co-ordinate axes as shown. they decide that the road must
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Question
A team of road design and construction engineers have been given the task of building a road
that will bypass the town shown on the map below and yet not require any tunnelling through
the mountains. To help them in their task, they cover the map with a square grid, axes and
co-ordinate axes as shown. They decide that the road must begin at the point (1,2) and end at
the point (4,0), and must also pass through the points (2,4) and (3,3). The route of the road
between these points is given by a set of equations called cubic splines. For this road, the
splines are:
a. Show that the construction of the new road will not require the demolition of the
Chief Examiner’s house at point A with co-ordinates (1.5, 4)
1 mark
b. Show that the three sections of road, as defined by f, g and h, have no breaks in them.
3 marks
c. Find the exact co-ordinates of the most northerly point on the road.
3 marks
d. Explain why there will be no sudden bends in the new road where the sections meet
each other at the points (2,4) and (3,3)
4 marks
2015MATHMETH(CAS)EXAM2 12
SECTION 2 – Question 1–continued
Question 1 (9marks)
Let f R R f x x x: ( ) .→ = −( ) −( ), 15
2 52 ThepointP 1 45
,
isonthegraphoff,asshownbelow.
ThetangentatPcutsthey-axisatSandthex-axisatQ.
y
x
4
S
QO
P 1 45
,
a. Writedownthederivativef ′(x)off(x). 1mark
SECTION 2
Instructions for Section 2Answerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
13 2015MATHMETH(CAS)EXAM2
SECTION 2 – Question 1–continuedTURN OVER
b. i. Findtheequationofthetangenttothegraphoffatthepoint P 1 45
, .
1mark
ii. FindthecoordinatesofpointsQandS. 2marks
c. FindthedistancePS andexpressitintheform bc,whereb andcarepositiveintegers. 2marks
2015MATHMETH(CAS)EXAM2 16
SECTION 2 – Question 2–continued
Question 2 (14marks)Acityislocatedonariverthatrunsthroughagorge.Thegorgeis80macross,40mhighononesideand30mhighontheotherside.Abridgeistobebuiltthatcrossestheriverandthegorge.Adiagramforthedesignofthebridgeisshownbelow.
y
X (–40, 40) E
F Y (40, 30)
A–40
B40
x
60Q N
MP
θ O
Themainframeofthebridgehastheshapeofaparabola.Theparabolicframeismodelledby
y x= −60 380
2 andisconnectedtoconcretepadsatB(40,0)andA(–40,0).
Theroadacrossthegorgeismodelledbyacubicpolynomialfunction.
a. Findtheangle,θ,betweenthetangenttotheparabolicframeandthehorizontalatthepointA(–40,0)tothenearestdegree. 2marks
17 2015MATHMETH(CAS)EXAM2
SECTION 2 – Question 2–continuedTURN OVER
TheroadfromX to YacrossthegorgehasgradientzeroatX(–40,40)andatY(40,30),andhas
equation y x x= − +3
25600316
35.
b. Findthemaximumdownwardsslopeoftheroad.Giveyouranswerintheform−mn wheremandnarepositiveintegers. 2marks
Twoverticalsupportingcolumns,MNandPQ,connecttheroadwiththeparabolicframe.Thesupportingcolumn,MN,isatthepointwheretheverticaldistancebetweentheroadandtheparabolicframeisamaximum.
c. Findthecoordinates(u,v)ofthepointM,statingyouranswerscorrecttotwodecimalplaces. 3marks
Thesecondsupportingcolumn,PQ,hasitslowestpointatP(–u,w).
d. Find,correcttotwodecimalplaces,thevalueofwandthelengthsofthesupportingcolumnsMNandPQ. 3marks
2015 MATHMETH (CAS) EXAM 2 18
SECTION 2 – continued
For the opening of the bridge, a banner is erected on the bridge, as shown by the shaded region in the diagram below.
y
60
E
F
x40–40 O
e. Find the x-coordinates, correct to two decimal places, of E and F, the points at whichthe road meets the parabolic frame of the bridge. 3 marks
21 2014MATHMETH(CAS)EXAM2
SECTION 2 – Question 5–continuedTURN OVER
Question 5 (13marks)
Let and f R R f x x x x g R R g x x x: , : , .→ ( ) = −( ) −( ) +( ) → ( ) = −3 1 3 82 4
a. Expressx4 – 8x intheform x x a x b c−( ) + +( )( )2 . 2marks
b. Describethetranslationthatmapsthegraphof y f x= ( ) ontothegraphof y g x= ( ) . 1mark
c. Findthevaluesofdsuchthatthegraphof y f x d= +( ) has i. onepositivex-axisintercept 1mark
ii. twopositivex-axisintercepts. 1mark
d. Findthevalueofn forwhichtheequation g x n( ) = hasonesolution. 1mark
2014MATHMETH(CAS)EXAM2 22
END OF QUESTION AND ANSWER BOOK
e. Atthepoint u g u, ( ) ,( ) thegradientof y g x= ( ) ismandatthepoint v g v, ( ) ,( ) thegradientis–m,wheremisapositiverealnumber.
i. Findthevalueof u3 + v3. 2marks
ii. Finduandvif u + v=1. 1mark
f. i. Findtheequationofthetangenttothegraphof y g x= ( ) atthepoint p g p, ( )( ) . 1mark
ii. Findtheequationsofthetangentstothegraphof y g x= ( ) thatpassthroughthepointwithcoordinates 3
212, −
. 3marks
2013MATHMETH(CAS)EXAM2 18
SECTION 2 – Question 3–continued
Question 3 (19marks)TasmaniaJonesisinSwitzerland.Heisworkingasaconstructionengineerandheisdevelopingathrillingtrainrideinthemountains.Hechoosesaregionofamountainlandscape,thecross-sectionofwhichisshowninthediagrambelow.
y
xO
y = f (x)
E
FG
B(4, 0)
A
D
lake
0 12
,
C(2, 0)
Thecross-sectionofthemountainandthevalleyshowninthediagram(includingalakebed)ismodelledbythefunctionwithrule
f x x x( ) = − +364
732
12
3 2.
TasmaniaknowsthatA 0, 12
isthehighestpointonthemountainandthatC(2,0)andB(4,0)are
thepointsattheedgeofthelake,situatedinthevalley.Alldistancesaremeasuredinkilometres.a. FindthecoordinatesofG,thedeepestpointinthelake. 3marks
19 2013MATHMETH(CAS)EXAM2
SECTION 2 – Question 3–continuedTURN OVER
Tasmania’strainrideismadebyconstructingastraightrailwaylineABfromthetopofthemountain,A,totheedgeofthelake,B.ThesectionoftherailwaylinefromA to Dpassesthroughatunnelinthemountain.b. WritedowntheequationofthelinethatpassesthroughAandB. 2marks
c. i. Showthatthex-coordinateofD,theendpointofthetunnel,is23 . 1mark
ii. FindthelengthofthetunnelAD. 2marks
2013MATHMETH(CAS)EXAM2 20
SECTION 2 – Question 3–continued
InordertoensurethatthesectionoftherailwaylinefromD to Bremainsstable,Tasmaniaconstructsverticalcolumnsfromthelakebedtotherailwayline.ThecolumnEFisthelongestofallpossiblecolumns.(Refertothediagramonpage18.)d. i. Findthex-coordinateofE. 2marks
ii. FindthelengthofthecolumnEFinmetres,correcttothenearestmetre. 2marks
Tasmania’straintravelsdowntherailwaylinefromA to B.Thespeed,inkm/h,ofthetrainasitmovesdowntherailwaylineisdescribedbythefunction
V:[0,4]→ R,V x k x mx( ) = − 2,
wherexisthex-coordinateofapointonthefrontofthetrainasitmovesdowntherailwayline,andkandmarepositiverealconstants.
ThetrainbeginsitsjourneyatA 0, 12
.Itincreasesitsspeedasittravelsdowntherailwayline.
ThetrainthenslowstoastopatB(4,0),thatisV(4)=0.e. Findkintermsofm. 1mark
21 2013MATHMETH(CAS)EXAM2
SECTION 2–continuedTURN OVER
f. Findthevalueofxforwhichthespeed,V,isamaximum. 2marks
Tasmaniaisabletochangethevalueofmonanyparticularday.Asmchanges,therelationshipbetweenkandmremainsthesame.g. If,ononeparticularday,m =10,findthemaximumspeedofthetrain,correcttoonedecimal
place. 2marks
h. If,onanotherday,themaximumvalueofVis120,findthevalueofm. 2marks
2012MATHMETH(CAS)EXAM2 12
SECTION 2 – Question 2–continued
Question 2
Letf :R\{2} → R, f (x)= 12 4x −
+3.
a. Sketchthegraphofy=f (x)onthesetofaxesbelow.Labeltheaxesinterceptswiththeircoordinatesandlabeleachoftheasymptoteswithitsequation.
x
y
O
3marks
b. i. Findf ′(x).
ii. Statetherangeoff ′.
iii. Usingtheresultof part ii. explainwhyfhasnostationarypoints.
1+1+1=3marks
13 2012MATHMETH(CAS)EXAM2
SECTION 2 – Question 2–continuedTURN OVER
c. If(p,q)isanypointonthegraphofy=f (x),showthattheequationofthetangenttoy=f (x)atthispointcanbewrittenas(2p–4)2(y–3)=–2x+4p–4.
2marks
2012MATHMETH(CAS)EXAM2 14
SECTION 2 – Question 2–continued
d. Findthecoordinatesofthepointsonthegraphofy=f (x)suchthatthetangentstothegraphatthese
pointsintersectat −
1, 7
2.
4marks
15 2012 MATHMETH(CAS) EXAM 2
SECTION 2 – continuedTURN OVER
e. A transformation T: R2 → R2 that maps the graph of f to the graph of the function
g: R\{0} → R, g (x) = 1x
has rule Txy
a xy
cd
= +0
0 1, where a, c and d are non-zero real numbers.
Find the values of a, c and d.
2 marks
SECTION 2 – Question 3
Question 3a. f R R f x x3 + 5x i. f x
ii. f x x
b. p p R R p x ax3 + bx2 + cx + k a b c k i. p m m
ii. p m
c. q q R R q x x3
i. q–1 x
21
SECTION 2TURN OVER
ii. y q x y q–1 x
d. g g R R g x x3 + 2x2 + cx + k c k i. g c
ii. y g x y g–1 x k
19 2010 MATHMETH(CAS) EXAM 2
SECTION 2 – Question 4 – continuedTURN OVER
Question 4
Consider the function f : R → R, f (x) = 127
(2x – 1)3(6 – 3x) + 1.
a. Find the x-coordinate of each of the stationary points of f and state the nature of each of these stationary points.
4 marks
In the following, f is the function f : R → R, f (x) = 127
(ax – 1)3(b – 3x) + 1 where a and b are real constants.
b. Write down, in terms of a and b, the possible values of x for which (x, f (x)) is a stationary point of f.
3 marks
c. For what value of a does f have no stationary points?
1 mark
2010 MATHMETH(CAS) EXAM 2 20
SECTION 2 – Question 4 – continued
d. Find a in terms of b if f has one stationary point.
2 marks
e. What is the maximum number of stationary points that f can have?
1 mark
21 2010 MATHMETH(CAS) EXAM 2
f. Assume that there is a stationary point at (1, 1) and another stationary point (p, p) where p ≠ 1. Find the value of p.
3 marks
Total 14 marks
END OF QUESTION AND ANSWER BOOK
2009 MATHMETH(CAS) EXAM 2 10
SECTION 2 – Question 1 – continued
Question 1Let f : R+ {0} → R, f (x) = 6 x – x – 5.The graph of y = f (x) is shown below.
y
x
y = f (x)
5O 10 15 20 25 30
a. State the interval for which the graph of f is strictly decreasing.
2 marks
SECTION 2
Instructions for Section 2Answer all questions in the spaces provided.A decimal approximation will not be accepted if an exact answer is required to a question.In questions where more than one mark is available, appropriate working must be shown.Unless otherwise indicated, the diagrams in this book are not drawn to scale.
2009 MATHMETH(CAS) EXAM 2 12
SECTION 2 – Question 1 – continued
d. The points P (16, 3) and B (25, 0) are labelled on the diagram.
y
x
P(16, 3)
5O 10 15 20 25 30
B(25, 0)
i. Find m, the gradient of the chord PB. (Exact value to be given.)
ii. Find a [16, 25] such that f ′ (a) = m. (Exact value to be given.)
1 + 2 = 3 marks
2009 MATHMETH(CAS) EXAM 2 14
SECTION 2 – Question 2 – continued
Question 2
Q P NO Mx
y
direction of train
valley
mountain
A train is travelling at a constant speed of w km/h along a straight level track from M towards Q.The train will travel along a section of track MNPQ.
Section MN passes along a bridge over a valley.Section NP passes through a tunnel in a mountain.Section PQ is 6.2 km long.
From M to P, the curve of the valley and the mountain, directly below and above the train track, is modelled by the graph of
y ax bx c1200
3 2( ) where a, b and c are real numbers.
All measurements are in kilometres.
a. The curve defined from M to P passes through N (2, 0). The gradient of the curve at N is –0.06 and the curve has a turning point at x = 4.
i. From this information write down three simultaneous equations in a, b and c.
15 2009 MATHMETH(CAS) EXAM 2
SECTION 2 – Question 2 – continuedTURN OVER
ii. Hence show that a = 1, b = – 6 and c = 16.
3 + 2 = 5 marks
b. Find, giving exact values i. the coordinates of M and P
ii. the length of the tunnel
iii. the maximum depth of the valley below the train track.
2 + 1 + 1 = 4 marks
19 2006 MATHMETH(CAS) EXAM 2
Question 4
A part of the track for Tim�s model train follows the curve passing through A, B, C, D, E and F shown above. Tim has designed it by putting axes on the drawing as shown. The track is made up of two curves, one to the left of the y-axis and the other to the right.B is the point (0, 7).The curve from B to F is part of the graph of f (x) = px3 + qx2 + rx + s where p, q, r and s are constants and f ′(0) = 4.25.a. i. Show that s = 7.
ii. Show that r = 4.25.
1 + 1 = 2 marks
supertrain
A
B
y
x
C
F
E
O–2 D
SECTION 2 – Question 4 � continuedTURN OVER
2006 MATHMETH(CAS) EXAM 2 20
The furthest point reached by the track in the positive y direction occurs when x = 1. Assume p > 0.b. i. Use this information to Þ nd q in terms of p.
ii. Find f (1) in terms of p.
iii. Find the value of a in terms of p for which f ′(a) = 0 where a > 1.
iv. If a = 173
, show that p = 0.25 and q = �2.5.
2 + 1 + 1 + 2 = 6 marks
For the following assume f (x) = 0.25x3 � 2.5x2 + 4.25x + 7.c. Find the exact coordinates of D and F.
2 marks
SECTION 2 – Question 4 � continued
21 2006 MATHMETH(CAS) EXAM 2
d. Find the greatest distance that the track is from the x-axis, when it is below the x-axis, correct to two decimal places.
1 mark
The curve from A to B is part of the graph with equation g x abx
( ) =−1
, where a and b are positive real constants.
The track passes smoothly from one section of the track to the other at B (that is, the gradients of the curves are equal at B).e. Find the exact values of a and b.
3 marks
4 marks
Total 18 marks
END OF QUESTION AND ANSWER BOOK
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