
ld al
so in
crea
se, h
ence
, the
expe
cted
pro
fit/m
etre
wou
ld d
ecea
se. (
In th
is ca
se it
wou
ld re
duce
to $
1.21
per
met
re.)
43a
0.4
b 0.
096
c 0.2
25d
0.63
5
Ord
er 1
Ord
er 3
1 4---3 8---
pX
x=
()
5 x
10 55
–
15 5
----
--------
--------
------
x,0
12
34
5,
,,
,,
==
5 3---50 63----
--
21 40------
40 89------
9 23------
2 11------
2 77------
4 7---
λ2 3---
=
128
850
---------
0.15
06≈
xP
Xx
=(
),
()
EX(
)0.
70v
arX(
),
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
=(
),
() 2 18----
--4 3---
3 7--- xP
Xx
=(
),
()
03 16------
,
1
7 16------
,
2
5 16------
,
3
1 16------
,
1 19------
μ0.
9586
σ,0.
0252
==
10 21------
EX(
)0.
8v
arX(
),
14 25------
==
1 8---47 72----
--1 8---
47 72------
9 47------
189
8192
--------
----
43 60------
0.71
67≈
117
145
---------
0.80
69≈ xP
Xx
=(
),
()
01 6--- ,
11 3--- ,
21 2--- ,
,,
EX(
)4 3---
var
X()
,5 9---
==
2 3---5 24------
b 4
c Ord
er 3
: 9, O
rder
4: 1
6 &
Ord
er 5
: 25
d e Ord
er 1
: 3, O
rder
2: 6
, Ord
er 3
: 10,
Ord
er 4
: 15
f a =
1, b
= 3
, c =
2
n2
HL
Mat
hs 4
e.bo
ok P
age
1070
Tue
sday
, May
15,
201
2 8
:54
AM
1071
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
44a
b v
alue
s are
: ,
,
c d
45a
i i
i ii
i i
v v
b
46
47a
0.13
59b
137.
22c 1
37d
a =
141.
21
48a
b c n
ot in
depe
nden
t
49a
W =
{2, 3
, 4, 5
}b
c
50a
b
51a
4b
0.90
8452
a 0.
081
b 53
a 0.
0169
b i 0
.934
2ii
127
iii 0
.008
54a 0
.158
7b
0.77
45c $
0.23
55
a i 0
.077
ii 0.
756
iii 0
.167
b $7
.61
56a
i
ii
57a
i 0.2
4ii
0.36
b c
Q >
29.
17
58a
P(X
is od
d) =
b
P(X
is ev
en) =
c
Revision
Exe
rcises
– S
et D
1a
b
2a
19.8
°Cb
1.6°
C pe
r min
ute
c 17.
3 m
in
3a
b
410
m
5a
b (o
r )
6a
i 0ii
2b
c x ≥
0d
7a
b –2
81.
455
ms–1
9
a Ab
solu
te m
ax. a
t ; l
ocal
min
. at (
0, 0
); x-
inte
rcep
t at (
±1, 0
)
b Lo
cal m
in. a
t ; a
sym
ptot
es at
, y
= 0
.
10a
b
11a
b 98
°
12b
i 2ii
72 cm
3 13
a b
3 5---x
PX
x=
()
,(
)0
4 25------
,
1
12 25------
,
2
9 25------
,
EX(
)1.
2v
arX(
),
0.48
==
3 7---
8 15------
7 15------
1 5---4 5---
4 7---x
pq
–(
)10
0q
+10
0----
--------
--------
--------
--------
--
2 3---
1 3---
2 3---2 9---
10 3------
17 42------
b6
a+
0b
1 3---≤
≤
4 13------
1 2
3 4
5 6
7 8
9 10
11
12 1
3 14
15
16
6 5 4 3 2 1
freq
uenc
y
x
1 2
3 4
5 6
7 8
9 10
11
12 1
3 14
15
16
25 20 15 10 5
cum
ulat
ive
freq
uenc
y
x
172
0.96
Q+
eλ–
λλ3 3!----
--λ5 5!----
--…
++
+
1 2---
1e
2λ
–+
()
λ1.
122
=
x
x24
+----
--------
-------
22
x2
2x
1–
()
2x
sin
–co
s
x1
0
]0
∞ [
,∪
[,
–[∈
x ]
∞0
2
∞ [
,[
∪[
,–
∈
4x
x21
+(
)2----
--------
--------
--4
2x
2x
cos
sin
–2
4x
sin
–
x2
2,–[
]∈
x
4x2
–----
--------
------
2x
2<
<–,
–
2h
+
1h
+(
)2----
--------
--------
h0
≠,
–
1 2----
---1,
±
1 2----
---1,
±
x1±
=
62
xsin
22
xco
sx
3+
2x
3+
()3
2/----
--------
--------
--------
(3, 3
) (4, 0
)
(0, 0
)
y
x
3x
13
x2–
--------
--------
------
–ex
1ex
+(
)2----
--------
--------
--
b i 1
1.44
ii 2.
6695
c i
, , m
ed =
12,
,
ii
med
= 1
2, m
ode =
12
iii 4
x min
4=
x max
16=
Q3
13.5
=Q
19.
5=
1 2
3 4
5 6
7 8
9 10
11
12 1
3 14
15
16x
HL
Mat
hs 4
e.bo
ok P
age
1071
Tue
sday
, May
15,
201
2 8
:54
AM
1072
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
14
15a
b ra
dius
= 5
cm, h
eigh
t = 1
2.7
cm
16a
b c
17
a b
~ 38
.3 m
illio
nc
i dec
reas
ing
ii 0.
1 m
illio
n/ye
ard
50 y
ears
tim
e, i.e
. 203
0; 4
2.2
mill
ion
1876
222
cm3
19a
b
20a
b 12
21
a i 2
83 s
ii 25
0 s
c 244
s 22
a =
–1, b
= 6
, c =
–9
23a
b i
ii
24
25M
ax. =
64,
min
. =
26a
b c
27a
b c
d
e
28b
c cu
bic u
nits
30a
b
31
320.
5 cm
s–1
33a
[0, 5
]b
Use
gra
phic
s cal
cula
tor.
c 0.6
25d
34c M
in.
; max
.
35a
b 1
36a
b 64
sc f
allin
g at
1.1
5 cm
min
–1
37a
or
b o
r
38
39b
i ii
iii P
oint
Aiv
Yes
40a
b
41a
b an
d x
= 2
c (1
, 4) a
nd (3
, 0)
d
42a
b
43a
, b
and
44
a i
ii f i
s not
a on
e-on
e fun
ctio
nb
c = –
2
c i
, ran
ge =
[–2,
0],
dom
ain
= ii
45
0.16
rad
per s
econ
d46
3 –1
–2
–1
x
y
h10
00
πr2
--------
----=
3x2
h3
xh2
h3+
+3
x23
xhh2
++
3x2
p't()
0.8
10.
02t
–(
)e0.
02t
–=
1x
xx
sin
+co
s+
1x
cos
+(
)2----
--------
--------
--------
--------
-----
x
x21
+----
--------
--
126
hh2
h0
≠,
++
43
3----
------
63
x3
xco
ssi
n–
x 2---1 12------
6x
c+
sin
+
yx
2–
=21
8725
6----
--------
–
3x
cos
22
3x
sin
–----
--------
--------
--------
--–
21 2---
x1 2---
2x
sin
–co
s
e2
x1
xln
– x2----
--------
-----
x0
>,
Vπr
2h
4 3---πr
3+
=P
2πk
rh6
πkr2
+=
P2
kV r----
------
10π 3
--------
- kr2
+=
0r
3V 4π
-------
1
3/<
<
r3
V10
π----
-----
1
3/= a 4---
a 2--- ,3
36-------
a3
62
xcos
22
xsi
n–
12
x2– 1
x2–
--------
--------
--
y1 2---
x–
2+
=
a1 2---
1 5---t
0t
5≤
≤,
–=
3πa
25 3---
1
3/3
πa2
9 4---
1
3/
ex–
xx
sin
+co
s(
)– V
4 27------ π
h3
=
x1 2---
<x
1 2--->
1 2---x
0<
<–
0x
1 2---<
<
4 3---
AB
200
θta
n=
20se
c2θ
Sin
1–x(
)x
1x2
–----
--------
------
+y′
2x2
x21
+(
)ln
– x2x2
1+
()
--------
--------
--------
--------
--------
=
09 2--- ,
y4
x–
=y
x
128
x2–
2x2
3+
()2
--------
--------
--------
--8 9---
3
y3 2-------
x1
+=
y3 2-------
x–
1+
=2
34,
()
2–3
4,(
)
–2
2
–2
y
x
fog
x()
2x
sin
–=
nπ
n∈
,{
}–
ye
1–x
=
HL
Mat
hs 4
e.bo
ok P
age
1072
Tue
sday
, May
15,
201
2 8
:54
AM
1073
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
47a
x =
1, x
= –
1, y
= 0
c
48a
b
49a
b 50
Dom
=
51
~2.7
7 km
h–1
52
532.
83 m
s–1
54a
27 s
b 74
6 m
55
a b
x =
14, y
= 2
1
56a
57a
i b
i ii
Loca
l min
.iii
Use
a gr
aphi
cs ca
lcul
ator
.
c i
ii x
= 0
58a
i ii
b c
d m
s–1
59a
b c
, som
e val
ues w
ill
be n
egat
ive i
ndic
atin
g th
at th
e tid
e is g
oing
out
.d
60b
k =
250
c d
212
me 9
6.23
f ii ~
–0.
68
610.
032
rad
per s
econ
d
Revision
Exe
rcises
– S
et E
1a
i ii
b i
ii iii
2a
b 3
mc
3 4a
b –8
c
5a
b c 2
6b
i ii
1.13
cubi
c uni
ts
732
7 cm
8 9
a b
c (10
, 10,
3)
10a
b c
4 sq
. uni
ts
1172
0 m
3 12
a i
ii
b i (
–5, 3
, 2)
ii (8
, 11,
0)
c
13a
= 1,
b =
614
015
a b
c sq
. uni
ts
16a
Are
a =
, Vol
ume =
b
17a
(–1,
4),
(1, –
2), (
3, 0
)b
Use
gra
phic
s cal
cula
tor.
c sq
. uni
ts
18a
b
19a
b i y
= x
ii iii
c i
ii cu
bic u
nits
20a
b i
, ii
c i U
se g
raph
ics
calc
ulat
or.
ii sq
. uni
tsiii
cu
bic.
units
21
k =
1 o
r 3
22t =
2, (
16, –
8, 4
)
y
x
42
x3
2x
cos
–si
n(
)e
x–⋅
23
x2–
x21
x2–
()3
--------
--------
--------
--------
2C
os1–
x
x2----
--------
--------
-+
y3
3ln
()x
33
3ln
–+
=y
1 2ln----
----x
1+
()
=
xx
4–<
{}
x4
x0
<<
–{
}x
x0
>{
}∪
∪
125
π----
----- m
s1–
A31
86
x–
1176 x2----
--------
4x
49≤
≤,
–=
a4 3---
b,1
c,–
4=
==
0x
1 3---L
≤≤
x3
L
3π
9+
--------
--------
----=
x3
L
3π
9+
--------
--------
----= 2
x
x21
+(
)2----
--------
--------
--66
3x
2+
()3
--------
--------
-------
–20
5 elo
g(
)yx
–1
400
5 elo
g+
+=
1 2ln----
----1.
52
ln
a8
b,2
π 11------
==
11 5
5.5
11
y
x
6π 11------
2π 11------
t
si
n–
82
π 11------
t
si
n–
I25
0θ
2co
s2θ
sin2
θ–
[]
sin
=
91 3------
4ln
+4 15------
12
+(
)2
1x2
––
c+
1 3---T
an1–
4x 3------
c+
1 2---S
in1–
2x
()
c+
Nke
t2
N2
>,
+=
yT
ant
π 4---–
=
v2
t1
+t
0≥
,–
=
7i
–6
jk
++
a1 3
-------
ij
k+
+(
)=
1 48------
π 3---
π4
y2yd
00.5
π
1 y---1
–
yd
0.5
1 +
dm dt
-------
7m
100
3t
+----
--------
--------
–=
4x
y–
4z
–2–
=x
–7
y19
z+
+42
=
π 3---4
π 3------
,x
π 3---x
4π 3------
<<
x7 2---
λy,
+λ
z,9 2---
λλ
∈,
+=
==
x3.
5– 1
--------
--------
y 1---z
4.5
– 5----
--------
---=
=
2x
5y
z–
+8–
=
8i
–11
j9
k+
+a
2 5---b,
–4 5---
==
53
A8 15------ h
32/
=V
0.48
h3
2/=
5 144
---------
m/m
in
16 3------
33 e
log
–
23 4---
3–
3 elo
g
A2
2e
1–,
()
≡d dx
------
xex
k/–
()
1x 2---
–
ex
2/–
=4
22
a+
()e
a2/
––
1 2---a2
–
2x
x2–
()e
x–π
210
e2–
–(
)
fx(
)1 5---
2x
1–
()5
2/1 3---
2x
1–
()3
2/c
++
=π 12------
π 48------
1 9---
π 3---π 2---
3ln
HL
Mat
hs 4
e.bo
ok P
age
1073
Tue
sday
, May
15,
201
2 8
:54
AM
1074
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
23a
i ii
k ¦ –
2, 3
b k
= 3
c i k
= –
2ii
, ,
,
24; (
–1, –
3, 4
)
25,
,
26a
b sq
. uni
tsc
i At (
0, 1
): y =
–2x
+ 1
;
At
: ii
iii
sq. u
nits
d ii
cubi
c uni
ts
27b
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
28A
= 0,
B =
0.5
29
a sq
. uni
tsb
cubi
c uni
ts
30a
a =
2;
, b
cubi
c uni
ts
31a
b c
sq. u
nits
32a
sq. u
nits
b cu
bic u
nits
33a
b 19
min
utes
34,
35a
b m
36a
b 10
0°c
37
b
38a
, b
i ii
c
39 cu
bic u
nits
40a
b t =
5, b
= 0
.4
41a
b 0.
32 k
g pe
r litr
e
42a
b a
= 2
c x =
–3,
y =
5
43a
a =
1b
1.25
c d
44a
i 90°
ii
b i
ii iii
iv
45a
1b
c
46a
27°
b
x3
k3
–----
-------
y,–
8k
–k
3–
--------
---z,
k5
–k
3–
--------
---=
==
x2
λ–
=y
2–=
zλ
=λ
∈A
2I 3
3×
=
BX
1 2---A
=X
1 18------ B
A=
AX
1 18------ A
BA
=
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–
()
π 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+
7 12------
7 15------ π
f1–
x()
2x
+=
x0
≥13
6 3---------
π
yex
–e
e1–
++
=
x
y
norm
alcurv
e1 2---
ee
2–+
4 3---64 15----
-- π
x1 3---
2014
e0.
1t
––
[]
=
Tan
1–x
22
--------
--------
------
1 2---1
1x
+----
--------
⋅+
23
ππ 2---
–2
ln–
dh dt
------
Vk
h– πh
2----
--------
--------
=V k---
2
3i
j–
2k
–4
i3
j–
3k
–
A1–
1 3---A
2I
–(
)=
1x
1x2
–----
--------
------
Sin
1–x
⋅–
3 2-------
π 6---–
2 5---1
x–
()5
2/2 3---
1x
–(
)32/
–c
+ex
ex1
+(
)c
+ln
–
1 3---2
ln
64 15------ π 2
2
dx dt
------
50x
– 10----
--------
--t,
0x,
6=
==
1a
2–
--------
----3–
2a
1–
13 48------
19 64------
7 2---26
unit
2s
3p
+s
2p
+1 2---
s2
p+
1 2---–
s2
p+
An
2n
2n1
–(
)a0
1=
a2 9---
–=
1 2---17
unit
2
HL
Mat
hs 4
e.bo
ok P
age
1074
Tue
sday
, May
15,
201
2 8
:54
AM
1075
MATH
EMATI
CS –
Highe
r Le
vel (C
ore)
ANSW
ERS
47a
i ii
iii
b c
i No
ii Li
nes a
re sk
ew.
48a
i ii
90°
b i
ii 49
(–8,
11,
–6)
50
a b
51
52a
c
53k
= –1
, 1 o
r 2
54a
i ii
b c
55a
i ii
b cu
bic u
nits
c i
ii sq
. uni
ts
56a
See p
age3
78.
b sq
. uni
ts
57b
i ii
58 cu
bic u
nits
59a
b cu
bic u
nits
60a
5b
61
b 76
sec
62
63a
i a =
3, b
= –
2ii
b (–
12, –
10, –
2)c
i
ii iii
64a
i ii
b i
ii 65
a t =
1 &
t =
5.ii
Use
gra
phic
s cal
cula
tor.
b t =
2.2
1
6640
.5°C
x y
2 3–
λ3 7
λ∈
,+
=x
23
λy,
+3
–7
λλ
∈,
+=
=x
2– 3
--------
---y
3+ 7
--------
----=
i–
11j
+2
x3
y4
z–
+2
=28
°35
′32
°25
′
a3 2---
=b
3 2---c,
1 3---=
=
x y z
0 10 3
λ
1 19–
5–
+
=
c1 77
--------
--5
i–
4j
6k
++
()
=2
xy
3z
++
4=
2 5
1 2---1
0
22
xy
z–
+1
3x
4y
6z
++
,7
==
x y z
3– 4 0
λ
10 9– 1
+
=
π 2---2
π4
–π2 2----
--π
+y
x–
2+
=3 2---
e1–
–
π3
3----
------
1 2---ln
+
a3
=b
1 6---=
π 2---c2
e2–
–(
)
y
x–2
0
2
2
2π2
8π 3------
+
5 3---5
I 1
I 1 I 2----
a1 3---
2λ
1–
()
b,λ
λ∈
,=
=1 2---
629
x12
y–
22z
+0
=r
λ1 12– 22
=
1 2---x2
1+
()a
rcx
1 2---x
–c
+ta
nx
arcs
inx
1x2
–c
++
⋅
1 4---1
2a2
–(
)2
a2
2a
1–
sin
+co
s[
]5 27------ e
32 27------
–
HL
Mat
hs 4
e.bo
ok P
age
1075
Tue
sday
, May
15,
201
2 8
:54
AM