Western Mathematics Exams
2022
TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION
Mathematics Extension 1
General
Instructions
• Reading time – 10 minutes
• Working time – 2 hours
• Write using black pen
• Calculators approved by NESA may be used
• A reference sheet is provided at the back of this paper
• In Questions in Section II, show relevant mathematical reasoning
and/or calculations
•
Total
Marks: 70
Write your Name and Student Number on the Question XX
Writing Booklet attached
Section I – 10 marks (pages 2 – 5)
• Attempt Questions 1 – 10
• Allow about 15 minutes for this section
Section II – 60 marks (pages 6 – 11)
• Attempt Questions 11 – 14
• Allow about 1 hour and 45 minutes for this section
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.
1
What is the derivative of
with respect to x?
A.
B.
C.
D.
2
The probability that it will rain on any given day in February 2024 is 0.2.
What is the probability that February 2024 (a leap year) will have exactly 8 rainy days?
A.
B. 9 %
C. 10 %
D. 29 %
2
3
4
–1
What is the domain and range of the function y = 6 cos (3x) ?
A. Domain
; Range
B.
Domain
; Range
C.
Domain
; Range
D. Domain
; Range
What is the value of
, correct to 3 significant figures?
A. –1.05
B. –0.955
C. 0.955
D. 1.05
5
between x = 0 and x = p is rotated around
3
Find the volume generated when
the x-axis.
2
A.
p – 3 cubic units.
6 8
B.
p – 3 p cubic units.
6
8
C.
p – 3p cubic units.
3 4
D.
p – 3 p cubic units.
6
4
2
2
2
3
6
A Year 12 Biology class consists of 10 girls and 15 boys.
In how many ways can a group of three boys and two girls be chosen from this class for a
group to work on an investigation project?
A. 250
B. 900
C. 12 600
D. 20 475
7
A particle moves in a straight line. Its position at any time t is given by
What is the acceleration of the particle in terms of x?
A.
..
x=
B.
–3
x
–4
..
x= x
2
..
C. x = –16x
D.
8
..
x=
–6
cos 2x + 8 sin 2x
A curve is described by the parametric equations below.
What is the cartesian equation of the curve?
A.
B.
C.
D.
4
9
Which set of values for a and b that satisfy the following equation?
A. a = 0 and b = 4
B. a = 1 and b =
C. a =
and b = 1
D. a = 4 and b = 0
10
A Bernoulli distribution is described below:
.
What are the mean and variance of the distribution?
A.
B.
C.
D.
5
Section II
60 marks
Attempt Questions 11 – 14.
Allow about 1 hour and 45 minutes for this section.
Answer each question in the appropriate writing booklet. Extra writing booklets are available.
In Questions 11 – 14, your responses should include relevant mathematical reasoning and/or
calculations.
Question 11 (15 marks) Use the Question 11 writing booklet.
(a)
Show that
satisfies the differential equation
2
for any real values of A and B.
(b)
In a large population of birds, the proportion of Cockatoos is
Let
2
be the random variable that represents the sample proportion of Cockatoos for
samples of size n drawn from the population.
Find the smallest integer value of n such that the standard deviation of
is less than
or equal to 1%.
(c)
When the polynomial P(x) is divided by
the remainder is
2
What is the remainder when P(x) is divided by x – 1?
(d)
Using the substitution
find
Question 11 continues on page 7
6
3
Question 11 (continued)
(e)
(f)
A function is defined by
.
(i)
Find the vertical and horizontal asymptotes for
.
(ii)
Find the inverse function
(iii)
State the domain of
2
1
.
.
1
Find the equation of the curve f(x) that passes through the point
End of Question 11
7
and has
2
Question 12 (14 marks) Use the Question 12 writing booklet.
Rylie has a boat which moves at a top speed of 12 ms-1 in still water.
(a)
From point R, he wants to go due north to point D on the opposite side of the river, as
shown in the diagram below.
Today the current in the river is flowing at 5 ms-1. From R, he steers the boat due north
toward D at top speed. Due to the current he drifts down the river and arrives at
point E.
(i)
and
2
(ii)
What is the bearing of Rylie’s velocity vector and how far does he travel from R to E ?
2
(iii)
On what bearing should Rylie have pointed the boat, so that he arrived at the D, with
the boat travelling at its top speed?
2
Find
2
Taking R as the origin, write down Rylie’s velocity vector in the form
find the magnitude of this vector.
(b)
.
(c)
Solve the differential equation below, to give an equation for y in terms of x, given that
y (0) = 0.
3
(d)
Prove by mathematical induction that
integers n.
3
is divisible by 3 for all positive
End of Question 12
8
Question 13 (15 marks) Use the Question 13 writing booklet.
(a)
Solve
(b)
The diagram below shows a vector
(c)
A circular metal plate is heated so that its diameter is increasing at a constant rate of
0.005 m/s.
2
and a point P that divides
in a ratio m : n.
2
2
At what rate is the area of the circular surface of the plate increasing when its diameter
is 6 metres? (Answer in m2/s, correct to 2 significant figures.)
Megan has taken a frozen quiche from the freezer with an initial temperature T = 0o C
and placed it in an oven set to 200o C. After ten minutes she checks the quiche and now
it has a temperature of T = 20o C. The rate that the temperature of the quiche increases
is proportional to the difference in temperature between the oven and the quiche.
(d)
(e)
(i)
Write a differential equation to model this scenario and use it to show that
𝑇 = 200 + 𝐴𝑒 !"# .
2
(ii)
Find the value of k in exact form.
2
(iii)
The quiche is ready when the internal temperature reaches 60o C. How much longer
after Megan checked the quiche will it be ready? Answer to the nearest minute.
2
Sahara arranges 2n + 2 different objects taken n at a time and Casey arranges 2n
different objects taken n at a time.
3
If the number of objects are in the ratio of 18 : 5, find the value of n.
End of Question 13
9
Question 14 (16 marks) Use the Question 14 writing booklet.
(a)
A region is bounded by the line
and a curve of the form
where n can be an even integer and
,
.
This region is rotated about the x-axis to form a solid.
3
(i)
Show the volume of the solid formed is
(ii)
Find
(iii)
Explain this result in geometric terms.
.
1
Vn .
1
Question 14 continues on page 11
10
Question 14 (continued)
(b)
A particle is projected horizontally from a point P, h metres above O, with a velocity
of V metres per second. The only acceleration that the particle undergoes is the
acceleration due to gravity (-g) vertically in a downward direction. There is no
horizontal acceleration.
(i)
Find
(ii)
A package of supplies is dropped from a plane to a disabled sailboat (S) in the ocean.
The plane is travelling at a constant velocity of 252 km/h and is 150 m above sea level
and directly due west of the disabled sailboat.
the displacement vector of the particle at time t.
2
2
How long will it take the package to hit the water? (Take g = -10 m/s2)
(iii)
(c)
A current is causing the sailboat to drift at a speed of 1.8 km/h in the same direction as
the plane is travelling. Between what two values (distances) can the plane drop the
package so that it lands at most 40 m from the disabled sailboat?
3
An archery game is played by shooting arrows at a target. Each person can choose to
have two or three shots at the target. Oscar decides to play two games. In the first game
he chooses to shoot two arrows and he wins if he hits the target at least once. In the
second game, he chooses to shoot three arrows and wins if he hits the target at least
twice. The probability that Oscar can hit the target on any shot is p, where 0 < p < 1.
4
The probability that Oscar wins Game 1 is
and the probability that he wins
Game 2 is
Prove that Oscar is more likely to win Game 1 than Game 2 and find the exact value
of p for which Oscar is twice as likely to win Game 1 than he is to win Game 2.
End of paper
11
12
Student Name
Student Number
Western Mathematics Exams
2022
TRIAL HIGHER SCHOOL CERTIFICATE
EXAMINATION
Mathematics Extension 1
Writing Booklet
Question XX
Instructions
•
Use this booklet for Question XX
§
Write the number of this booklet and the number of
booklets that you have used for this question (e.g. 1 of 3 )
XX
of
§
Write your name and Student Number at the top of this
page.
§
Write in a black pen
§
You may ask for an extra writing booklet if you need more
space
§
If you have not attempted the question(s), you must still hand in the writing
booklet with “NOT ATTEMPTED” written clearly on the front cover.
§
You may not take any writing booklets, used or unused, from the examination
room.
this
booklet
number of
booklets
for this
question
Start here for
Question Number XX
Tick this box if you have continued this answer in another writing booklet.
2022
TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION
Mathematics Advanced
Mathematics Extension 1
Mathematics Extension 2
REFERENCE SHEET
2022 Trial HSC Examination
Mathematics Extension 1 Course
Name ________________________________
Teacher ________________________
Section I – Multiple Choice Answer Sheet
Allow about 15 minutes for this section
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
A
B
C
D
If you think you have made a mistake, put a cross through the incorrect answer and fill in the new
answer.
A
B
C
D
If you change your mind and have crossed out what you consider to be the correct answer, then
indicate the correct answer by writing the word correct and drawing an arrow as follows.
A
B
1.
A
B
C
D
2.
A
B
C
D
3.
A
B
C
D
4.
A
B
C
D
5.
A
B
C
D
6.
A
B
C
D
7.
A
B
C
D
8.
A
B
C
D
9.
A
B
C
D
10.
A
B
C
D
C
D