2021 Ace Math Adv Paper PDF

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! ! ACE EXAM PAPER

!

! Student name: ______________________!

!

2021

! ! ! ! YEAR 12 YEARLY EXAMINATION

!

!

! ! !

Mathematics Advanced

General Instructions

! ! ! ! ! !

Total marks: 100

Section I – 10 marks ! Attempt Questions 1-10 ! Allow about 15 minutes for this section

Working time - 180 minutes Write using black pen NESA approved calculators may be used A reference sheet is provided at the back of this paper In section II, show relevant mathematical reasoning and/or calculations

Section II – 90 marks ! Attempt all questions ! Allow about 2 hours and 45 minutes for this section !

1!!

Year 12 Mathematics Advanced

! Section(I( ( 10(marks( Attempt(questions(1(-(10( Allow(about(15(minutes(for(this(section( ! Use!the!multiple-choice!answer!sheet!for!questions!1-10! ! !

1. What!is!the!derivative!of!cos! 3%!with!respect!to!x?! ! (A)! ! (B)! ! (C)! ! (D)!

−6sin3%cos3%!

−2sin3%cos3% ! 2sin3%cos3% ! 6sin3%cos3%!

! 2. ! The!height!(in!cm)!and!foot!length!(in!cm)!for!nine!students!are!displayed!in!the! scatterplot!below.!A!least-squares!line!of!best!fit!has!been!fitted!to!the!data!as!shown.! !

! !

What!is!the!equation!of!the!least-squares!line!of!best!fit?!

! (A)! ! (B)! ! (C)! ! (D)!

ℎ,-.ℎ/ = 0.80 × 566/78,9./ℎ + 165! ℎ,-.ℎ/ = 0.80 × 566/78,9./ℎ + 140! ℎ,-.ℎ/ = 1.25 × 566/78,9./ℎ + 165!

ℎ,-.ℎ/ = 1.25 × 566/78,9./ℎ + 140!

2!!

Year 12 Mathematics Advanced

! 3. What7is7the7amplitude7and7period7for7the7function75(%) = 4sin L

%+π N ?! 3

"

! (A)!

Amplitude!3!and!period! !

! (B)!

Amplitude!3!and!period!6π!

! (C)!

Amplitude!4!and!period! ! !

! (D)!

Amplitude!4!and!period!6π!

!

"

! 4. What!type!of!relation!is!shown?! !

! ! (A)!

Many-to!many!

! (B)!

One-to-many!

! (C)!

One-to-one!

! (D)!

Many-to-one!

! 5. The!probability!distribution!of!random!variable!X#is!shown!below.! !

! x#

–3#

–2!

–1!

0!

1!

2!

3!

P(X=!x)!

0.05!

0.05!

a#

0.20!

0.15!

a!

0.05!

! Find!the!value!of!a.! ! (A)!

0.15!

! (B)!

0.20!

! (C)!

0.25!

! (D)!

0.30!

3!!

Year 12 Mathematics Advanced

! 6.

! !

Which!expression!will!give!the!area!of!the!shaded!region!bounded!by!the!curve!! P = % ! − % − 2,!the!x-axis!and!the!lines!x!=!0!and!x!=!5?! %

#

! (A)!

! (B)!

Q = RS (% ! − % − 2)T% R + S (% ! − % − 2)T% ! $

#

#

%

Q = S (% ! − % − 2)T% + RS (% ! − % − 2)T%R! $

#

%

!

! (C)!

! (D)!

Q = RS (% ! − % − 2)T% R + S (% ! − % − 2)T% ! $

!

!

%

Q = S (% ! − % − 2)T% + RS (% ! − % − 2)T%R! $

!

! 7. An!infinite!geometric!series!has!a!first!term!of!12!and!a!limiting!sum!of!15.!What!is!the! common!ratio?! ! (A)!

! (B)! ! (C)!

! (D)!

1 ! 5

1 ! 4 1 ! 3 1 ! 2

4!!

Year 12 Mathematics Advanced

! 8. At!which!point!on!this!curve!are!the!first!and!second!derivatives!both!negative?! !

! ! (A)!

A#

! (B)! ! (C)!

B# C#

! (D)!

D#

!

9. The!curve!P = 2% ! + U% + 97has!a!stationary!point!at!x!=!–2.!What!is!the!value!of!a?! ! (A)!

–8!

! (B)!

–4!

! (C)!

4!

! (D)!

8!

!

10. Which!of!the!following!statements!is!true!for!the!function!5(%) = , |'| − 1!?! ! (A)!

The!function!is!not!differentiable!at!x!=!0.!

! (B)!

The!function!is!not!continuous!at!x!=!0.!

! (C)!

The!function!has!a!stationary!point!at!x!=!0.!

! (D)!

The!function!has!an!asymptote!at!y!=!–1.!

!

5!!

Year 12 Mathematics Advanced

! Section(II(( ( 90(marks( Attempt(all(questions( Allow(about(2(hours(and(45(minutes(for(this(section( ! Answer!each!question!in!the!spaces!provided.! Your!responses!should!include!relevant!mathematical!reasoning!and/or!calculations.! Extra!writing!space!is!provided!at!the!back!of!the!examination!paper.!! ! Question(11((3!marks)!

!

Marks(

The!cost!of!making!a!hat!is!linearly!related!to!the!number!of!hats!sold.!To!predict! the!costs,!data!was!collected!on!the!cost!of!making!a!hat!for!different!levels!of! demands.!The!data!is!shown!below.! Number#(hats)# 10! 15! 207 257 307 35! 40! 45! 50! 55! 60!

(

(

Cost#(dollars)# 38! 54! 59! 82! 98! 98! 114! 108! 138! 134! 144! (a)!

Find!the!equation!of!the!least-squares!line!of!best!fit.!Answer!correct!to!one! decimal!place.!

2(

!

!

(

!

!

(

!

!

(

(b)!

What!is!the!predicted!cost!of!making!36!hats?!

1(

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(

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!

(

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!

(

( Question(12((3!marks)!

(

Find!the!area!bounded!by!the!curve!P = 1 + , $·%' ,!x-axis!and!the!lines!x!=!1!and!! x!=!2!by!using!the!trapezoidal!rule!with!five!trapezia.!Answer!correct!to!two!decimal! places.! !

3(

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( 6!!

Year 12 Mathematics Advanced

Question(13((4!marks)!

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Marks(

Pasta!boxes!are!labelled!with!a!net!weight!of!375!g.!The!business!selling!the!pasta! boxes!fills!the!boxes!to!a!net!weight!of!380!g,!to!avoid!customer!complaints.!The! weight!of!the!pasta!boxes!follows!a!normal!distribution!with!a!mean!weight!of!380!g! and!a!standard!deviation!of!2.5!g.!

(

(a)!

What!is!the!probability!that!a!box!selected!randomly!will!contain!less!than! 375!g!of!pasta?!

2(

!

!

(

!

!

(

!

!

(

!

!

(

!

!

(

(b)!

The!business!sets!a!target!of!less!than!16%!of!all!boxes!will!contain!more!than! 385!g!of!pasta.!The!business!can!change!the!mean!weight!of!the!boxes!to!meet! this!target.!What!is!mean!weight!required!to!meet!the!target!if!the!standard! deviation!remains!at!2.5?!

2(

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Question(14((2!marks)!

(

Solve!|5 − 3% | ≥ 11 ! !

2(

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Question(15((2!marks)!

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Differentiate!% ) , !' !with!respect!to!x.!

2(

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( 7!!

Year 12 Mathematics Advanced

Question(16((5!marks)!

!

Marks(

A!particle!moves!such!that!its!velocity!at!a!given!time!is!given!by:! %Y = 8 − 16sin/!

( (

(a)!

What!is!the!initial!acceleration!of!the!particle?!

1(

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(b)!

When!is!the!particle!first!at!rest?!

1(

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!

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(

(c)!

Given!that!the!particle!is!initially!at!the!origin,!find!an!equation!for!the! displacement!(x),!at!a!time!t.!

1(

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(d)!

Find!the!distance!travelled!between!the!first!two!times!when!the!particle!is!at! rest.!

2(

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Question(17((2!marks)!

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Differentiate7, *+,("') !

2(

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( 8!!

Year 12 Mathematics Advanced

Question(18((5!marks)!

!

Marks(

The!probability!density!function!for!the!continuous!random!variable!X!is!given!by:!

(

\%(4 − %) 1 ≤ % ≤ 4 7! 5 ( %) = [ 0 otherwise

(

(a)!

What!is!the!value!of!k?!

2(

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( 3(

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Find!_( / )7 the!cumulative!distribution!function.! !

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(b)!

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( Question(19((2!marks)!

(

Solve7the7equation72sin% + √3 = 07where70 ≤ % ≤ 2π.! !

2( (

!

(

!

(

!

(

!

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9!!

Year 12 Mathematics Advanced

Question(20((6!marks)!

!

Marks(

James!has!some!sticks!that!are!all!of!the!same!length.!He!arranges!them!in!squares! and!has!made!the!following!3!rows!of!patterns:!

( (

! He!notices!that!4!sticks!are!required!to!make!the!single!square!in!the!first!row,!! 7!sticks!to!make!2!squares!in!the!second!row!and!in!the!third!row!he!needs!10! sticks!to!make!3!squares.!

(

(a)!

Find!an!expression,!in!terms!of!n,!for!the!number!of!sticks!required!to!make!a! similar!arrangement!of!n!squares!in!the!nth!row.!

2(

!

!

(

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!

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!

!

(

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!

(

!

!

(

(b)!

James!continues!to!make!squares!following!the!same!pattern.!He!makes!! 4!squares!in!the!4th!row!and!so!on!until!he!has!completed!10!rows.!Find!the! total!number!of!sticks!James!uses!in!making!these!10!rows.!

1(

!

!

(

!

!

(

!

!

(

!

!

(

!

(

!

! James!started!with!1750!sticks.!Given!that!James!continues!the!pattern!to! complete!k!rows!but!does!not!have!sufficient!sticks!to!complete!the!(k#+!1)th! row,!show!that!k!satisfies!(3\ − 100)(\ + 35) < 0.! !

!

!

(

!

!

(

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!

(

!

!

(

(d)!

What!is!the!value!of!k?!

1(

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(

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!

(

! (

!

(

(c)!

! 10!

2(

(

Year 12 Mathematics Advanced

Question(21((8!marks)!

(

Marks(

Let!5( % ) = ( % + 3)( % − 1)! !

(

(a)!

Find!the!coordinates!of!the!intercepts.!

2(

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(

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!

(

!

!

(

!

!

(

(b)!

3(

!

Find!the!coordinates!of!the!stationary!points!on!the!curve!with!the!equation! P = 5(% )!and!determine!their!nature.! !

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(c)!

What!are!the!coordinates!of!any!points!of!inflexion?!

2(

!

!

(

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(

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(

!

!

(

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!

(

!

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(

(d)! !

Sketch!the!graph!of!!P = 5(%)!over!a!suitable!domain.!

(

1(

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11! !

Year 12 Mathematics Advanced

Question(22((4!marks)!

!

Marks(

A!continuous!function!P = 5(%)!defined!in!the!domain![–4,!5].!The!function!consists! of!a!line!segment,!a!quarter!of!a!circle!centred!at!(0,!0)!with!radius!2!and!a! logarithmic!function!5(%) = ln(% − 1)!in!the!domain![2,!5]!

(

(

!

!

Find!the!exact!area!bounded!by!the!function! P = 5(%)!and!the!x-axis!in!the! domain![–4,!2]! !

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(b)!

Hence, find7the7exact7value7of7 S ln(% − 1) T%!

(a)!

%

2( (

2(

!

%

given7that7 S 5(%)T% = 4ln4 − 7 − π ! /0

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( 12! !

Year 12 Mathematics Advanced

Question(23((4!marks)!

!

Marks(

The!table!below!shows!the!future!value!of!a!$1!annuity.!

(

!

( Future#value#of#$1# End!of!year!

1%!

2%!

3%!

4%!

5%!

6%!

1!

1.0000!

1.0000!

1.0000!

1.0000!

1.0000!

1.0000!

2!

2.0100!

2.0200!

2.0300!

2.0400!

2.0500!

2.0600!

3!

3.0301!

3.0604!

3.0909!

3.1216!

3.1525!

3.1836!

4!

4.0604!

4.1216!

4.1836!

4.2465!

4.3101!

4.3746!

(a)!

What!would!be!the!future!value!of!a!$24!000!per!year!annuity!at!5%!per! annum!for!2!years,!with!interest!compounding!annually?!

1(

!

!

(

!

!

(

!

!

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(b)!

An!annuity!of!$31!800!is!invested!every!half-year!at!4%!per!annum,! compounded!six-monthly!for!2!years.!What!is!the!future!value!of!the!annuity?!

1(

!

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!

(

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!

(

(c)!

Sophia!aims!to!have!a!deposit!for!an!apartment!of!at!least!$92!000!in!! 4!years-time!by!investing!in!an!annuity.!The!annuity!has!an!interest!rate!of! 6%!p.a.!compounded!annually.!Calculate!Sophia’s!yearly!contribution!to! achieve!the!deposit.!Answer!to!the!nearest!dollar.!

2(

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Question(24((2!marks)! )

Find7k7given7that7 S !

!

(

1 T% = lnk! 1−%

2( (

!...