Unit 2 Maths Methods (CAS) Exam

Transcription

1 Name: Teacher: Unit 2 Maths Methods (CAS) Exam Monday November 20 (9.05 am) 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 /50 1

2 Question 1 Shifty car rentals charge $60.00 per day to hire a car, with an extra cost of $10 per 100 km. a) Write a linear relation that describes the cost (c) as a function of the distance travelled (x). (1 mark) b) Calculate the total cost of hiring the car for a day & driving: (3 marks) 100 km 200 km 500 km c) Use this data to draw an appropriate graph of the cost vs distance. (2 marks) Low Budget rentals offer the same car at a fixed cost of $100 per day. d) Draw this cost on the graph. (1 mark) e) Use an equation to find the distance at which the Low Budget rental becomes cheaper than Shifty. (2 marks) 2

3 Question 2 The quadratic function y =2x 2 +2x 40 describes a parabola. a) Fully factorise the quadratic function. (2 marks) b) Find the two x intercepts and the y intercept of the parabola. (2 marks) x = x = y = c) Find the stationary point of the parabola. (2 marks) 3

4 Question 3 Use long division to fully factorise the cubic function f (x )=2x 3 7x 2 +2x + 3. (4 marks) Question 4 Solve each of the following equations for x. (4 marks) 25 x+2 = 5 6 x=log 3 9 log 3 2+log 3 6 4

5 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

6 Question 6 For the circular function y =2sin( x 2 )+ 3 : a) State the period, minimum and maximum values of the function. (2 marks) Period: Minimum: Maximum: b) Draw the graph of the function over the interval (0,4π ). (4 marks) c) Find the solution(s) to the equation 2=2sin( x )+ 3 over the interval (0,4π ). (2 marks) 2 6

7 Question 7 For the quadratic function y = x 2 6x a) Find the derivative of the function. (1 mark) b) Find the gradient of the curve at the point x = 3. (1 mark) c) Find a point on the curve at the point where the gradient is 8. (2 marks) 7

8 Question 8 The staff at Happy Valley High School is made up of 55% females (F) and 45% males (F ). 20% of the staff are males under % of the staff are 40 or over. a) Complete the Probability table for the events F (female) and Y (young) to help answer the questions. (4 marks) F F Y Y 100% If a staff member is selected at random, what is the probability that: b) a male is selected? (1 mark) c) a male 40 years or over is selected? (1 mark) d) a female under the age of 40 is selected? (1 mark) e) a person under 40 years of age is selected? (1 mark) f) the person is a female given that the person selected is under 40 years of age? (2 marks) 8

9 Name: Teacher: VIJ Unit 2 Maths Methods (CAS) Exam Monday November 20 (9.05 am) 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 /50 1

10 Question 1 Shifty car rentals charge $60.00 per day to hire a car, with an extra cost of $10 per 100 km. a) Write a linear relation that describes the cost (c) as a function of the distance travelled (x). (1 mark) C = $60+ $0.1x b) Calculate the total cost of hiring the car for a day & driving: (3 marks) 100 km C = $ $0.1= $ km C = $ $0.1= $ km C = $ $0.1= $110 c) Use this data to draw an appropriate graph of the cost vs distance. (2 marks) $110 Low Budget rentals offer the same car at a fixed cost of $100 per day. d) Draw this cost on the graph. (1 mark) e) Use an equation to find the distance at which the Low Budget rental becomes cheaper than Shifty. (2 marks) $100 = $60+ $0.1x $40 = $0.1x x = 400 km 500 km 2

11 Question 2 The quadratic function y =2x 2 +2x 40 describes a parabola. a) Fully factorise the quadratic function. (2 marks) y =2x 2 +2x 40 y =2(x 2 + x 20) y =2(x 4)(x +5) y =2(x 4)(x +5) b) Find the two x intercepts and the y intercept of the parabola. (2 marks) 0 =2(x 4)(x +5) x = 4 or x = 5 y = y = 40 x = -5 x = 4 y = -40 c) Find the stationary point of the parabola. (2 marks) (Any of these methods) y =2x 2 +2x 40 dy = 4x +2= 0 dx 2= 4x x = 1 2 y =2( )( 1 2 4) y =2( 9 2 )( 9 81 )= 2 2 ( 1 81, 2 2 ) y =2(x 2 + x 20) y =2 (x 2 + x ) y =2 (x 2 + x ) 4 4 ) y =2 (x ) ) y =2(x ) ( 1 81, 2 2 ) x = b 2a = = 1 2 y =2( )( 1 2 4) y =2( 9 2 )( 9 81 )= 2 2 ( 1 81, 2 2 ) 3

12 Question 3 Use long division to fully factorise the cubic function f (x )=2x 3 7x 2 +2x + 3. (4 marks) f (1)= f (1)= = 0 (x 1) is a factor 2x 2 5x 3 x 1 2x 3 7x 2 + 2x + 3 2x 3 2x 2-5x 2 + 2x + 3 5x 2 + 5x - 3x + 3-3x x 2 5x 3 = (2x +1)(x 3) f (x )= (2x +1)(x 1)(x 3) Question 4 Solve each of the following equations for x. (4 marks) 25 x+2 = 5 6 x=log 3 9 log 3 2+log 3 6 (5 2 ) x+2 = x+4 = 5 6 2x + 4 = 6 2x =2 x = log x = log 3 27 ( ) ( ) x = log x = 3log 3 ( 3) x = 3 1 x = 1 x = 3 4

13 Question 5 For the function f (x )= x +2 : a) State the implied domain of the function. (1 mark) x 0, so the domain must be (, 0 (, 0 b) State the range of the function. (1 mark) 2, ) c) Sketch the graph of the function, including any endpoints. (1 mark) y f (x) (0,2) (2, 0) x f 1 (x) d) On the same axes, sketch the graph of the inverse function f 1 (x), including any endpoints. (2 marks) 5

14 Question 6 For the circular function y =2sin( x 2 )+ 3 : a) State the period, minimum and maximum values of the function. (2 marks) Period: = 2π 1 = 4π 2 Minimum: 3-2 = 1 Maximum: = 5 b) Draw the graph of the function over the interval (0,4π ). (2 marks) c) Find the the solution(s) to the equation 2=2sin( x )+ 3 over the interval (0,4π ). (2 marks) 2 2=2sin( x 2 )+ 3 1=2sin( x 2 ) x = π 3 (not in the domain),x = 4π π 3 = 11π 3 x = 7π = sin( x 2 ) π 6 = x 2 and 7π 6 = x 2 x = 7π 3,11π 3 6

15 Question 7 For the cubic function y = x 2 6x a) Find the derivative of the function. (1 mark) dy dx =2x 6 b) Find the gradient of the curve at the point x = 3. (1 mark) f '(x )=2x 6 f '(3)=2 3 6 = 0 0 c) Find a point on the curve at the point where the gradient is 8. (2 marks) 8 =2x 6 14 =2x x =7 y = y = y =7 (7,7) 7

16 Question 8 The staff at Happy Valley High School is made up of 55% females (F) and 45% males (F ). 20% of the staff are males under % of the staff are 40 or over. a) Complete the Probability table for the events F (female) and Y (young) to help answer the questions. (4 marks) F F Y 50% 20% 70% Y 5% 25% 30% 55% 45% 100% If a staff member is selected at random, what is the probability that: b) A male is selected? (1 mark) Pr(F ')= 45% c) a male 40 years or over is selected? (1 mark) Pr(F ' Y ')=25% d) a female under the age of 40 is selected? (1 mark) Pr(F Y )= 50% e) a person under 40 years of age is selected? (1 mark) Pr(Y )=70% f) the person is a female given that the person selected is under 40 years of age? (2 marks) Pr(F Y )= Pr(F Y ) = 50% Pr(F Y )= 5 Pr(Y ) 70% 7 8