2023normanhurst BHS Year 12 - Ext 1 Assessment Task 4 Yearly Trial Questions
2023normanhurst BHS Year 12 - Ext 1 Assessment Task 4 Yearly Trial Questions
2023normanhurst BHS Year 12 - Ext 1 Assessment Task 4 Yearly Trial Questions
Class (please ✔)
MARKS
10 15 17 17 11 70
2 2023 Year 12 Mathematics Extension 1 Course Assessment Task 4 (Trial Examination)
Section I
10 marks
Attempt Question 1 to 10
Allow approximately 15 minutes for this section
Mark your answers on the answer grid provided (labelled as page 11).
Questions Marks
2. It is given that f (x) = ex+2 . Which of the following is the inverse function f −1 (x)? 1
3. How many numbers greater than 2100 can be formed with the digits 1, 2, 3, 4 if no 1
digit is to be used more than once?
1
4. How many solutions does the equation x 3 = |x − 2| − 3 have? 1
5. P QRST U is a regular hexagon with side lengths of 4 cm and is divided into six 1
# » # » # »
equilateral triangles. It is given that P Q = a, P X = b and P U = c, as shown in
the diagram below. e e e
Q R
a
e
b
P e S
X
c
e
U T
Which of the following is the value of a · (a + b + c)?
e e e e
(A) 8 (B) 16 (C) 32 (D) 48
7. Which of the following differential equations best corresponds to the slope field 1
shown below?
1 1
(A) y ′ = (−x2 + y 2 ) (C) y ′ = (x2 − y 2 )
4 4
1 1
(B) y ′ = − (x2 + y 2 ) (D) y ′ = (x2 + y 2 )
4 4
cos x
8. Which diagram best represents the graph y = ? 1
x
(A) (C)
y y
x
x
(B) (D)
y y
x
x
9. A particle is moving in a straight line such that its displacement, x, from the origin 1
after t seconds is given by x = −3 cos2 t.
5π
Which of the following best describes the motion of the particle when t =
6
(A) The particle is moving in the positive direction with an increasing speed
(B) The particle is moving in the positive direction with a decreasing speed
(C) The particle is moving in the negative direction with an increasing speed
(D) The particle is moving in the negative direction with a decreasing speed
Section II
60 marks
Attempt Questions 11 to 14
Allow approximately 1 hours and 45 minutes for this section.
Write your answers in the writing booklets supplied. Additional writing booklets are available.
Your responses should include relevant mathematical reasoning and/or calculations.
4
3
2
1
x
−2 −1 1 2 3 4
−1
−2
i. Without using calculus, write down a possible equation of y = P (x). 1
ii. Solve P (x) ≤ 0 for x. 2
(e) There are twelve people that need to be divided into groups.
i. In how many ways can they be divided into two groups of six people? 1
ii. In how many ways can they be divided into two groups of six people if two 1
particular people must be in the same group?
iii. In how many ways can they be seated around one circular table, if the two 2
groups of six people from (i) must be seated among their own groups?
(b) It is given that t = tan A and p cos 2A − sin 2A = 1, where p ∈ R. Show that 3
p−1
t=
p+1
π
where 0 ≤ A ≤ .
2
Z
1 √
(c) By using the substitution x = (u2 + 5), find 3x 4x − 5 dx. 3
4
# » # »
(d) The diagram below shows OA = a and OB = b and θ is the acute angle between
# » # » e e
OA and OB.
A
a
e
B
θ b
b
O P e
projb a
ee
i. By using the fact that a · b = a b cos θ, or otherwise, prove that 2
e e e e
a·b
projb a = e e b
ee b·b e
e e
−2 3
It is given that a = and b = .
e 5 e 6
ii. Find a perpendicular vector to b. 1
e
iii. Hence, or otherwise, find the shortest distance from the point A to the line 2
OB by using vector methods.
(e) A fish at point A wants to swim across the river that is 70 m wide. The banks of 3
the river are parallel and the points A and B are on opposite sides of the river.
B b
N
1.2 m/s
70 m
A
The fish swims at a speed of 1.8 m/s and the river is flowing downstream at 1.2
m/s. However, a 0.5 m/s wind blows constantly to the north-westerly direction.
What is the bearing that the fish should head to in order to land directly at
point B from point A?
(a) Prove by mathematical induction that for all positive integer values of n, 3
12 × 21 + 22 × 22 + 32 × 23 + · · · + n2 × 2n = (n2 − 2n + 3) × 2n+1 − 6
Initially, the beaker of chemical had a temperature of T1 = 120◦ C and the barrel of water
had a temperature of T2 = 22◦ C. Ten minutes later, the temperature of the beaker of
chemical had fallen to 90◦ C.
iv. Show that 2
7
T1 = 64 + 56e− 4 kt
v. Show that eventually the beaker of chemical and the surrounding barrel of 2
water reach the same temperature.
Mr Kim found out that the number of molecules N in the beaker of chemical is inversely
1000
proportional to the temperature of the beaker, T1 , which is given by N = .
T1
vi. Show that the rate of growth of N is 3
dN 7k 64
= N 1− N
dt 4 1000
vii. Hence, or otherwise, find the maximum rate of growth of N given that 2
k = 0.001.
(a) The diagram shows an inclined plane that makes an angle of β with the 3
horizontal. A projectile is fired from O, at the bottom of the incline, with a
speed of V m/s at an angle of elevation α to the horizontal, as shown below.
y
V
T
α
β
b
x
O
b
3π
3π
2
b x
−1 1
Find the exact volume of the solid of revolution formed if the area bounded by
y = 3 cos−1 x, the coordinate axes and the line x = −1 is rotated about the
y-axis.
y = sin x cos x
y = mx
where m < 0 .
End of paper.
NESA STUDENT #: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Class (please ✔)
• Mark only one circle per question. There is only one correct choice per question.
• Fill in the response circle completely, using blue or black pen, e.g.
A B ● D
• If you think you have made a mistake, put a cross through the incorrect answer and fill in
the new answer.
A B ● ●
• If you continue to change your mind, write the word correct and clearly indicate your
final choice with an arrow as shown below:
correct
A B ● ●
1– A B C D 6– A B C D
2– A B C D 7– A B C D
3– A B C D 8– A B C D
4– A B C D 9– A B C D
5– A B C D 10 – A B C D