2023, Paper 2

QUESTION 1 The acceleration (m s-2) of an object moving with simple harmonic motion is modelled by a = -2.95x, where x is its displacement (m) from the origin. Determine the period of the motion in seconds. (A) 1.72 (B) 2.13 (C) 2.95 (D) 3.66

QUESTION 2 The standard deviation for the scores of 1000 students completing an entry test at a certain university is 13. A researcher takes repeated random samples of the test results, with each sample comprising 40 scores, and calculates the mean score for each sample. Determine the standard deviation of the distribution of the sample mean scores. (A) 3.08 (B) 2.06 (C) 0.41 (D) 0.33

QUESTION 3 Given that 2i is a root of z2 - pz - q = 0, where p, q ∈ R, determine the values of p and q. (A) p = -4 and q = -4 (B) p = -4 and q = 4 (C) p = 0 and q = -4 (D) p = 0 and q = 4

QUESTION 4 The position of a particle can be modelled using Which curve best represents the path of the particle? (A) 0 y x 2 2 −2 −2 (B) y x 2 −2 2 −2 0 (C) y x 2 −2 2 −2 0 (D) y x 2 2 −2 −2 0

QUESTION 5 A plane contains the origin and the points (1, 2, 3) and (3, 2, 1). A vector normal to the plane is (A) (B) (C) (D)

QUESTION 6 Two coplanar forces of magnitudes 12 N and 10 N act on an object in the directions shown. 22° y x 10 N Not to scale 12 N Determine the magnitude of the resultant force acting on the object. (A) 12.41 N (B) 15.55 N

QUESTION 7 Matrix N represents the results for a competition involving four teams. P Q R S P 0 0 1 1 Q 1 0 0 0 R 0 1 0 0 S 0 1 1 0 Key: Team P lost to team Q but won against teams R and S. Using the ranking model N + 0.5N2, the teams that placed first, second and third respectively are (A) P, S and Q. (B) P, S and R. (C) S, P and Q. (D) S, P and R. N = Winning teams Losing teams

QUESTION 8 Given f (x) = tan-1 (2x), determine f '(3). (A) 0.05 (B) 0.15 (C) 2.17 (D) 3.10

QUESTION 9 The time in minutes between the arrival of customers at a certain shop is assumed to be a random variable X with an exponential distribution that has the probability density function A customer arrives at the shop. The probability that the next customer arrives within 30 to 60 seconds, to the nearest percent, is (A) 3% (B) 5% (C) 7% (D) 11%

QUESTION 10 The Argand diagram that represents the solutions to is (A) Im(z) Re(z) −4 −4 4 4 (B) Im(z) Re(z) −4 −4 4 4 (C) Im(z) Re(z) −2 −2 2 2 (D) Im(z) Re(z) −2 −2 2 2

QUESTION 11 (4 marks) The bounded region between the graphs of the functions and y = 0.1x2 over a certain domain is shaded as shown. The two functions intersect at the origin and point A. x y O A a) Determine the coordinates of point A. [1 mark] b) Calculate the area of the shaded region.  [1 mark] The shaded region is rotated about the x-axis to form a solid of revolution. c) Determine the volume of the solid formed. [2 marks]

QUESTION 12 (7 marks) Consider the complex number z = -3 + 2i. a) Determine z3 using the binomial theorem. Leave your answer in the form a+bi, where a, b ∈ R.  [2 marks] b) Convert z into the form of , where . [1 mark] c) Use the result from Question 12b) to determine z3 using De Moivre’s theorem. Leave your answer in the form of , where . [2 marks] d) Evaluate the reasonableness of your results from Questions 12a) and 12c), noting that the two methods to determine z3 should produce the same result. [2 marks]

QUESTION 13 (4 marks) The wait time for customers put on hold when calling complaint departments is assumed to be normally distributed. A company claims that the mean wait time for their customers is 7.6 minutes. The following data represents the wait time (minutes) from a random sample of 12 customers who called the complaint department of this company. 8.3 12.7 9.1 7.3 10.3 5.4 8.5 10.7 6.9 12.5 7.2 11.9 a) Determine the mean of this data. [1 mark] The standard deviation of this data is calculated to be 2.384 minutes. b) Use an approximate 95% confidence interval for the mean to evaluate the reasonableness of the company’s claim. Justify your decision using mathematical reasoning. [3 marks] DO NOT WRITE ON THIS PAGE THIS PAGE WILL NOT BE MARKED CONTINUE TO THE NEXT PAGE

QUESTION 14 (4 marks) At a certain location, a biologist measures the width of a river to be 12 m. She also records the depth of the river at regular 2 m interval widths as shown. Width (m) 0 2 4 6 8 10 12 Depth (m) 0.52 2.15 3.70 4.27 3.32 1.28 0.59 The biologist estimates the cross-sectional area of the river at this location to be 15 m2. Use Simpson’s rule to evaluate the reasonableness of this estimation. Justify your area calculation and decision regarding reasonableness using mathematical reasoning.

QUESTION 15 (7 marks) The travel time for students attending a certain university is assumed to be normally distributed, with a population mean of 25.2 minutes and standard deviation of 4.7 minutes. Travel times are collected from a random sample of 120 of these students and used to calculate a sample mean, , in minutes. a) Determine . [2 marks] b) Given , determine the value of k. [1 mark] Travel times are collected from a second random sample of the university’s students and used to calculate a second sample mean, , in minutes. c) Given , determine the number of students in the second sample. [4 marks]

QUESTION 16 (6 marks) A curve modelled by the relation xy2 - y + cos-1(2x) = 1, where -0.35 ≤ x ≤ 0.27 and 0 ≤ y ≤ 1, intersects the y-axis at point A. Determine the equation of the tangent to the curve at point A.

QUESTION 17 (6 marks) An object is projected upwards from ground level with an initial velocity of 15 m s-1 at an angle of 54° to the horizontal. The object just passes over a drone hovering in the air. An observer is positioned directly below the drone and at a horizontal distance of 20 m from where the object is projected. The observer commented that: • it took the object around 2 to 2.5 seconds after its projection to reach the drone • the object was still moving in an upwards direction as it passed the drone. Assuming that air resistance is negligible, use a vector calculus approach to evaluate the reasonableness of the observer’s comments.

QUESTION 18 (5 marks) Consider the complex solutions to the following equation, where 0 < arg(z) < π. Let w1 be the solution with the maximum possible real part and w2 be the solution with the maximum possible imaginary part. Show that .

QUESTION 19 (7 marks) The height of Year 9 students at a school is assumed to be normally distributed with a population mean height of μ cm. A teacher at the school measured the height of all the students in her Year 9 class. This data was used to calculate an approximate 95% confidence interval for μ of (163.7, 166.9) cm. The teacher repeated the procedure using data from another Year 9 class. Although this class had the same number of students, its data produced an approximate 95% confidence interval for μ of (167.8, 172.4) cm. Using the same data, the teacher recalculated the approximate confidence intervals for μ for each class using a confidence level of x %. She observed that the upper bound of the confidence interval from her Year 9 class now equalled the lower bound of the confidence interval from the other Year 9 class. Determine the value of x. Give your answer rounded to one decimal place.