a

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ABBOTSLEIGH
General Instructions Total marks – 66
• Working time – 70 minutes
• Write using black pen.
• NESA approved calculators may be used.
• A NESA Reference sheet is provided.
• Answer all questions in the spaces
provided.
• Answer the Multiple-Choice questions in
Section I on the answer sheet provided.
• Extra writing spaces are provided at the
end of the task. Please indicate the
question you are answering if you use
this space.
• In Questions 6 – 8, show relevant
mathematical reasoning and / or
calculations. Bold answers will not be
awarded full marks.
• Attempt Sections I and II
• Marks are as indicated.
5 marks
• Attempt Questions 1–5
• Allow about 5 minutes for this section.
61 marks
• Attempt Questions 6 - 9
• Allow about 65 minutes for this section.
Student Name:
Student Number:
Teacher Name:
Mathematics Advanced
Section I Pages 3 - 4
Section II Pages 5 - 28
2021
Year 11 Preliminary
Assessment Task 1

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Outcomes to be assessed:
Advanced Mathematics
Preliminary:
A student
MA11-1 uses algebraic and graphical techniques to solve, and where appropriate, compare alternative
solutions to problems
MA11-2 uses the concepts of functions and relations to model, analyse and solve practical problems
MA11-8 uses appropriate technology to investigate, organise, model and interpret information in a
range of contexts
MA11-9 provides reasoning to support conclusions which are appropriate to the context

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Section I
5 marks
Attempt Questions 1 – 5
Use the multiple-choice answer sheet
Select the alternative A, B, C or D that best answers the question. Fill in the response oval
completely.
Sample 2 + 4 = (A) 2 (B) 6 (C) 8 (D) 9
If you think you have made a mistake, put a cross through the incorrect answer and fill in the new
answer.
If you change your mind and have crossed out what you consider to be the correct answer, then
indicate this by writing the word correct and drawing an arrow as follows.
1 What is the gradient of the line 1
3 2
x y
+ =
?
(A)
3
2
− (B)
2
3
− (C)
2
3
(D)
3
2
2 Which of the following is equal to
1
3 2 5
−
?
(A) 3 2 5
− (B) 3 2 5
+ (C)
3 2 5
13
−
(D)
3 2 5
13
+
Section I continues on page 4
(A) (B) (C) (D)
(A) (B) (C) (D)
correct
(A) (B) (C) (D)

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Section I (continued)
3 What is the domain of the function
1
( )
2
f x x
x
= +
−
?
(A) ( )
0,2 (B) [ )
0,2 (C) ( ]
0,2 (D) [ ]
0,2
4 Which of the following is equal to
4 1
2 1
n
n
−
−
?
(A) 2 1
n
+ (B) 2 1
n
− (C) 2
2 (D) 2
5 Below is a graph of 2 3
2 4
x xy y
+ − =
.
What type of relation is this?
(A) one-to-one (B) one-to-many
(C) many-to-one (D) many-to-many
End of Section I

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Section II
61 Marks
Attempt Questions 6 – 9
Marks are as indicated
In Questions 6 – 9, your responses should include relevant mathematical reasoning and/or
calculations.
Question 6 (17 marks) Marks
(a) Expand and simplify 2 ( 3 ) 4 (2 3 ).
a a b a a b
− + − 2
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(b) What angle does the line 4 12 0
x y
− + + = make with the positive x-axis? 2
Round to the nearest minute.
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Question 6 continues on page 6
Student Number:

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Question 6 (continued) Marks
(c) Find any values of k such that the polynomial 2 3 2
( ) (1 ) 2 1
P x k x kx x
= − + − + 2
is monic with degree 3.
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(d) If
( 3)(2 1)
( )
x x
f x
x
+ +
= , show that ( )
f x can be written in the form
3 1 1
2 2 2 .
Ax Bx Cx
−
+ + 4
Find the values of A, B and C.
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Question 6 continues on page 7

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Question 6 (continued) Marks
(e) Simplify
2 8
20 .
5 80
+ + 2
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(f) For what value of k are the lines ,
y kx
= 1 0
x y
+ − = and 3 0
x y
− + = concurrent? 2
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Question 6 continues on page 8

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Question 6 (continued) Marks
(g) Sarah has a small manufacturing business.
The cost of manufacturing is given by the equation 50 200
C x
= + and the income earned
is given by the equation 100 ,
I x
= where x is the number of items that the business has
manufactured.
(i) Graph each of these equations on the grid below. 2
(ii) How many items need to be manufactured for the business to break even? 1
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End of question 6

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Question 7 (15 marks)
Marks
(a) Simplify
2
1
2 1 .
1 1
t
t t
 
+ − +
 
+ +
 
2
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(b) A(‒1, 16), B(‒3, 2) and C(5, ‒2) are three points. D is the midpoint of AB 3
and E is the midpoint of AC. Show that DE is parallel to BC.
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Question 7 continues on page 12
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Question 7 (continued) Marks
(c) A rectangular pool is 50% longer than it is wide. It is surrounded by a path 2 m wide. 3
If the area of the path is 116 m2
, find the area of the pool.
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(d) A piecewise function is defined below: 3
( )
2
( 2) when 0
4 when 0 2
4 when 2
c x x
f x ax x
bx x
 − <

= + ≤ ≤

 − >

Find the values of a, b and c if (3) 5, (1) 3
f f
= = and ( 2) 8.
f − =
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Question 7 continues on page 13

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Question 7 (continued) Marks
(e) Calculate a and b if
5 3
.
5 3
a b
−
= −
+
2
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(f) Solve: 2
2 2
( 3 10)( 3 4) 0
x x x x
− − − − =
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End of question 7

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Question 8 (15 marks)
(a) The diagram below shows a straight-line l that passes through two points A (‒2, 9) Marks
and M (1, 3). M is the midpoint of AB.
(i) Determine the gradient of the line l. 1
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(ii) Show that the equation of the line l is 2 5 0
x y
+ − =. 2
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(iii) Find the coordinates of B. 2
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Question 8 continues on page 18
NOT TO
SCALE
Student Number:
y
x

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Question 8 (continued) Marks
(iv) Line p is perpendicular to line l and has a y-intercept at ‒6. 2
Does line p pass through point B ? Show working to justify your answer.
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Question 8 continues on page 19

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Question 8 (continued) Marks
(b) Solve the following equations simultaneously: 3
1
2 3
5
5 4
x y
x y
− =
+ =
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Question 8 continues on page 20

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Question 8 (continued) Marks
(c) (i) Factorise the function ( ) 2
2 3.
f x x x
= + − 2
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(ii) In the space below graph the function ( ) 2
2 3.
f x x x
= + − . 3
Show all important features.
End of Question 8

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Question 9 (14 marks)
Marks
(a) By expressing 2
( ) 2 6 9
f x x x
= + − in the form 2
( )
a x h c
− + find the minimum value 2
of the function.
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Question 9 continues on page 24
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Question 9 (continued) Marks
(b) Find the value(s) of k for which the equation 2
( 3) 4 0
x k x k
− + + =
has:
(i) Equal roots. 2
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(ii) Roots that are equal in magnitude but opposite in sign. 1
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Question 9 continues on page 25

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Question 9 (continued) Marks
(c) Given that ( ) 2
( 6 )( 3) 2 .
f x x x x x
= − − +
(i) Express f (x) in the form 2
( ),
x ax bx c
+ + where a, b and c are constants. 2
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(ii) Hence factorise f (x) completely. 1
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(iii) Sketch the graph of ( ),
y f x
= showing all x and y intercepts. 2
Question 8
Question 9 continues on page 26

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Question 9 (continued) Marks
(d) The shaded shape is called an arbelos. It consists of two smaller semicircles, with
centres P and Q, removed from a large semicircle, radius R and centre O.
The ratio of the radii of the semicircles is 1 : 2 : 3.
(i) Show that the equation for the area A of the arbelos in simplest form is given 3
by
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(ii) Hence, find the radius R if 2
98 cm .
A π
= 1
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End of Paper
NOT TO
SCALE
2
2
.
9
R
A
π
=

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