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Clear zone — margin trimmed off after completion of assessment LUI School code School name Given name/s Family name Attach your barcode ID label here Book of books used External assessment Specialist Mathematics Paper 1 — Technology-free Time allowed • Perusal time — 5 minutes • Working time — 90 minutes General instructions • Answer all questions in this question and response book. • Calculators are not permitted. • QCAA formula book provided. • Planning paper will not be marked. Section 1 (10 marks) • 10 multiple choice questions Section 2 (55 marks) • 9 short response questions Question and response book

External assessment 2020: Specialist Mathematics Paper 1

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Page 1: External assessment 2020: Specialist Mathematics Paper 1

Cle

ar z

one

— m

argi

n tr

imm

ed o

ff af

ter

com

plet

ion

of a

sses

smen

t

LUI School code

School name

Given name/s

Family name

Attach your barcode ID label here

Book of books used

External assessment

Specialist MathematicsPaper 1 — Technology-free

Time allowed• Perusal time — 5 minutes

• Working time — 90 minutes

General instructions• Answer all questions in this question and

response book.

• Calculators are not permitted.

• QCAA formula book provided.

• Planning paper will not be marked.

Section 1 (10 marks)• 10 multiple choice questions

Section 2 (55 marks)• 9 short response questions

Question and response book

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Section 1

Instructions• Choose the best answer for Questions 1–10.

• This section has 10 questions and is worth 10 marks.

• Use a 2B pencil to fill in the A, B, C or D answer bubble completely.

• If you change your mind or make a mistake, use an eraser to remove your response and fill in the new answer bubble completely.

A B C DExample:

A B C D1.2.3.4.5.6.7.8.9.

10.

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Section 2

Instructions• Write using black or blue pen.

• Questions worth more than one mark require mathematical reasoning and/or working to be shown to support answers.

• If you need more space for a response, use the additional pages at the back of this book.

− On the additional pages, write the question number you are responding to.

− Cancel any incorrect response by ruling a single diagonal line through your work.

− Write the page number of your alternative/additional response, i.e. See page …

− If you do not do this, your original response will be marked.

• This section has nine questions and is worth 55 marks.

DO NOT WRITE ON THIS PAGE

THIS PAGE WILL NOT BE MARKED

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QUESTION 11 (7 marks)The vertices of a regular hexagon are positioned on the circumference of a unit circle as shown on the Argand plane.

Consider the complex number w, as shown on the plane.

a) Determine w, expressing your answer in the form r cis(θ). [1 mark]

b) Convert w into Cartesian form. [2 marks]

w

Im(z)

Re(z)

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Each vertex of the hexagon is a solution of an equation of the form z an = where z CÎ .

c) State the value of n. [1 mark]

d) State the value of a. [1 mark]

e) Verify that w satisfies the equation z an = using the results from 11c) and 11d). [2 marks]

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QUESTION 12 (8 marks)Consider the vertices A, B and C of the rectangular prism as shown.

z

y

B

C

0

2

3

1

xA

a) State the coordinates of A, B and C. [1 mark]

b) Determine a unit vector, , that is normal to the plane containing A, B and C. [3 marks]

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c) Verify that n is perpendicular to AB� ���

. [2 marks]

d) Determine the Cartesian equation of the plane that contains A, B and C. [2 marks]

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QUESTION 13 (4 marks)The expected value of an exponential random variable X with parameter l > 0 can be determined using the rule

E X x e dxx( )=

∞ −∫0

l l

Use integration by parts to determine E (X).

Express your answer in simplest form.

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QUESTION 14 (4 marks)

The motion of an object that moves in a straight line is given by v x x( )= ( )−cos

12 where v is the

velocity (m s–1 ) and x is the displacement (m) from the origin.

a) Determine a(x) where a is the acceleration (m s–2 ) of the object. [2 marks]

b) Use the result from 14a) to determine a (0), given –2π ≤ a(0) ≤ 0. Express your answer in simplest form. [2 marks]

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QUESTION 15 (6 marks)The points O (0, 0, 0), A (–6, 2, –2) and C (3, 1, 2) are represented in three-dimensional space in the diagram.

B

y

x

z

A

C

NM

Not drawn to scale

O

OABC forms a parallelogram in three-dimensional space.

a) Determine the coordinates of B. [1 mark]

M is the midpoint of BC.

b) Determine the vector that represents OM� ���

. [1 mark]

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N divides AM in the ratio 2:1.

c) Determine the vector that represents ON� ���

. [2 marks]

d) Use a vector method to show that O, B and N lie on a straight line. [2 marks]

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QUESTION 16 (6 marks)

Given cos q( )≠ ∀ ∈ +0 n Z , use mathematical induction to prove

cos cos cos coscos

q q q qq

( )− ( )+ ( )−…+ −( ) −( )( )=− −( )+

3 5 1 2 11 1 21n

n

nn(( )

( )2cos q

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QUESTION 17 (7 marks)Determine the smallest positive value of a given

−∫ +( )+ ( )( )+

=a

a x xx

dx12 2

2 11

2

sec tan

cosec

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QUESTION 18 (6 marks)

Consider the function P z z az z az b( )= + + + +2 64 3 2 where a b Z,� ∈ +

One of the roots of P z z i( ) =−is P z z i( ) =−is

Determine the possible value/s for a and b such that all remaining roots of P z( ) have an imaginary component.

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QUESTION 19 (7 marks)A circular-based bowl has been positioned symmetrically on a Cartesian plane as shown in the diagram.

y

x

The bowl has a shape that can be generated by rotating the curve yx

=−−

4

81 about the y-axis

for 4 7 6£ £x . cm.

The bowl is being filled with a liquid at the rate of 7 3 1p cm s- .

Determine the rate at which the depth of liquid is increasing when the depth of liquid reaches one-third of the height of the bowl.

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END OF PAPER

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ADDITIONAL PAGE FOR STUDENT RESPONSESWrite the question number you are responding to.

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ADDITIONAL PAGE FOR STUDENT RESPONSESWrite the question number you are responding to.

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ADDITIONAL PAGE FOR STUDENT RESPONSESWrite the question number you are responding to.

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ADDITIONAL PAGE FOR STUDENT RESPONSESWrite the question number you are responding to.

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Clear zone —

margin trim

med off after com

pletion of assessment

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 © State of Queensland (QCAA) 2020Licence: https://creativecommons.org/licenses/by/4.0 | Copyright notice: www.qcaa.qld.edu.au/copyright — lists the full terms and conditions, which specify certain exceptions to the licence. | Attribution: © State of Queensland (QCAA) 2020