2025, Prova 1

QUESTION 1 Use the substitution 2 u x = to express 2 2∫ x xe dx in terms of u. (A) ∫ u e du (B) 1 2 u e du ∫ (C) u ue du ∫ (D) 2 u u e du ∫

QUESTION 2 The position vector of a particle at time t is given by ( ) ( ) ˆ ˆ sin cos = + t t r i j. The path of the particle is (A) hyperbolic. (B) parabolic. (C) elliptical. (D) circular.

QUESTION 3 The polynomial 3 2 ( ) 1 P z z pz qz = + + + has complex conjugate roots when (A) p = 1 and q = 1 (B) p = 1 and q = i (C) p = i and q = 1 (D) p = i and q = i

QUESTION 4 X is a random variable with mean µ and standard deviation σ. From random samples of X values, each of size n, the sample mean is calculated. This sampling and calculation is repeated a large number of times. The mean of the distribution of the sample means would be approximately (A) x n (B) n µ (C) x (D) µ

QUESTION 5 Within the method of proof using mathematical induction, for which sum is the initial statement true? (A) 3 2 2 1 ( 1) = = + ∑ n i i n n (B) 3 2 2 1 2 ( 1) = = + ∑ n i i n n (C) 2 2 3 1 ( 1) 2 = + = ∑ n i n n i (D) 2 2 3 1 ( 1) 4 = + = ∑ n i n n i

QUESTION 6 At time t, a particle travels with a velocity of 2 2 ˆ ˆ 2 . 1 t t   = −   +   v i j Determine a general expression for the position vector, r, of the particle during this motion. (A) ( ) 1 ˆ ˆ 2tan 2 − = − + t c r i j (B) ( ) 1 2 ˆ ˆ 2tan− = − + t t c r i j (C) ( ) 1 1 ˆ ˆ tan 2 2 − = − + t c r i j (D) ( ) 1 2 1 ˆ ˆ tan 2 − = − + t t c r i j

QUESTION 7 For , z C ∈ a subset of the complex plane of the form ( ) 1 Arg z z θ − = is shown. 1 0 −2 −1 −1 2 2 1 −2 Im(z) Re(z) The values of z1 and θ respectively are (A) − i and 4 −π (B) − i and 4 π (C) i and 4 −π (D) i and 4 π

QUESTION 8 The position vectors of two objects over time, t, where 0 t ≥, are given by ( ) ( ) ( ) 2 1 2 ˆ ˆ 2 ˆ ˆ 4 where t t t at a R = − = + ∈ r i + j r i j Given that the two objects collide, the value of a is (A) 2 (B) 1 (C) −1 (D) −2

QUESTION 9 An approximate confidence interval for a population mean is calculated based on a sample size of 16 and is found to have width, w. A second confidence interval is calculated from another sample with the same sample standard deviation. Given that the same z-value was used for both intervals and the width of the second interval is 2w, what is the size of the second sample? (A) 2 (B) 4 (C) 8 (D) 64

QUESTION 10 Given z1 and z2 are complex numbers, which statement is always true? (A) 1 2 1 2 z z z z + = + (B) 1 2 1 2 z z z z + ≠ + (C) 1 2 1 2 z z z z + ≤ + (D) 1 2 1 2 z z z z + ≥ +

QUESTION 11 (5 marks) Consider the system of linear equations represented using the augmented matrix shown. 1 2 3 1 1 1 6 2 1 1 1 0 0 4 4 R R R  − − −   −       Key: R1 represents the row 1 values. a) Modify the augmented matrix using the row operation shown. [1 mark] R1 R3 1 –1 –1 – 6 4 0 0 4 R 2′ = R 2 + 2R1 Key: R2′ = R2 + 2R1 indicates that the new row 2 values are equal to the sum of the existing row 2 and twice row 1 values. b) Given the row 1 values represent the equation x − y − z = −6, use your result from Question 11a) to determine the solution of the system of linear equations. [3 marks] The system of linear equations is geometrically represented by three planes. c) Use your result from Question 11b) to describe a geometrical interpretation of your solution of the system of linear equations. [1 mark]

QUESTION 12 (6 marks) A line, l, is given by the equation 1 3 2 2 2 x y z + − = = − − . a) Given the point (−1, 3, a) lies on the line, determine the value of a. [1 mark] b) Determine a vector, d, in the direction of the line.  [1 mark] A plane, φ, is given by the equation x − y + 4z = 8. c) Given the point (b, b, −2b) lies on the plane, determine the value of b. [1 mark] d) Determine a vector, n, that is normal to the plane. [1 mark] e) Show that d · n = 0. [1 mark] Consider the statement: The line, l, is perpendicular to the plane, φ. f) Use your result from Question 12e) to comment on the reasonableness of the statement. [1 mark]

QUESTION 13 (5 marks) The velocity (m s−1) of a 3 kg object moving in a straight line is given by 1 2cos , 0 3 3 x v x −  = ≤ <     where x is its position (m) from the origin. a) Determine the momentum (kg m s−1) of the object when it is at the origin. [2 marks] b) Determine the acceleration (m s−2) of the object when it is at the origin. [3 marks]

QUESTION 14 (5 marks) A factory produces cans of juice. Each can has a labelled volume of 500 mL. The factory manager conducted a random sample of 100 cans to assess whether the labelled volume was being met satisfactorily in production. The mean volume of the sample was 498.9 mL with a sample standard deviation of 4.0 mL. a) Based on this sample and using a z-value of 2, determine an approximate confidence interval for the population mean volume. [3 marks] Industry regulation requires that the mean volume must meet or exceed the labelled volume. b) Use your result from Question 14a) to state whether the current production meets the regulation. Justify your decision using mathematical reasoning. [2 marks]

QUESTION 15 (5 marks) a) Use integration by parts to show ( ) 2 2 2 2 − − = − + + + ∫ x x x e dx e x x c [3 marks] The area of the bounded region between the graphs of y = x2e−x and y = x2e−1 over the domain [0, 1] is given by ( ) 1 2 1 0 x x e e dx − − − ∫ b) Use your result from Question 15a) to determine this area. Simplify your answer.  [2 marks]

QUESTION 16 (5 marks) A parallelepiped is a three-dimensional figure where all six faces are parallelograms. It can be defined by vectors a, b and c, as shown. The origin O and points P, Q, R and S are vertices of the parallelepiped. O S R P Q a c b Use vectors a, b and c to prove that the diagonal from P to R and the diagonal from Q to S bisect each other.

QUESTION 17 (7 marks) The radius of a cylinder decreases at a constant rate of 0.5 m s−1, while maintaining a constant height of four metres. Given that the cylinder has an initial volume of 100π m3, determine the rate of change of the volume (m3 s−1) of the cylinder after four seconds.

QUESTION 18 (6 marks) An object is projected at an acute angle of θ below the horizontal, with an initial speed of 30 m s−1 from a position 90 m above ground level. The object hits the ground 90 m horizontally from its projection point. Use vector calculus to determine θ in its simplest form. Assume that the magnitude of mean acceleration due to gravity on Earth is 10 m s−2 and that there is no air resistance.

QUESTION 19 (6 marks) Let w C ∈ be any fifth root of unity, where w R ∉ Show that ( )( ) 3 3 1 1 w w w Z − + + ∈