2020, Prova 2

QUESTION 1 The position x (m) at time t (s) of a 7 kg particle moving in a straight line is given by x t t t t = − + − ≤≤ 3 5 2 4 0 10 3 2 for Determine the time when the particle has a momentum of 620 kg m s–1. (A) 1.73 s (B) 2.60 s (C) 3.66 s (D) 3.71 s

QUESTION 2 The Leslie matrix for a certain endangered species is given. L =       0 8 2 4 0 3 0 4 0 0 0 0 55 0 . . . . . A group of the species was moved into a secure property at the start of 2018. The initial female population is given. N0 150 80 40 =       The best estimate of the total female population at the start of 2025 is (A) 3000 (B) 4000 (C) 5000 (D) 6000

QUESTION 3 The masses of packages of cheese produced by a company are assumed to be normally distributed with a known mean of μ grams and a standard deviation of 7.37 grams. The packages of cheese are labelled to contain 500 grams. Given there is a 25% probability that the mean mass of 20 randomly selected packages will be less than the labelled amount, the value of μ is (A) 498.89 (B) 500.25 (C) 501.11 (D) 504.98

QUESTION 4 A particle is moving with simple harmonic motion described by the equation x t =       1 32 2 . cos p where x (m) is the displacement of the particle from a central position over time t t s ( ) ≥ , 0 The maximum speed of the particle is (A) 2.07 m s–1 (B) 4.15 m s–1 (C) 4.30 m s–1 (D) 5.28 m s–1

QUESTION 5 The gradient of the tangent at point A on the curve y x 2 4 1 36 = is . The x-coordinate of A is (A) 0.12 (B) 0.46 (C) 0.54 (D) 1.47

QUESTION 6 Solve the matrix equation for X. 0 1 2 3 4 5 6 7 8 9 0 1                   X = (A) − −       9 9 4 4 (B) 3 4 1 2 − −       (C) 13 14 11 12 − −       (D) 54 56 28 29 − −      

QUESTION 7 The heights of all students at a school were measured. A mean height of 157.0 cm was calculated from this data. A random sample of 35 students from this school was selected. The mean height of this sample was 159.7 cm with a standard deviation of 8.7 cm. The smallest confidence level that could be used to produce a confidence interval that contains μ, based on this sample, is (A) 85% (B) 90% (C) 95% (D) 99%

QUESTION 8 Let u = 1 + i and v = –12 + 5i Re u v 5 − ( )is (A) –17 (B) –4 (C) 8 (D) 9

QUESTION 9 Two objects, P and Q, move in three-dimensional space such that their positions, r, over time, t, are described by the following vectors until they collide. The objects will collide at (A) t = 0 (B) t = 1 (C) t = 2 (D) t = 3

QUESTION 10 The time taken by the Year 7 students at a particular school to complete a standardised test is known to be normally distributed. A researcher claims that the population mean is 8.2 minutes. The mean time taken to complete this test by a sample of 10 of these students is 8.1 minutes with a standard deviation of 1.2 minutes. The 95% confidence interval for μ based on this sample is (A) (7.36, 8.84) minutes (B) (7.46, 8.94) minutes (C) (7.86, 8.33) minutes (D) (7.96, 8.44) minutes

QUESTION 11 (5 marks) Teams A, B, C, D and E participated in a competition with the following results: • A defeated D. • B defeated A, C and E. • C defeated A and E. • D defeated B, C and E. • E defeated A. To rank the teams at the end of the competition, the organisers constructed a dominance matrix, N, that is partially completed. a) By allocating 1 to represent ‘defeated’ and 0 to represent either ‘was defeated by’ or ‘no result’, complete matrix N. [1 mark] A B C D E A B C D E Losing teams Winning teams N = Note: If you make a mistake in the matrix, cancel it by ruling a single diagonal line through your work and use the additional matrix on page 26 of this question and response book. The organisers need to rank the teams into individual places from first to fifth place. They decide to use the ranking model N + N2 to achieve this. b) Use the model N + N2 to rank the teams. [2 marks] c) Use the result from 11b) to identify a limitation of the organisers’ ranking model. [1 mark] d) State a mathematical refinement the organisers could consider to overcome the limitation of the ranking model identified in 11c). [1 mark]

QUESTION 12 (9 marks) For a certain experiment, the number of yeast cells, N, after t hours in a test tube can be modelled by the differential equation dN dt N N t = − ( ) ≥ 1 1000 1000 0 for a) Given 1000 1000 1 1 1000 N N N N − ( ) = + − , show that the general solution of the differential equation can be expressed as ln N N t c 1000− = +  [2 marks] A scientist commenced this experiment at 9:00 am on a certain day and observed that 100 yeast cells were present at this time. b) Show that the solution of the differential equation can be expressed as N e t = + − 1000 1 9  [3 marks] c) Determine the time of day when 900 yeast cells were present. [2 marks] The scientist predicted that the number of yeast cells would eventually exceed 1200. d) Evaluate the reasonableness of the scientist’s prediction. [2 marks]

QUESTION 13 (6 marks) Data records show that the speeds of cars at a particular location on a highway are normally distributed with a mean of 98.7 km h–1 and a standard deviation of 4.1 km h–1. The speed limit at this location is 100 km h–1. A police officer plans to record the speeds of 20 randomly selected cars at this location. a) Determine the expected number of cars in the sample that will be travelling within ± 1 km h–1 of the population mean. [2 marks] b) Determine the probability that the mean speed of this sample will exceed the speed limit. [2 marks] There is a 5% probability that the mean speed of this sample will exceed k. c) Determine the value of k.  [2 marks]

QUESTION 14 (5 marks) The time, t, (months) that it takes before a phone owner cracks the screen on their phone can be modelled by an exponentially distributed random variable f t e t t ( ) . , . = ≥     − 0 16 0 0 0 16 , otherwise a) Show that f t( ) is a probability density function. [1 mark] b) Determine the probability that a phone owner cracks the screen on their phone within 1 year. [2 marks] Three-quarters of phone owners take between 1 and m months before they crack the screen on their phone. c) Determine the value of m. [2 marks]

QUESTION 15 (4 marks) The position vectors of points P and Q are and respectively. Let O be the origin. a) Determine the angle POQ. [2 marks] Points O, P and Q are joined to form a triangle. b) Determine the area of triangle POQ. [2 marks]

QUESTION 16 (6 marks) Consider the identity cos 4q ( )= A 4 q( )+ B cos 2 q( )+C sin where A B C Z , and Î a) Determine the values of A, B and C using De Moivre’s theorem. [5 marks] b) State an appropriate method of verifying your results from 16a). [1 mark]

QUESTION 17 (7 marks) An object is released from rest at a height of 100 m above the ground. The motion of the vertical descent of the object is modelled by v v dv dx v = − ≥ 9 8 0 004 0 2 . . ( ) where v is the velocity (m s–1 ) and x is the displacement from the ground (m). Determine the velocity of the object when it strikes the ground.

QUESTION 18 (6 marks) The mass of a certain species of kangaroo is known to be normally distributed with a mean mass of μ kg and standard deviation of σ kg. When one of the kangaroos is randomly selected, the probability that its mass is greater than 83.2 kg is 0.145. When a sample of 12 kangaroos is randomly selected, the probability that the sample mean mass is less than 74.1 kg is 0.079. A 90% approximate confidence interval for μ is calculated using a random sample of n of the kangaroos that has a sample mean mass of 79.1 kg and a sample standard deviation equal to σ. Determine the possible range of values that n could have been, given that the confidence interval did not contain μ.

QUESTION 19 (7 marks) An object is swinging at the end of a 0.5 m length of string in a vertical circular path with a constant angular speed, completing each revolution in 0.24 seconds. The object is projected from a height of 0.3 m above the ground in a vertical plane and just passes over a narrow pole as shown in the diagram. The pole is 2.05 m high and its base is 14 m horizontally from where the object was projected. Not drawn to scale 14 m 0.6 m 2.05 m 0.3 m 0.5 m 0.4 m a m s–2 A flat-topped vehicle of length 0.6 m and height 0.4 m is initially at rest against the pole as shown in the diagram. At the instant that the object is projected, the vehicle moves in a horizontal direction away from the pole in the same vertical plane with an acceleration of magnitude of a m s–2. The object strikes the middle of the top of the vehicle. Assuming that air resistance is negligible, use vector calculus to model the motion of the projectile in order to determine the value of a.