Table Of Contentmathsquest11
mathematical methods
vce units 1 and 2
mathsquest11
mathematical methods
vce units 1 and 2
author
sue Michell
contributing authors
Genevieve Green | Jo Bradley | steven morris
raymond rozen | marGaret swale
suPPort material
maree GranGer | Kate BalKa | isaBelle lam
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 data
Creator: Michell, Sue, author.
Title: Maths quest. 11, VCE mathematical methods
units 1 & 2 / Sue Michell [and 8 others].
ISBN: 978 0 7303 2298 6 (set)
978 0 7303 2301 3 (eBook)
978 0 7303 2757 8 (paperback)
978 0 7303 2452 2 (studyON)
Notes: Includes index.
Target audience: For secondary school age.
Subjects: Mathematics — Textbooks.
Mathematics — Study and teaching (Secondary)
Mathematics — Problems, exercises, etc.
Victorian Certificate of Education examination —
Study guides.
Dewey number: 510
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10 9 8 7 6 5 4 3 2 1
contents
Introduction vii 4.3 the remainder and factor
About eBookPLUS and studyON x theorems 177
Acknowledgements xi 4.4 Graphs of cubic polynomials 185
4.5 equations of cubic polynomials 195
Topic 1 Lines and linear relationships 2
4.6 cubic models and applications 202
1.1 Kick off with cas 3
4.7 Review 209
1.2 linearly related variables, linear
answers 210
equations and inequations 4
1.3 systems of 3 × 3 simultaneous linear Topic 5 Higher-degree polynomials 222
equations 16
5.1 Kick off with cas 223
1.4 linear graphs and their equations 21
5.2 Quartic polynomials 224
1.5 intersections of lines and their 5.3 Families of polynomials 231
applications 33
5.4 numerical approximations to roots
1.6 coordinate geometry of the straight line 40
of polynomial equations 240
1.7 Review 52 5.5 Review 250
answers 53 answers 251
Topic 2 Algebraic foundations 58 Topic 6 Functions and relations 258
2.1 Kick off with cas 59 6.1 Kick off with cas 259
2.2 algebraic skills 60 6.2 Functions and relations 260
2.3 Pascal’s triangle and binomial 6.3 the circle 269
expansions 67
6.4 the rectangular hyperbola and
2.4 the binomial theorem 72 the truncus 278
2.5 sets of real numbers 80 6.5 the relation y2 = x 294
2.6 surds 86 6.6 Other functions and relations 305
2.7 Review 97 6.7 transformations of functions 319
answers 98 6.8 Review 328
answers 329
Topic 3 Quadratic relationships 104
3.1 Kick off with cas 105 Topic 7 Matrices and applications to
3.2 Quadratic equations with transformations 348
rational roots 106 7.1 Kick off with cas 349
3.3 Quadratics over R 110 7.2 addition, subtraction and scalar
3.4 applications of quadratic equations 122 multiplication of matrices 350
3.5 Graphs of quadratic polynomials 127 7.3 Matrix multiplication 357
3.6 determining the rule of a quadratic 7.4 determinants and inverses of
polynomial from a graph 138 2 × 2 matrices 362
3.7 Quadratic inequations 142 7.5 Matrix equations and solving 2 × 2
linear simultaneous equations 367
3.8 Quadratic models and applications 149
7.6 translations 376
3.9 Review 154
7.7 Reflections 383
answers 155
7.8 dilations 389
Topic 4 cubic polynomials 164 7.9 combinations of transformations 394
4.1 Kick off with cas 165 7.10 Review 397
4.2 Polynomials 166 answers 398
Topic 8 probability 404 11.5 applications of exponential functions 596
8.1 Kick off with cas 405 11.6 inverses of exponential functions 604
8.2 Probability review 406 11.7 Review 618
8.3 conditional probability 417 answers 619
8.4 independence 425
Topic 12 introduction to differential
8.5 counting techniques 430
calculus 630
8.6 Binomial coefficients and Pascal’s
12.1 Kick off with cas 631
triangle 443
12.2 Rates of change 632
8.7 Review 451
12.3 Gradients of secants 640
answers 452
12.4 the derivative function 645
Topic 9 Trigonometric functions 1 456 12.5 differentiation of polynomials by rule 652
9.1 Kick off with cas 457 12.6 Review 662
9.2 trigonometric ratios 458 answers 663
9.3 circular measure 465
Topic 13 Differentiation and applications 668
9.4 unit circle definitions 473
13.1 Kick off with cas 669
9.5 symmetry properties 482
13.2 limits, continuity and differentiability 670
9.6 Graphs of the sine and cosine
functions 493 13.3 derivatives of power functions 679
9.7 Review 503 13.4 coordinate geometry applications
of differentiation 686
answers 504
13.5 curve sketching 695
Topic 10 Trigonometric functions 2 510 13.6 Optimisation problems 704
10.1 Kick off with cas 511 13.7 Rates of change and kinematics 710
10.2 trigonometric equations 512 13.8 Review 720
10.3 transformations of sine and answers 721
cosine graphs 521
10.4 applications of sine and cosine Topic 14 Anti-differentiation and
functions 533 introduction to integral calculus 728
10.5 the tangent function 540 14.1 Kick off with cas 729
10.6 trigonometric relationships 550 14.2 anti-derivatives 730
10.7 Review 558 14.3 anti-derivative functions and graphs 736
answers 559 14.4 applications of anti-differentiation 742
14.5 the definite integral 748
Topic 11 Exponential functions 570
14.6 Review 758
11.1 Kick off with cas 571
answers 759
11.2 indices as exponents 572
11.3 indices as logarithms 579 Index 765
11.4 Graphs of exponential functions 587 Glossary
ONLINE ONLY
vi cOntents
Introduction
At Jacaranda, we are deeply committed to the ideal that learning brings life-changing benefi ts 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 11 Mathematical Methods VCE Units 1 and 2 comprehensively covers the
requirements of the revised Study Design 2016–2018.
Features of the new Maths Quest series
CAS technology
13.1 Kick off with CAS
The limit does not exist
1 Using CAS technology, find the appropriate template
and evaluate the expressions below.
a lxi→m1x+3
b xli→m0 1x
c xli→m2 (x−12)2
d xli→m1 (x2(−1−3xx+)2)
e xli→m2 hx, where h(x)= xx2+−93,, xx<≥22
f lxi→m∞ (x−x1)
g xli→m0 3−x12
h lim(x sin (x))
x→∞
2 Use CAS technology to sketch graphs of the
following functions.
HOLLAND | GRIFFITH | ROZEN MARK BARNES a f(x) = (x + 3)
b f(x) = 1x
c g(x)=(x−12)2
d g(x)=(x2(−1−3xx+)2)
MANMUAATLH FSOQRU TEHSET MANMUAATLH FSOQRU TEHSET e h(x)= xx2+−93,, xx<≥22
TI-Nspire CAS CASIO ClassPad II f h(x)=(x−x1)
CALCULATOR CALCULATOR
5TH EDITION 3RD EDITION g f(x)= 3−x12
h f(x) = (x sin (x))
3 Compare the graphs for which the limit in question 1 exists to those for which the
limit is undefined. Describe the difference with reference to the function value at
the point(s) of interest.
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.
9.2 Trigonometric ratios
Theprocessofcalculatingallsidelengthsandallanglemagnitudesofatriangleis
called solvingthetriangle.Herewereviewtheuseoftrigonometrytosolveright-
angledtriangles. ssssssttttttuuuuuuddddddyyyyyyOOOOOONNNNNN lllllliiiiiinnnnnnkkkkkkssssss
Units1 & 2
Right-angledtriangles
AOS1
TTTooppiicc 6 Thehypotenuseisthelongestsideofaright-angled
Concept 1 triangleanditliesoppositethe 90°angle,thelargest
angleinthetriangle.Theothertwosidesarelabelled
TTTrriiggoonnoommeettrriicc rraattiiooss Hypotenuse Opposite
Concept summary relativetooneoftheotheranglesinthetriangle,an
Practice questions exampleofwhichisshowninthediagram.
Itislikelythatthetrigonometricratiosofsine,cosine θ
Adjacent
andtangent,possiblytogetherwithPythagoras’theorem,
Interactivity willberequiredtosolvearight-angledtriangle.
TTTrriiggonometric ratios
int-2577 opposite adddjjacent opposite
sin(θ)= ,cos(θ)= andtan(θ)= ,
hypotenuse hypotenuse adddjjacent
Theseratioscannotbeappliedtotriangleswhichdonothavearightangle.However,
isoscelesandequilateraltrianglescaneasilybedividedintotworight-angledtriangles
INTRODUCTION vii
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2 Substitute the pre-image point The pre-image point is (3, 1) and the value of a = 1 13 Find the image of the point (–
and the value of a into the from the line x = 1. a x = 1 b y = –2.
matrix equation. x′ −1 0 3 2 14 Find the image of the point (7, –
y′ = 0 1 1 + 0 a x = –4 b y = 3.
3 Cimaalcguel aptoei ntht.e coordinates of the Txyh′′e i=mag−e10 po01int 31is (+–1, 201).= −11 1156 tthhee ccoooorrddiinnaatteess ooff PP?? yx == 2– 1t ot og igviev ea na ni mimagaeg ep opionitn tP P′(′–(3–2, ,– 15)).. WWhhaatt aarree
b21 StSaontu adbbt esett hitutehus eevt edam .ltuhaeter iopxfr eeb-q iiumnatatoigo etnh peo int b Tthxyhe′′e l ipn=ree -yi10m=a−g−e011 p.oxyint +is (320,b 1) and tyh e= v–a1lu.e of b=1 from 17 Taeq huexya l′′tiinoen= sw gi10itvhe e−nq.01 uFaitnixyodn t hye =eq−uxat+ion3 oisf tthraen ismfobar gmexye f′′do ra =cecaoc−rhd 10itnrag01n tsofo xytrhme amtioatnr.ix
matrix equation. x′ 1 0 3 0 c The line with equation x = 4
y′ = 0 −1 1 + −2 18 The parabola with equation y=x2+2x+1 is transformed according to the
3 Cimaalcguel aptoei ntht.e coordinates of the Txyh′′e i=mag10e p−o01int 31is (+3, –−302). = −33 am atxyri′′x e=qua10tio−ns01 givxyen. Find bth e xye′′qua=tion− 10of t01he ixymage for ce acxyh′′ tra=nsf01orm10atioxyn.
EXERCISE 7.7 Reflections MASTER 19 x – y = x.
20 Find the coordinates of the vertices of the image of pentagon ABCDE with
PRACTISE 1 WE21 Find the imxa-agxei so.f the point (–3, –1) after a A(2, 5), B(4, 4), C(4, 1), D(2, 0) and E(0, 3) y-axis.
Work without CAS 2 Find the image of the point (5, –2) after a 7.8 Dilations
y-axis.
3 WE22 Find the image equation after y=(x−2)2 A dilation is a linear transformation that
y-axis. enlarged or reduced parallel to either axis or
4 Find the image equation after y=x2+1 is Units1 & 2 both. A dilation requires a centre point and a
x-axis. AOS2 scale factor.
5 WE23 Find the image of the point (–2, 5) after a Topic 2 Original Reduction Enlargement
y=x. Concept 7 denoted by k.
6 Find the image of the point (8, –3) yaf=terx .a DCProialnaccttieicopent qssuuemsmtioanrsy IIff 0k >< k <
7 WE24 Find the image of the point (2, –1) after a One-way dilation
x=−3. A one-way dilation is a dilation from or parallel to one of the axes.
CprOocNeAspmSspeaOalsytp h LaptenhrImodeDp amtrtAoiiaocoTtsalestEl 1890 aFaFF iiinnntMhdddex ttt hhhx-eeea iiixmmmisaaagggeee ooofff ttthhheee pppoooiiinnnttt (((–8–, – bb tMhey. y-axis. y=2. IDinnitlta-e6tiro2an9cs7tivity omADr ia ldptarialitraxiato ileolnqenslu aitnfotri ooothnnme:e xdt-hiarxeeic sty iio-san r xefrpiosr meos ertn hptee ady- rbaaxyli lsteh le to thye x-aPx(xis, y) P'(x', y') = (k1 x, y)
1112 FFiinndd tthhee iimmaaggee ooff tthhee p pooinint t( 9(0, ,– – y = x. xy′′ = 0k1 01 xy = k1y x
equation y = x. where k1 is the dilation factor. 0 x
388 MATHS QUEST 11 MATHEMATICAL METHODS VCE Units 1 and 2 Topic 7 MATRICES AND APPLICATIONS TO TRANSFORMATIONS 389
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viii intROductiOn
The eBookPLUS is available for students and
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Sue MICHELL
task and Modelling and Problem-solving tasks,
including fully worked solutions MATHSQUEST11
• two tests per topic with fully worked solutions.
MATHEMATICAL METHODS
SOLUTIONS MANUAL
Maths Quest 11 Mathematical
VCE UNITS 1 AND 2
Methods Solutions Manual
VCE Units 1 and 2
Available to students and teachers to purchase
separately, the Solutions Manual provides
fully worked solutions to every question in the
corresponding student text. The Solutions Manual
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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