2021, Paper 2

QUESTION 1 The time taken to complete orders at a pizza store is normally distributed with a mean time ( ) of 10 minutes. The owner of the pizza store records the time taken to complete orders for a random sample of 20 pizzas each day over a 30-day period. From this data, an approximate 90% confidence interval for is calculated at the end of each day. How many of these confidence intervals would be expected to contain ? (A) 3 (B) 18 (C) 27 (D) 30

QUESTION 2 Determine the area of the shaded region between the graphs of the functions and , as shown. y x −3 0 3 Not to scale (A) 5.29 (B) 5.51 (C) 5.65

QUESTION 3 Given , for which proposition can the initial statement for mathematical induction be proven? (A) is divisible by (B) (C) is divisible by 3 (D)

QUESTION 4 The mean time that visitors spend at an art exhibition is 39 minutes and the standard deviation is 6 minutes. Determine the approximate probability that the mean time spent at the exhibition by a random sample of 35 visitors is between 38 and 40 minutes. (A) 0.13 (B) 0.16 (C) 0.68 (D) 0.84

QUESTION 5 A vector normal to the plane that contains the vectors and is (A) (B) (C) (D)

QUESTION 6 The Cartesian equation of a sphere is given by . The centre and radius of the sphere are (A) and 3 respectively. (B) and 9 respectively. (C) and 3 respectively. (D) and 9 respectively.

QUESTION 7 The altitude angle of is represented as . z O A x y φ Not to scale Given the coordinates of A are , the altitude angle of in radians is (A) 0.93 (B) 0.88 (C) 0.69 (D) 0.66

QUESTION 8 The imaginary part of is (A) −6.83 (B) −0.71 (C) 0 71

QUESTION 9 Two vertical forces act on a skydiver with a mass of 85 kg, as shown. Air resistance Weight When the magnitude of the air resistance is 62 N, the magnitude of the acceleration of the skydiver is (A) 0.73 (B) 2.65 (C) 9.07 (D) 12.44

QUESTION 10 A random variable is normally distributed with a mean . An approximate 95% confidence interval for from a sample from this distribution is . An approximate confidence interval for based on the same sample, using a confidence level greater than 95%, could be (A) (B) (C) (D)

QUESTION 11 (4 marks) OABC is a triangular-based pyramid, as shown. x z C(0, 5, 2) A(1, 2, 5) B(–1, 2, –2) O y Not to scale Use a vector method to determine the area of the shaded face of the pyramid.

QUESTION 12 (6 marks) The life span of batteries manufactured by a company is assumed to be normally distributed with an unknown mean and standard deviation. A supervisor at the company randomly selects n batteries and uses the life spans from this sample to calculate an approximate 95% confidence interval for the population mean of hours. a) Determine the mean life span for this sample of batteries. [1 mark] The standard deviation of the life spans of batteries in this sample is 125 hours. b) Determine n. [3 marks] c) Use the result from Question 12b) to explain whether the assumption that the life span of batteries is normally distributed is required to support the supervisor’s calculations. [2 marks]

QUESTION 13 (6 marks) The area under the graph of the function for is shaded. f(x) x 9 1 Not to scale a) Use Simpson’s rule with four intervals to determine an approximation for this area.  [4 marks] b) Use a calculus approach to evaluate the reasonableness of your area approximation from Question 13a). [2 marks]

QUESTION 14 (5 marks) The Tasmanian thornbill is a species of bird that has an average life span of three years. Female thornbills do not reproduce in their first year, but produce an average of four female offspring in each of their second and third years. The survival rate of each age group is estimated as 25% in their first year and 30% in their second year. A Leslie matrix, L, modelling the population distribution of the Tasmanian thornbill, has been partially completed. a) State the values of x and y. [1 mark] At the start of 2021, a study began into the population of Tasmanian thornbills. The study: • estimated that the initial female population was 510 in their first year, 480 in their second year and 420 in their third year • found that the ratio of male to female was approximately 1:2. b) Estimate the total population of Tasmanian thornbills at the start of 2025. [4 marks]

QUESTION 15 (8 marks) Water is poured into a cone-shaped cup at a rate of 2 . The cup has a height of 12 cm and a radius of 6 cm, as shown. Not to scale r 6 cm h 12 cm As the cup fills, the ratio of the height of the water h to the surface radius of the water r remains constant. a) Given that , determine a function for the volume of water in the cup, V, in terms of h. Express your answer in simplified form. [1 mark] b) Use the results from Question 15a) to show that the rate at which the height of water in the cup is increasing with respect to time is given by . [3 marks] c) Determine the rate at which the height of water in the cup is increasing with respect to time when the volume of water in the cup reaches half of the total capacity of the cup.  [4 marks]

QUESTION 16 (6 marks) Given , use matrix algebra to determine the value of the scalars p, q and r.

QUESTION 17 (7 marks) An object with a mass of 2 kg is released from rest at the top of a 1 metre long frictionless plane inclined at 30° to the horizontal. A force of P newtons acting parallel to the plane opposes the motion of the object as it travels down the plane. When the object is x metres from the top of the plane, its velocity is v . Given , determine x when v = 2.

QUESTION 18 (6 marks) Consider the polynomial , where and . Two of the roots of are also roots of . The remaining root of is . Given , determine a possible expression for . Leave your answer in expanded form.

QUESTION 19 (7 marks) Consider the following information. Continuous random variable X mean variance The waiting time (minutes) until workers at a certain call centre receive their nth phone call, where , is a random variable T with probability density function where k is a positive constant. The waiting time until workers receive their 5th call is collected from a random sample of 80 workers. Determine the probability that the mean waiting time from this sample is more than 16 minutes.