2024, Prova 1

QUESTION 1 Determine (A) (B) (C) (D)

QUESTION 2 Determine for the function (A) (B) (C) (D)

QUESTION 3 A sample of size n can be used to obtain a sample proportion . An approximate margin of error for the population proportion can be obtained using the formula If the level of confidence is increased from 95% to 99%, then (A) the associated z-value would decrease, so E would increase. (B) the associated z-value would increase, so E would increase. (C) the associated z-value would decrease, so E would decrease. (D) the associated z-value would increase, so E would decrease.

QUESTION 4 Simplify (A) (B) (C) (D)

QUESTION 5 Determine , where and (A) (B) (C) (D)

QUESTION 6 Differentiate with respect to x. (A) (B) (C) (D)

QUESTION 7 Twenty families are selected to participate in a lifestyle study related to family size. The number of children in these families is uniformly distributed as shown. 2 Number of children Frequency 3 4 5 1 0 4 A random sample of five families is chosen from this group, without replacement. A possible mean number of children in the sample is (A) 5.0 (B) 2.0 (C) 1.0 (D) 0.0

QUESTION 8 The graph of f (x) is shown. y 2 3 1 −2 2 4 6 8 10 −2 −3 x 0 −1 −1 Identify the graph of the second derivative f " (x). (A) y 2 3 −1 −2 2 6 8 10 −2 −3 x 0 4 1 4 (B) y 2 3 1 −1 2 −2 −8 −10 −2 −3 x 0 −4 −6 −4 −6 (C) y 2 3 1 −1 2 −2 −4 −6 −2 −3 x 0 −8 −8 (D) y 2 3 1 −1 −2 2 4 6 −2 −3 x 0 8

QUESTION 9 At a certain location, the temperature (°C) can be modelled by the function , where x is the number of hours after sunrise. Determine the rate of change of temperature (°C/hour) when x = 4 (A) (B) (C) (D)

QUESTION 10 Given that , determine the value of (A) 77.800 (B) 10.778 (C) 2.778 (D) 1.556

QUESTION 11 (6 marks) a) Determine the second derivative of y = x3 – 3x2. [2 marks] b) Use your result from Question 11a) to calculate the value of the second derivative when x = –1.  [1 mark] c) Determine the x- and y-coordinates of the point on the graph of y = x3 – 3x2 for which the rate of change of the first derivative is zero.  [3 marks]

QUESTION 12 (6 marks) Each day over a three-day long weekend, a family spins a pointer on a circular board to decide whether they will spend the day at the beach or bushwalking. The circular board consists of three equal sections. Beach Bushwalking Beach a) Determine the probability that the family will spend all three days bushwalking. [1 mark] b) Determine the following binomial probabilities, expressed as fully simplified fractions. i. Exactly two days will be spent at the beach. [2 marks] ii. Fewer than three days will be spent at the beach. [3 marks]

QUESTION 13 (6 marks) a) . Use integration to determine F(x), if F(0) = 5.  [3 marks] b) If , determine y. [3 marks]

QUESTION 14 (5 marks) At a particular game at a local sporting venue, 60% of spectators support the home team and the remainder support the away team. A researcher asked six groups of 10 spectators which team they supported. Each spectator was recorded as either H (supports home team) or A (supports away team). The results were: Group 1: H A H H H A H H H H Group 2: A A H A A H A A A A Group 3: H A A H H A A A H H Group 4: H H H H H A H H H H Group 5: A A H H A H H A H A Group 6: A H A A A A A H A A a) The researcher would like to see the distribution of the sample proportions of home team supporters obtained by entering the information into a column graph. The sample proportion for group 1 is shown. Complete the column graph by including the sample proportions for the remaining five groups. [3 marks] Sample proportions of home team supporters 0.2 0.1 0 0.0 0.5 0.4 0.3 1 2 3 Frequency 0.7 0.6 0.8 0.9 1.0 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. b) If the researcher had interviewed more groups, each containing 100 spectators, describe two ways the distribution of the resulting sample proportions would be expected to differ from the distribution shown in Question 14a). [2 marks]

QUESTION 15 (4 marks) A survey was conducted to understand whether people support a new policy. Using a z-score of 2, the approximate confidence interval for the population proportion of people who support the policy was calculated as . a) Determine the margin of error. [1 mark] b) Determine the number of people surveyed. [3 marks]

QUESTION 16 (4 marks) The graph is of the form . A point on the graph (4, 3) is labelled. The line x = − 4 is an asymptote. y x 0 (4, 3) There is a point P (xP , yP) on the graph where yP is twice the value of the y-intercept of the curve. Determine the value of xP.

QUESTION 17 (3 marks) A community group that uses social media created a new post on the internet on a day when they had 1000 members. The rate of change in their number of members (members/day) is given by , where t represents days after the new post. Determine the time it will take for the community group to achieve seven times the initial number of members. Express your answer in the form a ln(b). DO NOT WRITE ON THIS PAGE THIS PAGE WILL NOT BE MARKED CONTINUE TO THE NEXT PAGE

QUESTION 18 (5 marks) The diagram shows some dimensions of a large storage container that is a rectangular prism. The angle ABC is 60°. A person requires a container that is at least 4 metres in height. Not to scale 4 m 3 m A B C Make a justified decision about whether this storage container meets the person’s requirements.

QUESTION 19 (6 marks) A permanent ice glacier is in a valley in New Zealand. Due to the temperature changes of the seasons each year, the glacier expands for six months and recedes for six months. The changing distance of a point on the front edge of the glacier to a car park can be modelled by a sine function. During the colder months, when the glacier expands, the front edge of the glacier moves to within 270 m of the car park. However, in the warmer months, when the glacier recedes, the front edge moves to a maximum distance of 280 m away from the car park. The erosion effects of the glacier on the ground are of most interest to geologists when the absolute value of the acceleration of the front edge is greater than metres/month2. During these times, a team of geologists sets up a camp site nearby to perform field work. Whenever the acceleration is less than this, the geologists leave camp. The following claim is made. The geologists will spend a total of between seven and eight months at the camp site each calendar year. Evaluate the reasonableness of this claim.