2020, Prova 1

QUESTION 1 The indefinite integral 3 1 2 x A x dx − − ∫ can be determined using the partial fractions − + + − 1 1 2 1 x x The value of A is (A) –3 (B) –1 (C) 1 (D) 3

QUESTION 2 When using proof by mathematical induction to show that n n n 2 1 2 1 − ( ) + ( ) is divisible by 3 ∀ ∈ + n Z , the inductive step requires proving (A) k k k + ( )( ) + ( ) 1 2 2 2 is divisible by 3. (B) k k k + ( )( ) + ( ) 1 2 2 3 is divisible by 3. (C) k k k + ( ) + ( ) + ( ) 1 2 1 2 2 is divisible by 3. (D) k k k + ( ) + ( ) + ( ) 1 2 1 2 3 is divisible by 3.

QUESTION 3 According to a recent census, the mean hours worked per week by all Australian workers is 35.6 hours. A mean of 36.1 hours worked per week is calculated from a random selection of 500 Australian workers. Based on this data, which of the following is correct? (A) x = = 35 6 36 1 . , . µ (B) x X = = 35 6 36 1 . , . (C) x = = 36 1 35 6 . , . µ (D) x X = = 36 1 35 6 . , .

QUESTION 4 Consider points A and B as shown. 20 10 B 12 5 y x z o A The position vector representing the midpoint of AB is (A) 5 8 5 10 .       (B) 5 10 8 5.       (C) 10 8 5 5 .       (D) 10 5 8 5.      

QUESTION 5 Determine 4 3 5 2 3 x x dx ( ) + ∫ (A) 1 6 3 5 2 4 x c + ( ) + (B) 2 3 3 5 2 4 x c + ( ) + (C) 2 3 5 2 2 x c + ( ) + (D) 72 3 5 2 2 2 x x c + ( ) +

QUESTION 6 Given z = 2 – 2i and w = – 3 + i , calculate z w 2 - (A) 3 – 9i (B) 3 – 7i (C) 11 – 9i (D) 11 – 7i

QUESTION 7 The diagram shows a slope field. y x The differential equation represented by the slope field is (A) dy dx y x = 5 (B) dy dx y x = 5 2 (C) dy dx y x = 5 2 (D) dy dx y x = 5 2 2

QUESTION 8 An equation of the line passing through the points A(2, 4, 5) and B(3, –2, 1) is (A) (B) (C) x y z −= + = + 1 2 6 4 4 5 (D) x y z − − = + = − 3 1 2 6 1 4

QUESTION 9 The scores on a test are assumed to be normally distributed. Researchers use the results from a random sample of scores to calculate a confidence interval for the population mean. However, a shorter confidence interval width is required so the researchers decide to use a second sample for their calculations. Assuming that the standard deviations for both samples are the same, the researchers can ensure that a shorter confidence interval width is produced by (A) decreasing the sample size and decreasing the confidence level. (B) decreasing the sample size and increasing the confidence level. (C) increasing the sample size and decreasing the confidence level. (D) increasing the sample size and increasing the confidence level.

QUESTION 10 The subset of the complex plane that represents arg z i z C + − [ ]+ = ∈ 1 4 0 p for is (A) Im(z) Re(z) o (B) Im(z) Re(z) o (C) Im(z) Re(z) o (D) Im(z) Re(z) o

QUESTION 11 (7 marks) The vertices of a regular hexagon are positioned on the circumference of a unit circle as shown on the Argand plane. Consider the complex number w, as shown on the plane. a) Determine w, expressing your answer in the form r cis(θ). [1 mark] b) Convert w into Cartesian form. [2 marks] w Im(z) Re(z) Each vertex of the hexagon is a solution of an equation of the form z a n = where z C Î . c) State the value of n. [1 mark] d) State the value of a. [1 mark] e) Verify that w satisfies the equation z a n = using the results from 11c) and 11d). [2 marks]

QUESTION 12 (8 marks) Consider the vertices A, B and C of the rectangular prism as shown. z y B C 0 2 3 1 x A a) State the coordinates of A, B and C. [1 mark] b) Determine a unit vector, ÖQ, that is normal to the plane containing A, B and C. [3 marks] c) Verify that n is perpendicular to AB . [2 marks] d) Determine the Cartesian equation of the plane that contains A, B and C. [2 marks]

QUESTION 13 (4 marks) The expected value of an exponential random variable X with parameter l > 0 can be determined using the rule E X x e dx x ( ) = ∞ − ∫0 l l Use integration by parts to determine E (X). Express your answer in simplest form.

QUESTION 14 (4 marks) The motion of an object that moves in a straight line is given by v x x ( )= ( ) − cos 1 2 where v is the velocity (m s–1 ) and x is the displacement (m) from the origin. a) Determine a(x) where a is the acceleration (m s–2 ) of the object. [2 marks] b) Use the result from 14a) to determine a (0), given –2π ≤ a(0) ≤ 0. Express your answer in simplest form. [2 marks]

QUESTION 15 (6 marks) The points O (0, 0, 0), A (–6, 2, –2) and C (3, 1, 2) are represented in three-dimensional space in the diagram. B y x z A C N M Not drawn to scale O OABC forms a parallelogram in three-dimensional space. a) Determine the coordinates of B. [1 mark] M is the midpoint of BC. b) Determine the vector that represents OM . [1 mark] N divides AM in the ratio 2:1. c) Determine the vector that represents ON . [2 marks] d) Use a vector method to show that O, B and N lie on a straight line. [2 marks]

QUESTION 16 (6 marks) Given cos q( )≠ ∀ ∈ + 0 n Z , use mathematical induction to prove cos cos cos cos cos q q q q q ( )− ( )+ ( )−…+ − ( ) − ( ) ( )= −− ( ) + 3 5 1 2 1 1 1 2 1 n n n n ( ) ( ) 2cos q

QUESTION 17 (7 marks) Determine the smallest positive value of a given −∫ + ( )+ ( ) ( )+       = a a x x x dx 1 2 2 2 1 1 2 sec tan cosec

QUESTION 18 (6 marks) Consider the function P z z az z az b ( )= + + + + 2 6 4 3 2 where a b Z ,
∈ + One of the roots of P z( ) z i = − is Determine the possible value/s for a and b such that all remaining roots of P z( ) have an imaginary component.

QUESTION 19 (7 marks) A circular-based bowl has been positioned symmetrically on a Cartesian plane as shown in the diagram. y x The bowl has a shape that can be generated by rotating the curve y x = − − 4 8 1 about the y-axis for 4 7 6 £ £ x . cm. The bowl is being filled with a liquid at the rate of 7 3 1 p cm s- . Determine the rate at which the depth of liquid is increasing when the depth of liquid reaches one-third of the height of the bowl.