2024, Prova 1

QUESTION 1 Repeated random samples will be used to calculate a large number of 90% confidence intervals for a population mean µ. Which statement best describes the possible outcomes? (A) Approximately 90% of the intervals will contain µ. (B) More than 90% of the intervals will contain µ. (C) Less than 90% of the intervals will contain µ. (D) Exactly 90% of the intervals will contain µ.

QUESTION 2 Given that , determine the value of A. (A) − 4 (B) − 2 (C) 2 (D) 4

QUESTION 3 Consider a proof of the proposition using mathematical induction. Within the proof of the inductive step, the proposition for could be expressed as (A) (B) (C) (D)

QUESTION 4 A plane contains the point (1, 3, 1) and is normal to the vector . The vector equation of the plane is (A) (B) (C) (D)

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. Given that row 1 values of the matrix represent , the unique solution for y is (A) (B) (C) (D)

QUESTION 6 Players P, Q, R and S played each other once in a competition where there were no draws. Only the following results are known. • Player P defeated players Q and R. • Player Q defeated two players. • Players R and S each defeated one player. Based on these results, a dominance matrix N was partially constructed as shown. N = P Q R S P 0 1 1 0 Q R 0 0 0 1 S 1 0 0 0 The completed matrix N is (A) 0 1 1 0 1 0 0 1 0 0 0 1 1 0 0 0 (B) 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 (C) 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 (D) 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 0

QUESTION 7 A, B and C are points in three-dimensional space. If , then (A) is twice the value of . (B) and  BC are perpendicular. (C) only one plane contains A, B and C. (D) a straight line passes through A, B and C.

QUESTION 8 Given , determine 3 z . (A) − 8 (B) − 6 (C) 6 (D) 8

QUESTION 9 Use a suitable double-angle identity to determine . (A) (B) (C) (D)

QUESTION 10 The polynomial can be expressed in factorised form as , where b ∈ Z. Determine the value of b. (A) 2 (B) 1 (C) −1 (D) −2

QUESTION 11 (4 marks) The vector equation of a straight line is given by , where k is a scalar. a) Express the equation of the line as a pair of parametric equations.  [1 mark] b) Use your result from Question 11a) to express the equation of the line as a Cartesian equation. [1 mark] c) Determine the coordinates of the point that the line passes through when k = 5.  [1 mark] d) Determine the value of k when the line intersects the y-axis. [1 mark]

QUESTION 12 (7 marks) Point A lies on a section of the ellipse as shown. The coordinates of A are . x y 0 A a) Determine the value of y1. [2 marks] b) Use implicit differentiation to determine an expression for in terms of x and y. [2 marks] c) Use your results from Questions 12a) and 12b) to determine the gradient of the tangent to the curve at A. [1 mark] Consider the region between the given section of the ellipse, the x-axis and the lines x = 0 and . d) Determine the volume of the solid of revolution formed by rotating this region about the x-axis. Express your answer in simplest form. [2 marks]

QUESTION 13 (4 marks) and , where , have the same remainder when divided by z − i. Use the remainder theorem to determine the value of a.

QUESTION 14 (5 marks) The displacement (cm) of a particle from the origin as it travels in two-dimensional space at time t for seconds is given by a) Express the path of the particle as a pair of parametric equations. [1 mark] A general Cartesian form of a hyperbola with centre (h, k) is , where . b) Use a suitable Pythagorean identity to show that the path of the particle can be expressed in this general Cartesian form. [3 marks] c) Determine the centre of the hyperbolic path of the particle. [1 mark]

QUESTION 15 (6 marks) A sketch of a partially completed slope field for the differential equation is shown. a) Complete the slope field by sketching the slopes at the 10 points indicated. [2 marks] y x 2 1 0 1 −1 −1 −2 −2 −1 −1 −2 −2 2 b) Use the completed slope field to sketch the solution curve for , given the point (–1, 0) lies on this curve. [1 mark] 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. The differential equation can be solved by rearranging it into the form . c) Determine the equation of the solution curve sketched in Question 15b) by solving the differential equation, given the point (–1, 0) lies on this curve. Leave your answer in the form . [3 marks]

QUESTION 16 (6 marks) Use mathematical induction to prove that is a multiple of 7 for .

QUESTION 17 (6 marks) The acceleration (m s–2) of an object that moves in a straight line in an easterly direction over time t for seconds is given by , where v is its velocity (m s–1). The object is initially at rest at a position that is metres west of the origin. A student uses this information to calculate that the object is positioned at the origin when seconds. Evaluate the reasonableness of the student’s calculation.

QUESTION 18 (6 marks) A random variable X has a probability density function given by where k is a positive constant. Determine the value of k.

QUESTION 19 (6 marks) Consider complex numbers of the form , where x is a positive real number. If , determine all possible values of x.