2025, Paper 2

QUESTION 1 Rounded to the nearest degree, an angle equivalent to the arctangent of 2 is (A) 1° (B) 2° (C) 63° (D) 64°

QUESTION 2 Which unit vector product is correct? (A) ˆ ˆ 1 × = − i i (B) ˆ ˆ 0 × = i k (C) ˆ ˆ ˆ × = j i k (D) ˆ ˆ ˆ × = j k i

QUESTION 3 If all roots of the equation z3 = 1 are plotted on an Argand plane and then joined by straight lines, the shape formed is (A) a right-angled triangle. (B) an equilateral triangle. (C) an isosceles triangle. (D) a scalene triangle.

QUESTION 4 The expression 1 9 2 2 n n + × + , where n Z + ∈ is divisible by (A) 10 (B) 11 (C) 18 (D) 19

QUESTION 5 The height (cm) of people in a certain population is normally distributed with a standard deviation of 7.42 cm. A researcher takes repeated random samples of 15 people and calculates the mean height for each sample. The expected standard deviation (cm) of the distribution of these sample mean heights would be approximately (A) 0.49 (B) 1.92 (C) 2.02 (D) 5.51

QUESTION 6 Determine the gradient of the tangent to y2 = 4x when y = 1. (A) 4 (B) 2 (C) 0.5 (D) 0.25

QUESTION 7 Given the complex number cis 2 ð = π z –      , determine ( ) 6 Arg z . (A) π (B) 0 (C) −π (D) −3π

QUESTION 8 Consider the plane that contains both the x-axis and the y-axis. A sphere centred at (3, 4, 6) touches this plane. The length of the radius of the sphere is (A) 3 units. (B) 4 units. (C) 5 units. (D) 6 units.

QUESTION 9 The masses (grams) of a random sample of 40 chocolate muffins produced by a local bakery are recorded. Using the sample standard deviation of 0.845 grams, an approximate confidence interval for the population mean mass of chocolate muffins produced by this bakery is (149.68, 150.12) grams. The z-value used in this calculation is (A) 1.04 (B) 1.65 (C) 1.96 (D) 2.58

QUESTION 10 Consider the solid of revolution formed by rotating a section of the curve y = sin(x) around the x-axis. 0 1 x y 2 π Not to scale Determine the volume of the solid of revolution. (A) 0.79 units3 (B) 0.86 units3 (C) 2.47 units3 (D) 2.60 units3

QUESTION 11 (4 marks) The mass of checked bags that passengers take on a Brisbane–Sydney flight is normally distributed with a mean of 21.3 kg and a standard deviation of 4.2 kg. A random sample of 16 checked bags was conducted. a) Determine the probability that the mean mass of the checked bags for this sample exceeds 23 kg. [2 marks] There is a 40% probability that the mean mass of the checked bags for this sample is within ± m kg of the population mean. b) Determine the value of m. [2 marks]

QUESTION 12 (7 marks) A 10 kg object is travelling at ground level with a constant velocity. At an instant, two forces of 60 N and 42 N act simultaneously on the object parallel to the ground in the directions shown. North South East 42 N West Not to scale 60 N 18° 32° Let unit vectors in the east and north directions be ˆi and ˆj respectively. a) Determine the resultant force acting on the object expressed in Cartesian form. Simplify your answer. [2 marks] b) Determine the acceleration of the object after the forces act. Leave your answer in Cartesian form. [1 mark] The object is initially travelling at 5 m s−1 in a northerly direction when the forces act. c) Determine an expression for the object’s velocity in terms of time, t, after the forces act. Leave your answer in Cartesian form. [2 marks] d) Calculate the speed of the object after the forces have been acting for two seconds. [2 marks]

QUESTION 13 (7 marks) The sketch shows sections of the functions 2 ( ) 0.5 7.5 18 f x x x = − + − and . Two points of intersection at (12, 0) and point A are shown. x Not to scale x ( ) y f x = ( ) y g x = y A (12, 0) a) Determine the coordinates of point A. [1 mark] Consider the shaded bounded region between the functions. b) Determine an approximate area of this region using Simpson’s rule with four intervals. Show evidence of the values substituted into this rule in your solution.  [3 marks] c) State a definite integral that represents the area of the shaded bounded region.  [1 mark] d) Determine the value of your result from Question 13c). Give your answer to two decimal places. [1 mark] e) Other than working to more decimal places, state a strategy involving Simpson’s rule that could be used to improve the accuracy of your result from Question 13b). [1 mark]

QUESTION 14 (8 marks) The origin, O, is joined to points A(1, 2, 5) and B(−3, 4, 0) to form triangle OAB. Point C is the point on OB such that AC is perpendicular to OB, as shown. Not to scale O A B C x z y a) Given the length of side OB is 5 units, show that the vector projection of  OA on OB  is 3 1 4 5 0 −           . [2 marks] b) Use your result from Question 14a) to determine the length of OC. [1 mark] c) Determine the length of side OA. [1 mark] d) Use Pythagoras’ theorem to determine the length of AC. [1 mark] e) Use your result from Question 14d) to determine the area of triangle OAB. [1 mark] f) Use a vector product method to verify your result from Question 14e). [2 marks]

QUESTION 15 (6 marks) De Moivre’s theorem can be expressed as ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos sin cos sin n n r i r n i n n Z θ θ θ θ + + = + ∀∈ Prove De Moivre’s theorem using mathematical induction.

QUESTION 16 (6 marks) A certain population can be approximately modelled by the differential equation 0.5 (1 0.2 ) dP P P dt = − where P is the population in millions and t is the number of years since 1 January 2025. Given that the population on 1 January 2025 was estimated at 0.3 million, use a calculus approach to estimate the population on 1 January 2030.

QUESTION 17 (5 marks) A variable, X, is assumed to be normally distributed with µ = 24.311 and σ = 5.102. Two 90% confidence intervals for µ were calculated from two different random samples from X, with the second sample being smaller than the first sample by 60. Both confidence intervals were calculated using the population standard deviation rather than their respective sample standard deviations. The confidence interval produced from the first sample was (23.560, 25.498). Determine the probability that the confidence interval produced from the second sample overlaps the confidence interval from the first sample.

QUESTION 18 (7 marks) Polar curves are defined by points that are a variable distance of r units from the origin and dependent on the angle θ (in radians) measured from the positive x-axis. Consider the polar curve r = 1 + cos(θ). A table of four polar coordinates on this curve is shown. θ r 0 2 3 1 2 + 1.5 1 The graph shows the polar curve r = 1 + cos(θ) for 0 ≤ θ ≤ 2π on a Cartesian plane. The polar coordinates from the table have been plotted on the curve. 1 0 −2 −1 −1 2 2 1 −2 y x The length of a polar curve, L, from θ = a to θ = b can be determined using the rule 2 2 b a dr L r d d θ θ   = +     ∫ Use a complete algebraic method to determine the length of the section of the given polar curve that lies above the x-axis.