ACE
EXAM
PAPER
2020
Student name: ______________________
YEAR 12
HALF YEARLY
EXAMINATION
Mathematics Advanced
General
Instructions
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Working time - 90 minutes
Write using black pen
NESA approved calculators may be used
A reference sheet is provided at the back of this paper
In section II, show relevant mathematical reasoning and/or calculations
Total marks:
50
Section I – 5 marks
– Attempt Questions 1-5
– Allow about 8 minutes for this section
Section II – 45 marks
– Attempt all questions
– Allow about 82 minutes for this section
1
Year 12 Mathematics Advanced
Section I
5 marks
Attempt questions 1 - 5
Allow about 8 minutes for this section
Use the multiple-choice answer sheet for questions 1-5
1.
The correlation coefficient (r) was calculated for the data displayed in the above scatterplot.
It was found to be r = –0.6. If the point (10, 2) was replaced with the point (70, 2) and the
correlation coefficient, r, recalculated, then the value of r would be:
(A)
Unchanged
(B)
Negative but closer to 0
(C)
Negative but closer to –1
(D) Negative but closer to 1
2. The diagram below shows the graph of y = f(x).
Which of the following statements is false?
(A)
The horizontal asymptote is y = 1.
(B)
The curve is continuous.
(C)
The curve is concave up for x < –1.
!"#
(D) The equation of the function is 𝑦 = !$%.
2
Year 12 Mathematics Advanced
3. The following cumulative frequency table shows the results of a test out of 15.
Score (x)
9
10
11
12
13
14
15
Frequency (f)
5
4
7
2
2
8
4
Cumulative Frequency
5
9
16
18
20
28
32
What is the median?
(A)
11.5
(B)
12
(C)
13
(D)
14
4.
What is a possible equation for the above graph?
π
π
𝑦 = 2cos3 (𝑥 + - − 4
(A) 𝑦 = 2cos3 (𝑥 + - − 2
(B)
4
4
π
π
𝑦 = 3cos2 (𝑥 + - − 4
(C) 𝑦 = 3cos2 (𝑥 + - − 2
(D)
4
4
5. For 𝑦 = /1 − 𝑓(𝑥)
(A)
(C)
−1
/1 − 𝑓′(𝑥)
3
2(1 − 𝑓(𝑥 ))
𝑑𝑦
is equal to:
𝑑𝑥
2𝑓′(𝑥)
(B)
/1 − 𝑓 (𝑥)
−𝑓′(𝑥)
(D)
2/1 − 𝑓(𝑥 )
3
Year 12 Mathematics Advanced
Section II
45 marks
Attempt all questions
Allow about 1 hour and 22 minutes for this section
Answer each question in the spaces provided.
Your responses should include relevant mathematical reasoning and/or calculations.
Extra writing space is provided at the back of the examination paper.
Question 6 (2 marks)
Marks
π
Solve the equation 2sin (𝑥 + - = 1 where 0 ≤ 𝑥 ≤ 2π.
6
2
Question 7 (3 marks)
Sketch the curve of the following equations on separate diagrams. Show on each
sketch the coordinates of each point at which the graph meets the axes.
(a) 𝑦 = (𝑥 + 2)#
(b)
𝑦 = (𝑥 + 2)# + 𝑘 where k is a positive constant.
4
1
2
Year 12 Mathematics Advanced
Question 8 (3 marks)
Marks
The number of students absent from year 12 for the past nine days was as follows:
17, 23, 20, 21, 16, 15, 32, 18, 21
(a)
What is the mean? Answer correct to one decimal place.
1
(b)
Find the interquartile range?
1
(c)
Is 32 an outlier for this set of data?
Justify your answer with calculations.
1
Question 9 (5 marks)
Differentiate with respect to x.
(a)
𝑥 & + 6√𝑥
1
(b)
(𝑥 + 4)#
𝑥
2
(c)
𝑒 #! ln𝑥
2
5
Year 12 Mathematics Advanced
Question 10 (4 marks)
Marks
For the trigonometric function 𝑦 = 4sin(2𝑥 + π) − 3:
(a)
Find the amplitude, period, phrase and vertical shift.
2
(b)
Draw a sketch of the curve for one period.
2
Question 11 (3 marks)
Given 𝑦 = 𝑒 ! sin𝑥.
𝑑𝑦
(a) Find
𝑑𝑥
(b) Find
(c)
1
𝑑# 𝑦
𝑑𝑥 #
Show that
1
𝑑# 𝑦
𝑑𝑦
−2
+ 2𝑦 = 0
#
𝑑𝑥
𝑑𝑥
1
6
Year 12 Mathematics Advanced
Question 12 (4 marks)
Marks
The shoe size and height of eight students is recorded in the table below.
Shoe size, s
Height, h
6
7
7.5
8
8
8.5
9
9.5
150
153
163
165
172
174
184
188
A scatterplot of the data is shown below.
(a)
What is Pearson’s correlation coefficient? Answer correct to 4 decimal places.
1
(b)
Find the equation of the least-squares line of best fit in terms of show size (s)
and height (h). Answer correct to 2 decimal places.
1
(c)
What is the height difference between a student who wears a size 6 shoe and
one who wears a size 9 shoe. Answer correct to the nearest whole number.
2
7
Year 12 Mathematics Advanced
Question 13 (3 marks)
Marks
Find how many solutions exist for the equation 𝑥 ' − 𝑥 # − 𝑥 = 0, by drawing graphs.
3
Question 14 (3 marks)
The diagram above shows the curves with equations 𝑦 = 4! and 𝑦 = 3#"!
intersecting at point A.
(a)
Show that the x-coordinate of A can be written in the form:
log ( 𝑝
for all a > 1
log ( 𝑞
2
(b)
Calculate the y-coordinate of A. Answer correct to one decimal place.
1
8
Year 12 Mathematics Advanced
Question 15 (3 marks)
Marks
(a)
Draw graphs to find the number of solutions for tan𝑥 = −√3, in the domain
π ≤ 𝑥 ≤ π.
2
(b)
What are the solutions to the equation tan𝑥 = −√3?
1
Question 16 (3 marks)
Draw and clearly label the graphs of 𝑦 = 𝑒 ! , 𝑦 = 𝑒 !"% and 𝑦 = 2𝑒 ! .
9
3
Year 12 Mathematics Advanced
Question 17 (3 marks)
Marks
The speeds of a vehicle are shown below in the cumulative frequency histogram.
(a)
What is the frequency of 75 km/h?
1
(b)
Construct a cumulative frequency polygon (or ogive) on this graph.
1
(c)
Use the graph to estimate the median.
1
Question 18 (2 marks)
If 𝑓(𝑥) = (sin𝑥 − 1)) find 𝑓′(𝜋).
2
10
Year 12 Mathematics Advanced
Question 19 (4 marks)
Marks
The curve 𝑦 = cos2𝑥 in the domain 0 ≤ 𝑥 ≤ π is shown above. It cuts the x-axis at A
and B. The line 𝑦 =
√'
#
is also shown and it meets the curve at C and D.
(a)
What are the coordinates of A and B?
2
(b)
What are the coordinates of C and D?
2
End of paper
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Year 12 Mathematics Advanced
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