2023, Paper 1

QUESTION 1 The position of a particle is given by 2 ˆ ˆ ( 2) t t = + + r i j for Determine the corresponding Cartesian equation. (A) (B) (C) (D)

QUESTION 2 Consider the proof of the following proposition using mathematical induction. An appropriate assumption statement within the proof is (A) (B) (C) (D)

QUESTION 3 One solution of z3 − z2 − 7z - 2 = 0 is z = -2. Which equation could be used to determine the remaining solutions? (A) z2 − 3z − 1 = 0 (B) z2 − 3z + 1 = 0 (C) z2 − z − 1 = 0 (D) z2 − z + 1 = 0

QUESTION 4 The age-specific population distribution of a particular species of animal is shown. Age (years) 0–1 1–2 2–3 3–4 Female population 94 82 37 6 Breeding rate 0 1.3 0.9 0.2 Survival rate 0.6 0.8 0.4 0 The Leslie matrix based on this data is (A) 94 82 37 6 0.6 0 0 0 0 0.8 0 0 0 0 0.4 0 (B) 1 2 3 4 1.3 0 0 0 0 0.9 0 0 0 0 0.2 0 (C) 0.6 0.8 0.4 0 1.3 0 0 0 0 0.9 0 0 0 0 0.2 0 (D) 0 1.3 0.9 0.2 0.6 0 0 0 0 0.8 0 0

QUESTION 5 A confidence interval for a parameter is a range of values within which the (A) sample estimate of the parameter always lies. (B) sample estimate of the parameter never lies. (C) parameter always lies. (D) parameter never lies.

QUESTION 6 The shaded region defined as is best represented by (A) Im(z) Re(z) (B) Im(z) Re(z) (C) Im(z) Re(z) (D) Im(z) Re(z)

QUESTION 7 The differential equation for which the solution is a logistic equation of the form where a, b and C are constants is (A) (B) (C) (D)

QUESTION 8 Point A is the centre of a sphere and point B lies on its surface as shown. x y z B(1, −3, −5) A(1, 0, −1) The equation of the sphere is (A) x2 − 2x + y2 + z2 + 2z = 23 (B) x2 + 2x + y2 + z2 − 2z = 23 (C) x2 − 2x + y2 + z2 + 2z = 25 (D) x2 + 2x + y2 + z2 − 2z = 25

QUESTION 9 The geometric interpretation of a certain system of three equations with no solution is shown. Given two of the equations are x + y - z = 0.5 and x - y - z = 0.5, the third equation could be (A) 2x - 2y - 2z = 1 (B) 2x + 2y - 2z = 1 (C) 2x - 2y + 2z = 3 (D) 2x + 2y - 2z = 3

QUESTION 10 A random variable is drawn from a population with the distribution shown in the histogram. Frequency 0 4 8 12 16 20 A number of samples of size 10 were randomly selected from this distribution and the sample means, , were recorded. The histogram that most likely represents the distribution of the sample means is (A) Frequency 0 4 8 12 16 20 (B) Frequency 0 4 8 12 16 20 (C) Frequency (D) Frequency

QUESTION 11 (5 marks) Determine the following definite integrals.  a)  [2 marks] b)  [3 marks]

QUESTION 12 (5 marks) Given and , determine X in the matrix equation XA-XC = B.

QUESTION 13 (5 marks) Given , where , prove .

QUESTION 14 (6 marks) Consider a cube with three edges positioned along the x-, y- and z-axes on the Cartesian plane as shown. Points O, A and B are vertices of the cube. x y z O A B Not to scale a) Given , determine . Express your answer in terms of ˆ and . [1 mark] b) Calculate × . [1 mark] Consider the triangle formed by joining points O, A and B. c) Use the result from Question 14b) to determine the area of the triangle. [2 marks] Let points M and N be the midpoints of the triangle’s sides OA and OB respectively.  d) Determine .  [1 mark] e) Use the result from Question 14d) to show that the length of AB is twice the length of MN.  [1 mark]

QUESTION 15 (5 marks) The sum of a geometric progression with n terms, where the first term is 1 and the common ratio is r, is given by Prove that this rule is true using mathematical induction by completing the steps of the proof as indicated. a) Initial statement:  [1 mark] Assuming the rule is true for , . b) Inductive step: [3 marks] c) Conclusion: [1 mark]

QUESTION 16 (5 marks) A curve is defined by the parametric equations and , where . Given that can be expressed in the form , where , determine the values of a and b. DO NOT WRITE ON THIS PAGE THIS PAGE WILL NOT BE MARKED CONTINUE TO THE NEXT PAGE

QUESTION 17 (7 marks) An object of mass 2 kg is moving with a constant velocity (m s-1) of . At an instant, two forces (N), and , act simultaneously on the object for t seconds, where . Determine the magnitude of the momentum of the object when t =1.

QUESTION 18 (6 marks) A particular solution to the differential equation , where and , passes through the origin. Determine this solution in the form x = f (y). Leave your answer in simplified form.

QUESTION 19 (6 marks) Object A is released from the origin with constant velocity, , such that its position after t seconds is given by At a later time, object B is released from point and travels towards point with constant velocity, , such that Given that objects A and B collide, determine the time between the release of the two objects. Assume all positions are given in metres and all velocities are given in metres per second.