Name: Teacher: Unit 2 Maths Methods (CAS) Exam 2 2014 Monday November 17 (1.50 pm) Reading time: 15 Minutes Writing time: 60 Minutes Instruction to candidates: Students are only permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers. No calculator or notes are allowed. Materials Supplied: 8 page question and answer booklet. Instructions: Write your name and that of your teacher in the spaces provided. Answer all short answer questions in this booklet where indicated. Always show your full working where spaces are provided. Total exam /40 1
Question 1 a) On the axes below, draw the graph of the line y = x 1 (1 mark) b) On the axes below, draw the graph of the line 3x +2y =18 (2 marks) c) Use simultaneous equations to find the co-ordinates of the point of intersection of the two lines. (2 marks) x = y = 2
Question 2 The quadratic function y =2x 2 + 6x 36 describes a parabola. a) Factorise the quadratic function. (2 marks) b) Find the two x intercepts and the y intercept of the parabola. (2 marks) x = x = y = 3
Question 3 Use long division to fully factorise the cubic function f (x )= x 3 6x 2 13x + 42. (4 marks) Question 4 Solve each of the following equations for x. (4 marks) 16 x+3 = 4 8 x=log 3 9 log 3 2+log 3 6 4
Question 5 For the function f (x )= x 2 : a) State the implied domain of the function. (1 mark) b) State the range of the function. (1 mark) c) Sketch the graph of the function, including any endpoints. (1 mark) y x d) On the same axes, sketch the graph of the inverse function f 1 (x), including any endpoints. (2 marks) 5
Question 6 For the circular function y =2sin(2x )+1: a) State the period, minimum and maximum values of the function. (2 marks) Period: Minimum: Maximum: Draw the graph of the function over the interval (0,2π ). (2 marks) Find the the solution(s) to the equation 2=2sin ( 2x )+1 over the interval (0,2π ). (2 marks) 6
Question 7 For the cubic function y = x 3 3x 2 a) Find the derivative of the function. (1 mark) b) Find the gradient of the curve at the point x =1. (1 mark) Question 8 For the matrix A = 2 4 1 1 : a) Find the matrix 2A. (1 mark) b) Find the matrix A 2. (2 marks) 7
Question 9 A survey of 120 car drivers was conducted about the one car that they drive the most. It was found that 50 of the drivers have a Holden. 40 of the Holden cars have automatic transmissions. In total, there were 35 drivers with manual (not automatic) transmissions. a) Complete the table below, showing the number of cars in each set. (4 marks) H H A A 120 b) What is the probability that a randomly selected Holden driver has a manual car? (1 mark) c) What is the probability that a randomly selected automatic car is a Holden? (1 mark) 8
Unit 2 Maths Methods (CAS) Exam 2 2014 Solutions Question 1 a) On the axes below, draw the graph of the line y = x 1 (1 mark) b) On the axes below, draw the graph of the line 3x +2y =18 (2 marks) c) Use simultaneous equations to find the co-ordinates of the point of intersection of the two lines. (2 marks) 3x +2y =18 and y = x 1 3x +2(x 1)=18 3x +2x 2=18 5x =20 x = 4 y = x 1 and x = 4 y = 4 1 y = 3 x = 4 y = 3 1
Unit 2 Maths Methods (CAS) Exam 2 2014 Solutions Question 2 The quadratic function y =2x 2 + 6x 36 describes a parabola. a) Factorise the quadratic function. (2 marks) y =2x 2 + 6x 36 y =2(x 2 + 3x 18) y =2(x + 6)(x 3) y =2(x + 6)(x 3) b) Find the two x intercepts and the y intercept of the parabola. (2 marks) 0 =2(x + 6)(x 3) x = 3 or x = 6 y =2 0 2 + 6 0 36 y = 36 x = 3 x = -6 y = -36 2
Unit 2 Maths Methods (CAS) Exam 2 2014 Solutions Question 3 Use long division to fully factorise the cubic function f (x )= x 3 6x 2 13x + 42. (4 marks) f (2)=2 3 6 2 2 13 2+ 42 f (2)= 8 24 26 = 42= 0 (x 2) is a factor x 2 4x 21 x 2 x 3 6x 2 13x + 42 x 3 2x 2-4x 2 13x + 42 4x 2 + 8x -21x + 42-21x + 42 0 x 2 4x 21= (x 7)(x + 3) f (x )= (x 7)(x 2)(x + 3) Question 4 Solve each of the following equations for x. (4 marks) 16 x+3 = 4 8 x = log 3 9 log 3 2+log 3 6 16 x+3 = 4 8 (4 2 ) x+3 = 4 8 4 2x+6 = 4 8 2x + 6 = 8 2x =2 x = log 3 9 log 3 2+log 3 6 x = log 9 6 3 2 x = log 3 27 ( ) ( ) x = log 3 3 3 x = 3log 3 ( 3) x = 1 x = 3 3
Unit 2 Maths Methods (CAS) Exam 2 2014 Solutions Question 5 For the function f (x )= x 2 : a) State the implied domain of the function. (1 mark) x 2 0, so the domain must be 2, ) 2, ) b) State the range of the function. (1 mark) 0, ) c) Sketch the graph of the function, including any endpoints. (1 mark) y f -1 (x) y = x 2 f(x) 2 x d) On the same axes, sketch the graph of the inverse function f 1 (x), including any endpoints. (2 marks) 4
Unit 2 Maths Methods (CAS) Exam 2 2014 Solutions Question 6 For the circular function y =2sin(2x )+1: a) State the period, minimum and maximum values of the function. (2 marks) Period: = 2π 2 = π Minimum: 1-2 = -1 Maximum: 1+ 2 = 3 Draw the graph of the function over the interval (0,2π ). (2 marks) Find the the solution(s) to the equation 2=2sin ( 2x )+1 over the interval (0,2π ). (2 marks) 2=2sin(2x )+1 1=2sin(2x ) 1 2 = sin(2x ) π 6 =2x x = π 12 Other solutions: x = 6π 12 π 12 = 5π 12 x = π 12 + 12π 12 = 13π 12 x = 5π 12 + 12π 12 = 17π 12 x = π 12,5π 12,13π 12,17π 12 5
Unit 2 Maths Methods (CAS) Exam 2 2014 Solutions Question 7 For the cubic function y = x 3 3x 2 a) Find the derivative of the function. (1 mark) dy dx = 3x 2 6x b) Find the gradient of the curve at the point x =1. (1 mark) f '(1)= 3 1 2 6 1 f '(1)= 3 3 Question 8 For the matrix A = 2 4 1 1 : a) Find the matrix 2A. (1 mark) 2A =2 2 4 1 1 = 4 8 2 2 b) Find the matrix A 2. (2 marks) A 2 = 2 4 1 1 = = 4+4 8 4 2 1 4+1 8 4 1 5 2 4 1 1 6
Unit 2 Maths Methods (CAS) Exam 2 2014 Solutions Question 9 A survey of 120 car drivers was conducted about the one car that they drive the most. It was found that 50 of the drivers have a Holden. 40 of the Holden cars have automatic transmissions. In total, there were 35 drivers with manual (not automatic) transmissions. a) Complete the table below, showing the number of cars in each set. (4 marks) H H A 40 45 85 A 10 25 35 50 70 120 b) What is the probability that a randomly selected Holden driver has a manual car? (1 mark) n(a' H) Pr(A' H) = n(h) = 10 50 = 1 5 1 5 c) What is the probability that a randomly selected automatic car is a Holden? (1 mark) n(h A) Pr(H A) = n(a) = 40 85 = 8 17 8 17 7