2022, Paper 2

QUESTION 1 A solution of the equation 2= z ai, where a R ∈, is 2 2 = - - z i. The other solution is (A) 8 - i (B) 2 2i - + (C) 2 2 + i (D) 8i

QUESTION 2 The win/draw/loss results after a netball competition involving five teams is represented in matrix M. P Q R Losing teams S T P 0 1 2 0 2 Q 1 0 0 1 1 R M = Winning teams 0 2 0 0 0 S 2 1 2 0 2 T 0 1 2 0 0 Key: Team P drew with Team Q, defeated Team R and Team T, and lost to Team S The model M + M2 + M3 is used to rank the teams. The final positions from first to fifth are (A) S, Q, P, R, T (B) S, Q, P, T, R (C) S, P, Q, T, R (D) S, P, Q, R, T

QUESTION 3 Determine the solution of the differential equation     sin 2 cos 2 x dy dx x  given 0 y  when 5 π x  . (A) ( ) 2ln cos 2 2.35 y x = - - (B) ( ) 2ln cos 2 2.35 y x = - + (C) (D) ( ) 1 ln cos 2 0.59 2 y x = - - ( ) 1 ln cos 2 0.59 2 y x = - +

QUESTION 4 The time taken for students to answer questions in a class is assumed to be a random variable X with an exponential distribution that has the probability density function  , 0 , λx λe f x    , otherwise x  The mean of X is 1 λ . The mean length of time taken for students to answer questions in this class is 15 seconds. The probability that the next question in this class is answered between 8 seconds and 17 seconds is (A) 0.05 (B) 0.12 (C) 0.22 (D) 0.26

QUESTION 5 A random sample of the petrol price per litre at 50 petrol stations produced a sample mean of $1.52 and a standard deviation of $0.14. Based on this sample and using a z-value of 1.5, an approximate confidence interval for μ is (A) ($1.47, $1.57) (B) ($1.48, $1.56) (C) ($1.49, $1.55) (D) ($1.50, $1.54)

QUESTION 6 A 4 kg object moves in a straight line over time, ( )s t , where 0 5 t ≤≤ with velocity ( ) 2 1 9 8 m s v t t - = + - . Determine the momentum of the object when 3 t = . (A) 24 kg m s-1 (B) 27 kg m s-1 (C) 96 kg m s-1 (D) 100 kg m s-1

QUESTION 7 Given ( ) ( ) ˆ ˆ ˆ 3 2 2 , 2 n n = + + = - a i j b j and ( ) ˆ 1 2n × = - a b k, the possible values of n are (A) (B) (C) (D) 1 5 and 3 - 5 1 and 3 - 5 1 and 3 - 1 5 and 3 -

QUESTION 8 Determine the gradient of the tangent to the curve 2 3 5 y x - = at the point ( ) 1, 2 2 . (A) 0.41 (B) 0.53 (C) 1.06 (D) 8.49

QUESTION 9 Consider the matrix equation. 0 0 1 0 1 1 1 1 1 1 2 2 2 1 2 2 2 1 = X Matrix X is (A) (B) (C) (D) 0 1 1 −1 1 2 −1 0 2 0 −1 1 1 0 −1 1 2 2 2 3 2 3 1 5 5 5 4 2 2 3 4 5 5 1 3 5

QUESTION 10 In a town, the mean number of residents per household is 3.79 people with a standard deviation of 1.47 people. Using a random sample of 45 households from the town, determine the probability that the mean number of residents per household will be more than 4. (A) 0.17 (B) 0.33 (C) 0.83 (D) 0.96

QUESTION 11 (6 marks) An aerial view of the surface of a dam, 6 km in length, is symmetrically positioned on a Cartesian plane as shown. A dam wall is located along the y-axis. The surrounding edge of the dam can be modelled by the ellipse ( ) 2 2 2 1 16 9 x y - + = , for 0 6 x ≤ ≤. Dam wall Not to scale 6 km y x a) Use Simpson’s rule with four strips to determine an approximate area of the surface of the dam. [4 marks] b) Evaluate the reasonableness of this approximation. [2 marks]

QUESTION 12 (5 marks) A scientist collects data for a species of tree frog in a protected area. Details for the female tree frog population are shown in the table. Age (years) 0–1 1–2 2–3 3–4 Population in Year 1 150 101 84 62 Birth (breeding) rate 0.4 0.7 0.5 0.1 Survival rate 0.6 0.3 0.2 0 The scientist uses a Leslie matrix model to make predictions about the female tree frog population. a) State the initial population matrix. [1 mark] b) Determine the Leslie matrix. [1 mark] A species is considered to be endangered if the female population in a restricted area is predicted to fall to less than 125 in the next 20 years. c) Determine whether this species of tree frog is considered to be endangered. [3 marks]

QUESTION 13 (5 marks) An article claims that the mean starting salary of graduates in Australia is currently $64 800 with a standard deviation of $4500. To check the validity of this claim, an employment agent intends to collect data on the starting salaries of a random sample of 360 graduates. a) Determine the probability that the sample mean starting salary will be between $64 000 and $65 000. [2 marks] From the data, the agent calculates a confidence interval for the population mean starting salary of ($64 589, $65 811). b) Determine the sample mean. [1 mark] c) Comment on the reasonableness of the article’s claim based on this confidence interval. [2 marks]

QUESTION 14 (5 marks) An object is moving in a straight line with an acceleration represented by the differential equation ( ) 2 4 dv v dt = - + , where v is the object’s velocity ( ) 1 m s- over time, ( )s t , where 0 t ≥, until it comes to rest. a) Determine the general solution of the differential equation. [3 marks] The initial velocity of the object is 1.5 m s-1. b) Determine the time when the particle comes to rest. [2 marks]

QUESTION 15 (5 marks) Consider points ( ) A 3, 1, 3 - and ( ) B 1, 1, 6 . a) Determine . [1 mark] b) Determine the Cartesian equation of the line that passes through points A and B. [2 marks] Point A lies on the plane, φ, and is perpendicular to this plane. c) Determine the Cartesian equation of the plane. [2 marks] DO NOT WRITE ON THIS PAGE THIS PAGE WILL NOT BE MARKED CONTINUE TO THE NEXT PAGE

QUESTION 16 (6 marks) An object with a mass of 12 kg lies on a frictionless inclined plane. A rope is attached to the object at an angle of 25° above the plane, as shown. Rope Not to scale 25° θ The force of the rope, T N, prevents the object from moving. When the rope is detached, the object moves down the plane with an acceleration of 5.6 m s-2. Determine the magnitude of T.

QUESTION 17 (6 marks) The mass of a population of elephants is known to be normally distributed. A biologist randomly selects a number of elephants from this population and measures their masses. The mean mass of the sample is 5206 kg with a standard deviation of 356 kg. The biologist uses the data to calculate a 90% confidence interval for the population mean mass of (5159.1, 5252.9) kg. Determine a 99% confidence interval for the population mean mass based on the same data.

QUESTION 18 (5 marks) Consider the polynomials ( ) ( ) 3 2 2 3 P z z i a z biz i = + - - + and ( ) 2 Q z z i = - , where , a b R ∈. Given ( ) ( ) P z Q z has a remainder of a bi - , evaluate the reasonableness that ( ) ( ) z a bi - - is a factor of ( ) P z .

QUESTION 19 (7 marks) A research organisation plans to use a drone to drop a scientific instrument vertically from a stationary position above the ocean surface. The acceleration ( ) 2 m s- of the falling instrument can be modelled by 9.8 0.1v - , where v is its velocity ( ) 1 m s- . In order for the instrument sensors to activate, its speed as it hits the ocean surface must reach at least 20 m s-1. However, if it hits with a speed above 50 m s-1, the sensors will be damaged. Determine the range of the drone’s flying height above the ocean surface to ensure that the sensors are activated but not damaged.