ACE Papeer 1 ADV - Due to the implementation of the new syllabus, past paper resources that cover PDF

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Due to the implementation of the new syllabus, past paper resources that cover the new content is almost impossible to find. HOWEVER, I have collated 35+ past papers from different schools for their 2020 trials :)...

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! ! ACE EXAM PAPER

!

! Student name: ______________________!

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! ! ! !

PAPER 1

!

YEAR 12 YEARLY EXAMINATION

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! ! !

Mathematics Advanced

General Instructions

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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 questions 11-16, show relevant mathematical reasoning and/or calculations

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Total marks: 100

Section I – 10 marks ! Attempt Questions 1-10 ! Allow about 15 minutes for this section Section II – 90 marks ! Attempt questions 11-16 ! Allow about 2 hours and 45 minutes for this section

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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!solution!to!the!equation!2cos% & − 1 = 0!in!the!domain!0 ≤ & ≤ 2π!?! ! (A)! ! (B)! ! (C)! ! (D)! !

π 11π ! &= , 6 6 π 7π &= , ! 4 4

π 5π 7π 11π , , ! &= , 4 4 4 4 π 3π 5π 7π &= , , , ! 4 4 4 4

2.

! ! Which!of!the!following!properties!matches!the!above!graph?! ! (A)! ! (B)! ! (C)! ! (D)!

3′(&) > 08and83′′(&) < 0! 3′(&) > 08and83′′(&) > 0!

3′(&) < 08and83′′(& ) < 0! 3′(&) > 08and83′′(&) > 0!

2!!

Year 12 Mathematics Advanced

! 3. ! A!factory!produces!bags!of!cashews.!The!weights!of!the!bags!are!normally!distributed,! with!a!mean!of!900!g!and!a!standard!deviation!of!50!g.!What!is!the!best!approximation!for! the!percentage!of!bags!that!weigh!more!than!1000!g?! ! (A)!

0%!

! (B)!

2.5%!

! (C)!

5%!

! (D)!

16%!

!

C

4. What!is!the!value!of! = (> ?@ + 1)B&8?! ! (A)! ! (B)! ! (C)! ! (D)! !

>?!

D

1 ? > ! 3

1 ? (> + 1)! 3 1 ? (> + 2)! 3

5. What!is!the!gradient!to!the!curve!F = (& − G)(& % − 1)!at!the!point!when!x!=!–2?! ! (A)! −3G − 6! ! (B)! ! (C)! ! (D)! ! 6.

−5G − 1! 4G + 11! 5G + 4!

! ! What!is!the!correlation!between!the!variables!in!this!scatterplot?! ! (A)!

Weak!negative!

! (B)!

Weak!Positive!

! (C)!

Moderate!negative!

! (D)!

Moderate!positive! 3!!

Year 12 Mathematics Advanced

!

7. A!section!of!the!graph!F = 3(&)!is!shown!below.! !

! ! Which!of!the!following!is!the!correct!function!for!the!above!graph?! ! (A)!

! (B)!

! (C)!

! (D)!

1 π 3(&) = tan I J& − KL! 4 2

π 3(&) = tan I2 J& − KL! 4 π 1 3(&) = tan I J& − KL! 2 2 π 3( &) = tan I2 J& − KL! 2

! 8. The!graph!of!the!derivative!function!is!shown!below.! !

!

! Where!is!the!function!F = 3(&)!increasing?! ! (A)! ! (B)! ! (C)! ! (D)!

{& ∶ & > 0}! {& ∶ & > 2}! {& ∶ −3 < & < 2}! {& ∶ & < −3 }!or!{& ∶ & > 2 }!

4!!

Year 12 Mathematics Advanced

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

! !

Present(value(of($1( End!of!year!

3%!

4%!

5%!

6%!

5!

4.5797!

4.4518!

4.3295!

4.2124!

6!

5.4172!

5.2421!

5.0757!

4.9173!

7!

6.2303!

6.0021!

5.7864!

5.5824!

8!

7.0197!

6.7327!

6.4632!

6.2098!

! What!is!the!present!value!of!an!annuity!where!$12,000!is!contributed!each!year!for!six! years!into!an!account!earning!3%!per!annum!compound!interest?! ! (A)!

$15!183.83!

! (B)!

$54!956.40!

! (C)!

$65!006.40!

! (D)!

$72!000.00!

! 10 Which!of!the!following!graphs!could!not!represent!a!probability!density!function!f(x)?! ! (A)!

(B)!

! (C)!

(D)!

!

! !

5!!

Year 12 Mathematics Advanced

! Section(II(( ( 90(marks( Attempt(questions(11(-(16( 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.! ! Question(11((2!marks)!

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Marks(

Differentiate!the!following!functions!with!respect!to!x.! (a)!

3(&) = sin& + & !

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%

3(&) = ln(& % + 1)?!

1(

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(b)! !

( Question(12((3!marks)!

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For!the!arithmetic!sequence!4,!9,!14,!19,!….!

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(a)!

Write!the!rule!to!describe!the!nth!term.!

1(

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(b)!

What!is!the!25th!term?!

1(

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(c)!

Find!the!sum!of!the!first!100!terms.!

1(

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( 6!!

Year 12 Mathematics Advanced

Question(13((4!marks)!

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Marks(

A!continuous!random!variable!X!has!a!function!f!given!by! 3(&) = R

|3 − &| 0

2≤&≤4

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8! otherwise

Find!X(2 ≤ Y ≤ 3.5)!

2(

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(a)! !

Find!X(2 ≤ Y ≤ 2.5)!

2(

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(b)! !

Question(14((4!marks)!

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Differentiate! (a)!

2> cos&!

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(b)!

@

Question(15((1!mark)!

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Find8 =(2& + 3)CD B&!

1(

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( 7!!

Year 12 Mathematics Advanced

Question(16((2!marks)!

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Marks(

Tran’s!industrial!unit!produces!aluminium!rods.!In!the!past!week!the!industrial!unit! has!produced!aluminium!rods!with!a!mean!weight!of!12.5!kilograms!and!a!standard! deviation!of!0.5!kilograms.!

(

(a)!

1(

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Quality!control!requires!any!aluminium!rod!with!a!z-score!less!than!–1!to!be! rejected.!What!is!the!minimum!weight!that!will!be!accepted?! !

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(b)!

Aluminium!rods!with!a!z-score!greater!than!2!are!also!rejected.!What!is!the! maximum!weight!that!will!be!accepted?!

1(

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Question(17((2!marks)!

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What!is!the!area!enclosed!between!the!curves!F = & % + 1!and!F = 3& + 1!?!

2( (

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8!!

Year 12 Mathematics Advanced

Question(18((3!marks)!

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Find!the!value!of!k!if!!F = > \@ sin&!!and! !

Marks( BF − 3F = > \@ cos&.! B&

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Question(19((3!marks)!

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The!diagram!below!shows!a!native!garden.!All!measurements!are!in!metres.!

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! Use!the!Trapezoidal!Rule!with!4!intervals!to!find!an!approximate!value!for!the! area!of!the!native!garden.! !

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(b)!

If!25!millimetres!of!rain!fell!overnight,!how!many!litres!of!rain!fell!on!the! native!garden?!Assume!18m? = 10008L.!

1(

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(a)!

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9!!

2( (

Year 12 Mathematics Advanced

Question(20((3!marks)!

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Marks(

Consider!the!functions!F = & % !and!F = & % − 3& + 2.!

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(a)!

Sketch!the!two!functions!on!the!same!axes.!

2(

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Hence!or!otherwise!find!the!values!of!x!such!that!& > (& − 1)(& − 2).! !

1(

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(b)!

%

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( Question(21((2!marks)!

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State!the!amplitude!and!period!of!the!function! 3(&) = 4 + 3cos J !

π& K! 2

2( (

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! 10!

Year 12 Mathematics Advanced

Question(22((2!marks)!

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Marks(

The!normal!distribution!shows!the!results!of!a!mathematics!assessment!task.!It!has! a!mean!of!60!and!a!standard!deviation!of!10.!

( (

! (a)!

What!is!the!mathematics!assessment!result!with!a!z-score!of!–2?!

1(

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(b)!

What!is!the!z-score!of!a!mathematics!assessment!result!of!65?!

1(

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( Question(23((2!marks)!

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Find8 = (sec% 2&)B& ! !

2(

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Question(24((2!marks)!

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How!many!solutions!does!the!equation!|cos(2&)| = 1!have!for!0 ≤ & ≤ 2π?!

2(

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( 11! !

Year 12 Mathematics Advanced

Question(25((5!marks)!

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Marks(

A!function!3(&)!is!defined!by!3(&) = & % (3 − &)!.!

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Find!the!stationary!points!for!the!curve!F = 3(&)!and!determine!their!nature.!

2(

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(a)! !

Sketch!the!graph!of!F = 3(&)!showing!the!stationary!points!and!x-intercepts.!

2(

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(c)!

Find!the!equation!of!the!tangent!to!the!curve!at!the!point!X(1,2)!.!

1(

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(b)! !

Question(26((2!marks)!

(

Construct!a!recurrence!relation!in!the!form!abcC = ab × (1 + e) − f!to!model!the! balance!of!a!loan!of!$58!000!borrowed!at!6%!per!annum,!compounding!monthly,! with!payments!of!$810!per!month.!

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12! !

Year 12 Mathematics Advanced

Question(27((4!marks)!

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Marks(

Ten!kilograms!of!chlorine!is!placed!in!water!and!begins!to!dissolve.!After!t!hours!the! amount!A!kg!of!undissolved!chlorine!is!given!by!g = 10> h\i ! (a)! Calculate!the!value!of!k!given!that!A!=!3.6!when!t!=!5.!Answer!correct!to!three! decimal!places.!

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(b)!

After!how!many!hours!does!one!kilogram!of!chlorine!remain!undissolved?! Answer!correct!to!one!decimal!place.!

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( Question(28((2!marks)!

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The!third!and!seventh!terms!of!a!geometric!series!are!1.25!and!20!respectively.! What!is!the!first!term?!

2(

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( 13! !

Year 12 Mathematics Advanced

Question(29((5!marks)!

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Marks(

The!table!below!shows!forearm!length!and!hand!length.!

(

Forearm!(in!cm)! 25.0! 25.6! 26.0! 26.6! 27.0! 27.4! 28.0! 28.6! 29.0! 29.2! Hand!(in!cm)!

(

17.2! 17.6! 18.2! 18.4! 19.0! 19.0! 19.8! 19.8! 20.4! 20.6!

! !

(a)!

Draw!a!scatterplot!using!the!above!table.!

1(

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! (b)!

Draw!a!line!of!best!fit!on!the!scatterplot.!

1(

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(c)!

Charlotte!has!a!forearm!whose!length!is!27.8!cm.!What!is!her!expected!hand! length?!

1(

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(d)!

2(

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Calculate!the!value!of!the!Pearson’s!correlation!coefficient.!Answer!correct!to! four!decimal!places.! !

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Year 12 Mathematics Advanced

Question(30((3!marks)!

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Marks(

Florence!left!$1000!in!her!will!for!World!Vision.!Her!instructions!were!that!this! money!be!invested!at!5%!interest,!compounded!annually.!

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(a)!

How!much!money!would!be!given!to!World!Vision!after!100!years?!Give!your! answer!to!the!nearest!dollar.!

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(b)!

Florence!has!requested!her!family!invest!a!further!$1000!at!the!beginning!of! each!subsequent!year!at!the!same!interest!rate.!How!much!money!would!be! given!to!World!Vision!after!100!years!if!her!family!fo...