Alternative Education Equivalency (AEE) Testsdfraeea.com/download/year12-cpk-advanced-maths-supplement.pdfAlternative Education Equivalency Scheme (AEES) ... The questions in the Year

Alternative Education Equivalency (AEE) Tests

Year 12

Advanced Mathematics

Candidate Preparation Kit Supplement

Alternative Education Equivalency Scheme (AEES) Year 12 Candidate Preparation Kit

© 2017 VETASSESS, Level 5, 478 Albert Street, East Melbourne Victoria 3002.

All rights reserved. No part of this book may be reproduced without written permission from

VETASSESS.

20170901

Year 12 Advanced Mathematics Candidate Preparation Kit Supplement

Alternative Education Equivalency Scheme (AEES) Tests

INFORMATION ABOUT THE AEES TESTS

1. Year 12 Advanced Mathematics Candidate Preparation Kit Supplement

2. Test Details

3. Non-Established Venue Testing

4. Candidate Test Attempts

5. Candidate Reminders

6. Results

7. Types of Questions in the AEES Tests

8. How to Prepare for the Year 12 Advanced Mathematics Test

9. Reference Materials

10. Sample Questions

11. Attending the Test Session

12. Stationery and Personal Belongings

13. Test Rules

14. Breaching the Test Rules

15. Consequences of Breaching the Test Rules

16. Accessing your Personal Records and Appeals

APPENDICES

A. Reference Materials

B. Sample Questions

© 2017 VETASSESS- Leve l 5 , 478 A lber t St ree t , Eas t Melbourne V ic tor i a 3002 .

A l l r igh ts reserved . No par t of th is book may be reproduced w ithout wr i t ten perm iss ion f rom VETASSESS .

5

C. Answers to Sample Questions



7

1. Year 12 Advanced Mathematics Candidate Preparation Kit Supplement

This supplement accompanies the Year 12 Candidate Preparation Kit and includes the

information relevant to candidates booked to sit the Year 12 Advanced Mathematics test.

Please read this supplement together with the Year 12 Candidate Preparation Kit.

2. Test Details

Tests Time Allowed Items


Extended response

* all 3 test components must be completed at

the same test session one after another with no

designated break

30 minutes x

3 test components

(1.5 hours in total)

11 items x

3 test components

(33 items in total)

3. Non-Established Venue Testing

Please refer to the Year 12 Candidate Preparation Kit for information about non-established

venue testing.

4. Candidate Test Attempts

• You have 92 days to complete a Test Attempt.

• You will be given two Test Attempts per subject.

• If you do not sit the test/s within the 92 day Test Attempt period, you forfeit the Test

Attempt and this counts as one of your Test Attempts.

• You will be given a further 92 day period to complete a second Test Attempt.

• If you need to defer your testing within the 92 day Test Attempt period, we will attempt to

re-book you within the 92 day period.

Please refer to the confirmation letter or candidate voucher sent from VETASSESS for:

• Test Attempt number

• Test Attempt start date

• Test Attempt expiry date

5. Candidate Reminders

Please take note of your ‘Test Attempt expiry date/s.’ (refer confirmation letter / candidate voucher).



8

Department of Defence, Canberra. ACT 2600. Australia.

You are required to confirm you are attending the test session booked for you. On receipt of a

text message or email from VETASSESS, please reply to confirm or cancel your testing.

Please be aware if you cancel your testing by email or text message, we cannot guarantee

there will be another test date available during the 92 day Test Attempt period. If your testing

cannot be rebooked within this 92 day Test Attempt period, you will forfeit the Test Attempt.

This will count as one of your Test Attempts.



9

6. Results

Your result for the Year 12 Advanced Mathematics test is the number of correct answers given.

VETASSESS will report your results to Defence Force Recruiting. You will need to contact

Defence Force Recruiting on 13 19 02 to be advised of your results. Please note, we are unable

to advise you directly of your results or provide you with a results statement.

Advice on your performance against individual topics in the test/s is not available as the results

processing does not include this analysis.

7. Types of Questions in the AEES Tests

The questions in the Year 12 Advanced Mathematics test are based on the ‘Year 12 New South

Wales Mathematics 2 Unit curriculum.’ Please refer to the ‘2 Unit' content in the ‘Year 12 New

South Wales Mathematics 2/3 Unit Years 11-12 Syllabus’ available from the website

www.dfraeea.com. Further information, including HSC exam papers, is available from the Board

of Studies New South Wales website.

The test assesses mathematical knowledge and skills in the following areas:

• basic arithmetic & algebra

• real functions

• linear functions

• the quadratic polynomial & the parabola

• plane geometry – geometrical properties

• tangent to a curve & derivative of a function

• coordinate methods in geometry

• applications of geometrical properties

• geometrical applications of differentiation

• integration

• trigonometric functions (including applications of trigonometric ratios)

• logarithmic & exponential functions

• applications of calculus to the physical world

• probability

• series & series applications

http://www.dfraeea.com/

http://www.boardofstudies.nsw.edu.au/

http://www.boardofstudies.nsw.edu.au/



10


8. How to Prepare for the Year 12 Advanced Mathematics Test

To assist you in your preparation for the AEES tests, useful revision tips and helpful information

is provided, see Appendix A in the Year 12 Candidate Preparation Kit.



11

9. Reference Materials

It is recommended that you access reference books from your local library or consider using a

tutor from within your local community.

There are many relevant text books that would provide reference material for the Year 12

Advanced Mathematics test. A list of Mathematics text books is provided, see Appendix A in

this supplement.

The text books listed in Appendix A in this supplement can be viewed online and ordered

through the corresponding publisher website. There are also other relevant text books

available from these and other publishers that would provide suitable reference material.

10. Sample Questions

Year 12 Advanced Mathematics sample questions are provided to show the types of items in

the test but they do not necessarily indicate the full range of questions or item difficulty, see

Appendix B in this supplement.

11. Attending the Test Session

On the day of the AEES tests, you must present at the test venue at the time specified in the

confirmation letter received from VETASSESS.

When you arrive at the test venue on the day, you must register before the test session.

You must bring the following:

• Candidate Voucher (sent from VETASSESS)

• one form of photographic identification (includes drivers licence, learner’s permit, boat

licence, passport (current), student identification card, employment identification card,

proof of age card or other identification with a photograph that shows your full name)

If you do NOT have photographic identification, call VETASSESS on 03 9655 4849.

Please note: you will NOT be admitted to the test room without the above items.

12. Stationery and Personal Belongings

STATIONERY

You will need to bring:

• stationery: pens, 2B pencils, soft eraser, sharpener



12


• calculator: silent, battery-operated, non-programmable scientific calculator

• ruler (for use in the Year 12 Advanced Mathematics test only)

• bottled water (recommended)



13

Please note:

• Mobile phones with calculator functionality and CAS/graphics calculators are not

permitted

• You will not be permitted to borrow a calculator from another candidate after entering

the test room

• It is your responsibility to ensure the calculator is in good working order

• We recommend you take extra batteries to the test session

• Spare calculators and spare batteries will not be available at the test session

PERSONAL BELONGINGS

• no dictionary of any kind is permitted in the test room

• mobile phones, music players and other electronic devices must be switched off

• personal items brought into the test room must be stored in the designated area or

under the desk

MATERIALS PROVIDED ON THE DAY

Once the test session commences, you will be issued with the required test materials. These

include:

• test booklets

• answer sheet

• working space sheets (note paper)

13. Test Rules

Please refer to the Year 12 Candidate Preparation Kit for information about the test rules.

14. Breaching the Test Rules

Please refer to the Year 12 Candidate Preparation Kit for information about breaching the test

rules.

15. Consequences of Breaching the Test Rules

Please refer to the Year 12 Candidate Preparation Kit for information about the consequences

of breaching the test rules.



14


16. Accessing your Personal Records and Appeals

Please refer to the Year 12 Candidate Preparation Kit for full information about accessing your

personal records and appeals.



15

Appendix A Reference Materials

Following is a list of some of the leading publications for Year 12 Advanced Mathematics.

This list provides suggested reference material only and VETASSESS does not guarantee these

books are available or that the topic areas in the test are covered in these text books.

Year 12 Advanced Mathematics

Title: Oxford Insight Mathematics General HSC CEC General 1

Author: John Ley, Michael Fuller

Publisher: Oxford University Press

Year of Publication: 2014

Title: Oxford Insight Mathematics General HSC General 2

Author: John Ley, Michael Fuller

Publisher: Oxford University Press


Title: Maths Quest HSC Mathematics General 2

Author: Rowland

Publisher: Jacaranda


Title: StudyOn HSC Mathematics General 2

Author: Rowland

Publisher: Jacaranda


Title: 2 Unit Mathematics Book 1

Author: S. B. Jones et al



16


Publisher: Pearson Australia




17













Title: Maths in Focus: Mathematics Extension 1 HSC Course Revised

Author: Margaret Grove

Publisher: Nelson Cengage Learning


Title: Nelson Senior Maths Specialist 12 for the Australian Curriculum

Author: Allason McNamara et al

Publisher: Nelson Cengage Learning


Publisher websites

Publisher Website

Jacaranda Publishing www.jaconline.com.au

Oxford University Press www.oup.com.au

http://www.jaconline.com.au/

http://www.oup.com.au/



18


Publisher Website

Pearson Australia www.pearson.com.au

Nelson Cengage Australia www.cengage.com.au/secondary

Macmillan Education Australia www.macmillan.com.au

http://www.pearson.com.au/

http://www.cengage.com.au/secondary

http://www.macmillan.com.au/



19

Appendix B Sample Questions


The following examples show the types of items in the test, but do not necessarily indicate the

full range of items or test difficulty. For the Advanced Mathematics test, you may use a silent,

battery-operated, non-programmable scientific calculator (not CAS or graphing calculator) and

a ruler. For answers to these sample questions, see Appendix C.

The following formulae may be used in your calculations:

Formulae

Please note: drawings are not to scale.

The following formulae may be used in your calculations:

QUADRATIC EQUATIONS

If ax2 bx c 0 then x =

−𝑏 ± √(𝑏2−4𝑎𝑐)

2𝑎

SERIES

Where a is the first term, L is the last, d is the common difference and r is the common ratio

ARITHMETIC

a (a d ) (a 2d ) ... (a (n 1 ) d ) n

2 ( 2a + (n − 1)d ) =

n

2 (a + L)

GEOMETRIC

a ar ar 2 ... arn− 1=𝑎(1−𝑟𝑛)

1−𝑟, r ≠ 1

SPACE & MEASUREMENT

In any triangle ABC,

𝑎

𝑠𝑖𝑛𝐴=

𝑏

𝑠𝑖𝑛𝐵=

𝑐

𝑠𝑖𝑛𝐶

Area = 1

2 ab sinC

a2 = b2 + c2 – 2bc cosA

cosA = 𝑏2+ 𝑐2− 𝑎2

2𝑏𝑐



20


TRAPEZIUM

Area = 1

2 (a+b) x height, where a and b are the lengths of the parallel sides.

PRISM

Volume = Area of the base x height

CYLINDER

Total surface area = 2π r h + 2π r 2

Volume = π r 2 x h

PYRAMID

Volume = 1

3 x area of base x height

CONE

Total surface area = π r s + π r2, where s is the slant height

Volume = 1

3 x π r2 x h

SPHERE

Total surface area = 4π r2

Volume = 4

3 π r3

VOLUME OF SOLIDS OF REVOLUTION ABOUT THE AXES

∫ π y2 dx and ∫ π x2 dy

RATE

If y = ky, then y = Aekx

TEMPERATURE CONVERSION FORMULA

Degrees Celsius to degrees Fahrenheit: ˚ F (˚ C ×1.8) + 32

THEOREM OF PYTHAGORAS

In any right-angled triangle: 2 2 2c a b

INDEX LAWS

For a, b>0 and m, n real,

m n m na a a

( )

m m ma b a b ( )

m n m na a



21

For m an integer and n a positive integer

1m

ma

a

mm n

n

aa

a

0

1a

m

mn nmna a a



22


CALCULUS

Function notation Leibniz Notation

Product rule

Quotient rule

Chain rule

FUNDAMENTAL THEOREM OF CALCULUS:

and

STANDARD DERIVATIVES

If y f (x ) x

n, then y '

dy

dx f '(x ) nx

n1

If y f (x ) e

x, then

dy

dx f '(x ) e

x

If y f (x ) log

ex then y '

dy

dx f '(x )

1

x

If y f (x ) sin(ax ), then y '

dy

dx f '(x ) a cos(ax )

If y f (x ) cos(ax ), then y '

dy

dx f '(x ) a sin(ax )

STANDARD INTEGRALS

= , , and if

= = ,

= , = ,

y y y y

( ) ( )f x g x ( ) ( ) ( ) ( )f x g x f x g x u vdu dv

v udx dx

( )

( )

f x

g x2

( ) ( ) ( ) ( )

( ( ))

f x g x f x g x

g x

u

v 2

du dvv u

dx dx

v

( ( ))f g x ( ( )) ( )f g x g x ( ) ( )andy f u u g x dy du

du dx

( ) ( )x

a

df t dt f x

dx ( ) ( ) ( )

b

af x dx f b f a

dxxn

1

1

1

nx

n1n 0x 0n

dxx

1,ln x 0x dxe

ax

ax

ea

10a

axcos dx axa

sin1

0a axsin dx axa

cos1

0a



23

PROBABILITY LAWS

Pr (A / B) = Pr ( A ⋂B ) =

Pr (B)

TRIGONOMETRY

In any right-angled triangle:

sin θ = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

cos θ = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

tan θ = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

( ) ( ) 1P A P A

( ) ( ) ( ) ( )P A B P A P B P A B

( ) ( ) ( / ) ( ) ( / )P A B P A P B A P B P A B

Opposite side

Adjacent side

Hypotenuse

θ



24


GROWTH AND DECAY FORMULAE

• Simple growth or decay: A = P (1± ni )

• Compound growth or decay: A = P (1± i )n

Where:

A = amount at the end of n years

P = principal

n = number of years

i = interest rate per year, r % =r

100

• Compound interest, where the interest is compounded t times per year: A = P (1 + it )

nt

Where:

t = number of interest periods per year

• Future value of an annuity: F = x[(1 + i)

n - 1]

i Contributions at end of each period

or F = x[(1 + i)

n - 1] x (1 ÷ i)

i Contributions at beginning of each period

Where:

F = future value of annuity

i = interest rate per compounding period, as a decimal fraction

n = number of compounding periods



25

REAL FUNCTIONS

Example 1 For the basic following functions: f (x) = 2x -1

1 + x and h(x) = 1 – 2x find the

composite function, f (h (x)) in simplest terms. (2 marks)

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

LINEAR FUNCTIONS

Example 2 The line 2y + x = 4 is reflected across the x axis. Sketch the original line and its

reflection (clearly marking coordinates of any intercepts) then find the equation

of the reflected line. (3

marks)



26


THE QUADRATIC POLYNOMIAL AND THE PARABOLA

Example 3 Find the coordinates of the turning point of the parabola y = x2 – 4x – 5.

(2marks)

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

PLANE GEOMETRY – GEOMETRICAL PROPERTIES

Example 4 An equilateral triangle is inscribed in a circle of radius 3cm. Calculate the

unshaded area as shown below (correct to 2 decimal places). (3

marks)

TANGENT TO A CURVE AND DERIVATIVE OF A FUNCTION

Example 5 Find the gradient of the curve f(x) = 2e3

x at the point where x = 1

(correct to 2 decimal places).

(2 marks)



27

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

COORDINATE METHODS IN GEOMETRY

Example 6 The vertices of ∆ ABC are A(1,2), B(6,-1) and C(2,-2). Use your knowledge of the

properties of a right angled triangle to show that ∆ ABC is a right angled

triangle.

(2 marks)

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

APPLICATIONS OF GEOMETRICAL PROPERTIES

Example 7 Given AB = 5 units, ED = 3 units and AD = 4 units, find the length of DC in the

diagram below. (2

marks)



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(diagram not drawn to scale)

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________



29

GEOMETRICAL APPLICATIONS OF DIFFERENTIATION

Example 8 Find the equation of the normal to the curve y = (x – 2)2 at the point where x =

3

(4 marks)

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

INTEGRATION

Example 9 Find the exact value of ∫ 1

(2x+1)2 dx (2

marks)

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

TRIGONOMETRIC FUNCTIONS

(including applications of trigonometric ratios)

Example 10 The function f(Ѳ) = 1 + sin2Ѳ is defined for Ѳ є [0,2π]. Write down the

maximum value of f(Ѳ) and the values of Ѳ for which it occurs.

(3 marks)

2

0



30


_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________



31

LOGARITHMIC AND EXPONENTIAL FUNCTIONS

Example 11 An insect population grows according to the rule P = 2loge(t + 2 ) where P is the

population, in millions, t years after the population was first estimated.

According to this rule:

a) What was the population when first estimated?

(1 mark)

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

b) How long will it take for the population to reach 5 million? (correct to 2

decimal places) (2

marks)

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

APPLICATIONS OF CALCULUS TO THE PHYSICAL WORLD

Example 12 With wind assistance, a balloon ascends at an acceleration of 2t m sec-2 (where t

is the time in seconds after release). If the balloon is stationary until it is

released from a height of 1 metre above ground level, how long will it take to

reach a height of 100 metres? (correct to 2 decimal places)



32


(4 marks)

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________



33

PROBABILITY

Example 13 A tennis player wins 80% of her matches. To the nearest %, what is the

probability she will win at least 4 of her next 5 matches?

(2 marks)

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

SERIES AND SERIES APPLICATIONS

Example 14 A “not so wise” boss agreed to pay a worker $1 on the 1st day, $2 on the 2nd day,

$4 on the 3rd day and so on.

a) Show that this form of payment is a geometric sequence. (1

mark)

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

b) How many dollars would the boss have to pay on the 20th day?

(2 marks)

__________________________________________________________________________



34


__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________



35

Appendix C Year 12 Advanced Mathematics Answers to Sample Questions

Example 1 solution

f(h(x)) = F(1 – 2x) = 2(1 - 2x)-1

1 + (1 - 2x) =

1- 4x

2 - 2x

Example 2 solution

New equation : 2y – x = –4 or x – 2y = 4 or –x + 2y = –4

Both intercepts (or 2 points) must be shown on graphs

Example 3 solution

y = (x – 2)2 – 5 – 4 OR y = (x –5)(x + 1)

y = (x – 2)2 – 9 TP at x = 5 - 1

2 = 2, y = (2 – 5)(2 + 1) = − 9

TP (2, –9) TP (2, –9)



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Example 4 solution

Area of 3 triangles side length 3cm, α = 120° = 3 x 1

2 (3)(3)sin120° = 11.6913

Area of circle A = πr2 = π x 9 = 28.2743

Unshaded area = 28-2743 – 11.6913 = 16.58cm2

Example 5 solution

f’(x) = 6e 3x

f’(1) = 6e 3 = 120.51

Example 6 solution

m(AC) = -2 - 2

2 - 1 = − 4, m(BC) =

-2 - -1

-4 =

1

4

− 4 x 1

4 = − 1 so sides AC and BC are at right angles ∆ ABC is a right angled triangle

OR

AC 2 = (2-1)2 + (-2-2)2 = 17 AB2 = (6-1)2 + (-1-2)2 = 34 BC2 = (2-6)2 + (-2- -1)2 = 17

BC2 + AC2 = AB2 so ∆ ABC must be a right angled triangle



37

Example 7 solution

Let DC = x

53 =

4 + xx

5x = 12 + 3x

2x = 12

x = 6

Example 8 solution

dy

dx = 2(x − 2)

Gradient of tangent is 2(3 – 2) = 2 at x = 3

Gradient of the normal is:

m = −

1

2 ,

y - 1

x - 3 = −

1

2 ,

Equation of the normal 2y + x = 5

Example 9 solution

∫ 1

(2x+1)2 dx = −

1

2 (2x + 1)

− 1 ]

= − (

1

10 ) − (−

1

2 )

=

2

5

Example 10 solution

Max value of 2

when:

2Ѳ = π2 , π2 + 2π

Ѳ = π4 , π4 + π

Ѳ = π4 ,

5π4

(no working or sketch required - marks only for answers)

2

0

2

0



38


Example 11 solution

1.39 million

5 = 2loge(t + 2)

e2.5 = t + 2

t = 10.18 years

Example 12 solution

Let x = height above ground level

a = x = 2t

v = x = ∫ 2tdt = t 2 + c

At t = 0, v = 0 c = 0

x = t 2

x = ∫t 2 dt =

1

3 t

3 + c1

At t = 0, x = 1 c1 = 1

x = 1

3 t

3 + 1

At x = 100 = 1

3 t

3 + 1

t = √2973

= 6.67 seconds

Example 13 solution

(54) (0.8)4(0.2)1 + (0.8)5 (1 or both terms correct)

= 0.4096 + 0.3277 = 74%

Example 14 solution

a) Common ratio = 2 i.e. 2

1 =

4

2

b) a = 1, r = 2, t20

= ar19 =1 x 219

= $524,288



39



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Alternative Education Equivalency (AEE) Testsdfraeea.com/download/year12-cpk-advanced-maths-supplement.pdfAlternative Education Equivalency Scheme (AEES) ... The questions in the Year