2025, Prova 1

QUESTION 1 Determine the value of 2 2 1 3 ∫ x dx (A) 9 (B) 7 (C) 6 (D) 5

QUESTION 2 State the domain of the function ln( ) = y x (A) 0 < < ∞ x (B) 0 x ≤ < ∞ (C) 0 −∞< < x (D) x −∞< < ∞

QUESTION 3 A council wants to survey residents about a new dog park. Which sampling method would best minimise bias in the survey? (A) Questioning every third resident entering a supermarket near an existing dog park. (B) Collecting responses from residents who clicked a survey link on the website. (C) Asking residents visiting a dog park on a randomly selected day. (D) Selecting residents using a random number generator.

QUESTION 4 Solve for x in the equation ln (x) + ln (2) = ln (8). (A) 3 (B) 4 (C) 6 (D) 10

QUESTION 5 A box contains 100 balls, each of which is either green, blue or red. A teacher asks students to select a ball at random, record its colour and return it to the box. This process is repeated until each student has selected and recorded the colours of a sample of balls. Each student chooses their own sample size. The sample size, n, and sample proportion of green balls, , are shown for four students. n 10 1 2 15 1 3 20 1 4 80 1 5 Which sample proportion is expected to best approximate the proportion of green balls in the box? (A) 1 2 (B) 1 3 (C) 1 4 (D) 1 5

QUESTION 6 Calculate the area (m2) of the triangle. 4 m 4 m Not to scale 120º (A) 8 (B) 4 (C) 8 3 (D) 4 3

QUESTION 7 Determine ( ) 4 3 x d dx e + (A) ( ) 4 3 x e + (B) ( ) 4 3 1 4 x e + (C) ( ) 4 3 4 x e + (D) ( ) ( ) 4 3 4 3 x x e + +

QUESTION 8 Determine the area (units2) between the graph of 3 4 = y x and the x-axis from x = −1 to x = 2 (A) 12 (B) 15 (C) 17 (D) 20

QUESTION 9 Determine 1 (7 4) − ∫ dx x , where 4 7 x > (A) ( ) 1 ln 7 4 7 x c − + (B) ( ) 4 ln 7 4 7 − + x c (C) ( ) 7ln 7 4 − + x c (D) ( ) ln 7 4 x c − +

QUESTION 10 Researchers asked 100 customers at a cafe about their daily coffee consumption. The results are shown. Coffees per day Frequency 0 10 20 30 40 50 1 2 3 4 5 Determine the probability that a randomly selected customer from this group drinks more than three coffees a day. (A) 77 100 (B) 21 50 (C) 7 20 (D) 1 4

QUESTION 11 (5 marks) Determine the following integrals. a) 1∫ dx x  [1 mark] b) 4 ∫ x e dx  [1 mark] c) ( ) 3 8 x x dx + ∫  [3 marks]

QUESTION 12 (4 marks) A teacher conducted a survey about student involvement in school activity groups. They found that of a random sample of 25 students, five students were members of the environment group. Determine the approximate confidence interval for the proportion of students at the school who are members of the environment group, using z = 1. Express your answer using fractions in simplified form.

QUESTION 13 (4 marks) Use ( ) 2 6 ′′ = − f x x to demonstrate that the function ( ) = y f x has a point of inflection at 3 = x .

QUESTION 14 (4 marks) Determine all solutions to ( ) 2cos 1 0, θ −= where 0 720 θ ° ≤ ≤ °.

QUESTION 15 (6 marks) Determine all possible solutions for the equation ( ) ( ) 2 2 2 2 log ( ) log 8 = + x x . Evaluate the reasonableness of one solution.

QUESTION 16 (5 marks) The energy produced by solar panels on a house is stored in a battery. Energy consumed in the house comes out of the battery. The rate of energy produced by the solar panels during daylight hours on a particular day is approximately modelled by ( ) 2.5sin( ( 6)) 12 = − R t t π , 6 18 ≤≤ t , where t is the number of hours after midnight. ( ) 0 R t = for any hours after sunset and before sunrise. The rate of energy consumed in the house is 4 π units per hour. Determine the total energy change in the battery during daylight hours, assuming that sunrise is at 6:00 am and sunset is at 6:00 pm.

QUESTION 17 (6 marks) In a computer fishing game, a player repeatedly casts their hook into either a blue pond or a red pond. Each fish they catch scores points. In the blue pond, the probability of catching a fish on a cast is 2 3 and each fish caught scores 10 points. In the red pond, the probability of catching a fish on a cast is 1 3 and each fish caught scores 15 points. A player has three casts left and needs to score at least 30 points to win. All remaining casts must be in the same pond. It is claimed that the probability of winning if casting in the blue pond is 1 27 more than the probability of winning if casting in the red pond. Evaluate the reasonableness of the claim.

QUESTION 18 (6 marks) Two objects are launched simultaneously from different positions and travel along the same straight-line path. The objects are launched towards each other with the same initial speed. The first object’s displacement (m) from the origin is given by 3 2 1 1 1 3 2 = − + d t t kt, where t is the time (s) since the objects were launched and k is a constant, k ≠ 0. The second object is moving with a constant acceleration of 4 m s−2. The second object changes its direction, and at time t = 1 s the objects have equal velocities and continue to travel in the same direction. Compared to the first object, how much further does the second object travel between t = 1 s and the next time the objects have equal velocities?

QUESTION 19 (5 marks) A farmer wishes to construct the shortest possible fence to enclose a triangular area in the corner of a paddock. There are two existing fences, the western fence and the southern fence. The new fence must pass through the old homestead gate, as shown. 1 km N 27 km Old homestead gate Western fence New fence Not to scale Southern fence θ Determine the length of the shortest possible fence. Verifying that the fence length is a minimum is not required.