2024, Paper 2

QUESTION 1 Given that z = –2 + 3i is a root of , where , another root is (A) − 2 − 3i (B) − 2 + 3i (C) 2 − 3i (D) 2 + 3i

QUESTION 2 Rounded to two decimal places, the z-value used in the calculation of an approximate 95% confidence interval for µ is (A) 0.95 (B) 1.64 (C) 1.96 (D) 2.58

QUESTION 3 Given and , determine a × b. (A) (B) (C) (D)

QUESTION 4 The mass of biscuit packets produced by a company is normally distributed with a mean of 250 g and a standard deviation of 1.5 g. The distribution of the sample mean mass of these biscuit packets is formed using repeated random sampling of size 5. The mean and standard deviation of this distribution of sample means are (A) 50 g and 0.3 g (B) 50 g and 0.67 g (C) 250 g and 0.3 g (D) 250 g and 0.67 g

QUESTION 5 The equation of a plane is . Determine the point where the plane intersects the z-axis. (A) (0, 0, – 4) (B) (0, 0, – 2) (C) (0, 0, 2) (D) (0, 0, 4)

QUESTION 6 Two concurrent forces represented in the polar form of F1 = (1.21 N, 120°) and F2 = (1.30 N, − 160°) act on an object. Determine the magnitude of the resultant force. (A) 0.50 N (B) 1.92 N (C) 2.51 N (D) 3.70 N

QUESTION 7 Determine the number of roots of w8 = 1 that can be expressed in the form a + bi, where . (A) 0 (B) 1 (C) 2 (D) 3

QUESTION 8 T is a random variable. A random sample of four values of T is collected and used to produce an approximate confidence interval for the population mean of (3.3, 4.1). Given that three of the sample values are 3.4, 3.6 and 3.9, the remaining sample value is (A) 3.6 (B) 3.7 (C) 3.8 (D) 3.9

QUESTION 9 Random variable X has an exponential distribution with the probability density function Given that , determine k. (A) 0.10 (B) 0.69 (C) 2.03 (D) 3.47

QUESTION 10 The acceleration (m s-2) of an object at time, t, for seconds is given by . Given that the object is initially at rest, its velocity–time graph is (A) Velocity (m s−1) 0 Time (s) 4 1 2 −4 −2 2 (B) Velocity (m s−1) 0 Time (s) 4 1 2 −4 −2 2 (C) Velocity (m s−1) 0 Time (s) 4 1 2 −4 −2 2 (D) Velocity (m s−1) 0 Time (s) 4 1 2 −4 −2 2

QUESTION 11 (4 marks) A company claims that the mean battery life of their latest model of smartphone is 9.5 hours. To test this claim, the battery lives of a random sample of 40 of the smartphones were measured. A sample mean of 9.31 hours and a standard deviation of 0.52 hours were calculated from this data. a) Determine an approximate 95% confidence interval for µ. Give your answer to at least two decimal places. [1 mark] b) Determine an approximate 99% confidence interval for µ. Give your answer to at least two decimal places. [1 mark] A manager comments that either confidence interval could be used to support the company’s claim. c) Use your results from Questions 11a) and 11b) to evaluate the reasonableness of the manager’s comment. Justify your decision using mathematical reasoning. [2 marks]

QUESTION 12 (4 marks) A system of linear equations is given by x - 2y - 2z = – 6 – 3x – y + z = 2 2x + 3y - 5z = 10 a) Express the system of equations as a matrix equation of the form AX = B, where A is a 3 × 3 matrix and both X and B are 3 × 1 column vectors. [1 mark] b) Use matrix algebra to express X in terms of A and B. [1 mark] c) Use your result from Question 12b) to determine the solution of the system of equations. [1 mark] d) Verify your result from Question 12c) using one of the given linear equations. [1 mark]

QUESTION 13 (5 marks) A drone travels vertically from point A at a constant speed of 8 m s–1 over time t for t ≥ 0 seconds. Observation of the drone is made from point B, which is 50 m horizontally from point A. When the drone is y metres above point A, it is z metres from point B as shown. z y 50 m Not to scale B A a) Determine an equation expressing z2 in terms of y2. [1 mark] b) State the value of .  [1 mark] c) Use your results from Questions 13a) and 13b) to determine the rate at which z is increasing with respect to time when the drone is 20 metres above point A. [3 marks]

QUESTION 14 (5 marks) The height of Year 12 students at a school is normally distributed, with a mean height of 168.6 cm and standard deviation of 12.7 cm. The heights of a random sample of 20 of these students are recorded. a) Explain why it can be assumed that the sample means for random samples of the heights of students from this school are normally distributed. [1 mark] b) Determine the probability that the mean height of this sample will be greater than 170 cm. [2 marks] There is a 75% probability that the mean height of this sample will lie within ± h cm of the population mean. c) Determine . [1 mark] d) Use your result from Question 14c) to determine the value of h.  [1 mark] DO NOT WRITE ON THIS PAGE THIS PAGE WILL NOT BE MARKED CONTINUE TO THE NEXT PAGE

QUESTION 15 (8 marks) The vectors representing the position (m) of particles A and B are given by and respectively, where t is the time of motion for seconds. a) Show that particle A passes through the point . [2 marks] b) Given that particle B also passes through point P, determine the value of a. [2 marks] c) Determine the vector that represents the displacement of particle B relative to particle A during the given time of motion. Express your answer in simplest form. [1 mark] d) Use your result from Question 15c) to determine the shortest distance between particles A and B during the given time of motion. [3 marks]

QUESTION 16 (6 marks) Two subsets of the complex plane are and , where . Determine the complex number/s where S and T intersect. Leave your answer/s in Cartesian form. Provide an Argand diagram with a sketch of subsets S and T as part of your solution. 1 −1 1 0 Im(z) Re(z) −2 −1 −3 −4 2 2 3 4 5 6 3 4 5 6 −5 −6 −4 −3 −2 −5 −6 Note: If you make a mistake in the diagram, cancel it by ruling a single diagonal line through your work and use the additional response space at the back of this question and response book.

QUESTION 17 (6 marks) An object moves with a constant speed of v in a circular path. The position vector of the object is given by where • r is the radius (metres) of the circle • ω is the angular velocity (radians per second) • t is the time (seconds) of motion for t ≥ 0. Use vector calculus to prove that the magnitude of the acceleration of the object is .

QUESTION 18 (6 marks) A random variable X is normally distributed, with a known mean µ and standard deviation σ. In figure 1, the shaded region between 4 and µ represents 30% of the distribution of X. Not to scale 4 μ Figure 1 Consider the distribution of based on repeated random sampling of X using a certain sample size. In figure 2, the shaded region between µ and 6 represents 30% of the distribution of . Not to scale 6 μ Figure 2 Given , determine .

QUESTION 19 (6 marks) An experiment researching the population changes of a certain species of insect was conducted over a four-week period. The insect has two distinct stages in its two-week lifespan. Each stage is approximately one week in length. A constant proportion of females survive from stage 1 into stage 2. The ratio of the reproduction rate for females in stage 2 to females in stage 1 is 2:1. All offspring are born into stage 1. The number of females in each stage was measured initially and then again after two weeks as shown. Female population Stage 1 Stage 2 Initially 48 32 After two weeks 25 21 Use a matrix approach to estimate the total number of females after four weeks.