co-ordinate axes as shown. They decide that the road must ...

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



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



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



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


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



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



Tasmania’strainrideismadebyconstructingastraightrailwaylineABfromthetopofthemountain,A,totheedgeofthelake,B.ThesectionoftherailwaylinefromA to Dpassesthroughatunnelinthemountain.b. WritedowntheequationofthelinethatpassesthroughAandB. 2marks

c. i. Showthatthex-coordinateofD,theendpointofthetunnel,is23 . 1mark

ii. FindthelengthofthetunnelAD. 2marks



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


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



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



c. If(p,q)isanypointonthegraphofy=f (x),showthattheequationofthetangenttoy=f (x)atthispointcanbewrittenas(2p–4)2(y–3)=–2x+4p–4.

2marks



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


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


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




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.



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



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.


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


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


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


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


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