ACE
EXAM
PAPER
Student name:
______________________
YEAR 11
YEARLY
EXAMINATION
PAPER 4
Mathematics Advanced
General
Instructions
–
–
–
–
–
Working time - 120 minutes
Write using black pen
NESA approved calculators may be used
A reference sheet is provided at the back of this paper
For questions in Section II, show relevant mathematical reasoning
and/or calculations
Total marks:
80
Section I – 10 marks
– Attempt Questions 1-10
– Allow about 15 minutes for this section
Section II – 70 marks
– Attempt all questions
– Allow about 1 hour and 45 minutes for this section
1
Year 11 Mathematics Advanced
Section I
10 marks
Attempt questions 1 - 10
Allow about 15 minutes for this section
Use the multiple-choice answer sheet for questions 1-10
2" × 5" is equal to:
(A)
7"
(B)
7'"
(C)
10"
(D)
10'"
There are 200 tickets sold in a raffle. There are two prizes. Ian buys 5 tickets. Which
expression gives the probability that Ian wins both prizes?
5
4
(A)
+
200 199
5
4
(B)
+
200 200
5
4
(C)
×
200 199
5
4
(D)
×
200 199
A spinner is equally divided into n segments and each segment contains a value from
1 to n. If the expected value is 3, then n is equal to which of the following values?
(A) 4
(B)
5
(C)
6
(D)
7
Which of the following is not a function?
(A)
𝑦 = √𝑥
(B)
(C)
𝑥 = 1𝑦
y=2
(D)
x=2
Which of the following is an expression for
(A)
sin𝑥cos𝑥
(B)
tan𝑥
(C)
cos ' 𝑥
(D)
tan𝑥 − sin𝑥cos𝑥
1 − sin' 𝑥
?
cot𝑥
2
Year 11 Mathematics Advanced
Which of the following is the correct expression for differentiating 𝑓(𝑥) = 𝑥 ' − 2𝑥 from
first principles?
(A)
(𝑥 − ℎ)' − 2(𝑥 − ℎ) + (𝑥 ' − 2𝑥)
@→B
ℎ
(B)
(𝑥 − ℎ)' − 2(𝑥 − ℎ) − (𝑥 ' − 2𝑥)
@→B
ℎ
(C)
(𝑥 + ℎ)' − 2(𝑥 + ℎ) + (𝑥 ' − 2𝑥)
@→B
ℎ
(D)
(𝑥 + ℎ)' − 2(𝑥 + ℎ) − (𝑥 ' − 2𝑥)
@→B
ℎ
lim
lim
lim
lim
The line 2𝑥 + 4𝑦 + 8 = 0 cuts the x-axis at A.
The coordinates of A are:
(A)
(–4, 0)
(B)
(0, –4)
(C)
(–2, 0)
(D)
(0, –2)
What is the gradient of the tangent to the curve 𝑦 = 5 + 2𝑥 − 𝑥 ' at the point (–2, –3)?
(A)
–4
(B)
–3
(C)
6
(D)
8
3
Year 11 Mathematics Advanced
What is the value cos𝜃 if sin𝜃 =
(A)
±
(B)
±
(C)
(D)
F
G
and 0 ≤ 𝜃 ≤ 2𝜋 ?
3
7
√33
49
√33
±
49
√33
±
7
What is the maximum value of −𝑥 ' + 𝑥 + 12 ?
(A)
12
(B)
12.25
(C)
12.75
(D)
14
4
Year 11 Mathematics Advanced
Section II
70 marks
Attempt all questions
Allow about 1 hour and 45 minutes for this section
Answer the questions 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 11 (1 mark)
Marks
1
Simplify √32 − √18 + √2
Question 12 (3 marks)
The probability distribution of random variable X is shown below.
x
1
2
3
4
5
P(X= x)
0.1
a
b
0.4
0.2
3
Find the values of a and b if 𝐸(𝑋) = 3.5
Question 13 (2 marks)
Simplify
cos(90° − 𝜃)
sin(90° − 𝜃 )
2
5
Year 11 Mathematics Advanced
Question 14 (4 marks)
Marks
The function 𝑦 = 𝑓(𝑥) is defined as follows:
𝑥−3
𝑓(𝑥) =
for 𝑥 ≤ −3
2𝑥 + 2 for − 3 < 𝑥 < 0
𝑥'
for 𝑥 ≥ 0
(a)
Evaluate 𝑓(2) + 𝑓(−2) + 𝑓(−5).
2
(b)
Draw a sketch of the graph of 𝑦 = 𝑓(𝑥).
2
Question 15 (3 marks)
Amelia tosses two dice with faces numbered 1 to 6. She records the maximum of the
two uppermost faces as a score.
(a) Find the probability that she records the score 2 in a single throw of the two
dice.
(b)
Find the probability that she records the scores 1, 1, 1 in three tosses of the
two dice.
6
2
1
Year 11 Mathematics Advanced
Question 16 (2 marks)
Marks
Find the equation of the axis of symmetry of the parabola 𝑦 = 𝑥 ' − 4𝑥 − 5 and the
minimum value of the expression 𝑥 ' − 4𝑥 − 5.
2
Question 17 (4 marks)
Isla observes a cliff from her boat at position C. She then sails 500 metres closer to
the cliff to position D. The angle of elevation of the cliff-top from C is 5˚ and from D
is 8˚.
(a)
Find ∠𝐶𝐵𝐷.
1
(b)
Use the sine rule to calculate BD to the nearest metre.
2
(c)
Hence or otherwise, find AD, correct to the nearest metre.
1
7
Year 11 Mathematics Advanced
Question 18 (1 mark)
Marks
Find all the values of x with 0˚ ≤ x ≤ 360˚ for which tan𝑥 =
1
.
√3
1
Question 19 (3 marks)
(a)
Rationalise the denominator of:
2
2
2 − √3
(b)
Find integers a and b such that:
2
2 − √3
1
= 𝑎 + √𝑏
Question 20 (2 marks)
Make neat sketches of the following equations on separate sets of axes.
Mark clearly the essential features of each graph.
(a) 𝑦 = −2𝑥 ' + 2
(b)
𝑦 = 2" − 1
1
1
8
Year 11 Mathematics Advanced
Question 21 (4 marks)
Marks
The probability distribution of random variable Z is shown below.
z
–1
0
1
2
3
P(Z= z)
0.25
0.1
0.25
0.3
0.1
Find the
(a)
Expected value
1
(b)
Variance
3
Question 22 (2 marks)
Express
𝑥 − 1 2𝑥 − 3
−
5
9
as a fraction in its simplest form.
2
Question 23 (2 marks)
Find the area of the sector below. Answer to the nearest square centimetre.
9
2
Year 11 Mathematics Advanced
Question 24 (6 marks)
Marks
The points A(6, 4), B(2, –2) and C(–1, 3) are plotted on a number plane.
The point D lies on the y-axis such that AB is parallel to CD.
(a)
Find the length of AB.
1
(b)
Find the gradient of AB.
1
(c)
Show that the equation of AB is 3𝑥 − 2𝑦 − 10 = 0.
1
(d)
Find the equation of DC.
2
(e)
Find the coordinates of D.
1
10
Year 11 Mathematics Advanced
Question 25 (2 marks)
Marks
What is the radius of the circle 𝑥 ' + 𝑦 ' − 4𝑥 + 8𝑦 + 11 = 0 ?
2
Question 26 (2 marks)
A box contains 8 blue and 11 red balls. Oscar randomly selects three balls one at a
time and without replacement.
(a)
Draw a tree diagram showing the sample space.
1
(b)
What is the probability that he selects “red, blue, red” in that order?
1
Question 27 (3 marks)
Consider the parabola 8𝑦 = 𝑥 ' − 2𝑥 − 7
(a)
Find the coordinates of the vertex.
2
(b)
Find the coordinates of the focus.
1
11
Year 11 Mathematics Advanced
Question 28 (2 marks)
Marks
Prove that sin 𝜃 cos 𝜃 tan 𝜃 = 1 − cos ' 𝜃.
2
Question 29 (6 marks)
Differentiate
(a)
6𝑥 ' − 𝑥 + 2
(b)
[
1
3
1
√𝑥
(c)
12𝑥 ' − 2
1
(d)
(𝑥 ' − 3)(𝑥 − 4)
1
(e)
𝑥
7 − 3𝑥
2
12
Year 11 Mathematics Advanced
Question 30 (4 marks)
Marks
Solve
(a)
2'"\] = 32
1
(b)
log _ 𝑥 = 4
1
(c)
log " 4 + 2log " 8 = 4
2
Question 31 (2 marks)
For what values of d does the equation 𝑥 ' + (𝑑 − 6)𝑥 + 1 = 0 have no real roots?
2
Question 32 (3 marks)
A particle is moving in a straight line such that its displacement x m from a fixed
point O on the line at time t seconds is given by 𝑥 = 𝑡 _ − 12𝑡 + 11. What is the
particle’s initial displacement, velocity and acceleration?
13
3
Year 11 Mathematics Advanced
Question 33 (4 marks)
Marks
The number of frogs, N, in a park at time t weeks, is given by the formula:
𝑁 = 𝑁B 𝑒 B._Fdd e where 𝑁B and k are constants.
Initially there were 100 frogs in the park.
1
(a)
What is the value of 𝑁B ?
(b)
How many frogs were in the park after 10 weeks? Answer correct to two
significant figures.
1
(c)
Find the rate of increase in the number of frogs at 10 weeks. Answer correct
to the nearest whole number.
2
Question 34 (3 marks)
Evaluate the following limits
(a)
(b)
lim (16 − 𝑥 ' )
1
"→F
lim g
"→f
𝑥−5
h
2𝑥 ' − 9𝑥 − 5
2
End of paper
14
Year 11 Mathematics Advanced
Section II extra writing space
If you use this space, clearly indicate which question you are answering.
15
Year 11 Mathematics Advanced
16
Year 11 Mathematics Advanced
17
Year 11 Mathematics Advanced
18
Year 11 Mathematics Advanced
19