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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