1 18 a
16 a 36 (l mark) 3
(l mark)
(l mark) 3 Tower hears reply —
(l mark)
Ship hears 4 5
b6 1
36 6
1 1 Tower does not hear reply
36 18 2
5
1
(2 marks)
36
Ship does not hear
1
4
or using a lattice diagram
(2 marks) b —x— 9 (2 marks) 50 1 (l mark)
(l mark)
36 6 20 100 2 (l mark)
90 9
—1 since independent 9 11 (2 marks) 10
20 20 ii 2
6 (l mark) d
100 5
g p (R5u P (ship not hear Itower has no reply) =
40
100
——— —— —1 + 1 nP (ship not hear tower has no reply) 40 4
1 11 (2 marks)
P (tower has no reply) e If independent then
= or using a
6 6 36 36 5 (2 marks)
11 11
lattice diagram (2 marks)
h Considering the list in 20
ne B) = O so events (l mark)
are mutually exclusive. —1 60
so not independent
e — = — or using conditional (2 marks) 4 100
probability formula 19 (2 marks) (l mark)
(2 marks) 5
(2 marks) 21 a
Independent 5 catch 20%
P(FnR) = P(R)
(l mark) 5
—1 —x— so not (2 marks) injected 30% not catch 80%
not Injected 70%
independent (l mark) 20 a Let x be the number catch
b n R) speaking both English
and French.
P(Fu R) =
(60 —x) +x
5 (l mark) (2 marks) not catch 10%
(l mark) b
111 12
(4 marks) (2 marks layout
346 2 marks numbers)
c P(exactly one team) =
1 70 90 63
(2 marks) b
12 6 50 10 30 10 100 100 100 (l mark)
Could also use a Venn —x—30 80 6
diagram in (b) and (c). c (l mark)
100 100 25
d p(Fn = P(FnR) 1 30 20 70 90 69
2 d
(1 mark shape 2 marks 100 100 100 100 100
numbers)
(2 marks)
(2 marks)
689
ANSWERS
63 21 122 61 (l mark) Exercise 9A
100
P(1'nC) 100 ii 3 2a7b3c 2 14x7y8
2u 2 4 21112
¯ 23 200
—8x12y3z15 6 27r3s9
92 23 (l mark)
200 50 1 8 X14y4
(2 marks) x11
iv I— 82 118 59 9
24 200 200 100 x7y9 10
p (IIC') = P(/nC') — 100 24 (2 marks) ii 9x4y2 10a4
31 12
31 — —d 20 = 5 or by using the Exercise 9B
100 b
(2 marks) 48 12
formula (2 marks)
22 a
23 a
Rowing Kayaking —x— (2 marks)
40
22 30 53358 7 14
ii RG orGR -x-+-x-=1—5
8 8 7 8 7 28 1 1
12 10
(2 marks) 3 (3x)3
18
Surfing (60) b 25
(2 marks)
64
(4 marks) (A1 shape, 3 marks, RG or GR —x—+—3 x—5 15 3 b 6
7 numbers, (2 marks, d
8 8 32 2 102 1
4 numbers, I mark 2 numbers)
(2 marks) 7 5x)Z
c 1
b 200 —140 = 60 (2 marks) 5 1
(l mark)
3 (2d)ü
ci
Exercise 9C
200 20
24 a P(AnB) d2 b6 C5
0.5 1 f4
p (A 1B) = 0B') = 0.2+0.4= 0.6 2
p(B)
(2 marks) 1 i -2
(2 marks)
h
8
3
iii 0.6+0 . 5- 0.2 = 0.9 2 1 5
7 b5 d
c
2 3
P(AnB') 04 (2 marks) Exercise 9D b Decay
(2 marks) i a Growth d Decay
0.5
c Decay
b 1B) P (A so not independent (2 marks) e Growth
2
i(x)=B j(x) = E
3 a i ii, iv
Chapter 9 3
Skills check
1 a 32 b 1000
1 125 yiii
d
c
216
243
2 a 3 b4 c4 d3
690
b i, ii, iv 2 Exercise 9F
x Ai a vertical translation of
a It is a straight line. 4 units up
b In between g(x) = 2x and Ab horizontal translation of
i(x) = 3x
3 units right
3
Ac vertical stretch of scale
-25 -2 -15 -1 -05 0 OS 1 IS 2 25
factor 2
A reflection in the y-axis
y =iii 2a
4 a 4500cm2 b 90400 cm2.
c i, ii, iv 5 $6749 1.5
6 In 2011 the population of the
(1.0) 05
world was approximately 7 -05
billion, and it is growing at a
iiiy=2 rate of 1.1% -15 12
a +0.011)t
4 900C b 48.0 oc b
c 4 min d 250C b 8.16 billion
15
c During the year 2043
1
05
-05
-1
-15
c
5 $30 ooo b $21870 2
1.5
c 6.58 years
1
6 a 40 b 90 0.5
cp -0.5
-1
250
200 d
150
100
50
12 345 12
10
The population will reach
200 in 3.97 years. 8
6
7 a 88.6 g 4
2
laa 3
years ate•r 11
20 5030 8000 (-1.98
0
a mm 5 10 15 20 25 30 35 40 x = -1.98, 2.72
Exercise 9G
b i 2378 years 7 The value of a car decreases
ii 5730 years
by 15% every year. If Tatiana
c 88.7 g bought a new car for $25 000,
a y-25(1 -0.15)t, in
thousands of dollars
b $15400
c After 5.64 years
Exercise 9E b 7.389 25
d 8.155 20
2.718 f 8.244 15
c 0.135
(5.64 10) e lox-II
1.359 10
g 5.873 5
2 4 6 8 10
691
ANSWERS
2 a logrt = s b log864 = 2 d 2 a —5 - 2ex b 2x+—e 1
1 x2 x
c 10811025 = x d log C 1 + 3X2 3
9 ———5e-5X x
2 x
e log279 = _ 1 1
3 4 d
3 a o bO c O do logy x x
4 1 b1 c 1 d1 1 e ——5 + 24e4x
c3
5 a2 b5 5 x
1
log e 2x 3x2 -1
3
d4 1 6 "Show that" is to use numbers x2+l —X
—3
2 to demonstrate a certain
property and that it works for
the numbers that you are using. 3
To prove is to use variables to
prove that the system works for 3 a 6X2e2X
all numbers.
Exercise 9H
i a log 15 b log 8 c log 125 4x3 +5
b 24x2 (4x3 + 5) e
d log— e log x f log 236 5 3 (In
i log 24 d
81
g log 800 h log 2 xx
Exercise 9L e (1 + 2x)e2x
j log 9 k log 2592 I log 81 i 2.32 b 2.58 c 1.77
d 3.85 1.52 f 0.65 f 6x2(l —x)e-3X
g 0.712 h 0.235 1.61
Exercise 91
j 230
c 2x g 2xe3X+ 3e3X(x2 + 1)
2 0.792
b f
c 2m h eax2+l + 2ax2eax2+I
d 3x 2m
i b 0.258 c 5.43 Inx+ 1 j x2(l+31nx)
2a 3n d 2.91 2.77 f 3.30 —+2x2
h 0.330 i 9.70
d 3n e 2m —n g 2.58 k 2x In (2x +3)
-8.02 2X+3
g m + 2n h m
j b 0.258 c 0807 e3x (3X —2)
n x3
3 1.22
d -0.535 f 1.87 2e4x (4 — -2ex
3 g 0.0524 h 6.92 )
4a n
1 4 a In 24 1 In x —l
b b (Inx)2
2
16 p
x
5 a 0, ln4 b In 3 In 2 q x4
b c 4 Turning point is at (l, —l) and
9
2 is a maximum
Exercise 9J 1 6 a 10 cm b 0.630 5
6
a3 b4 2 c 15.2 days
-1 11
1 -2 7 $21 OOO b 0.910
b3 8 —,——ln2
d cx c 16.8 years
3 8 84 b 60 c 29 days 5
1
22 9 a 499 kg b 58 048 years 9 In 2, In—
3
d 16 In 3 2
10
3 14.9 b 5.42 10 Tangent fix) = ex — e. Normal
In 2 —4
4 11
Exercise 9K Exercise 9M f(x)
i a 1.89 b 1.77 d 1 9 ee
1.29 d 3.10 x
-0.712 f -0.737 x b
1
2q 4
x
1
x
1 h 2 e2x
g
x
bc j 5e5x
692
Cha pter review 12 a 23 a
24=16 b 53=125 140
C 92 = 81 d 122= 144 120
100
104 = 10000
80
log381 = 4 b log15225=2 60
40
_1 b 20 T
c log819 = d log c = 14 Ac reflection in the y-axis t(hours)
2 and a horizontal translation 2 4 6 8 10 12 14
e In x = 4 of 2 units to the right. (shape: I mark; domain: 1 mark)
3b 1 d2 d b
6
1 1 ooo 36.1 (1 d.p.) (l mark)
b c Solve 25+ e -100
24 a
60 7
13 g(x) = -(2-x) - 2 (l mark)
(l mark)
1 d 25 14 g(x) = 3 In(X+ 5)
1000 t = 10.793...
27 IS a 8ex + —7 c Inx 10 hours 48 minutes
2x
b 3e3x
eo f 310g3(79) g 19 x (l mark)
7
d 5 ) e6x2+5x (l mark)
4.09 b -0.226 b (l mark)
0.229 d -0.106
e —2.44 7eX 3
-1.15 -1.07
h -1.13 2(3x - 2) C 3111 (x — 2) +1 = O
(2ex +1) 1
log 12 b log 3 —e (2 marks)
—x2 d log+8y2 h + eXlnx 3
x
c log —x4 (l mark)
In xU f In 16 x — 49
y3z2
C 211 + 3q f' (x) = I for tangent
b
2.58 4.5 parallel to y = x
c 2.55 d 5q — 4p
10 1.02, 5.65 18 a 30000 b 35 205 c 2029 3 x = 5 (l mark)
b -2.32 19 2028
=
20 0.502 mg
x-2
f (5) = 31113+1 so tangent
Exam-style questions is at the point (5, 31n3+l)
21 a
(l mark) (l mark)
b -2 (l mark)
(l mark) Tangent line is
c 12 (l mark)
(l mark) (l mark)
d Y<16 (l mark)
(or y=x+31n3—4)
f y=16
25 a Combining any two of the
three terms, e.g.
(l mark) log2 3x — log2 (x — 3) (l mark)
b N = 409.97; 410 Combining the remaining
bacteria (2 marks) terms, e.g.
b N (t) > 1000 (l mark) 3x (l mark)
(l mark)
c Vertical stretch of scale factor t > 5.449... (l mark) log2
d OSt<5.45
5 and a vertical translation of b In x3 — In (x
2 units down.
(3 marks)
693
ANSWERS
26 a i c -73.205—73205 29 29 a
taxis (2 marks) 25 + = 29 3
2
(l mark) 2
3
(l mark) b =-0.142 (3 s.f.)
This is in the 9th year after (1 mark) (2 marks)
2010, which is 2019 (l mark) (l mark)
b = 2.1873705... b T(30) = 25 +69e -o. 14200 (2.67,0) (l mark)
= 26.00C (3 s.f.) (2 marks)
(2 marks) 3
b
2187 000 people (nearest c y=25
2
thousand) (1 mark) (l mark)
c Adjusting units in d The temperature
(i) or (ii) (l mark) of the room. (l mark) 3
— = -9.82...e -0.142...t (l mark)
2
= 28.4 people dt
(2 marks) y = logb (x —2)+1
per taxi; (2 marks) —d2T
= 1.399...e-0.142...t
(2 marks)
dt2
ii 110 x106 = 18.6 people
g(x) = log 3 (x —2)+1
(l mark)
2
per taxi. (l mark) g The rate of change is
always negative which
d The model predicts a reduc- means the temperature is (l mark)
tion in the number of people
xii (l mark)
per taxi which may mean
decreasing; (1 mark)
that the taxis are in use for
less hours, or less taxis are as the second derivative d (l mark)
used everyday. (l mark) is always positive, the e Use GDC solver or inter-
temperature will not
Choosing a counterexample; section of graphs (l mark)
have a minimum but will
e.g. for x = e, (l mark) x =2.16 (l mark)
approach the value 25
given by the horizontal
In (e2) = (In asymptote. (l mark)
(2 marks)
(In — 2 In (x 30 a f (4) = log2 4+ log2 (15) — log2 (5) (l mark)
(l mark)
(l mark) = log2 = log2 12
(l mark)
(In x— 5) (In x +3) = 0 5 (2 marks)
(l mark)
(l mark) log2 x + log2 (x 2 —l) —log2 (x + l)
In x —5=0, In x +3=0 log2 x + log2 (x — l) (x + l) — log2 (x + l)
In x = 5, In x = —3 (l mark) x(x—l)(x+l)
f (x) = log2
-3
log2 x (x —l)
(2 marks) log2 (x2 —x)
28 a i (l mark)
(l mark)
a = 69
694
Chapter 10 C1-1 Exercise IOE
1 h(t) = 2t3 + t— 10
11 a 5x
4 b 2e2 +—t2 +2
Skills check
—In 18x —71 +
12
Exercise IOF
i a 1800 cm2 b 18 m2 Exercise IOC
C+i 6 In IXI —xdx= 12
c 87t mm2 2 5eU+C
—(6x4) = 12
2 312 + 15x x2-25 5dx=15
c 9x2+6x+1 d 2x2— —In IXI + C
+ 6X2 + + C
—t4 + C
Exercise IOA 10 —en -211 +C
Note: throughout this chapter, Exercise IOD
denotes an arbitrary constant. 35
F(x) +C
F(x) +C
3 F(x) -¯ CA"26 21
26 3 —Inli0x +131+C
10 Ttr2
—x +3 dx=24
F(x) = -L+C —In13x +81 + C 25Tt
——e4-2x + C
11 A = —Ttr2 — 2 25Tt
—(2x -9)5 +C Tt(5)
10
10 dx = 16
12 = 16
4(4x +3)
Exercise 10B
10 Exercise IOG
2
3 — +— — 31 15 21
4
12 2 -3 8 16
6116 48
13 -12 6 20 b 14
14 9 12 10 10
12 a 10
14 a —(3x +10)6 +C Exercise IOH
18 i 15 2 3 5 In—
12 ? -32
(12x +7)2
—1I2 nli2x+71+c
695
ANSWERS —256 1.417 29
(—x2 + 2x)dx = dx ((3x +4) -2ex)dx 3.68
-0.5x — 0.5x
10 a 20 b 28 10 10
11 a 21nIxI+C
Exercise 101 10
In 7
27 49 8 19
—10 a— 2x(x2 — 4)dx c8 -10
dx k=13 2x)) dx -16
Exercise IOJ 4 10
32
10
1358 89 7) dx-27.5
-7.35889
10
2 46
8.21093 8.78
In (x) -10 -8
0.145551
246
-3 -2 -1
4 —(x2 —5x 125 7 8 9 10
=
-10
3.26556 10 G — —x dx
-0.765 564
-5x+1) dx-21.8
-4 -3 -2 -1
bi
ii —or 1.33
123456 Gc i x dx
696 1 23456
ii
iii 1.510
Exercise 10K =—adv = —3m -2
x —2x — s
2 10 a
—3x dx+ x —2x) dx = 24 dt
(2 marks)
2 —3x2 +3x+ l) —(x + +1) ( -3x +3x+1) dx=O.5 b
((x3 — 2x) — 3xe- 1.51677 x3 — 4.65 3t2
—1.51677 dX+ 3t) dt = 40t
2
—1.414 21 (2 marks)
—x4 + 16x 2) —(x4 — 20x2 +64)) dx +C
1.414 21 10 40
((x4 - +64) —(—x4 102
x4 -20x2 338
-1.414 21
4 x4 +16x 57
1.41421 2 (l mark)
5a 3.5) ——3t2 57
b
s (t) = 40t (l mark)
y-3.5=-3x-9 (2 marks)
2
3x-5.5
(2 marks)
2
-1.09807 1 5.5) dX+ —1 x (l mark)
—3
2 0-(-3x- —1.09807 2 12 a Use GDC to obtain value of
definite integral (l mark)
ii 1.81
Cha pter review 28 f (x) dx = 1.1202
(l mark)
1 b
2 10 3 c 12
9
d 20 e 80 20 b i 2.24 (3 s.f.) (l mark)
b X5 — 2X3 + 7X C 3 20
4 e —e12 ii
13
4
10
c c 22 2f(x —l) dx = 2110 f (x) dx
13 = 2.24 (3 s.f.)
1 S a —87t b 5Tt
d 32 13 a (2 marks)
2x8 b Either: (2 marks)
6
x4 + 2 In IXI + C 3 b 21.1
f 4eX+ C a 7530
3 8a 3x-2 b
g 4xE +2x +C c 6.75 • (x2 -1)+ = 3x2 -1
1 Exam-style questions (2 marks)
h —X5 + 2X3 9X 2(x2 +1) -2x.2x OR
5 (x2 +02 (l mark)
1 2-2x2 f'(x) = (l mark)
28 (3 marks) 1
(x2 +1)2
j 2e3X+2 + C C
J —?-L dx = In (x2 + l) + C 3
1
(3 marks) = ±0.577
k —In 16x —71 + C
(2 marks)
6
3
2
ANSWERS
d J 11 f (x) dx=0 14 3xex -0.020x dx = 2.63
(2 marks)
(2 marks) (2 marks)
e The function changes sign = g(-3) 18 a
in the interval _ l, II, so the (l mark)
areas above an below the x1 3
2 x3
x-axis cancel out. 3x x 1
(l mark) 3 22
-3
b 15 21
(3 marks) 3
27 32 f(x)
33
f Either:S = d.x = 0.5 88
(l mark)
(2 marks) (3—2x— (l correct: I mark; all correct:
Or (by symmetry) 2 marks)
= 311--
S = 2JO f (x) dx= 0.5 b
3
4
(2 marks) 3
2
15 a (x-2)4 x + (-2)4 1
= x4 -8x3 +24x2 -32x+16 (3 marks) x
0.5 1 1.5 2
b J (x — 2)4dx = Jx4 — 8x3 +24x2 —32x+16 dx (l mark)
(2 marks) (l mark for shape; I mark for
(2 marks) domain; I mark for intercepts)
(l mark)
(l mark) a Either
x5 —xOx + 15 —1 + x —1
= — 2x4 + 8x3 — 16x2 + 16x + C 22
5 21 1
16 Use GDC to obtain graph of y =
4
Attempt to calculate area of both triangles (4 marks)
—J21 dx = 2x2 4
3
2.5 2
22 1
8 AB contains the origin 0.5 1 1.5 2
7
6 (l mark) Or 21 1
5 2
4 y = 0.0202x (l mark) 15 1
3 4
2 d f' (x) = 3e-x + 3x + Ox—1 (4 marks)
1 = (3- 3x)e-x 2
x
(2 marks)
1 23 45
e Solve
—0.0202... x = 0.98201...
(l mark for shape; I mark (l mark) 4
for domain; I mark for end- 3
f (0.98201 .. 1.10345... 2
points coordinates; I mark
(l mark) 1
for maximum point and its
0.5 1 1.5 2
coordinates.)
b 09<1.104 (l mark) y -1.10 = 0.020(x -0.982)
0.1011 (l mark)
= 0.0202... f Use of GDC to calculate
5 (l mark)
(l mark)
698
2 x4 5 a 56.5cm2 b 84.8cm2 Exercise IIF
4 c 56.5cm3
i 17.00 b 27.20
(2 marks) 6 a 302 cm3 c 18.80 d 15.00
b 236 cm2
c J22 (l mark) 36 700 litres e 43.10 f 31.20
b 8.05
dx = 8 (l mark) 8 170 crn3 2 a 33.4 cm d 13.7
f 7.24
The graph of y = If (x)l is Exercise IIC c 6.02
4 9.35 km
symmetric about the y —axis. 44.70 b 30.50 e 22.6
(l mark) c 67.40 d 56.60 m6 9.94
2 a 25.0 b 42.5 m3 18.1
8 4.90 mm2
5 9.31 km
m15.3
c 6.76 d 15.0 Exercise 1 IG
Chapter 11 3 h = 5.60 a = 29.20 46.00 and 134
Skills check 4 2 34.80 and 145
i a 15.8 b 4.90 s 7.45 3 62.10 and 118
2 a i 12600cm2 6 10.4 cm 4 53.50 and 1270
ii 1.26m2
a 3000 cm3 b 32.5cm 5 1500
b 12 566 litres c 67.40 d 31.6
Exercise 1 IH
Exercise 1 IA e 71.70
a 8 2 592274.48 rn3 cmi a I I. I b 36.4cm
d 10.4m
mb 186 c 23.9m f 45.0m
d e 18.6 cm
(1.5, 2, 1) 5.4 c answer given 2 97.30 b 1010
d 139 OOOm2
42.00
2 (0.5, 1.5, 3) C 1590 d 34.90
c (0.5, -0.5, -6) Exercise 1 ID 51.30 f 26.90
d (-1.85, -0.15, 10)
5.60m 2 41 3 29.0 b 26. I cm2
3 a 4.47 b 6.56 3 a 15.50 b 2.89m 4 89 km
c 10.2 d 4.47 4 97.1m S 8.24m S a 29.9m and 8.56m
4 14.1 b 6.40 b 77.0m
6 a 20.2 km b 14.7 km
C 5.39 d5 47.3 km 6 22.6 km
m8 301
Exercise 11B 9 260m 10 1857m Exercise 111
i a 1440 cm2 b 66.4cm2 Exercise 1 IE b 61.4 m2 i 82.6 cm2
c 155 cm2 d 283 cm2 i a 23.6 cm2
e 377 cm2 f 201 cm2 d 13.9cm2 2 1230.8 million km
c 2.73cm2 f 84.3cm2
2 a SA = 314cm2 e 104 cm2 3 a 11.8m b 6.68m
Volume = 524crn3 g 47.6 cm2 b 300
2 a 12.6 cm2 c 94.81 m2
b SA = 28.3 3 208 cm2
Volume = 14.1 cm3 4 173 cm2 4 a 1200 b 104 m2
S 38.0 m2.
3 a 64.0 cm3 b 240 cm3 62 C 53.10
c 105 cm3
d Angle ABC = 1050
4 a 204 cm2 b 283 cm2
c 314 cm3 S 19.70 b 60.8m
6 a 6.67 cm b 48.40
c 1000 d 8.78cm
e 25.5 cm2
699
ANSWERS
7.06m 300 - e CP = m600 = 10.4
8 a 1240 b 178 km (2 marks)
c 2810 23 a
9 a 278 km b 2920 V = base area x height = (2 marks)
10 680 km, 2360 v = 3x2 300 -3x2 ... 1 10 cm2
a 21.2 cm b 76.00
c 95.60 3 (2 marks)
12 a 11 4 2) 32 x 5
24 9 x 100 %
= —x(100 —x2) TtX3.12XIO.1
4
b The cosine is negative = 30.9% (2 marks)
Chapter review 14 3.6 km h-l 22-12
IS 60.1m 2
24 a (l mark)
i volume = 641113 16 a 46.60 b 20.3 cm2
surface area = 144m2
i? 21.40 b 36.00 - 132-52-12
2 volume = 96Ttcm3
surface area = 96Ttcm2 c 259m2 b 25.6m (2 marks)
18 a 20.2m
b 242 m 22+12 — 204 cm2
3 16Ttm2 2 (2 marks)
4 a 168 cm3 b 1 5 1 cm2 86.70 or 93.30 —c C = cos-l 5 -67.40
md 24.1 13
S 1294.14cm3 19 1 300 b 2980 (2 marks)
c 431 cm3 d 0.000431 rn3 c 39.4 km b 57.1m
6 a 74.21113 b 108 m2 20 68.50
a 36 cm2 b 432 g c 56.8m (2 marks)
2 Exam-style questions 2S ABC=1350 (l mark)
AB2+BC2 2i a S — 4 x +52 = 95 cm2
_ NIF7Ö=8.49cm
2
(2 marks) (2 marks)
(l mark)
Then OC = — b AC = 41.6 km
2 2 (2 marks)
= 4.24 cm b sin C sin 1350 (l mark)
=... 54.5 crn3 20 41.61...
Then 3
(2 marks) & = 21.30 (l mark)
_ 32 + 4.242 22 a
= 5.20 cm
Therefore the bearing of C with
d 70.50 e 87.0 cm2 (2 marks)
6x9.539... Arespect to is 2340 (l mark)
8 a 5 cm (1, 2.5, 3)
b I Ocm
2 26 a
m=114.47 .. —114 2 (3 marks)
9 6 cm2
10 a 171 573 cm3 b 102 cm (2 marks) 0+0 6+6
bM
20m 222
42 2=51. Im (2 marks)
12 366 cm2
13 a A = 2xh+ (2 marks) (2 marks)
= 6x2 + 8xh —d arccos 3 -72.50
10
(2 marks)
(l mark) 30 30 (3 marks)
29 a tan 320 — (2 marks)
(2 marks) = 48.0 metres (2 marks)
(l mark) x
tan 320
(3+48.010...)2 +302 = 59.2 metres
Me tan = —6 = 3 30 = 30.50
2 c arctan
51.010...
Mcos = di¯ö 30 metres (3 marks)
10
2 0 1220sq metres (3 s.f.) (2 marks)
b (3 marks)
25 sin B sin 170 28.50 (3 marks)
2 48 89.650...
(2 marks)
(2 marks) 4 a 57.30 b 1150 c 36.10
d 80.80 e 88.80 f 1720
BD = 125—506 (1 mark) Chapter 12 g 20.60 h 73.30 i 0.5730
j 1230
BD = 25 (5 — 2$) (l mark) Skills check
a 2
2
sin CDB sin 450 2 a (-0.618, o), (1, o), (1.62, 0) Exercise 12B
b (0.633, 0)
13 5 5-26
3 a (-1.61, o. 199)
(2 marks) b (2.21, 0.792) anii 9Tt
sin CDB= i a i 7Ttcrn 707t
3
iii
2
(l mark) Exercise 12A b i 497t cm2 m2ii 54Tt
d The angle C DB can either a b 3Tt 1 5Tt
3
be acute or obtuse 4 2 m2 iv 1757t cm2
(l mark)
4
and the two possible
values add up to 1800. d 2 60 cm
(l mark) 10 4 mb 42.8
3a
2
4 Tt I Olt 27t
h 3 4 25.1 units2
99 mS 207t
28 a — X 33 + 32 X 7
37t 6 I .85 km
(3 marks) 4
= 8 lit = 254 (3 s.f.) 2 300 b 180 c 1500 Exercise 12C
(I mark) d 5400 e 630 1440 i a Negative b Negative
c Positive
b
g 3150 h 2800 3000
1 x 3 x 7 + It x 32 j 5850 2 1440 b 1300 c 950
(2 marks)
S x 2n x 32 + 3 a 0.175 b 0.698 c 0.436 27t
2 3
d 1000
d 5.24 e 1.92 f 1.31 5
= 60Tt = 188 cm2 (3 s.f.) g 1.48 h 0.223 i 0.654 57t 27t
j 0.0175 h
(l mark)
73
701
ANSWERS
3 a 3200 b 2500 c 600 It 27t 7 Tt 1 37t 147t Exercise 12G
2a
99 9 9 9 9
1 57t 197t ia
d 1400 8 10 3
b O, — —27t, , 27t 37t
33 22 2
h 37t 1
24 d
57t 77t 4 x
4a b
6 4' 4 Exercise 12F -1
c 1 a sin 100 b Sinn -2
c sin 8Tt d cos0.8
3' 3 -3
15 e cos 12 b
15 8
Sa 2 3 omo
17 2
Exercise 12D
2a
a 53.10 306.90 3 9 1
b 8.60, 171.40
c 11.30, 191.30 7 445 oanc x
d 142.80, 322.80
e 115.50, 244.50 c d -1
7
2 a 0.96, 2 18 9
b 2.39, 5 53 b
c 2.79, 3 49 2 -2
d 0.69, 5 59
6.05, 3 37 3a 31
32
57t 2 -3
3 1 3
66 2 2
4 0.93, 4 07
4a 1
b 2.68, 5 82
c 1.11, 2.36, 4.25, 5 50 c 512 x
d 1.57, 3.67, 5.76 31 -1 mvmvn
S a -1.27, 1.27, 5.02, 7.55
Sa 57t -2
b 1.01, 4.16, 7.30, 10.4
c -1.11, 2.03 33 -3
d -5.82, -4.39, -2.68, -1.25
6 310, 2110 7Tt I lit d
b 3
2 [immunø
Tt 27t 37t 2
c
IUIUIUID
3'2'3'2
1
Exercise 12E Tt 37t 57t 37t
x
d
-1
4'2'4'4'2 4
-2
27t 47t
, 27t mvnvnun!l
165 -150, 150, 1650 6a 16, -3
b -1300, -1100, -100,
57t —2 a Amplitude = l, period = 27t
100, 1100, 1300
c -1400, -1000, b 3
-200, 200, 1000, 140 12' 12 b Amplitude = 0.5, period = It
d —900
32 —27t
3
c Amplitude = 4, period =
3
d Amplitude = —1 period = 67t
2'
702
3 a Amplitude = l, ÄAuA 20
period = It, y = sin2x unoa 15
10
b Amplitude = 2, period = It 2n
Y = 3 cos 2x -10
2.5 -15
c Amplitude = 2, period = 2Tt -20
Y = —2 cosx 11011011
20
d Amplitude = 2, period = 2Tt 1.5 15
Y = 2 sin(—x) 10
0.5
4 -10
0.5 -15
Exercise 12H -20
s -2.38
iv b ii -1.29 20
d iii 15
10
y = 2 sin x + 3
1
C y = 2 cos(x) —2
y = 0.5 sin(3x) -1
57t 137t 177t
6'
b The graph of y = cosx may
—be translated to the
right to become the graph
of y = sinx.
0.5 b 0.6075, 1.571, 2.534, 4.712 -10
-15
0.5 10 a b 47t -20
-0.5
c o, 1.26, 3.77, 4.19 20
uoox-1.5 15
Exercise 121 10
20 -10
15 -15
10 -20
-10
-15
-20
703
ANSWERS
20 a y=50+10cos 12 a 5
3 5
15 b 43.3 cm c 0.515s 4
10 3
8 Answers will vary for each
5 2
student.
m6m6 x
Chapter review
-5
-10 57t 1
-15
-20 a x
2 1.24, 4.39 66 4
b 1.13, 4.53
27t 9 3 -1
c —It, (), It -2
d -0.903, 0.677, 1.98, 2.61 d
-3 oouu
3
-4
37t -5
25 13 0.755 cm2
2 2700 b 2100 c -1050
d 200 e 4200 f —660
g 3300 h 4080
Exercise 12J 37t
6 14 a 4 b 27t
i a Max = 12 m Min = 2 m 3a 5 b 15
b 8 am a O, 0.877 1.1
c 2 am
4 a cos 9=0.8 and tan 0 75
d 11.3m b 16 2.89 rad
33
i? 15.0 cm2
3:14am 12:46 pm 77t llTt b b 19.1 cm
3:14 pm 5a 18 a 14.7 cm d 29.1 cm2
4' 4
2 a 13000 on February 1st 57t c 25.6cm2
c
19 f(x) = 10sin6x b 0,—
b 7000 on August 9th 20 a
10212
3 20 b 10 6a f (0) = 2x (cos2 0 —
c Y = (10 sin 0.5x) +10 Tt 57t = 4 cos2 0 + cos0 +1
d 16 b bo
12',12' 12 12 240, 120, -240, -120
4 a 35 b5 5 119 5 2i 7.85m b 58.0 m2
15 a b
d a = 15 13
13 169
c 20m d 25.6m
4 2 1 22 a m4.39 f 1.18s
b
g 0.535 8
2
b sin 57t 4
2 66 P-3, q=—, 11=2,
3
12b —c=14 1.30, 3.41, 6.19
20 23 a 3
2Tt 9a b 1 b 10-2= period,
b c 13.3 s 4 8
3
10 a=30, 84
6 a y = —20 cos + 21 11 a It b3 C5
c3 d2
20 d y= —(5+ 3 sln(— 3))
b 10.64 s
704
Exam-style questions 32 = sin 2x+c0s2 2x 34 a i (l mark)
(l mark)
29 a A (t) = (2cost — l) (cost — l) + 2sin 2x cos 2x b (l mark)
(2 marks) = I + sin 4x (3 marks)
b (l mark)
2 cost —l = O cost b (l mark for correct shape, (2 marks)
I mark for 2 cycles, I 2
1 Tt 5Tt mark for correct max/min)
(l mark)
2
(3 marks) c —2COS Ttx + I = () 1
cost —l = I cost = 2 which 3 COSTtX =
has no solution (l mark) 2
2
1
x (l mark)
30 2 coex = -1 TtX e 33
2 cos2 x — 2sinx cosx = O 1 (l mark)
(l mark)
3' (l mark)
2cos x(cosx —sin x) = 0
(l mark)
cosx = O, cos x = sin x 2 (I mark) 35 a i x 2
(l mark) (l mark) (l mark)
osy<2 (l mark) —ii (l mark)
37t (l mark)
1 2 (l mark)
22
d Oiii
2
(l mark)
4' 4
ii p = n (or any b (l mark)
Tt + 2n1t,n € Z)
31 a Correct attempt to at least (l mark)
(l mark)
one parameter (l mark) (l mark)
14 8 (l mark) c The first point of
2 inflexion occurs at x =
4
(l mark)
(l mark)
A = 22 x
14+8 33 a
-11 3
(l mark) 3
2 (l mark)
d
b 8
33 a tan
15 (l mark)
10
(l mark)
5
bi (I mark)
3 a tan —— = —3
4 (l mark)
-5 (l mark) -1
2
(1 mark for trigonometric
scale and correct domain, Solve simultaneously
I mark for correct max/min,
I mark for two complete (l mark) (l mark)
cycles) r = 3 cm (l mark)
(l mark) 8 (l mark)
9 (l mark)
—— and
88
ANSWERS
36 a —2 cos2 x + sin x + 3 = —2 (l — sin2 x +sinx+3 (l mark) Exercise 13C
(l mark)
= 2sin x +sinx+l (l mark) f" (x) = 12 cos 4x—E +5
b —2 cos2 x (1 mark)
(l mark) 6
2sin2 x
2sin (2 marks) 2 y' = (12x2 1)e4X3+2V2+X
(2sinx — l) (sin x +1) = O 1
sin x = —1 , sin x = —l 3
2 cos x —l
4 f' (x) = eX(x + l) 2
I lit It 37t I lit =S h' (t) —2(sin t) (ecos t)
f6 = 3e5Xcos 3x+ 5e5Xsin 3x
6 2' 6'6'2'6 7 = xcos X
Award (l mark for two correct solutions) 8 f(x) = —2xeX2sin (ex2) —sin 3x
9 f'(x) = 31n
+
x
Y10 = 2 cos (4x) or 10 f' (x) — 3 cos 3x 3
Chapter 13 2 cos2(4x) — 2 sin2(4x) or
Skills check sin 3x tan 3x
= cos (4x3) 11 a f' (x) = —sin x(ecos x)
11 a
b b increasing: < X < 27T
decreasing: O < X < IT
ia b -1 c
2 2 12x sin c local maxima at (O, e) and
d eo 1 c Ax) = 2 x cos (4x3) (2m, e)
2 — 12x4sin (4x3)
2 local minimum at (IT, e-l)
12 a i f' (x) = cos x 12 a f'(x) = I — sin x; f''(x) =—cosx
2 a cos 4x b 3 sin 2x C ii f" (x) = —sin x
iii f"' (x) = —cos x
3a 2 f = xiv (4) (x) sin b Concave up when
3(4x3 +70 b 4, 8, 12 37t
b + 6xe2x = sin x
ii = —sin x
I — 21nx Concave down when f' '(x) < O
c —3Tt
, 27t
x3
22
Exercise 13A
i f'(x) = 4 cos x — 3 sin x Exercise 13B
f2 = 5 cos (5x) i Tangent: — — n) c Inflexion points: 22
37t 37t
1 —x
3 y' +3 sin(3x)
3
Normal: y— — 13 a
4 3 sin x 2 f'(x) = —2 sin (2x) + 2 cos sin x)
cos x x
2 2 = —2 sin (2x) — (2 sin x cosx)
5 h'(t) = 3 sin2 t cos t = —2 sin (2x) — sin (2x)
cos x 2 Tangent: l, normal: x = —
3 = —3 sin (2x)
7 y = —sin— 3a =b f' (x) Ttcos (Ttx) 2
c c Inflexion points:
x2 x 2
37t I
8 — 12 TtX 1
sin x
05 + 05-40
9
4
44
706
d f(x) Exercise 13E Exercise 13F
2
i The order size is 100 —5 cos x + C
1
e4'2¯1) items; justification: x 0 2 4sin x + 2cos x + C
n 4' x Cand (x) = 0 x = 100 1
3rr 20 ooo 3 —sin (7x) + C
1003 7
4 3sin (2x) + C
-2 minimum 1
Exercise 13D 2 a b = 2x = 8 cos O 5 ——cos(5x +3) + C
C1 (120) = 70; this means it 5
cost 70 Euros to produce the h = 2y = 8 sin O 61 1
121st table. A = bh = (8cos O) 4 2
2 = 32(2 sin O cos 0) = 32 sin 20 x
2 sin
—b = 64 cos 20; maximum
2
dt
8 —COS (21tx) + C
when O = Exercise 13G
4
c —=-128sin20; 3
d02
21
=—128sin — 2 —cos (3x5) + C
—1 1 d 02 2 1
b Velocity = ——9 speed = 4 3
2
e = —128 < O maximum 2x2 + I
c t = l; the particle changes 4 e.X2 + 7X
direction at I second, 3 —h) 2 4
because velocity changes
46b V = Ttr2h = (h) —(x4 —3x
from positive to negative
3 2
when t = l.
6 2ecx + C
—sin (2x2 — 2x) + C
3 1
2
8 ——cos5 (x) +C
120 e •200) = E(36h-12h2 +
5
9
4a -o 9 —cos (In x) + C
10 3
b P'(t) = 24e02t 10
9
c 177 bacteria per day; at day f(x) = eSinX+ 11
10 the number of bacteria
d2V 12 f(x) = In (2x2 + e2) + 3
are increasing at a rate of =
177 bacteria per day. + 2h)
3
V5 a (200) = 6; the profit gained Exercise 13H
by selling the 201st unit of 3 2 6+1
the chemical is 6 Euros. 8
2
b C(x) = 0.00005X3' - 2X+ 196 d
3 u3du = 324
c 4 Euros 3'
1
4 p(x)
4
2
2
6 3.19s
c v(t) = —9.8t+ 15.2 c x = —1 or 0.630 3 b 12.1
d 1.55 s; the ball reaches 4 6 sin u du =
its maximum height and d 2.38 thousand dollars or 2
changes direction at 1.55 s. $2381.10 4
5 a c=14X+ 5400 b $550 7 a k = or 3. 14
x 8 11.4
ANSWERS
Exercise 131 J09 sin x
v(t) = —t2+ 8t— 12 d
c f 9 +1—t2 8t— 12 ldt= 48.3m 2 cosx
sin x
t-9 t-2 2
x
-25 -8.67 c J0612t- 61dt= 18m
cos x ( In x)
2 a v(t) = 2t—6 b J6(2t— 6) 0m x
—sint — cost
e
gh (x) = 2e2Xcos + 2e2Vsin2x
-1 8 2 a —x5 + sin x + C
f0313(t-
3 a v(t)= 3(t— 5
b 1
——cos4x + C
4
8s 1
4 a Displacement = v(t) dt = 22m; distance — dt= 22m 2
b Displacement = v(t) dt = 6m; distance —
= 30m 5
(2x3 +5x) + C
c Displacement = J dt = 10m; distance — dt= 34m e sin(x3) + C
f 2ex2 + 5x + C
g sin(lnx) + C
5a m16.5 h —In (e4x +5) +C
6a
1 3
t
-16t+10 c 32m 3a
2
3
255
Exercise 13J 6 a —12.8 ms 2
i a v(t) = etcost — etsint b
b t = 0.696, 5.59
b a(t) = — 2etsin t c 13.2m 4
d 24.8m
6
d
2
b c 4m Exercise 13K 4a
5 (1025t2-
2 —x2 + 4xh = 432 4xh = 432-x2
t3)dt
=3 a i t 0, It, 27t 432-x2
1152 spectators 4x
< <ii It t 27t b
2 33.4 + J 432-x2 = 108x ——x3
4x 4
b a(t) = ecost (cos t— sin2 t) = 206 cm3
= — +c s(t) 20
ecos t zle
4 a 6.75m b 14.0m 3 3800 + -150 1- t -1175
0 80
5 i -2.37ms2 gallons c 12cm by 12 cm by 6cm
ii Speeding up
20.4e18dt 273 billions v(t) = —3e cosx sin x;
b 3.54 s and 5.01 s a(t) = 3 e cosxsin2x
of barrels — 3eCOSXcosx
c i —6.92m b
ii At 6 s, the particle is Chapter review c 0.905 5.38
m6.92 to the left of its i a f' (x) = 3 cosx— 4sinx md 14.1
b y' = —3sin(3x — 4)
initial position. c h' (t) = 4sin3xcosx 6 12.6 cm
d 18.6m
708
Exam-style questions c Find intersection of 13 a i -0.524 (l mark)
g' (x) = 2sinxcosx—5 graphs (l mark)
(2 marks) t-1.13 (l mark) -0.369 sy 1.76
b
(2 marks)
= sin 2x —5 S I —5 = —4 < O
—10 x —sin x dx —¯ + COS X iii f' (x) cosx— sin x
(2 marks)
2
(3 marks)
(3 marks)
Therefore g is decreasing x —sin x x2 (sin x + cosx) dx = 2.18
on all its domain. (l mark) —+cosx
2 (3 marks)
(l mark)
2
8 a f'(x) = 5 —sec2 x 2 mm14 a s (0) = 2
(3 marks)
(2 marks)
— (e ) cosx+ex (cosx) b iv = 15 cos 3t +2t
44 1 (2 marks)
f E =2E-1 2 = ex COS X —ex sin X ii a=v'=—45sin3t+2
(l mark)
44 (3 marks) (2 marks)
(l mark)
= 0.548,t = 1.50,
(l mark) c
(l mark) 2.74 (l mark)
4 24 (2 marks) = 0.548,t = 2.74
(l mark)
12 a I mark for shape, 1 mark for (2 marks)
domain, I mark for scale on
b The normal to the graph is
axes (l mark)
vertical when the tangent
is horizontal. (l mark)
f =ii (It) 5 (l mark)
f' (x) = O sec2x = 5
(l mark) 2 b (l mark for coordinates of
A, I mark for coordinates
cos x = —1 x = arccos— (0304,1) x
1 of B,
55
0105,0.909) I mark for zeros, I mark
(l mark) for shape and domain
h .36,0)
COS2 X = —1
-1
5
-2
tan x = =2
(l mark) b i Minimum points: 10 A 1.01,7.47
(0.785,0.909) and 5
(196)
—-2f arccos— = 5arccos (2.36,0) (2 marks)
55 x
Maximum points:
(l mark) (0.304,1) and (1.27,1) -5
-10
The coordinates of A are (2 marks) B(2S9t-690
(l mark)
—arccos , 5 arccos osx<l
55 sin I (l mark) Jabf(x) dx = 2.10
9a Isin 2t — sin (t —0.24)1 2.356... (2 marks)
a sin (l + sin 2x) dx d The graph crosses the
x-axis between a and b
(2 marks) (l mark)
b Use GDC to find the (2 marks)
(2 marks)
maximum (l mark) 1.76
t = 0.975 (l mark)
709
ANSWERS
e Either Chapter 14
dx = 7.39
(2 marks) Skills check
Or f (x) dx a 5.5 b 14.6 (3s.f.) 4 a x-1.719
2 a 5.5 b 14.6 (3s.f.) b x=2.98
1.961... 1.9601... 3 15 b 56 c 0.267 c x = 8.68
1.017...
(2 marks)
= 7.39
Exercise 14A
16 a
2
(l mark) s 2 345 6 8 9 10 11 12
In the 1st and 2nd quad-
rants sine is positive 36 36 36 36 36 36 36 36 36 36 36
(l mark)
Therefore f (x) 2 0 for all b
12
(l mark) 25 10 1
b 36 36
x c
—X) = sin
n
22
= sin —It x 1 2 345 6
11 9 7 5 3 1
22 36 36 36 36 36 36
x
d
(2 marks) 10
Therefore 1 22 3 24 2 12
36 36 36 36 36 36 36
(l mark) 12 15 16 18 20 24 25 30 36
A (x) = 2xsin 36 36 36 36 36 36 36
2
(2 marks) 2a
Find maximum point
(1.72,2.44) (2 marks) 2 34 5 6
10 12
and
36 36 36 36 36
B(4.86,0.652) (2 marks)
d p = 2AB+2xO.6520... = 8.19
(3 marks) —7
P(T > 4) =
12
3 3a
1.72,224 1 2 3 6 10
2 121 1 1
66666
1
1
x
b
-1
2
4a 1 1
6 2
710
1 P(blue then red) 3) -0.913
6 = 0.224
36
0.399
6k 1 10 10 25
16 a 3) = 0.0307
(k cannot be negative) b 125 0) = 0.463
5 d1
82 iii P(X22) = 0.171
27 10 10
b 0.215
8 a a = —1 and b = 5 8 0) = 0.7489
b E(Z) = 1.7 [P(X2 2)12 = 0.0292
8 24
iii
—25
-0.158
P(Sum > 7) =
576
9a = 3) = -1 and $1.70 is the expected
winnings on a ticket.
Exercise 14E
= 2 and B = 1) c Lose $0.30 per ticket
i 2 68 37
121 1 5 7
3336 18 Exercise 14C
b Exercise 14F
c 2 3456 ia i E(X) = 20
4 2
18 18 18 18 18 1 3
16
2 11=25
Exercise 14B 16 3a
15
15.2 (3 s.f.) 3 2 16 b E(X) = 3.75
3 c P(X2 10) = 0.000795
16 2 2) = 0.329
2 1.6 + k = 0.351
3 2) = 0.680
2) = 0.649
1 3 5) = 0.0389 4 a P(girl) = b 38.2
5) = 0.952 300
4a 25 5) = 0.00870
S 0.2 sk<l b 1) 0.932 Exercise 14G
61 Exercise 14D p = —3 and n = 16
45 i Most likely outcome is I red 4
face with probability 0.422.
8 2 X- B(20, 0.2)
2 = 5) = 0.257 b E(X) -4 var(X) = 3.2
45 3) = 0.260
7 c P(X2 10) = 0.002 59
3 = 0) = 0.851
45 b If 13 are not faulty 3 0.8 or 0.2
6 then 3 are faulty,
= 3) = 0.000491
45 Exercise 14H
2 2) = 0.0109
5 4 = 0.851
45 b P(13 not faulty) = P(X2 3) = 0.272
4 = 0.474
45 b P(0.5 1.5) + P(-1.5
3 < -0.5) = 0.483
45 2
2
2.4) = 0.00820
45
1 c P(-l < Z < 0) = 0.3413
45 Z < 0) = 0.4599
2 3 a 0.742 b 0.236
c1 c 0.0359 d 0.390
3
711
ANSWERS
4 0.306 b 0.595 4 P(520 < x < 570) = 0.673 313
c 0.285 b 0.215
b 582 3
S 0.311 c 0.186
d 0.5 S a d = 79.7 b f= 35.8 8 64
Exercise 141 b 0.748
1 0.655 b 0.977 4 12, 16
b 0.841 =0.933 Exercise 14M 121211
2 0.672
b
c 0.345
3 0.994 888888
c 0.494 33 c 7.5 d €62.50
Exercise 14J 2 {1=15.4 = 49 9 40
243
1a 3 0=423
4 138 {1=71.3 6 0.2
0=7.66 cm
a 85 b 0.023
6 {1=546.5 g 19 —5 1
8
0=0.389 kg 8a 27 19
27
b 35% 27
g = 54.3 cm
x
0=0.260 m
c P(80 125) = 0.736
2 477 10 0=33.7 125.66 9
9 a 0.254
3 > 20) = 0.0668 b > 117) = 0.601 = 60.1% b 0.448
Yes — this is consistent with
10) = 15.87% the normal distribution. 10 0.0243
4 53.50/0 7 34 11 a i 0.0881 ii 0.00637
475) = 0.106 c 14
- 0.001 18
Chapter review 12 .44
1a6 7 13 a 8.68 b 0.755
b
Exercise 14K
15
14 38.9; 8.63
b3
1 a a = 1.42 b a = 0.407 2a 1 15 a 33.3 b 0.328 c 0.263
c a = 2.58 b a = -1.00
35
2 a a = 1.77
c a = -0.841 Exam-style questions (l mark)
(2 marks)
3 a a = 0.385 16
(2 marks)
b 1.60 10 10 25 (3 marks)
18 18 81 (2 marks)
4 < a) = 0.95 x o1
< a) = 0.8 16 40 2
:. a = 0.841 81 81
25
81
Exercise 14L i?
1 5.64
2 a a=413 b a=433 18 0.05+0.22+0.27 0.46 (2 marks)
3 500) = 0.106
b 86.4% b E(x) = 2.46* +3a +4b = 2.46
(2 marks)
c a = 498.8 505.1
3a+4b=1.7 (l mark)
Solve simultaneously a +b = 0.46 and 3a+4b = 1.7 (2 marks)
a -0.14, b=O.32
712
19 Let Xbe the number of 23 a i Let X be the number of g = 79.8 and = 24.1
defective batteries in a pack of correct answers in the (2 marks)
10 selected at random. 12 questions answered
at random. > 100) = 0.20
(2 marks)
a (l mark) X -BO 2,0.5) (l mark) Let Y )
= 1) = 0.0478 (3 s.f.) 12 0.379.
(l mark)
p(x = 2) = 2 (0.5/2 (2 marks)
= 0.0161 d 630P(Y > 85) = 207
(2 marks)
(2 marks)
12
(2 marks) 1000 = 79.7573...
p(x = 12) = 12 (0.5/2 (l mark)
= llx s 1) = = 0.000244
p(x m=78.5
(I mark)
-0.0478
(l mark)
(3 marks) correct answers (l mark)
3 correct random answers 25 a
= 6 marks
20 Let X
9 incorrect random
np = 3 and npq = 1.2 answers = —9 marks
8 answers known = (2 marks)
(2 marks) 16 marks
Solve simultaneously If the student answers all > 55)
the questions the expected
(l mark) > 65)
number of marks is 13
p = 0.6 (l mark) marks which is 3 less than P(T255)
the total marks if he just
n=50 (l mark) 0.0131 3...
o. 13326...
21 x 2) -0.0986 (3 marks)
< 49.5) = 0.0668 (0.133...)3 = 0.00237
(2 marks)
answers the questions he
(3 s.f.) (2 marks) knows the correct answer.
< 50.5) = 0.775 (l mark)
(3 s.f.) (2 marks) 24 a 2) i E(N) = 6.66
—65) = 0.27 (2 marks)
> 491x < 49.5) 65-g = 0.27
xp(49 < < 49 5) -0.955 (l mark) 25) = I-P(N S 4)
•
> 96) = 0.254 = 0.814
p(x < 49.5) (3 marks)
(3 marks)
22 96 -g = 0.75 26 p(x > 82) = 0.1 p(x < 82)
82-g = 0.9
65 —g = -0.6128...,
< 5) = 0.754 (2 marks)
96-g = 0.6744...
= 0.754
(2 marks)
3 ii Solve simultaneously
(l mark) 65—g = -0.6128
= 0.6871...-4 g = 2.94 96 — g
= 0.6744...
3
(2 marks) (l mark)
(2 marks)
713
ANSWERS
< 40) = 0.2 c log x = (2 marks) Fair game
40 — g
= 0.2 (l mark) d log 100 + logy = 2+q
-g82 = 1.28..., (2 marks) = —0.8 (2 marks)
S y = 2x+l I has gradient of 2. —-3b = -1.5 b=O.5
Perpendicular gradient is —
40 — g c = 0.2 (3 marks)
= —0.841 (2 marks)
(l mark)
y = —+x+C through (4, 3)
9
Solve simultaneously (l mark)
(2 marks)
g = 56.6 and -19.8 (2 marks) b i Concave up quadratic,
(l mark) always positive, so no
(2 marks)
roots. (2 marks)
6 a This is an arithmetic so {D < O)
(l mark)
progression. Let the first
Paper 1: term be a. (l mark)
Section A 290 R{k € 1k > l} (2 marks)
(2 marks)
i a 4x3
b —2x -3 (l mark) 290 =10a+90 a = 20
c sin x + x cos x (l mark)
(2 marks) (l mark) (2 marks)
d 4e4x (l mark)
a too =
2 Wire length = 8 (2 marks)
arc length = ro (2 marks) (2 marks)
8 = 40 0 = 2rad Integral equals In (2x + l)] Since (x + 0, for all x
(2 marks)
3a = *In 9—0 (2 marks) Hence k > I as before.
= In 3 (l mark) (2 marks)
e for k = 4, f (x) =
(2 marks) Paper 1: So min point is (—3, 3)
b (2 marks)
Section B 10 a 2
(g 6x+1
(2 marks)
2
Ep(x
-1=2x = 0.4
2 (2 marks) PC=2-x
ii (2 marks)
(3 marks) Time in field is
4 a log x + logy = p +q = 1.1 (2 marks) 12
(2 marks)
(2 marks) Time on road is
b log x 2 —logy = 2 log x — logy 20
= 2p — q (2 marks)
(2 marks)
714
Using inverse normal
so T = 20 —-1 (2 marks) 5 = (0.198 35...)x
(I mark) = -0.499 99...
12
x = 25.206... (l mark)
So = 2 (l mark)
dT -1 h = 21.2m(3s.f.) (l mark)
dx 12 20 3 a Binomial (10, 0.8) Note: can also be solved
(3 marks) (2 marks) using the sine rule.
dT (l mark)
At a nun b = np = 5a (l mark)
x (I mark)
dx
(2 marks) p(x = 7) = 0.201 326...
x1 25x2
12NlfiTF 20 - = 0.201 (3 s.f.) b Total distance is
(l mark)
1 5 sin (t2)ldt = 8.51 m(3s.f.)
d (3 marks)
20
-0.879126 ... - 0.879(3 s.f.) c Displacement is
25x2 = + x2 (2 marks)
5 sin (t2)dt = +3.87 nt(3s.f.)
(3 marks)
= 36 +9X2 16X2 = 36 General term is C6x6 r 2
(2 marks) x
=71x > 6) =
require 6—2r = O so r = 3
2 36 3
(2 marks)
16 4 2
as x cannot be negative. p(x > 6) 3
(3 marks) (2 marks) 2 160
(2 marks)
0.201 326 ctx3 x
= 0.229(3s.f.)
Paper 2:
0.879 126...
Section A
(l mark)
4a Paper 2:
(l mark) Section B
ii x = —2 or —7 (2 marks) i 120.5 million GBP
ii 64.7 million GBP (3s.f.)
2 5 40
(2 marks) 5 (3 marks)
-3.414 or -0.5858 (4s.f.) (l mark) b i 67.7 (3s.f.) (3 marks)
(2 marks) b Let the monster's height be ii 19.2 (3 s.f.)
2 p(x > h and the original distance c i r = 0.950 (3s.f.)
(1 mark) between Maria and the (2 marks)
m()nster be x.
Applying substitution
— = tan 40 ii Strong, positive
z is )
x (2 marks)
(2 marks)
h iii y = 0.282x+33.7 (3s.f.)
(2 marks)
= tan 35
(l mark)
Pz 1 x tan 40 = (x + 5) tan 350
=0.30854 (2 marks)
(l mark) tan 400 = 1.19835... e Not very valid, should
x tan 350 use w on g line and
(2 marks) extrapolation. (2 marks)
715
ANSWERS d b i 3 years (290.7)
(2 marks)
8 ai x = 5
(2 marks)
ii Solving
2000.05)t 2 350 (0.94/
(2 marks) 6 years (268.02 > 241.45)
6 (2x -10) -(6x +3) 2 (2 marks)
c Solving
(2x-10)2
—66
(2x-10)2 (2 marks)
fii is always negative (3 marks) (2 marks)
Y = -1.83x+1.17 (3 s.f.)
so graph is always 8 years (248.32 > 242.08)
(2 marks) (l mark)
decreasing.
i -231.53 Note: could also be solved
(2 marks) (2 marks) with graphs or logs rather
than the preferred "table"
9 ii 15 years (415.79) method.
(2 marks)
716
Index
3D shapes 468-74, 477 asymptotes calculus 260—8, 450—4,
derivatives 246, 256 542-81, 563
absolute value 188 exponential functions 402—
3, 405 catanaries 182
acceleration 265—7, 567, limits at infinity 219—21 causation 322—5
572-4, 580 logarithms 407—8 central tendency measures
rational functions 189, 191,
Achilles and the tortoise 216 196-8, 201-4 287-95
reciprocal functions 189,
algebraic proofs 55 191, 196-8 chain rule
ambiguous case, sine rule 488
amplitudes 524-5, 527, 531 average costs 205 derivative applications 554,
average rates of change 222—
angles 557
3, 225, 552-4 derivatives with sine/cosine
between edge/base of 3D
shape 477 average speed 215, 222, 545—9
225-6
between line/plane 477 differentiation rules
complementary 482 average velocity 554
of depression 480 axes of symmetry 132—5, 142, 234-40
of elevation 480
standard position 514—15 144-8, 256 graphs of derivatives 248,
supplementary 488 253
terminal side 516, 518 bar charts 284—7
indefinite integrals 439—40
antiderivatives bases of logarithms 410, kinematics 574
natural logarithms 423—4
fundamental theorem of 416-17
calculus 452 bearings 480—1 circles
indefinite integrals 434—9 best fit lines see lines of best arcs 508, 511-12
integrals of sine/cosine 559 radians 508—10
substitution method 563—4 fit
sectors/segments 508,
see also derivatives bias, sampling 281 51 1-13
binomial coefficients 50—1,
antidifferentiation 435—6 surface area of cone 472
53 unit circle 513—17, 518,
approximation sign 72 binomial distributions
arcs 508, 511-12 563
area 592-602
characteristics 593 coefficients 50—1, 53, 98,
between two curves 455—9, expectation 599—601 163
565-6 functions 595
variance 601—2 collecting like terms 156—7
normal distribution 605—6, binomial expansions 49—53 combinations 46—51, 594—5
613-15 binomial experiments 594 commission 85
binomials, definition 49
sectors 511—12 binomial theorem common denominators 204
surface areas 472—4 44—5 5 common differences 14—16,
triangles 482—3, 512 bivariate data 318—51
under curve 444—8, 450—4, boundary conditions 442, 21, 24-6, 33-4
567
456, 565, 605-6, 613-15 box-and-whisker diagrams common factors 423—4
argument of logarithm 410 298-9 common ratios 17—21,
box plots 298
arithmetic mean 290 brackets 78 23-7
break-even points 155, 264
arithmetic sequences 6—8, 10, compass bearings 480
13-16, 21, 33 complementary angles 482
complements of sets 364—5
arithmetic series 12, 22—43
definition 22 completing the square 151,
modelling 36—43 155, 160-4, 165
simple interest 36—8
complex roots 168
composite functions 85—92,
234, 439-40, 559-63
compound interest 36—9
compressions 134, 136—40
INDEX
concavity data convergence 216—21
derivatives with sine/cosine bivariate data 318—51 derivative functions 222—33
549-50 first derivatives 240—59
univariate 276—317
graphs of derivatives decomposing functions 91 inverses of definite
249-51, 253-9
decreasing functions 240—6, integrals 451
parabolas 131, 144, 146—8, 255, 257-8 kinematics 260, 265—8
170, 172 limits 216—21
decreasing sequences/series optimization 260—4
conditional probability 375, 14 rules 234—40
second derivatives 240—59
377-81, 384-5 definite integrals 444—59 dilations 138
cones 471—4 area between two curves discrete data 77-8, 278, 284
constant function rule 436—7 455-9 discrete graphs 79
constant functions 71, 228—9 area under curve 444—8,
constant of integration 436 450—4, 456 discrete random variables
constant multiple rule 436—8, fundamental theorem of
calculus 450—4 584-6, 592, 595-7, 601
441, 548, 560 inverse of differentiation 451 discriminants 164, 168
constant rate of change I I I kinematics 567—76 disjoint events see mutually
constant terms 116—17, 163, properties 448—9
substitution method 561—6 exclusive events
3 34—5 dispersion measures
contingency tables 372 degrees 508—11
continuous data 77, 79, 278 demand 63, 69, 74, 76, 84 296-309
denominators 8, 204, 220,
continuous random variables displacement
239
5 84—5 dependent events 375 definite integrals
dependent variables 77, 92,
convenience sampling 281 567-74, 575
convergence 216—21 110, 116, 127 distance 568-71, 573-4
converging sequences 17, indefinite integrals 441—2
depreciation 20 kinematics 265, 267,
28-31, 221 depression angles 480
correlation 320—9 567-75, 580-1
derivatives
causation 322—5 distance
measuring 326—9 applications 552—8
scatter diagrams 320—5, constant functions 228—9 displacement 568—71,
curve sketching 25 5—9 5 73—4
329
Spearman rank correlation differentiating polynomials kinematics 265—7, 568—71,
226-7 5 73—4
350-1
strength of 321—2 exponential functions speed 215, 222, 225-6
distance formula 468—70
cosine (cos) 422-4 distinct roots 167—9
complementary angles 482 first 240-59, 549-50, diverging sequences 17, 28—9,
cosine rule 489—93, 505, 556-7 221
520 functions 222—33, 422—4 domains 77—84
curves 523-8, 530-1, 545 linear functions 228—9 composite functions 88
derivatives 544—51 natural logarithms 422—4
normals 230—3 exponential functions
double angle identity optimization 262—3
520-2, 546-7 power rule 226—7 402
second 240-59, 549-50, inverse functions 93
functions 524—8, 530—1, logarithms 407
544-51 556
sine/cosine 544—51 quadratic functions 142—3,
integrals 559—60 sums of differences of 148
sign in given quadrant
functions 228 rational functions 201,
515-17 tangents to curves 223—6,
203-4
trigonometric equations 230-3
517-19 reciprocal functions 187,
see also antiderivatives 189, 196-7, 198
trigonometric ratios 476—8,
difference rule 436 trigonometric equations
480 differential calculus 260—8 518
costs 205, 262-4, 552-3, 556 differentiation 214—7 5
cuboctahedrons 358—9 double angle identities
cumulative frequency 301—3 antidifferentiation 435—6 520-2, 546-7
cumulative proba bilities 608,
checking integration 440—1 drawing 75-6, 95, 123, 364
613, 617
cylinders 470 e (Euler's constant) 405—6,
415-16, 422-4, 439-40
elevation angles 480
718
empirical probability see raising a power to a power geometry 466—505
398, 399-400 3D shapes 468-74
experimental probability see also trigonometry
equal roots 156, 167—8 extrapolation 331—2
equating coefficients 98 extrema see maximums; global maximums/minimums
equations minimums 244-5
differing from identities
520 factorials 47—8 gradient-intercept equations
exponential 400—1 factorization 147—52, 155—66, 119, 122-4, 127-8
fitting to points on curve 170 gradients 1 11—24
182 factors 423—4 derivatives 223—5, 241,
families of curves 604 243-4, 247
functions 79, 81 fastest speed 215, 222 horizontal lines 1 18
horizontal lines 1 18 finite sequences 5, 15 linear functions
finite series 10—12, 26—7, 30, 111-24, 127-9
least squares regression
334-42 221 line through given points
1 12-14
logarithms 418—21 first derivatives
normals 230, 232, 238-40 normals 232
piecewise functions 81 graphical interpretation regression lines 334—5
rational functions 204 straight lines 1 11—24, 223
240-59
roots of 155—6, 164, notation 246 tangents to curves 230,
166-9
optimization 556—7 232
straight lines 118—24, 128, sine/cosine 549—50 vertical lines 118
230 five-number summaries 298
fractional exponents 399—400 graphic display calculators
tangents to curves 232—3 fractions, improper 186 (GDCs)
frequency
trigonometric 517—22, 525 cumulative 301—3 definite integrals 457, 459
vertical lines 118 limits and convergence
expected 589
y=mx+c116, 119 relative 358—60 218-19
tables 283-4, 286, 291 normal distribution 606—8,
see also quadratic equations functions 62—107
domains 77-84, 88, 93 611
equilateral triangles 3, 13, limits 217—19 optimization 262—3
35 notation 69-74, 85, 92 Pearson coefficient 327
ranges 77-84, 88, 93
errors in data 279, 299—300 points of intersection 124—5,
estimation 54, 291, 303 that are their own 158-9
Euler's constant (e) 405—6,
derivatives 438 quadratic inequalities 171
415-16, 422-4, 439-40 what they are 65—9 transferring graphs to paper
event, definition 355 in words 66
exact answers 512, 517 see also graphs of functions; 142-3
exchange rates 1 13
"exclusive or" 366 individual types graphing packages 182
expectation 356, 588—91, fundamental theorem of graphs
599-602, 626 calculus 450—4, 563 composite functions 89—90
cosine curves 523—8, 530—1,
see also means Gauss, Carl 22, 604 545
Gaussian distribution see discrete 79
experimental probability domains 79—83, 93
354-5, 358-62 normal distribution drawing 75-6, 95, 123
first derivatives 240—59
explicit formulae 61 GDCs see graphic display inverse functions 93—6
labelling key features 142
exponential growth/decay calculators piecewise functions 138—41
401
geometric means 19 quadratic functions 131—8,
exponentials 394—406 142-54, 170-2
derivatives 422—4 geometric sequences 7—8, 10,
13, 17-21, 29 quadratic inequalities
exponents 396—406 170-2
functions 401—4, 407—11, geometric series 22, 23—43
ranges 79-83, 93
422-4 compound interest 37—9 reciprocal functions 188—90
inverses 409 research 106—7
logarithms 407, 409—11 example of I I second derivatives 240—59
exponents 396—406 limits and convergence 216, straight lines 1 11—30
equations 400—1 tangent curves 528—9
fractional 399—400 221
functions 401—4 modelling 36—43
laws of 396—7
719
INDEX infinite sequences 5, 29 kinematics 567-76, 580-1
infinite series 10—12, 23—6, definite integrals 567—76
transferring from GDC to differential calculus 260,
28-30, 216, 221 265-8
paper 142—3 infinity I l, 188—9, 203, rates of change 567, 575—6
terminology 572
see also sine curves; 212-13, 219-21
Koch snowflake 3, 13, 35
sketching graphs inflection points see points of
grouped data 302 inflexion least common denominators
grouped frequency tables
initial conditions 442—3 204
286, 291 instantaneous rates of change
growth of population 40—3 least squares regression
552-4 332-42
hemispheres 471 insurance premiums 91 choosing lines of 337—9
hexagonal-based pyramids equations 334—42
integrals see definite integrals;
471 piecewise functions 339—42
histograms 284—7, 603, 606 indefinite integrals regression lines 333, 334—42
horizontal asymptotes integrands 436, 438, 441, life insurance 91
like terms 156—7
derivatives 246 560-1
limits at infinity 219—21 integration 432—65, 559—66 limits
rational functions 189,
antiderivatives 434—9, 452, and convergence 216—21
196-8, 201-3 559, 563—4 definite integrals 444, 449
horizontal compressions derivatives 246
area between two curves Euler's constant
137-40 455-9, 565-6
horizontal lines 93—5, II 8, 405
by inspection 561 integration 563—6
121 constant of 436
horizontal points of inflexion fundamental theorem of reciprocal functions 189,
191
252-4, 257 calculus 450—4, 563
horizontal shifts 527, 531 indefinite integrals 434—43, series 11
linear correlation 322—4,
horizontal stretches 1 37—8, 559-63, 567
140 kinematics 567—76 326-7
parabolic segments 464—5 linear functions 108—30
horizontal tangents 232—3 rules of 436
horizontal translations 132—5 with sine/cosine 559—60 definition 125
derivatives 228—9
138-9, 525 substitution method 559, indefinite integrals 439—40
hypothesis testing 316—17 560-6, 573 integration 559—60
linear graphs 116—17
icosahedrons 356 see also definite integrals models 127-30
parameters 110—17, 120
identities intercepts piecewise 138—41, 1 54,
gradient-intercept
differing from equations equations 119, 122—4, 339-42
520 straight lines II 1—30
127-8 linear trend lines 107
identity functions 97 lines of best fit 329—32
Pythagorean identity 516— see also x-intercept; lines of symmetry 605
17, 519, 521 y-intercept local maximums/minimums
self-inverse functions 97—8,
interest 36—40 244-5, 247, 250, 258,
192 549-50
trigonometric 5 17, 520—2 interpolation 331—2
improper fractions 186 logarithms 394, 407—21
"inclusive or" 366 interquartile range (IQR) binomial distributions 599
increasing functions 240—6, 296-9, 302 change of base 416—17
255, 257-8 functions 407-12, 422-4
intersections 124—5, 364, laws of 412-15, 419-20
increasing sequences/series 36 5-6 natural 415-16, 422-4
14 solving equations 418—21
see also points of
indefinite integrals 4 intersection London Eye 523, 530
lower quartiles 297—8, 303
34-43, 559-63, 567 interval notation 77—8
independent events 375—7, inverse functions 92—9
378, 384 exponential functions 409
independent variables 77, 92, indefinite integrals 434, 439
rational functions 205—8
110, 116, 127 reciprocal functions 192
indices see exponents inverse normal distribution
inequalities 77-8, 169-72 613-19
IQR see interquartile range
720
many-to-one mappings 66 quadratic functions 131, parabolic segments 464—5
mappings 65—7, 78, 89 134, 147-8
quadratic functions 131—7,
marginal costs/profits/revenue sine/cosine curves 524—5 142-54, 170-2, 182
552-3 minor arcs 511
missing data 279 parallel lines 1 14, 120
mathematical modelling 260
mode 287-9, 294-5, 604 parameters
maximums Monty Hall problem 35 3, linear functions 110—17,
120
derivatives 244, 247—8, 250, 362, 374, 380-1, 387-8 normal distribution 604—5
2 54—5 motion 260, 265—8 quadratic functions 148
moving averages 107 rational functions 203
global 244—5
multiplication laws/rules 375, reciprocal functions 190—1,
local 244-5, 247, 250, 258, 396, 398 195
549-50
mutually exclusive events regression lines 334—42
optimization 262—3, 557 370-1, 376 parent functions 116, 131,
quadratic functions 131,
natural logarithms 415—16, 196, 402
143, 146, 148 422—4 parentheses 71, 78
sine/cosine curves 524—5 Pascal's triangle 45—7, 595
negative correlation 321—2, paths 265
mean points 329—31, 334 324, 326 patterns, number 4—13
means 290—5 Pearson correlation
negative exponents 397—8
geometric 19 nets of polygons 358, 468, coefficient 326-7, 338, 350
normal distribution 604, percentiles 303—4
472 perfect squares 156—8, 160—3
616-19 non-linear correlation 320—4
normal distribution 603—19 periodic functions 5()8, 523,
percentiles 303
standard deviation 305—9 area under curve 605—6, 530-3
variance 305—9 613-15 periods 524-5, 527, 529, 531
perpendicular lines 114—15,
see also expectation characteristics 604—5
measures of central tendency distributions other than 117, 230-1, 477
287-95 standard 609—13 piecewise functions 81,
advantages 295 inverse 613—19 138-41, 154, 339-42
disadvantages 295 standard distribution 605,
median 293-9, 302-3, 307, point-gradient equations
606-9 119-20, 128, 331
604 normals 230-3, 238-40
notation points of inflexion 249—57,
mode 287-9, 294-5, 604 259, 549-50
see also means combinations 46
composite functions 85 points of intersection 124—5,
measures of dispersion derivatives 246, 434
296-309 functions 69-74, 85, 92 158-9, 457
interval 77—8 area between two curves
box-and-whisker diagrams inverse of function 92
298-9 linear functions 12 5—6 566
previous term in sequence 8 exponential functions 402
box plots 298 sigma 4, 11—12 quadratic equations 176
number patterns 4—13 rational functions 206
cumulative frequency numerators 8, 30, 220, 239 polynomials 226—7
301-3 populations 40—3,
one-to-many mappings 67 279-81, 306, 575
outliers 299-300, 306, 329 one-to-one functions 93—4
percentiles 303—4 one-to-one mappings 66—7 positive correlation 320—4,
quartiles 296—9, 302—3 optimization 260—4, 556—8 326-7
standard deviation 305—9, ordered pairs 65—7, 71, 79,
power rule 226—8, 232, 248,
604-5, 608-10, 616-19 89 436-8
variance 305—9, 601—2 ordering functions 86—7
see also ranges outliers 299-300, 306, 329 powers see exponents
median 293-9, 302-3, 307, presentation of data 283—7
604 parabolas principle amounts 37
midpoint formula 468—70 prisms 470
derivatives 243—4 probability 352—93
minimums
conditional 375, 377-81,
derivatives 243—4, 247—8, 384—5
250, 254-5
dependent events 375
global 244—5
local 244-5, 247, 250, 258,
549-50
optimization 263, 556
721
INDEX
distributions 582—627 quadratic formula 164—72 rational functions 184—213
quadratic functions 108,
experimental 354—5, equivalent representations
358-62 131-83
changing forms of 149—52 184-213
independent events 37 5—7, decreasing functions
378, 384 general equation 201—8
equations 155—83 inverse functions 20 5—8
repeated events 382—4 factorized form 147—52 limits and convergence 218
replacement situations fitting to graph 151—4 reciprocal functions 186—
general form 142, 145—6,
382-5 200, 203
representation 363—74 148-52, 166 sketching graphs 203—4
theoretical 354—7, 362 graphs 131-8, 142-54, solving equations 204
tree diagrams 381—6, 387—8 ratios, trigonometric 476—82,
170-2 513-17
probability distributions increasing functions
real-life problems
582-627 243 —4
binomial distribution inequalities 169—72 derivative applications
key features of graphs 148
592-602 normals 231 552-8
normal distribution 603—19 normal distribution
random variables 584—92, parent quadratic 131
tangents to curves 231 615-16
595-7, 600-1, 606, 619 transformations 131—41 optimization 260—4
vertex form 143—5, 148,
probability space diagrams quadratic applications
372-3 151-2 173-7
quantitative data 278
see also sample spaces quartiles 296—9, 302—3 reciprocal functions 192—5
production costs 262—4 quota sampling 282 real roots 166—9
product rule quotient rule 238—40, 248,
reciprocal functions
derivatives 546—7, 549, 423, 546-7
554, 557 186-200
radians 508—11, 544—5
differentiation rules random variables 584—91 asymptotes 189, 191,
196-8
236-40, 253 binomial distribution 592,
independent events 37 5—7 595-7, 600-1 domains 187, 189, 196-7,
kinematics 574 198
natural logarithms 423 expectation 588—91
profit 262-3, 552-3 normal distribution 606, inverse functions 192
progressions see sequences
proofs 55—6 619 limits 189, 191
pyramids 467, 471—4, 477, ranges 77-84, 296-8, 307 modelling 192—5
484, 498 parameters 190—1, 195
Pythagorean identity 516—17, composite functions 88 ranges 187-9, 196-7, 198
519, 521 exponential functions 402
Pythagorean theorem 468—9, inverse functions 93 rational functions 203
475, 489, 493 logarithms 407
quadratic functions 142—3, sketching graphs
quadrants of axes 514—17 189-90, 198-200
148
quadratic equations 155—83 rational functions 201, transformations 191,
applications 173—7
completing the square 155, reciprocal functions 187—9, 196-200
160-4 196-7, 198 reciprocals 126, 232
derivative functions 231 recursive formulae 60—1
discriminants 164, 168 ranking 350—1 recursive sequences 8—10
factorization 155—66 rates of change
inequalities 169—72 reflections
perfect squares 1 56—8, accumulating 567, 575—6
160-3 derivatives 222—3, 225, logarithmic functions 407—8
points of intersection 158—9 reciprocal functions 192
quadratic formula 164—72 552-4 sine/cosine curves 545
real-life problems 174—7 linear functions 110—11,
solving 1 5 5—64 in x-axis 132-6, 138, 140,
115 524, 545
regression lines 334
in y-axis 132, 137-9, 524
regression 332—42
relationships 62—107
relative extrema see local
maximums/minimums
relative frequency 358—60
reliable data 278—9
repeated events 382—4
722
replacement in probability segments 508, 511-13 trigonometric ratios 476—8,
382-5 self-inverse functions 97—8, 481
representation 192 sketching graphs 75—6, 93,
data 279, 280-1 sequences 2—61 95-6
curve sketching 255—9
functions 62—107 arithmetic sequences 6—8, quadratic functions 142
10, 13-16, 21, 33
probability 363—74 quadratic inequalities
rational functions 184—213 arithmetic series 12, 170-2
relationships 62—107
residuals 333—4 22-43 rational functions 203—4
revenue 262—4, 552—3 binomial theorem 44—55
Riemann sums 450—1 reciprocal functions 189—90,
general terms 5—7, 9—10, 198-200
right-angled pyramids 471—2 13, 21
right-angled triangles 47 5—84 straight-line graphs 123
geometric sequences 7—8, skewed data 285
angle between edge/base of 10, 13, 17-21, 29
3D shape 477 slopes see gradients
geometric series I l, 22—43,
angle between line/plane 216, 221 SOH-CAH-TOA mnemonic
477
limits 221 476, 493
applications 492—5 modelling series 36—43 sound waves 507, 540—1
bearings 480—1 sources of data 106
depression angles 480 number patterns 4—13 Spearman rank correlation
distance formula 469
elevation angles 480 proofs 55—6 50-1
solving problems 480—2 sigma notation 4, 11—12
special triangles 476 series 2—61 special angles 549
trigonometric ratios 476—82 arithmetic 12, 22—43
roots of equations geometric II, 22—43, 216, speed
decreasing functions 242 definite integrals 567,
derivative functions 231 221 573-4
increasing functions 242 derivative applications 555
sets differentiation 215, 222,
quadratic equations 15 5—6, 225-6, 266
164, 166-9 of numbers 77—8 kinematics 266, 567, 573—4
Venn diagrams 363—71
see also zeros spheres 471, 473—4
shifts 527, 531 square-based pyramids 471—3
rounding sigma notation 4, 11—12 square roots 157-8, 161, 399
functions 72—3 simple interest 36—8
standard deviation 305—9,
integration 566 simple random sampling 281 604-5, 608-10, 616-19
logarithms 420 simulation 354, 393
normal distribution 618 standardization 610—11 ,
sine curves 615-19
population growth 43 derivatives 544—5
sample spaces 355—6, 372—4, transformations of standardized values 610—11,
functions 1 38 615-19
375
trigonometric functions standard normal distribution
sampling 278—83 507-8, 510, 514-15, 605, 606-9
scale factors 133—40 523-8
standard position angles
scatter diagrams sine rule 484—9, 493 514-15
correlation 320—5 ambiguous case 488
lines of best fit 329 stationary points 249, 252
piecewise functions 340 sine (sin)
regression lines 338 see also maximums;
area of triangle 482—3 minimums
scatter plots 113, 186
secants 223—5 complementary angles statistics 276—351
second derivatives 240—59 482 straight lines II 1—30
graphs 240—59 derivatives 544—51 equations 118—24, 128,
notation 246 230
double angle identity
optimization 556 521-2 gradients 1 11—24, 223
sine/cosine 549—50 graphs 1 11—24
sectors 508, 511-13 functions 524—8, 530, horizontal 118, 121
544—51
intersections using GDC
integrals 559—60
sign in given quadrant 124-5
515-17 parameters of graphs
sine rule 484—9, 493 1 16-17
solving equations 517—20
723
INDEX
vertical 118 reciprocal functions 191, see also right-angled
y=mx+c116, 119 196-200 triangles; trigonometric...
shifts 527, 531 turning points 244—50, 252—3,
stratified sampling 281—2 stretches 133-5, 137-40,
stretches 133-5, 137-40, 610 256-8, 262
610 see also maximums;
substitution
summary 138 minimums
function notation 70—2 two-way tables 372
integration 559, 560—6, 573 see also reflections; unions of events 364, 366—7
sufficient data 278 translations
unit circle 513—17, 518, 563
sum rule 436—7 translations 1 32—6, 138—40 univariate data 276—317
definite integrals 449 universal sets 364, 367
sums of series 1 1, 22—7, 29—34, horizontal 132—5, 138—9, upper quartiles 297—8, 303
216, 221 525 u-substitution 561, 563
logarithmic functions 408
supplementary angles 488 normal distribution 610 see also substitution,
supply 63, 69, 74, 76, 84 reciprocal functions 198
surface areas 472—4 sine/cosine curves 525—7, integration
531, 545
symmetry vertical 132-5, 138-9 variable of integration
normal distribution 605,
607-8 tree diagrams 381—8, 586, 436-7
quadratic functions 132—5 594—5 variance 305—9, 601—2
142, 144-8 vectors 135
trend lines see lines of best fit
systematic sampling 281 trials 359-60, 362 velocity
tangents to curves 223—6, triangles definite integrals 567—75
230-3 derivative applications 554
equilateral 3, 13, 35 indefinite integrals 441—2
tangent (tan) Pascal's triangle 45—7, 595 kinematics 265—7, 567—75,
curves 528—9 see also trigonometry
580-1
double angle identity 522 "trident of Newton" curve Venn diagrams 363—71
functions 529, 547 240
sign in given quadrant addition rule 369—70
trigonometric equations complements 364—5
515-17 517-22, 525
conditional probability 377,
trigonometric ratios 476, trigonometric functions
506-41 378-9
478, 480 arcs 508, 511-12 drawing 364
temperature 1 1 5—16, 128—9 cosine functions 524—8, intersections 364, 365—6
"tending towards" 189 530-1, 544-51 mutually exclusive events
terminal side angles 516, 518 periodic functions 508,
terms, binomial 52—3 523, 530-3 370-1
term of sequence, definition 4 radians 508-11, 544-5 sample spaces 373
tetrahedrons 471 unions 364, 366—7
sectors/segments 508, vertex form of quadratic
theoretical probability 354—7, 511-13 function 143—5, 148,
362 151-2
sine functions 524—8, 530,
theta (0) 476 544—51 vertex of parabola
three-dimensional shapes
tangent functions 529, 547 increasing/decreasing
468-74 unit circle 513—17, 518
three-figure bearings 480 trigonometric identities 517, functions 243—4
three squares problem 520-2
quadratic functions 131—4,
504-5 trigonometric ratios 476—82, 142-8, 150-2
time 222, 225-6 513-17
Towers of Hanoi problem vertical asymptotes
trigonometry 466, 475—505 derivatives 246
60-1 applications 492—5 limits at infinity 220—1
transformations 131—41 cosine rule 489—92, 493, logarithmic functions 408
505, 520 rational functions 189,
compressions 134, 136—40 sine rule 484—9, 493 196-8, 201-3
dilations 138 three squares problem
exponential functions 402 504-5 vertical compressions 134,
logarithmic functions 136-8
407-8 vertical lines 68, 84, 93—5,
piecewise functions 138—41 118
724
vertical shifts 527, 531 quadratic functions 142, quadratic functions 142,
vertical stretches 1 33—5, 137, 147-50, 152
145-50
139-40 quadratic inequalities 170—2 rational functions 203—4
vertical translations 132—5 rational functions 203—4 reciprocal functions 191
reciprocal functions 191
138-9 straight line equations 123 zero
volumes 262, 473, 543
y-intercepts power of 397
x-intercepts reciprocal functions 186—7
derivatives 549
derivatives 256, 549 exponentials 402 zero-product property
points of intersection 159 linear functions 15 5-6
quadratic applications 176
quadratic formula 166—8 116-17, 120-3, 129 zeros 166-7, 251, 256, 556
see also roots of equations
725
MATHEMATICS: ANALYSIS AND APPROACHES
STANDARD LEVEL
Written by experienced practitioners and developed in cooperation with the 1B, this concept- Authors
based print and enhanced online course book pack offers the most comprehensive support for
Natasha Awada
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