BAULKHAM HILLS HIGH SCHOOL

YEAR 12 TASK 3 2021

Mathematics Extension 1

General Instructions Total marks – 52

• Reading time – 10 minutes Exam consists of 9 pages.
• Working time – 1 hour 30 minutes
• Write using black pen This paper consists of TWO sections.
• Calculators approved by NESA may
be used Section I – (5 marks) Pages (2-3)
• Show all necessary working in Questions 1-5
questions 6-9 • Attempt Questions 1-5
• Marks may be deducted for careless or • Answer on answer sheet provided
badly arranged work • Allow about 7 minutes for this section
• A reference sheet is provided at the Section II – (47 marks) Pages (4-9)
back of this paper • Attempt Questions 6-9
• Allow about 1 hour and 23 minutes
• Answer each question on the appropriate
page of your answer booklet

1
Section I - 5 marks
Use the multiple choice answer sheet for questions 1-5
1. 1
Which of the following functions is a primitive of ?
4 − 9x 2
1 −1 2 x
A) sin
3 3
1 −1 3 x
B) sin
9 2
1 −1 2 x
C) sin
9 3
1 −1 3 x
D) sin
3 2

2. A cricket ball is hit from the origin so that its position vector at time t is given by
r (t ) = 15ti + (20t − 5t 2 ) j for t ≥ 0, where i is the unit vector in the forward direction and j is a unit
    
vector vertically up.
When the cricket ball reaches its maximum height, what is its position vector?
r 30 i + 20 j
A) =
  
r 15i + 20 j
B) =
  
C) r = 60 i
 
r 30 i + 10 j
D) =
  
3. If sin(α
= + β ) a and sin(α
= cos β ?
− β ) b then sin α=

A) a 2 + b2

B) ab

C) a 2 − b2
a+b
D)
2

2
4. The direction (slope) field for a first order differential equation is shown.

Which of the following could be the differential equation represented?

dy
A) = ( x + 1)3
dx
dy
B) = x( y + 1)
dx
dy
C) = ( x + 1) y
dx
dy
D) = ( x − 1) y
dx

5. ax d 2 y dy
The equation y = e satisfies the differential equation + − 6y =
0. What are the possible values
dx 2 dx
of 𝑎𝑎?
A) a = 3or a = −2
B) a = −6 or a =−1
C) a = 1or a = −6
D) a = 2 or a = −3

End of Section I

3
Section II
47 marks
Attempt Questions 6-9
Allow about 1 hour and 23 minutes for this section
Answer each question on the appropriate page of your answer booklet. Extra paper is available. Your
responses should include relevant calculations and/or mathematical reasoning.

Question 6 (13 marks) Marks

a) Find the derivative of

1
i. y = tan −1 (2 x).

ii. y = e x ( cos −1 x ) 2

b) Use the substitution u= x + 1 to find x

dx .
3
∫ x +1

c) Find the solution to differential equation 3

dy 2 x
= = where y (0) 1
dx 3 y 2

d) What is the exact value of k , given that 2

k
dx π
∫ 9+ x
0
2
=
9
?

e) Evaluate 2
π
4

∫ sin 5 x sin x dx.

0

4
Question 7 (9 marks)
a) 1 for −π ≤ x ≤ π .
Consider the equation cos x − 2sin x =

t 2 + 2t =
0, where x 1
i. Show that the equation can be written as t = tan .
2
ii. Hence, solve the equation cos x − 2sin x =
1 for −π ≤ x ≤ π , giving solution to two 2
decimal places where necessary.

b) Solve=
sin 2θ cos θ for 0 ≤ θ ≤ 2π . 2

c) Using the substitution x = cos 2θ ,

4
1
1+ x
evaluate ∫ dx in simplest exact form.
0
1− x

5
Question 8 (14 marks) Marks
a) 2 3 sin θ − 2cos θ in the form π 2
i. Express R sin(θ − α ) where R > 0 and 0 < α < .
2

ii. =
Find the values of 𝑥𝑥 for which f ( x) 2 3 sin x − 2cos x attains its maximum 2

value over the domain [ −2π , 2π ].

b) x
The diagram below shows the graph of y = 2 cos −1 .
3
The area under the curve for 0 ≤ x ≤ 3 is shaded.

i. Find the y-intercept.

2
ii. Calculate the area of the shaded region.

Question 8 continued on next page

6
c)

3
The shaded region is bounded by the curve y = 1 + 2 x , the y − axis and the horizontal

interval BP. The x − coordinate of P is 4.

Calculate the exact volume of the solid of revolution formed when the shaded region is

rotated about the y − axis.

d) i. Show that 2

 π  1 + 3 tan θ
tan  θ +  = .
 6 3 − tan θ
2
ii. Hence, or otherwise, solve for 0 ≤ θ ≤ 2π

1 + 3 tan θ =
( 3 − tan θ ) tan(π − θ ).

7
Question 9 (11 marks)

a) A projectile is fired from O at an angle θ to the horizontal with initial velocity V ms−1 to

strike a target R metres right of O on the level ground. Assume the displacement of the

projectile is given by the vector

 gt 2 
r (t )= (Vt cos θ ) i +  Vt sin θ −  j. Do not prove it.
   2 

i. If the projectile hits the target, prove that: 2

 2V 2 
tan 2 θ −   tan θ + 1 =0
 gR 

ii. Show that the target will be hit from two angles of projection, say 𝜃𝜃1 and θ 2 , if
2

V2
R< .
g

iii. Let the respective times of flight for each path be t1 and t2 . By considering the roots 2

of the equation in part (i) or otherwise, find the difference in the two times in terms

of V , g and R.

Question 9 continued on next page

8
b) A colony of bacteria occupied the area A cm 2 . The colony has its growth modelled by the

dA 1
logistic equation = A(50 − A) where t ≥ 0 and t is measured in days. At time
dt 25

1
𝑡𝑡 = 0, the area occupied by the bacteria colony is cm 2 .
2

1 1 1 1  1
i) Show that =  + .
A(50 − A) 50  A 50 − A 
3
ii) Using the above result, solve the logistic equation and hence show that

50
A= .
1 + 99e −2t
1
iii) According to this model, what is the limiting area of the bacteria colony?

End of Exam