NAME:
Student Number:
SPECIALIST MATHEMATICS UNIT 3
School Assessed Coursework 1(SAC 1)
ITEM RESPONSE ANALYSIS TASK
Reading time: 5 minutes
Writing time: 70 minutes
QUESTION & ANSWER BOOK
Structure of Book
Number of Questions
8
Number of questions to
be answered
8
Number of marks
54

Students are permitted to bring into the SAC room: pens, pencils, highlighters, erasers,
sharpeners, rulers, a protractor, set-squares, aids for curve sketching, one bound
reference, one approved CAS calculator (memory DOES NOT need to be cleared) and,
if desired, one scientific calculator. For approved computer-based CAS, their full
functionality maybe used.

Students are NOT permitted to bring into the SAC room: blank sheets of paper and/or
white out liquid/tape.
Materials supplied

Question and answer book.
Instructions

Write your name in the space provided above on this page.

All written responses must be in English.
Students are NOT permitted to bring mobile phones and/or any other unauthorized
electronic devices into the test room.
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PRACTICE QUESTIONS
ITEM 1 (6 marks)
If z  rcis , then
A
rcis
B
rcis ( )
C
rcis (  )
D
1
cis ( )
r
E
1
cis
r
1
in polar form is:
z
a) Explain why responses A and B are incorrect.
(1 mark)
b) i) Find z in polar form.
(1 mark)
ii) Explain which response can be eliminated now.
c) i) Use de Moivre’s theorem to find
ii)
(1 mark)
1
.
z
Identify the correct response.
(2 marks)
d) A student has selected response D. What mistake may have been made?
(1 mark)
2
ITEM 2 (8 marks)
Consider the function f ( x)  3arccos  2 x  

2
. The domain and range of f ( x) are
respectively:
A
 1 1
 3 3 
x    ,  , y   , 
 2 2
 2 2 
B
 1 1
  5 
x   ,  , y   , 
 2 2
 2 2 
C
  5 
x  [1,1], y    , 
 2 2 
D
 3 3 
x  [1,1], y    , 
 2 2 
E
 1 1
  
x   ,  , y   , 
 2 2
 2 2
a) i) What steps do you need to take to determine the domain of f(x)?
(1 mark)
ii) Explain why response A is incorrect.
(1 mark)
iii) Which other two responses have incorrect domain?
(1 mark)
b) i) List the transformations which affect the range in their correct order.
(1 mark)
ii) Apply the transformations to the interval  0,  
(2 marks)
iii) Which responses have correct range?
(1 mark)
iv) What mistake was made to obtain the range in response D?
(1 mark)
3
ITEM 3 (7 marks)
Given that cis(3 )  (cos3   3cos  sin 2  )  i(3cos2   sin 3  ), it follows that cos(3 ) is
equal to
A
cos3 
B
4 cos   3cos3 
C
3cos 2  sin   sin 3 
D
4 cos3   3cos 
E
(2cos3   3cos  )
a) i) Is the identity (cis )3  cis(3 ) true?
ii) What theorem was applied?
iii)
Is the equation (cos  )3  cos(3 ) true? Explain.
iv)
Which answer can be eliminated?
(4 marks)
b) How can cos(3 ) be evaluated from the complex number given?
(1 mark)
c) What trigonometric identity do we need to use to arrive at cos(3 ) in terms of cos
only?
(1 mark)
d) Which is the correct response?
(1 mark)
4
ITEM 4 (6 marks)
The principal argument of
3 2  i 6
is
2  2i
13
12
A

B
7
12
C
11
12
D
13
12
E

11
12
a) Explain why responses A and D are incorrect
(1 mark)
b) i) Find 3 2  i 6 in polar form
ii) Find 2  2i in polar form.
(2 marks)
c) i) Find
ii)
3 2  i 6
in polar form.
2  2i
Write down the principal argument.
(2 marks)
d) Identify the correct response.
(1 mark)
5
ITEM 5 (5 marks)
Which of the following five expressions is not identical to any of the others?
A
cos 4   sin 4 
B
1  cos
C
cos 2
D
2 cos 2
E
1  cos

2
a) i) Simplify cos 4   sin 4 
ii) Which expression is it equal to?
(2 marks)
b) i) Use a double angle formula to express 2 cos
2
2
in terms of cos
ii) Pair up two other responses
(2 marks)
c) Identify the response with no pair.
(1 mark)
ITEM 6 (8 marks)
If z  1  i is one root of z 2  z  1  i  0 , then the other root is
A
1
B
i
C
1 i
D
1  i
E
1  i
a) i) Explain why the conjugate root theorem does not apply here.
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ii) Which response can be eliminated?
(2 marks)
b) i) Expand ( z  1)( z  1  i )
ii) Which response can be eliminated now?
(2 marks)
c) Use long division to find the other root.
(3 marks)
d) Identify the correct response.
(1 mark)
7
ITEM 7 (6 marks)
The equation representing the graph shown is
y2
1
4
A
x2 
B
x2
 y2  1
2
C
( x  2) 2
 y2  1
4
D
x2
 y2  1
4
E
x2
 y2  1
4
a) Which two responses can be eliminated and why?
(2 marks)
b) i) Find the distance between the vertices.
ii) Which response can be eliminated now?
(2 marks)
c) i) Write down the coordinates of the centre of the hyperbola.
ii) Identify the correct response.
(2 marks)
END OF PRACTICE ITEM RESPONSE
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