2020, Prova 1

QUESTION 1 The graphs of f(x) = e and g(x) = x – 1 are shown. x y 10 8 6 4 2 0 1 2 3 –2 –1 –2 12 The area of the shaded section bounded by these graphs between the lines x = 0 and x = 1 is (A) 1 – e (B) e – 2 (C) (D)

QUESTION 2 Determine (A) x – e + c (B) x + e + c (C) 1 + xe + c (D) x + xe + c

QUESTION 3 Determine (A) (B) (C) (D)

QUESTION 4 Pulse rates of adult men are approximately normally distributed with a mean of 70 and a standard deviation of 8. Which of the following choices correctly describes how to determine the proportion of men that have a pulse rate greater than 78? (A) Determine the area to the left of z = 1 under the standard normal curve. (B) Determine the area to the right of z = 1 under the standard normal curve. (C) Determine the area to the right of z = –1 under the standard normal curve. (D) Determine the area between z = –1 and z = 1 under the standard normal curve.

QUESTION 5 The equation of the tangent to the curve at t = 1 is (A) y = et (B) y = 2et – e (C) (D)

QUESTION 6 If the probability of success in a Bernoulli trial is 0.30, the variance is (A) 0.70 (B) 0.46 (C) 0.30 (D) 0.21

QUESTION 7 The life expectancy (in years) of an electronic component can be represented by the probability density function The probability that the component lasts between 1 and 10 years is (A) 0.010 (B) 0.100 (C) 0.900 (D) 0.990

QUESTION 8 A test includes six multiple choice questions. Each question has four options for the answer. If the answers are guessed, the probability of getting at most two questions correct is represented by (A) (B) (C) (D)

QUESTION 9 Determine (A) (B) ln(2x + 2) + c (C) (D)

QUESTION 10 Two types of material (A and B) are being tested for their ability to withstand different temperatures. A random selection of both materials was subjected to extreme temperature changes and then classified according to their condition after they were removed from the testing facility. The results are shown in the table. Material A B Total Broke completely 25 43 68 Showed defects 35 38 73 Remained intact 35 24 59 Total 95 105 200 An approximate 95% confidence interval for the probability that material A will break completely or show defects is given by The values of c and n are (A) and 95 (B) and 95 (C) and 95 (D) and 200 © State of Queensland (QCAA) 2020 Licence: https://creativecommons.org/licenses/by/4.0 | Copyright notice: www.qcaa.qld.edu.au/copyright — lists the full terms and conditions, which specify certain exceptions to the licence. | Attribution: © State of Queensland (QCAA) 2020

QUESTION 11 (3 marks) Determine the derivative of each of the following with respect to x. a)  [1 mark] b) Express your answer in factorised form. [2 marks]

QUESTION 12 (5 marks) An object is moving in a straight line from a fixed point. The object is at the origin initially. The acceleration a (in m s ) of the object is given by a(t) = cos( t) t ≥ 0, where t is time in seconds. The velocity at t = 1 is 0.5 m s a) Determine the initial acceleration. [1 mark] b) Determine the initial velocity. [2 marks] c) Determine the displacement after one second. [2 marks]

QUESTION 13 (7 marks) A function is defined as f(x) = x(ln(x)) , x > 0. The graph of the function is shown and has a local maximum at point A and a global minimum at point B. The derivative of the function is given by f'(x) = 2 ln(x) + (ln(x)) , x > 0. x y A B 2 1 0 0.5 1 1.5 a) Verify that there is a stationary point at x = 1. [2 marks] b) Determine the coordinates of A. [3 marks] The graph of the function has a point of inflection at x = e c) Determine p. [2 marks]

QUESTION 14 (3 marks) Determine the area of the triangle shown. B C 6 m 3 m 30° A Not drawn to scale

QUESTION 15 (4 marks) Solve the following equations. a) 4e = 100 [1 mark] b) 2 log x – log (x – 1) = 1 [3 marks]

QUESTION 16 (4 marks) Consider the following graph of f (x). y x 0 1 2 3 4 5 6 1 2 3 4 5 –1 –2 –3 –4 –5 Identify the graph of the second derivative f ''(x) from the graphs in Diagram 1, Diagram 2 and Diagram 3. y x 0 1 2 3 4 5 6 1 2 3 4 5 –1 –2 –3 –4 –5 y x 0 1 2 3 4 5 6 1 2 3 4 5 –1 –2 –3 –4 –5 y x 0 1 2 3 4 5 6 1 2 3 4 5 –1 –2 –3 –4 –5 Diagram 1 Diagram 2 Diagram 3 Justify your decisions using mathematical reasoning.

QUESTION 17 (6 marks) The volume of water in a tank is represented by a function of the form , where V is in litres and t is in minutes. Initially, the volume is 100 litres and it is decreasing by 50 litres per minute. Determine the time at which the volume is decreasing at the rate of litres per minute. Express your answer in the form ln(a).

QUESTION 18 (6 marks) The function f (x) has the form given by f (x) = 3 log (x + a) + b The function g(x) has the form given by g(x) = –log (x + c) + 5 A section of the graphs of the two functions is shown. y x f(x) g(x) 5 0 5 Determine the values of a, b and c.

QUESTION 19 (6 marks) A horizontal point of inflection is a point of inflection that is also a stationary point. Determine the value/s of k for which the graph of has only one horizontal point of inflection.

QUESTION 20 (6 marks) At the end of the first stage of its growth cycle, a species of tree has a height of 5 metres and a trunk radius of 15 cm. In the second stage of its growth cycle, the tree stays at this height for the next 10 years. However, the growth rate of the trunk radius (in cm per year) varies over the 10 years and is given by the function Assume the density (mass per unit volume) of the tree trunk is approximately 1 g/cm and the tree trunk is in the shape of a cylinder. Determine the ratio of the trunk’s mass at the end of the second stage to its mass at the end of the first stage.