Table Of Contentmathsquest12
specialist mathematics
vce units 3 and 4
mathsquest12
specialist mathematics
vce units 3 and 4
author
Raymond Rozen
contributing authors
catherine smith | Jo Bradley | sue michell
steven morris | margaret swale
support material
aileen toll | Kathryn marnell
First published 2016 by
John Wiley & Sons Australia, Ltd
42 McDougall Street, Milton, Qld 4064
Typeset in 12/14.5 pt Times LT Std
© John Wiley & Sons Australia, Ltd 2016
The moral rights of the authors have been asserted.
National Library of Australia Cataloguing-in-Publication entry
Creator: Rozen, Raymond, author.
Title: Maths quest. 12, Specialist mathematics, VCE
units 3 & 4 / Raymond Rozen, Sue Michell,
Steven Morris, Margaret Swale.
ISBN: 978 0 7303 2303 7 (set)
978 0 7303 2305 1 (ebook)
978 0 7303 2767 7 (paperback)
978 0 7303 2456 0 (studyON)
Notes: Includes index.
Target Audience: For secondary school age.
Subjects: Mathematics—Australia—Textbooks.
Mathematics—Problems, exercises, etc.
Victorian Certificate of Education examination—
Study guides.
Other Creators/
Contributors: Michell, Sue, author.
Morris, Steven P., author.
Swale, Margaret, author.
Dewey Number: 510
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10 9 8 7 6 5 4 3 2 1
contents
Introduction vii Topic 4 Kinematics 184
About eBookPLUS and studyON x 4.1 Kick off with cas 185
Acknowledgements xi 4.2 constant acceleration 186
4.3 motion under gravity 190
Topic 1 Sketching graphs 2
4.4 velocity–time graphs 193
1.1 Kick off with cas 3
4.5 variable acceleration 197
1.2 an introduction to the modulus
4.6 Review 204
function 4
answers 205
1.3 sketching graphs of reciprocal
functions 7
Topic 5 Vectors in three dimensions 208
1.4 sketching graphs of rational
5.1 Kick off with cas 209
functions 13
5.2 vectors 210
1.5 sketching graphs of y = |f (x )| and
y = f (|x |) from y = f (x ) 27 5.3 ∼i ∼j k∼ notation 215
5.4 scalar product and applications 228
1.6 circles, ellipses and hyperbolas 34
5.5 vector proofs using the scalar
1.7 Review 43
product 237
answers 44
5.6 Parametric equations 242
Topic 2 Trigonometry 56 5.7 Review 248
2.1 Kick off with cas 57 answers 249
2.2 Reciprocal trigonometric functions 58
Topic 6 Mechanics 256
2.3 trigonometric identities using
6.1 Kick off with cas 257
reciprocal trigonometric functions 66
6.2 statics of a particle 258
2.4 compound-angle formulas 69
6.3 inclined planes and connected
2.5 double-angle formulas 76
particles 272
2.6 inverse trigonometric functions 85
6.4 dynamics 279
2.7 General solutions of trigonometric
6.5 dynamics with connected
equations 97
particles 289
2.8 Graphs of reciprocal trigonometric
6.6 Review 299
functions 106
answers 300
2.9 Graphs of inverse trigonometric
functions 115
Topic 7 Differential calculus 302
2.10 Review 122
7.1 Kick off with cas 303
answers 123
7.2 Review of differentiation
techniques 304
Topic 3 complex numbers 130
7.3 applications of differentiation 314
3.1 Kick off with cas 131
7.4 implicit and parametric
3.2 complex numbers in rectangular
differentiation 324
form 132
7.5 second derivatives 331
3.3 complex numbers in polar form 142
7.6 curve sketching 338
3.4 Polynomial equations 156
7.7 derivatives of inverse trigonometric
3.5 subsets of the complex plane:
functions 351
circles, lines and rays 162
7.8 Related rate problems 360
3.6 Roots of complex numbers 170
7.9 Review 368
3.7 Review 177
answers 369
answers 178
Topic 8 integral calculus 378 11.3 other applications of first-order
8.1 Kick off with cas 379 differential equations 546
8.2 areas under and between curves 380 11.4 Bounded growth and newton’s
Law of cooling 553
8.3 Linear substitutions 391
11.5 chemical reactions and dilution
8.4 other substitutions 402
problems 560
8.5 integrals of powers of trigonometric
11.6 the logistic equation 572
functions 411
11.7 euler’s method 584
8.6 integrals involving inverse
trigonometric functions 418 11.8 slope fields 593
8.7 integrals involving partial fractions 428 11.9 Review 604
8.8 Review 439 answers 605
answers 440
Topic 12 Variable forces 612
Topic 9 Differential equations 446 12.1 Kick off with cas 613
9.1 Kick off with cas 447 12.2 Forces that depend on time 614
9.2 verifying solutions to a differential 12.3 Forces that depend on velocity 621
equation 448 12.4 Forces that depend on
9.3 solving type 1 differential equations, displacement 634
dy 12.5 Review 641
=f(x) 454
dddxxxxx answers 642
9.4 solving type 2 differential equations,
Topic 13 Vector calculus 644
dy
dddxxxxx =f(y) 461 13.1 Kick off with cas 645
9.5 solving type 3 differential equations, 13.2 Position vectors as functions of time 646
dy 13.3 differentiation of vectors 651
=f (x )g(y ) 468
dx 13.4 special parametric curves 662
9.6 solving type 4 differential equations, 13.5 integration of vectors 675
d2y 13.6 Projectile motion 681
=f(x) 472
dddxxxxx2 13.7 Review 696
9.7 Review 480 answers 697
answers 481
Topic 14 Statistical inference 704
Topic 10 Further applications of 14.1 Kick off with cas 705
integration 484
14.2 Linear combinations of random
10.1 Kick off with cas 485 variables 706
10.2 integration by recognition 486 14.3 sample means 713
10.3 solids of revolution 494 14.4 confidence intervals 719
10.4 volumes 504 14.5 Hypothesis testing 724
10.5 arc length, numerical integration 14.6 Review 730
and graphs of antiderivatives 512
answers 731
10.6 Water flow 524
10.7 Review 533 Index 733
answers 534
Topic 11 Applications of fi rst-order
differential equations 538
11.1 Kick off with cas 539
11.2 Growth and decay 540
vi contents
introduction
At Jacaranda, we are deeply committed to the ideal that learning brings life-changing benefits to all
students. By continuing to provide resources for Mathematics of exceptional and proven quality, we
ensure that all VCE students have the best opportunity to excel and to realise their full potential.
Maths Quest 12 Specialist Mathematics VCE Units 3 and 4 comprehensively covers the
requirements of the revised Study Design 2016–2018.
Features of the new Maths Quest series
cas technology
4.1 Kick off with CAS
Kinematics involves the study of position, displacement, velocity and acceleration.
From the study of calculus in Year 11, we know that if x is the position of an object
moving in a straight line at time t, then the velocity of the object at time t is given by
v=ddxt and the acceleration of the object at time t is given by a=ddvt.
1 If the position of a body moving in a straight line is given by
x(t = t3+ t2− t + 10 where x is in centimetres and t is in seconds,
a initial velocity and acceleration
b time when the body is at the origin
c velocity when the acceleration is zero
d acceleration when the velocity is zero.
2 A hot air balloon commences its descent at time t = 0 minutes. As it descends,
the height of the balloon above the ground, in metres, is given by the
equation, h(t)=600×2−1t0.
a Use CAS to sketch the height of the balloon above the ground at time t and the
rate at which the balloon is descending at time t.
b If the balloon is anchored by the crew on the ground when it is 2 metres above
the ground, how long did it take until the balloon was secured?
c Find expressions for the velocity and acceleration of the balloon.
HOLLAND | GRIFFITH | ROZEN MARK BARNES d Calculate the rate at which the balloon is descending after 20 minutes.
e Find the time when the balloon is descending at a rate of −9.0 m/minute.
MATHSQUEST MATHSQUEST
MANUAL FOR THE MANUAL FOR THE
TI-Nspire CAS CASIO ClassPad II
CALCULATOR CALCULATOR
5TH EDITION 3RD EDITION
Please refer to the Resources tab in the Prelims section of your eBookPLUS for a comprehensive
step-by-step guide on how to use your CAS technology.
4.2 Constant acceleration
Kinematics is the study of the motion of objects.
In this topic, the focus is on objects that move
along a straight line, which is also known as studyON links
rectilinear motion.
Units3 & 4
When we consider the motion of an object in a
AOS3
straight line with uniform acceleration, a number
Topic 5
of rules can be used.
Concept 4
The diagram below represents the motion of an
Constant
acceleration object with initial velocity u v
after t seconds.
of motion
Concept summary
Practice questions
Velocityuv t v–u
0 Time
The gradient of the line is calculated by v−u. On a velocity–time graph, the gradient
t
is the acceleration of the object. Hence, a=v−u.
t
Transposing a=v−u, we get v=u+at (equation 1). intRoduction vii
t
dv
Alternatively, by antidifferentiating =a with respect to t:
dt
dv
dt dt = a dt
v=at+c
When t=0, c=u (initial velocity):
v=at+u
The area under a velocity–time graph gives the displacement of the object. Therefore,
u12(suin+g tvh)e i sru tlhee f aovr etrhaeg ea rveeal oocf iaty t roafp tehzeiu omb,j ewcte. gDeits psla=ce12m(ue+nt vis) tt h(eeq auvaetrioagne 2 v),e wlohceitrye of
an object multiplied by time.
Substituting equation 1, v=u+at, into equation 2, s=12(u+v)t, gives:
s=12(u+u+at)t
s=12(2u+at)t
s=ut+12at2 (equation 3)
dx
Alternatively, by antidifferentiating =v=u+at with respect to t,
dt
x=ut+12at2+d, where d is the initial position.
186 MATHS QUEST 12 SPECIALIST MATHEMATICS VCE Units 3 and 4
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4 Sanudb sstoitluvtee. the quantities into the equation s=ut+12at2 13
s=0+12×9.8×3.52
s=60.025 14 A skyrocket is launched upwards with a velocity of 60 m/s. Two seconds later,
The rock travels a distance of 60.025 m to the another skyrocket is launched upwards from the same point with the same initial
bottom of the well. velocity. Find when and where the two skyrockets meet.
15 A stone, A, is projected vertically upwards with a velocity of 25 m/s. After
MASTER
EXERCISE 4.3 Motion under gravity point. Find when and where the two stones meet.
1 WE3 A basketball is thrown vertically upwards with a velocity of 5 m/s. What is 16 A worker climbs vertically up a tower to a certain height and accidently drops
PRACTISE the maximum height of the basketball in the air? a small bolt. The man ascends a further 45 m and drops another small bolt. The
2 A skyrocket is projected vertically upwards to a maximum
height of 154.34 m. Find the velocity of projection. a
b
3 WE4 A rock is dropped down a well. It reaches the bottom of
4.4 Velocity–time graphs
4 A stone is dropped from the top of a cliff 122.5 m high. How
long does it take for the stone to reach the bottom of the cliff? Velocity–time graphs are a useful visual representation of the motion of an object in
5 a straight line. We can use velocity–time graphs to solve kinematic problems. The
CONSOLIDAT to reach its maximum height? following properties of velocity–time graphs make this possible.
6 initial velocity of the shot. UAnOitSs33 & 4 • Because a=ddvt, the gradient of the velocity–time graph at time t gives the
Topic 5 instantaneous acceleration at time t.
78 AjA1u5 s bbtm oablu/lesl f diofser rorteh m friato l twlhhsien tfs rtv ootehmpre t o itgcfhar eaol l utybon upudi pl.odwfi naagr d c2sli 0fw fm i4the5t rame sve ethrlieogsch iht yaign ohdf. Find the boulder’s speed VtCPiCermoaolncoenctc ieccgipeetryt pa q–sptuu he3mssmtioanrsy • Bt1te2vc aduts=e vx2=−ddxxt1. The distance is found by determining the magnitude of the signed
area under the curve bounded by the graph and the t axis, v(t)dt. Distance
for the ball.
travelled cannot be a negative value.
9 A skyrocket is projected vertically upwards from the
ground. It runs out of fuel at a velocity of 52 m/s and a
height of 35 m. From this point on it is subject only to
acceleration due to gravity. Find its maximum height. Gradient = instantaneous acceleration
10 A stone is projected vertically upwards with a velocity of 12 m/s from the top of a Velocity
a the height of the cliff Area = displacement
b the velocity at which the stone must be projected to reach the bottom of the 0 Time
cliff in 4 s.
11 From a hot air balloon rising vertically upward with a speed of 8 m/s, a sandbag is
dropped which hits the ground in 4 seconds. Determine the height of the balloon • area of a triangle: A=12bh
when the sandbag was dropped. • area of a rectangle: A=LW
12 Ala temr iass sileec oisn dp rmojiescstieled ivse prtriocjaelclyte udp vweartridc awlliyth u ap wspaerde df roofm 7 3th.5e msa/ms,e a pnodi n3t swecitohn dthse • area of a trapezium: A=12a+bh.
same speed. Find when and where the two missiles collide.
192 MATHS QUEST 12 SPECIALIST MATHEMATICS VCE Units 3 and 4 Topic 4 KINEMATICS 193
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viii intRoduction
The eBookPLUS is available for students and
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Maths Quest 12 Specialist
Raymond ROZEN
Mathematics Solutions Manual
VCE Units 3 and 4 MATHSQUEST12
Available to students and teachers to purchase
SPECIALIST MATHEMATICS
separately, the Solutions Manual provides
fully worked solutions to every question in the SOLUTIONS MANUAL
corresponding student text. The Solutions Manual
VCE UNITS 3 AND 4
is designed to encourage student independence and
to model best practice. Teachers will benefi t by
saving preparation and correction time.
intRoduction ix
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