2025, Paper 2

QUESTION 1 Determine 3 10.4x dx ∫ (A) 4 2.6x c + (B) 4 6.4x c + (C) 4 14.4 + x c (D) 4 41.6x c +

QUESTION 2 Determine the approximate area (units2) between the curve and the x-axis for 0 8 ≤ ≤ x by calculating the area of the shaded rectangles. y x 0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 (A) 15 (B) 28 (C) 30 (D) 31

QUESTION 3 A streaming service surveyed a random sample of 100 customers, and 60 customers said that they prefer to watch movies rather than miniseries. Based on these results, the 95% confidence interval for the proportion of customers who prefer to watch movies was (0.5, 0.7). Which statement shows the correct interpretation of this outcome? (A) The streaming service can be 95% confident that the proportion of customers who prefer to watch movies is between 50% and 70%. (B) There is a probability between 50% and 70% that most customers prefer to watch movies 95% of the time. (C) 60% of the sampled customers prefer to watch movies between 50% and 70% of the time. (D) 60 customers in the random sample prefer to watch movies 95% of the time.

QUESTION 4 An object moves in a straight line such that its velocity (m s−1) is given by ( ) 2 − = − t v t t e for 0 5 ≤≤ t , where t represents time (s). Determine the time, t, when the acceleration of the object is 2 m s−2. (A) 0.768 (B) 1.157 (C) 3.865 (D) 4.135

QUESTION 5 Which statement best describes a feature of the graph of the exponential function , x y e x R = ∈ ? (A) ( ) lim x x e e →∞ = (B) When x = 0, y = e (C) The graph has an asymptote with the equation x = 0 (D) The gradient of the graph has the same value as the function at all points on the graph.

QUESTION 6 Solve the equation cos( ) 0.6 θ = − for 0 360 θ ≤ ≤ ° ° (A) 53.13° and 126.87° (B) 53.13° and 233.13° (C) 126.87° and 306.87° (D) 126.87° and 233.13°

QUESTION 7 An object starts from rest at the origin with acceleration (m s−2) given by ( ) 2 3 16 a t t t = − + for 0 t ≥, where t represents time (s). Determine the displacement from the origin of the object 1.5 seconds after it starts moving. (A) 7.0 (B) 17.125 (C) 17.958 (D) 24.141

QUESTION 8 The table shows the probability distribution for a random variable X in a Bernoulli experiment. The random variable has only two possible values: 0 represents failure and 1 represents success. x 0 1 P(X = x) 0.6 0.4 If six Bernoulli experiments are conducted, determine the probability of getting exactly two successes. (A) 0.138 (B) 0.160 (C) 0.311 (D) 0.360

QUESTION 9 Determine the gradient of the line perpendicular to the tangent to the graph of ( ) ln 4 y x = at the point (0.8, 1.163). (A) −1.25 (B) −0.80 (C) 0.80 (D) 1.25

QUESTION 10 The outside air temperature, T (°C), on a particular day at a certain location is modelled by the function 25 7cos , 0 24, 12 t T t π   = − ≤≤     where t is the time (hours) since 6:00 am. The rate of change of temperature at 11:30 am is (A) 1.833 (B) 1.817 (C) 0.334 (D) 0.239

QUESTION 11 (6 marks) a) Determine the equation of ( ) f x if ( ) ( ) cos( ) 4sin 2 ′ = − + f x x x π and ( ) 2 = f π . [4 marks] b) Use your result from Question 11a) to determine the value of x where ( ) 4.8 f x = , for 2 4. x ≤ ≤  [1 mark] c) Use your result from Question 11a) to determine the minimum value of ( ) f x , for 2 4. x ≤ ≤  [1 mark]

QUESTION 12 (7 marks) A cockroach population is modelled by the function ( ) 0 k t P t P e = , where P is the population after t weeks and k is a population constant. Initially, 100 cockroaches were counted. After three weeks, there were 120. a) Determine the constant k. [2 marks] b) Determine the function ( ) P t ′ . [1 mark] c) Determine when the rate of population growth would reach 10 cockroaches per week. [1 mark] To decrease the growth rate of the cockroach population, a pest control treatment was trialled. The new function to model the cockroach population t weeks after using the treatment is ( ) ( ) ln 8 172 I t c t = + + , where c is a constant. Three weeks after using the treatment, the rate of population growth was five cockroaches per week. d) Determine the value of c. [2 marks] e) Determine the cockroach population when the treatment began. [1 mark]

QUESTION 13 (4 marks) A school investigated how many hours students sleep per night. To obtain data, a random sample of students was surveyed. The results are shown. Hours of sleep per night 0 20 15 10 5 30 25 5 6 7 8 9 Number of students a) Use the data to determine the sample proportion of students who had seven or more hours of sleep per night. [3 marks] b) Use your result from Question 13a) to determine an 80% approximate confidence interval for the proportion of students who get seven or more hours of sleep per night. [1 mark]

QUESTION 14 (6 marks) The number of tourists visiting a country at any given time is modelled by , 0 12 ≤≤ t , where t is the time (months) from the start of the year. a) Determine when the number of tourists first reaches 36 000. Provide your response as a decimal number. [1 mark] b) Determine the equation of the second derivative of N ( t ). [2 marks] c) Determine the value of the second derivative when the number of tourists is a minimum. [2 marks] d) Use your result from Question 14c) to explain how the value of the second derivative is consistent with the number of tourists being a minimum.  [1 mark]

QUESTION 15 (3 marks) A tour operator offers day cruises off the Queensland coast. They advertise that on any given day, customers have a 65% probability of seeing at least one whale. The tour operator conducts cruises for 40 consecutive days. Let X be the binomial random variable representing the outcome — success or failure — over the 40 days. A success is customers seeing at least one whale. Calculate the probability that the number of days with a success will be within one standard deviation of the mean for X.

QUESTION 16 (4 marks) A species of fish is being raised in a fish pond. The number of fish, M, in the pond can be modelled by a function M = 100bt, where t is the time (days) since the fish were initially introduced into the pond and b is a constant to be determined. After seven days, there are 150 fish in the pond. Find the rate of population growth when the pond contains five times the initial number of fish.

QUESTION 17 (4 marks) The probability density function approximates the time (s) that people spend viewing a particular photo on a social media page. Determine the mean time people spend viewing the photo.

QUESTION 18 (5 marks) The results of an employee satisfaction survey of 500 employees at a large company are presented to board members. The results include a 95% confidence interval for the proportion of satisfied employees. The lower end of the confidence interval is 0.648. A board member would like to use the survey results to make the claim that the proportion of the satisfied employees in the entire company is larger than 75%. Evaluate the reasonableness of the claim.

QUESTION 19 (6 marks) A scientist is gathering data on two species of horned beetle, species A and B. Horn length is a method of distinguishing the species. Species A horn lengths are normally distributed with a mean of 20 mm and a standard deviation of 2 mm. It is known that 14.6% of species B beetles have horns shorter than 18 mm. In the particular population the scientist is studying, 30% of the beetles are species B and 70% are species A. The scientist captures a beetle with a horn length shorter than 18 mm. Determine the probability that the beetle is from species A.