2022, Prova 1

QUESTION 1 Let 3 z a i = + and 3 w bi = −+ , where , a b R ∈. If z w = , then (A) 3, 3 a b = − = − (B) 3, 3 a b = − = (C) 3, 3 a b = = − (D) 3, 3 a b = =

QUESTION 2 Which statement regarding sample means is true? (A) The distribution of X is always normally distributed. (B) The distribution of is always normally distributed. (C) The value of x̅ changes when different samples are selected. (D) The value of μ changes when different samples are selected.

QUESTION 3 A particle travels in a straight line over time, t, with a constant acceleration, ( ) a t . Which function could represent the particle’s displacement, ( ) x t ? (A) ( ) 3 x t t = (B) ( ) 2 x t t = (C) (D) ( ) x t t = ( ) 1 x t t =

QUESTION 4 When using proof by mathematical induction to prove De Moivre’s theorem expressed as      cis cis n n r θ r nθ n Z    , which statement would be correct in the proof of the inductive step? (A) (B) (C) (D)      cis cis k k r θ r kθ       1 cis cis k k r θ r k θ         1 1 cis cis 1 k k r θ r kθ            1 1 cis cis 1 k k r θ r k θ    

QUESTION 5 Four random samples of different sizes were taken to estimate a certain population mean, given a known population standard deviation. A 95% confidence interval was calculated for each sample. 10 15 20 25 30 35 40 Sample 1 Sample 2 Sample 3 Sample 4 Which sample used the largest sample size? (A) Sample 1 (B) Sample 2 (C) Sample 3 (D) Sample 4

QUESTION 6 The Cartesian equation for a sphere with centre ( ) 2, 3, 4 − − and radius 9 is (A) (B) (C) (D) ( ) ( ) ( ) 2 2 2 2 3 4 9 x y z − + + + − = ( ) ( ) ( ) 2 2 2 2 3 4 9 x y z + + − + + = ( ) ( ) ( ) 2 2 2 2 3 4 81 x y z − + + + − = ( ) ( ) ( ) 2 2 2 2 3 4 81 x y z + + − + + =

QUESTION 7 Two forces act concurrently on a 2 kg object placed at the origin. 30° 30° 4N y x 4N Not to scale The magnitude of the acceleration of the object is (A) 2 2 m s− (B) 2 2 3 m s− (C) 2 4 m s− (D) 2 4 3 m s−

QUESTION 8 Use the substitution ( ) tan u x = to determine ( ) ( ) 2 tan sec x x dx ∫ . (A) (B) (C) (D) ( ) 2 tan x c + ( ) 1 tan 2 x c + ( ) 2 1 tan 2 x c + ( ) tan x c +

QUESTION 9 A random variable X is normally distributed with a mean of 36 and a standard deviation of 4. The respective mean and standard deviation of the distribution of from repeated random samples of size 9 are (A) (B) (C) (D) 4 4 and 9 4 4 and 3 4 36 and 9 4 36 and 3

QUESTION 10 A plane is represented by the equation 2 5 x z − = . A vector normal to this plane is (A) (B) (C) (D) 1 2 5     −       1 0 2           1 0 2         −   1 2 −−      

QUESTION 11 (6 marks) The position vector of a particle, ( ) 1 cm r , over time, ( )s t , is given by ( ) ( ) ( ) ( ) 1 ˆ ˆ ˆ 2 1 3 2 3 t t t t = + + + − − r i j k a) Determine the velocity vector of the particle. [1 mark] b) Determine the time when the position vector of the particle is perpendicular to its velocity vector. [2 marks] The position vector of a second particle, r ( ) 2 cm , over time, ( )s t , is given by ( ) ( ) ( ) 2 ˆ ˆ ˆ 16 4 3 13 2 = − − − + t t t r i j k c) Determine whether the two particles collide.  [3 marks]

QUESTION 12 (6 marks) Given 1 2 , , , , z a bi z c di a b c d R = + = + ∀ ∈, and 2 0 z ≠, prove the identity 1 2 z z = 2z 1z 2z

QUESTION 13 (6 marks) a) Use partial fractions to determine  [4 marks] ( )( ) 22 2 3 4 dx x x − + ∫ b) Use the result from Question 13a) to determine ( )( ) 0 3 22 2 3 4 dx x x − − + ∫ Express your answer in simplest form. [2 marks]

QUESTION 14 (4 marks) The slope field for the differential equation ( ) 0.5 4 y dy dx x − − = , 0 x ≠ using 6 6 x −≤ ≤ and 6 6 y −≤ ≤ is shown. y x 6 5 4 3 2 1 0 −1 −1 −1 −1 1 2 3 4 5 6 −2 −2 −3 −3 −4 −4 −5 −5 −6 −6 −2 −2 −3 −3 −4 −4 −5 −5 −6 −6 0 A a) Determine the value of the slope at point A. [2 marks] b) Use the slope field to sketch the solution curve for ( ) 0.5 4 y dy dx x − − = given that when 6, 3.5 x y = − =  [2 marks] Note: If you make a mistake in the slope field, cancel it by ruling a single diagonal line through your work and use the additional response space on page 21 of this question and response book.

QUESTION 15 (4 marks) Consider the equation 3 1 z = where z C ∈ . a) Sketch the solutions to 3 1 z = on the Argand diagram. [2 marks] Im(z) Re(z) 2 1 −1 −1 1 2 −2 −2 Note: If you make a mistake in the Argand diagram, cancel it by ruling a single diagonal line through your work and use the additional response space on page 22 of this question and response book. The solutions to 3 1 z = can be expressed in the form z a bi = + , where , a b R ∈. b) Determine the largest possible positive value of ab. [2 marks]

QUESTION 16 (7 marks) Consider this system of equations that corresponds to three planes. 5 1 2 3 3 8 3 x y z x z y y λ z        a) Use a Gaussian technique to determine the value of λ for which this system of equations has infinitely many solutions. [4 marks] b) Use the result from Question 16a) to determine the infinitely many solutions. Express your answer in the form of a vector equation of a line. [3 marks]

QUESTION 17 (5 marks) The region between the x-axis and the curve of the function ( ) 1 sin 2 y x = + for 0 2 π x   is rotated about the x-axis to form a solid of revolution. Determine the volume of this solid. Express your answer in simplest form.

QUESTION 18 (5 marks) It is proposed that the following expression is divisible by    1 cis θ  for n    , 1 cis 0 Z θ    .   2 1 0 cis n r rθ   Evaluate the reasonableness of the proposition.

QUESTION 19 (7 marks) The function ( ) f x passes through the origin. The gradient function of ( ) f x is defined as ( ) ( ) 1 sin x g x e − = x e . Determine ( ) f x .