2021, Prova 1

QUESTION 1 Which of the following is a population parameter? (A) s (B) (C) (D) z

QUESTION 2 When the polynomial is divided by , the remainder is (A) –2i (B) 0 (C) 2i (D) 4i

QUESTION 3 An object has a velocity , where t represents time . The displacement of the object could be (A) (B) (C) (D)

QUESTION 4 The number of sunflower seeds in each packet produced by a company is known to be normally distributed with a standard deviation of 100. A worker counts the number of seeds in a random sample of four packets and calculates the sample mean. Based on this sampling, the standard deviation of the distribution of the sample mean is (A) 25 (B) 50 (C) 75 (D) 100

QUESTION 5 The augmented matrix shown is produced when a Gaussian elimination technique is used to solve a certain system of equations with three variables. The geometric interpretation of the solution to this system of equations is best represented by (A) (B) (C) (D)

QUESTION 6 The subset of the complex plane that represents for is (A) Im(z) Re(z) −2 −2 2 2 0 (B) Im(z) Re(z) −2 −2 2 2 0 (C) Im(z) Re(z) –2 –2 2 2 0 (D) Im(z) Re(z) –2 –2 2 2 0

QUESTION 7 The mass of a particular variety of cake is claimed to be normally distributed with a mean of 660 grams. A random sample of five of these cakes is found to have a mean mass of 600 grams. Which option represents an approximate confidence interval for based on this sample? (A) 600 grams 660 (B) 600 grams 660 (C) 600 grams 660 (D) 600 grams 660

QUESTION 8 Let be the proposition that Which option represents a correct formulation of the assumption that is true in a proof using mathematical induction? (A) (B) (C) (D)

QUESTION 9 The slope field for the differential equation is shown. y x P Q (3, –5) –5 5 –5 5 R S The solution curve to the differential equation that passes through the point also passes through point (A) P (B) Q (C) R (D) S

QUESTION 10 The 2016 Australian census recorded the number of bedrooms per household. The results are summarised in the histogram, as shown. Based on this data, the mean number of bedrooms per household was calculated to be 3.5. 0 1 2 3 4 5 6 Number of bedrooms Frequency Fifty samples of size 40 were randomly selected from the census data and the sample means recorded. The histogram that most likely represents the distribution of the sample means is (A) 1.5 20 10 0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 Number of bedrooms Frequency (B) 3.0 20 10 0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Number of bedrooms Frequency (C) 3.9 20 10 0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Number of bedrooms Frequency (D) 3.9 20 10 0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Number of bedrooms Frequency

QUESTION 11 (5 marks) Let for suitable values of x where . a) Determine .  [1 mark] b) Determine . [2 marks] c) Use the results from Questions 11a) and 11b) to determine the equation of the tangent to the graph of at . [2 marks]

QUESTION 12 (8 marks) Consider the plane . a) Determine a vector n that is perpendicular to the plane. [1 mark] b) Determine the vector equation of the line l that is perpendicular to the plane and contains the point . [1 mark] c) Use the result from Question 12b) to express the equation of the line l in parametric form. [1 mark] The line l and the plane intersect at point S. d) Show that the coordinates of S are .  [3 marks] e) Determine . [1 mark] f) Use a property of parallel vectors to verify that and n are parallel. [1 mark]

QUESTION 13 (6 marks) Use and , where , to prove

QUESTION 14 (6 marks) An object is projected vertically upwards from ground level. After the object has been in motion for t seconds, its position vector through the air, in metres, is modelled by a) Determine the velocity of the object through the air, , in metres per second. [2 marks] b) Determine the number of seconds until the object reaches its maximum height. [2 marks] c) Determine the maximum height that the object reaches, in metres. [2 marks]

QUESTION 15 (4 marks) Use partial fractions to determine , where . Express your answer in the form .

QUESTION 16 (6 marks) Use mathematical induction to prove that is divisible by .

QUESTION 17 (7 marks) The area between the graphs of the functions and is rotated about the y-axis to form a solid of revolution with a volume of V . Determine the exact value of V.

QUESTION 18 (6 marks) This differential equation can be used to determine the current I (amperes) at time t (seconds) with voltage V (volts) in an electric circuit containing a resistance R (ohms): where k, R and V are positive constants and . Assuming that there is no current in the electric circuit initially, show that the size of the current can never be greater than .

QUESTION 19 (7 marks) The velocity vectors of two objects A and B at time are given respectively by Objects A and B are initially at and respectively. Determine the position of Object A when it is 4 metres away from Object B for the first time.