Scotch 3cdmas Mock Exam

8/16/2019 Scotch 3cdmas Mock Exam

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Scotch CollegeSemester Two Examination, 2010

Question/Answer Booklet

MATEMAT!CS"C/"# S$ecialist

Section %ne&Calculator 'ree

Teachers (ame& !)) / *%BB+lease Circle-

(ame&

Time allowe. 'or this sectionReading time before commencing work: 5 minutesWorking time for this section: 50 minutes

Material reuire./recommen.e. 'or this sectionTo be provided by the supervisor

This Question/Answer BookletFormula heet

To be provided by the candidatetandard items: !ens" !encils" !encil shar!ener" eraser" correction fluid" ruler" highlighters

!ecial items: nil

!m$ortant note to can.i.ates#o other items ma$ be used in this section of the e%amination& 't is our res!onsibilit$ to ensurethat $ou do not ha(e an$ unauthorised notes or other items of a non)!ersonal nature in the

e%amination room& 'f $ou ha(e an$ unauthorised material with $ou" hand it to the su!er(isore'ore reading an$ further&


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Structure o' this $a$er

ection#umber of uestionsa(ailable

#umber of uestions to be

answered

Working time1minutes2

arksa(ailable

ection 3ne:,alculator)free 5 5 50 40

ection Two,alculator)assumed ++ ++ +00 0

+*0

Question + * - 4 5 Total

arks 6 7 +0 7 5 40

Awarded

!nstructions to can.i.ates

+& The rules for the conduct of Western Australian e%ternal e%aminations are detailed inthe Year 12 Information Handbook 2010 & itting this e%amination im!lies that $ou agreeto abide b$ these rules&

*& Write $our answers in the s!aces !ro(ided in this Question/Answer Booklet& !are

!ages are included at the end of this booklet& The$ can be used for !lanning $ourres!onses and/or as additional s!ace if reuired to continue an answer&

• 8lanning: 'f $ou use the s!are !ages for !lanning" indicate this clearl$ at the to! of the!age&

• ,ontinuing an answer: 'f $ou need to use the s!ace to continue an answer" indicate inthe original answer s!ace where the answer is continued" i&e& gi(e the !age number&Fill in the number of the uestion1s2 that $ou are continuing to answer at the to! of the!age&

-& Show all our working clearl 9our working should be in sufficient detail to allow $ouranswers to be checked readil$ and for marks to be awarded for reasoning& 'ncorrectanswers gi(en without su!!orting reasoning cannot be allocated an$ marks& For an$

uestion or !art uestion worth more than two marks" (alid working or ustification isreuired to recei(e full marks& 'f $ou re!eat an answer to an$ uestion" ensure that $oucancel the answer $ou do not wish to ha(e marked&

4& 't is recommended that $ou .o not use $encil e%ce!t in diagrams&


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Section %ne& Calculator3'ree +40 Marks-

This section has 'i5e +6- uestions& Answer all uestions& Write $our answers in the s!ace!ro(ided&!are !ages are included at the end of this booklet& The$ can be used for !lanning $ourres!onses and/or as additional s!ace if reuired to continue an answer&

• 8lanning: 'f $ou use the s!are !ages for !lanning" indicate this clearl$ at the to! of the !age&• ,ontinuing an answer: 'f $ou need to use the s!ace to continue an answer" indicate in the

original answer s!ace where the answer is continued" i&e& gi(e the !age number& Fill in thenumber of the uestion1s2 that $ou are continuing to answer at the to! of the !age&

uggested working time for this section is 50 minutes&

1 78 marks9

,onsider the functions 32)( += x x f and x x g 36)( −=

1a2 ketch both functions on the one set of a%es below& ;4<

%

$

1b2 =se algebra to find the %)coordinates of the !oint1s2 of intersection& ;-<

2 7: marks9


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1a2 ,onsider the conecture: 2( 2) 0a + ≥

1i2 8ro(ide an e%am!le that su!!orts this conecture& ;+<

1ii2 8ro(ide a counter)e%am!le that dis!ro(es this conecture& ;+<

1iii2 Amend the original conecture so that is alwa$s true& ;+<

Question * continued&&&


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1b2 =se the !rinci!al of mathematical induction to !ro(e that:

1 1 1...

2 5 5 8 (3 1)(3 2) 6 4

n

n n n+ + + =

× × − + + for all n +∈¢ . ;><

" 710 marks9


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1a2 Find:

1i2 duu

u

dt

d t

∫ +−

2

2

31

1;*<

1ii2 ∫ + dx x x 2)cos(sin

;-<

1b2 =se an a!!ro!riate substitution to e(aluate the integral: ;5<

1

20

1

4dx

x−∫ .


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4 7: marks9

'f cos sin z cis iθ θ θ = = + and cos sinw cis iφ φ φ = = + , where ,θ φ are acute:1a2 ?%!ress 1 z − in terms of real and imaginar$ com!onents& ;*<

1b2 how that . ( ) z w cis θ φ = + . ;-<

1c2 Find the modulus and argument of 1 z + in terms of θ . ;4<;@int: .raw a diagram


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6 76 marks9

1a2 3 z i= + is a !oint in the com!le% !lane that forms one of the (ertices of an euilateraltriangle& The triangle is inscribed within a circle of radius * units whose centre is theorigin& Find the other two (ertices of this triangle& ;-<

1b2 @ence or otherwise" e(aluate ( )3

3 i+ . ;*<

?#. 3F ?,T'3# 3#?


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A..itional working s$ace

Question number1s2:


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Question number1s2:


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Scotch CollegeSemester Two Examination, 2010

Question/Answer Booklet

MATEMAT!CS"C/"# S$ecialist

Section Two&

Calculator3assume.

Teachers (ame& !)) / *%BB+lease Circle-

(ame&

Time allowe. 'or this sectionReading time before commencing work: +0 minutesWorking time for this section: +00 minutes

Material reuire./recommen.e. 'or this sectionTo be provided by the supervisor This Question/Answer BookletFormula heet 1retained from ection 3ne2

To be provided by the candidatetandard items: !ens" !encils" !encil shar!ener" eraser" correction fluid" ruler" highlighters

!ecial items: drawing instruments" tem!lates" notes on two unfolded sheets of A4 !a!er"and u! to three calculators satisf$ing the conditions set b$ the ,urriculum,ouncil for this e%amination

!m$ortant note to can.i.ates#o other items ma$ be taken into the e%amination room& 't is our res!onsibilit$ to ensure that$ou do not ha(e an$ unauthorised notes or other items of a non)!ersonal nature in the

e%amination room& 'f $ou ha(e an$ unauthorised material with $ou" hand it to the su!er(isore'ore reading an$ further&


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Section Two& Calculator3assume. +;0 Marks-

This section has ele5en +11- uestions& Answer all uestions& Write $our answers in the s!ace!ro(ided&

!are !ages are included at the end of this booklet& The$ can be used for !lanning $ourres!onses and/or as additional s!ace if reuired to continue an answer&

• 8lanning: 'f $ou use the s!are !ages for !lanning" indicate this clearl$ at the to! of the !age&• ,ontinuing an answer: 'f $ou need to use the s!ace to continue an answer" indicate in the

original answer s!ace where the answer is continued" i&e& gi(e the !age number& Fill in thenumber of the uestion1s2 that $ou are continuing to answer at the to! of the !age&

uggested working time for this section is +00 minutes&


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; 710 marks9

1a2 Find m and n if mA B A+ = , where 2 5

3 4 A

= ÷−

and 10

6 2

n B

n

− − = ÷−

;-<

1b2 For what (alue1s2 of k will the matri% 2

3 1

k A

k

= ÷+

be singular ;-


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Question continued &&&

1c2 'f 2 0aX bX cI + + = and 1 Z Y XY −= , where X, Y and Z are suare" non)singularmatrices" !ro(e that 2 0aZ bZ cI + + = . ;4


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: 7< marks9

The table below shows the breeding rates" sur(i(al rates and the initial !o!ulation !rofileof the !o!ulation of uokkas on Bald 'sland&

Age 1$ears2 0)* *)4 4)> >) )+0

'nitial 8o!ulation +00 +50 *00 0 40

Breeding Rate 0 +&+ 0& 0&6 0&*ur(i(al Rate 0&4 0& 0&> 0&4 0

1a2 tate the Ceslie matri% C for the !o!ulation of uokkas& ;*<

1b2 Find the total number of uokkas within the 4th generation& ;*<

1c2 .etermine the a!!ro%imate long term intergenerational rate of decline&?%!lain $our answer& ;*


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10 76 marks9

A cur(e is defined im!licitl$ b$: ay x

y=

−1

sin , where a is a constant&

Find the (alue of the constant a in the cur(e when2

y π = and

4

dy

dx

π = − &

Cea(e $our answer as an e%act (alue&

11 78 marks9


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The tem!erature T ° C 1,elsius2 of a gold bar that is left to cool in cold storage is gi(en

b$ the euation ( 10)dT

k T dt

= − − where k is a !ositi(e constant and t is time measured

in minutes& 'n the first +0 minutes the tem!erature falls from +000 C ° to >00 C ° &

1a2 Find the (alue of k and hence a formula for T in terms of t . ;5<

1b2 What will be the tem!erature of the bar after one hour ;+<

1c2 @ow long will it take for the bar to reach -0 C ° ;+<

12 7< marks9


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A cameraman is asked to film an aircraft fl$ing o(erhead for a dramatic scene in amo(ie& The cameraman fi%es his camera at ground le(el and rotates the camera in a(ertical arc as the aircraft mo(es towards him& The aircraft a!!roaches the cameramanfl$ing at a s!eed of 650 km/h whilst maintaining a constant height of 000 m abo(e theground& At what rate is the camera rotating 1units in degrees/second2 when the

horiDontal distance of the aircraft is 5 km from the cameraman&

1" 712 marks9


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An obect e%hibits sim!le harmonic motion such that its dis!lacement" x metres" from afi%ed !oint 3 on the line after t seconds is gi(en b$ the euation:

cos( ) x A t Bλ = +

where A, B and λ are !ositi(e constants and 0 2 B π ≤ < .'f v and a are the (elocit$ 1m/s2 and acceleration 1m/s*2 res!ecti(el$" show that

1a2 2a xλ = − ;*<

1b2 2 2 2 2( )v A xλ = − ;-<

The obect !asses through !oint 3 for the first time after + second and for the second time after 5 seconds" and the ma%imum distance that the !article mo(es awa$ from the !oint 3 is metres&1c2 Find the (alues of A, B and λ ;4<

Question +- continued&&&


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1d2 When is the first time that the obect is furthest awa$ from 3 ;*<

1e2 What is the ma%imum s!eed of the obect ;+


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16 711 marks9

Two birds" ?ddie and Tweetie" are fl$ing with constant (elocities of 2, 1, 1− ms! and

4, 1, 3− ms! res!ecti(el$& At +0 am ?ddie and Tweetie" relati(e to a birdbath on the

ground" are at !osition (ectors of 0, 10, 10 m and 7, 3, 24− m res!ecti(el$&

1a2 how that the two birds collide" and gi(e the !osition at which this occurs& ;-<

1b2 Ei(e the (ector euation of the !lane shared b$ the !ath of the two birds& ;*<

Question +5 continued&&&


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1c2 Rock$ the cat lies on the ground *0 m from the birdbath and in the same !lane sharedb$ the birds& ,alculate the !osition (ector of Rock$ relati(e to the birdbath& ;><

1


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The line 1r is gi(en b$ the following (ector euation : 1

1 6

3 1

1 2

r λ

÷ ÷= + ÷ ÷ ÷ ÷−

%

1a2 Find the acute angle between the line 1r and the !lane

3

. 1 52

r

÷

= ÷ ÷ % . ;-<

1b2 Find the shortest distance from a !oint P(2, -2, 3) to the line 1r . ;4<

?#. 3F ?,T'3# TW3


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Scotch 3cdmas Mock Exam

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