2017 maths methods end of year exam 2 version 4 · unit 1 & 2 maths methods (cas) exam 2 2017...

Name:

Teacher:

Unit 1 & 2 Maths Methods (CAS) Exam 2 2017Monday November 20 (1.00pm - 3.15pm)

Reading time: 15 Minutes Writing time: 120 Minutes

Instruction to candidates:Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers, a single bound exercise book containing notes and class-work, CAS calculator.

Materials Supplied:Question and answer booklet, detachable multiple choice answer sheet at end of 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.• Answer the multiple choice questions on the separate answer sheet.

Section A Section B Total exam

/25 /65 /90

Unit 1 & 2 Maths Methods (CAS) Exam 2 2017

1

Section A – Multiple choice questions (25 marks)

Question 1A straight line has the equation 4y − 3x = 8 . Another form of this equation is:

a) y = 3x8

+ 4

b) y = 8x + 3

c) y = 3x4

+ 2

d) y = 3x4

+ 4

e) y = 3x4

+ 8

Question 2The equation of the linear function graphed here is:

a) y = 2x + 7b) y = −2x + 7

c) y = − 2x7

+ 2

d) y = 7x2

− 2

e) y = 2x7

− 2


2

Question 3A straight line segment joins the points (2,4) and (6,14). The midpoint of the segment and distance between the points are respectively:

a) (8,18) and 116

b) (4,9) and 116c) (4,9) and 116

d) (2,12) and 116

e) (4,9) and 1162

Question 4Which one of the following is perpendicular to the line y = 2x + 3 ?

a) y = x2− 5

b) y = x2− 13

c) y = x2− 3

d) y = 5 − x2

e) y = x3+ 12

Question 5Which of the following is the turning point form of the quadratic y = 2(x2 + 4x −12) ?

a) y = 2(x − 2)(x + 6)

b) y = 2(x + 2)2 − 32

c) y = (x + 2)2 −16

d) y = 2(x + 2)2 − 8

e) y = 2(x − 2)2 − 32


3

Question 6The complete set of solution(s) to the equation 27 = 64x3 is:

a) x = 43

b) x = − 43

c) x = 34

d) x = 34

and x = − 34

e) x = 34

and x = 43

Question 7The graph of a cubic function is shown below. Which of the following equations describes the graph?

a) y = −x2 (x + 4)

b) y = −x(x − 4)2

c) y = x2 (x + 4)

d) y = −x2 (x − 4)

e) y = x2 (x − 4)


4

Question 8In a mouse infested area, the population of mice is increasing by 25% every month. The time that it takes for the population to double is found by which equation?a) log21.25 = xb) log2 0.25 = x

c) log1.25 2 = x

d) log0.25 2 = x

e) log2.25 1.25 = x

Question 9

Which of the following is equal to x−32 ?

a) −x32

b) x23

c) x3

d)

1x3

e)

1x−3

Question 10Which of the sets correctly describes the interval shown on the number-line below?

a) (−5,−2)∪ (3,∞)b) [−5,−2]∪[3,∞]c) [−5,−2)∪[3,∞)d) (−5,−2]∪ (3,∞]

e) (−5,−2]∪[3,∞)

206 M a t h s Q u e s t 1 1 M a t h e m a t i c a l M e t h o d s C A S

Domain and range

1 Describe each of the following subsets of the real numbers using interval notation.

2 Illustrate each of the following number intervals on a number line.a [⇤6, 2) b (⇤9, ⇤3) c (⇤�, 2] d [5, �)e (1, 10] f (2, 7) g (⇤�, ⇤2) ⌅ [1, 3) h [⇤8, 0) ⌅ (2, 6]i R \ [1, 4] j R \ (⇤1, 5) k R \ (0, 2] l R \ [⇤2, 1)

3 Describe each of the following sets using interval notation.a {x: ⇤4 ⇥ x < 2} b {x: ⇤3 < x ⇥ 1} c {y: ⇤1 < y < }d {y: ⇤ < y ⇥ } e {x: x > 3} f {x: x ⇥ ⇤3}g R h R+

⌅ {0} i R \ {1}j R \ {⇤2} k R \ {x: 2 ⇥ x ⇥ 3} l R \ {x: ⇤2 < x < 0}

4

Consider the set described by R \ {x: 1 ⇥ x < 2}.a It is represented on a number line as:

a b

c d

e f

g h

A B

C D

E

1. The domain of a relation is the set of first elements of an ordered pair.2. The range of a relation is the set of second elements of an ordered pair.3. The implied domain of a relation is the set of first element values for which a

rule has meaning.4. In interval notation a square bracket means the end point is included in a set of

values, whereas a curved bracket means the end point is not included.

a b

(a, b]

remember

4CWORKEDExample

4

–2 10 50

40–3 90–8

0–1 0 1

30–5 –2 4210–3

WORKEDExample

5

31

2---

1

2-------

multiple choice

210 210

210 210

210


5

Question 11The function y = 2x − 4 has the inverse:

a) y = 2x − 4b) y = 4x − 2

c) y = x2+ 2

d) y = x2− 4

e) y = x2+ 4

Question 12What is the equation of the circle shown here?a) (x +1)2 + (y −1)2 = 4

b) (x +1)2 − (y +1)2 = 4

c) (x −1)2 + (y −1)2 = 4

d) (x +1)2 + (y +1)2 = 2

e) (x +1)2 + (y +1)2 = 4

Question 13Which of the following pairs of functions correctly describes the circle x2 + y2 = 25 ?

a) y = 25 − x2 and y = 25 + x2

b) y = 25 − x2 and y = − 25 − x2

c) y = x2 − 25 and y = − x2 − 25

d) y = 5 − x and y = x − 5

e) y = 25 − x2 and y = 25 + x2

Question 14The graph shown here could be described by the equation:a) y = −3cos4x

b) y = −3cos4x −1

c) y = −3sin4x −1d) y = −cos4x −1

e) y = 3cos4x −1


6

Question 15

Which of the following does not have the same value as sin π3

⎛⎝⎜

⎞⎠⎟ ?

a) cos π6

⎛⎝⎜

⎞⎠⎟

b) sin60°

c) −sin 5π3

⎛⎝⎜

⎞⎠⎟

d) cos 5π6

⎛⎝⎜

⎞⎠⎟

e) cos 13π6

⎛⎝⎜

⎞⎠⎟

Question 16

An angle θ in the first quadrant has sinθ = 45

⎛⎝⎜

⎞⎠⎟ . Which of the following is the value of cosθ ?

a) cosθ = 35

⎛⎝⎜

⎞⎠⎟

b) cosθ = 34

⎛⎝⎜

⎞⎠⎟

c) cosθ = 54

⎛⎝⎜

⎞⎠⎟

d) cosθ = 43

⎛⎝⎜

⎞⎠⎟

e) cosθ = 53

⎛⎝⎜

⎞⎠⎟

Question 17The function f (x) = ax3 + bx2 + cx + d has the derivative function:

a) f '(x) = −3ax2 − 2bx2 − cx − d

b) f '(x) = ax2 + bx + c

c) f '(x) = 3ax2 + 2bx2 + cx + d

d) f '(x) = 3ax2 + 2bx + ce) f '(x) = 0


7

Question 18The graph here shows the price of a particular share over 8 months from the start of January to the end of August.

The average increase in the price over the time period was:

a) $1.25 / monthb) $3.13 / monthc) $10 / monthd) $25 / monthe) $45 / month

Question 19

At the point x = 2 , the gradient of the curve y = 3x2 + 3 is equal to:

a) 0b) 7c) 9d) 12e) 27

Question 20

The cubic function y = x3

3− 3x

2

2+1has stationary points at:

a) 0,1( ) and 3,−312

⎛⎝⎜

⎞⎠⎟

b) 1,0( ) and 3,−312

⎛⎝⎜

⎞⎠⎟

c) 0,−1( ) and 3,312

⎛⎝⎜

⎞⎠⎟

d) 0,1( ) and −3,312

⎛⎝⎜

⎞⎠⎟

e) 0,1( ) and −3,−312

⎛⎝⎜

⎞⎠⎟


8

Question 21Which of the following graphs correctly shows the derivative of the function shown here?

a) b)

c) d)

e)


9

Question 22The chance of tossing 10 heads from 10 coin tosses is closest to:a) 0.1%b) 1%c) 10%d) 90%e) 99.9%

Question 23The Venn diagram shown here gives the results of a survey of 50 passengers on the 5.38 am train to Melbourne. Passengers were either in first or economy class, male or female.

Which of the following statements about the results is incorrect?a) There were 20 passengers in first class.b) 8 of the first class passengers were female.c) 16 of the passengers were males in economy class.d) There were 8 males in first class.e) 22 of the passengers were female.

Question 24Two events have the probabilities Pr(A) = 0.5, Pr(B) = 0.7, Pr(A⋂B) = 0.2. Which of the following statements is incorrect?a) Pr(A∪B) = 1

b) Pr(A’∩B’) = 0

c) Pr(A’∩B) = 0.5

d) Pr(A∩B’) = 0.5e) The most likely outcome is that only event B occurs.

Question 25In how many ways can the top three from ten competitors be arranged?a) 10b) 30c) 720d) 604880e) 3628800


10

Section B – Short answer questions (65 marks)Full workings must be shown.

Question 1

A potential buyer is investigating the overall costs of purchasing and running a new car. There are two models of the car:Model P (Petrol engine): $24,000 and costs $12 / 100 km for fuel.Model D (Diesel engine): $26,000 and costs $8 / 100 km for fuel.All other costs for the two models are the same.

a) What is the cost of fuel for driving each of the cars for 20,000 km and 100,000 km? (2 marks)

Model P Model D

20,000 km

100,000 km

b) Write an equation C(x) that gives the total cost (C) of purchase and fuel as a function of each kilometre travelled (x) for Model P. (1 mark)

c) Calculate how much further can be driven in the diesel car (rather than petrol) using $1000 worth of fuel. (2 marks)

d) Use simultaneous equations to calculate how many kilometres of driving it will take for the overall cost of the diesel car (car + fuel) to be less than the petrol model. (2 marks)


11

Question 2

The shape of a slide is defined by the cubic function y = − 1250

(x3 − 40x2 + 500x − 2000) , where y is the

height above ground level and x is the horizontal distance. The slide starts at x = 0 and ends where the curve goes below ground level.

a) Calculate the initial starting height of the curve. (1 mark)

b) Calculate the height of the slide where x = 10. (2 marks)

c) Using a CAS calculator or other means, find the value of x (to two decimal places) where the height of the slide is 4 m. (1 mark)

d) Using a CAS calculator or other means, find the value of x where the slide ends. (1 mark)


12

Question 3

Simplify the following expressions. (4 marks)

a)55 × 252 × a3

(52a)2

b)13log2 27( )− 1

2log2 36( )


13

Question 4

A population of rabbits can increase by 20% per month. At the start of the year, the number of rabbits in an area was 500.

a) Write an equation that describes the relationship between rabbit numbers and time. (2 marks)

b) Calculate the number of rabbits at the end of the sixth month. (2 marks)

c) Use a CAS calculator or other means to find the time (in months) at which the population of rabbits reaches 1200. (2 marks)


14

Question 5

The water level (in metres) at a harbour dock on a particular day is modelled by the equation:

h(t) = 1.5cos 4π t25

+ 4

(Time is measured from 12 midnight.)

a) Calculate the time period between successive high tides. (2 marks)

b) State the minimum and maximum heights that the water level reaches. (1 mark)

Minimum: m Maximum: m

c) Calculate the time of the first low tide. (2 marks)

d) Boats can only leave the harbour if the water level is above 3.0 m. Use a CAS calculator or another method to find the time periods when boats are stuck in the harbour. (2 marks)


15

Question 6

A stuntman is preparing for a stunt jump in a car. He has used a quadratic equation to model the path he will take, where h(x) is the height above the ramp and x is the horizontal distance covered. (Both distances are measured in metres.)

a) Find the horizontal distance (a) covered by the jump. (1 mark)

b) Find the derivative dhdx

. (1 mark)

h(x) = − x2

40+ x


16

c) Use calculus to show that the maximum height (b) occurs at x = 20m. (2 marks)

d) Calculate the maximum height reached during the jump. (2 marks)

e) Find the gradient of the launch ramp (when x = 0). (2 marks)

f) Calculate the angle of the launch ramp (c) above the horizontal. (1 mark)


17

Question 7

The graph below shows the area the curve y = −x(x − 4) .

a) Find the gradient of the curve at the point x = 1. (2 marks)

b) Find the equation of the tangent at x = 1. (2 mark)

c) Draw this tangent line on the graph. (1 mark)


18

Question 8

A function f (x) = cos(x) has a number of transformations applied to it.

a) Write the equations of the transformed functions. (4 marks)

i) f (−x) = iii) 3 f (x) =

ii) f (2x) iv) f (x −π )+1

b) The function g(x) = x2 is transformed into the function h(x) by :

dilating by a factor of 3 from the x-axis, then translated 2 units to the left and 4 down . What is the equation of h(x) ? (3 marks)

Question 9

Two cards are dealt (without replacement) from a standard deck of 52.

a) What is the probability that the first card dealt is a king? (1 mark)

b) If the first card dealt is a king, what that is the chance that the second card is also a king? (1 mark)

c) What is the chance that of the two cards dealt, exactly one is a king? (2 marks)

Five cards are dealt from the deck.

d) How many different combinations of cards can be made by dealing five cards? (2 marks)


19

Question 10

There are 6 Yr 11 English classes and 5 at Yr 12 at Happy Valley High School.A particular teacher (Mr X) has two Yr 11 classes and one Yr 12 class. Students are randomly assigned to these classes each year.

a) Draw a tree diagram showing the possible outcomes and probabilities of having the teacher Mr X. (5 marks)

b) What is the chance that a student has Mr X in Yr 11? (1 mark)

c) What is the chance that a student has Mr X in both Yr 11 and Yr 12? (1 mark)

d) What is the chance that a student has Mr X for only one of the two years? (2 marks)

e) What is the chance that a student does not have Mr X in either of the two years? (1 mark)


20

Answer sheet for section A

1. a b c d e

2. a b c d e

3. a b c d e

4. a b c d e

5. a b c d e

6. a b c d e

7. a b c d e

8. a b c d e

9. a b c d e

10. a b c d e

11. a b c d e

12. a b c d e

13. a b c d e

14. a b c d e

15. a b c d e

16. a b c d e

17. a b c d e

18. a b c d e

19. a b c d e

20. a b c d e

21. a b c d e

22. a b c d e

23. a b c d e

24. a b c d e

25. a b c d e


21

Answer sheet for section A

1. a b c d e

2. a b c d e

3. a b c d e

4. a b c d e

5. a b c d e

6. a b c d e

7. a b c d e

8. a b c d e

9. a b c d e

10. a b c d e

11. a b c d e

12. a b c d e

13. a b c d e

14. a b c d e

15. a b c d e

16. a b c d e

17. a b c d e

18. a b c d e

19. a b c d e

20. a b c d e

21. a b c d e

22. a b c d e

23. a b c d e

24. a b c d e

25. a b c d e


1

Section B – Short answer questions (50 marks)Full workings must be shown.

Question 1

A potential buyer is investigating the cost of purchasing and running a new car. There are two models of the car:Model P (Petrol engine): $24,000 and costs $12 / 100 km for fuel.Model D (Diesel engine): $26,000 and costs $8 / 100 km for fuel.All other costs for the two models are the same.

a) What is the cost of fuel for driving the cars for 20,000 km and 100,000 km? (2 marks)

Model P Model D

20,000 km

100,000 km

$2,400 $1600

$12,000 $8,000

b) Write an equation C(x) that gives the total cost (C) of purchase and fuel as a function of each kilometre travelled (x) for Model P. (1 mark)

C (x) = 24, 000+ 0.12x

c) Calculate how much further can be driven in the diesel car (rather than petrol) using $1000 worth of fuel. (2 marks)

Diesel: Petrol: The diesel car can travel 4167 more km.

1000 = 0.08x 1000 = 0.12x

x = 12,500km x = 8,333km

d) Use simultaneous equations to calculate how many kilometres of driving it will take for the overall cost of the diesel car (car + fuel) to be less than the petrol model. (2 marks)

24, 000 + 0.12x = 26, 000 + 0.08x

2, 000 = 0.04x

x = 50, 000


2

Question 2

The shape of a slide is defined by the cubic function y = − 1250

(x3 − 40x2 + 500x − 2000) , where y is the

height above ground level and x is the horizontal distance. The slide starts at x = 0 and ends where the curve goes below ground level.

a) Calculate the initial starting height of the curve. (1 mark)

y = − 1

250(−2000) = 8m

b) Calculate the height of the slide where x = 10. (2 marks)

y = − 1

250(103 − 40 × 102 +500 × 10 −2000) = 0m

c) Using a CAS calculator or other means, find the value of x (to two decimal places) where the height of the slide is 4 m. (1 mark)

4 = − 1

250(x 3 − 40x2 +500x −2000)

x = 2.45m

d) Using a CAS calculator or other means, find the value of x where the slide ends. (1 mark)

0 = − 1

250(x 3 − 40x2 +500x −2000)

x = 10m or 20m (The end is 20m)


3

Question 3

Simplify the following expressions. (4 marks)

a)55 × 252 × a3

(52a)2

55 ×54 × a3

54a2

= 55a

= 3125a

b)13log2 27( )− 1

2log2 36( )

= log

227( )

13 − log

236( )

12

= log23− log

26

= log

2

36= log

2

12

= −1


4

Question 4

A population of rabbits can increase by 20% per month. At the start of the year, the number of rabbits in an area was 500.

a) Write an equation that describes the relationship between rabbit numbers and time. (2 marks)

P =500×(1.2x )

b) Calculate the number of rabbits at the end of the sixth month. (2 marks)

P =500×(1.26 )=1493

c) Use a CAS calculator or other means to find the time (in months) at which the population of rabbits reaches 1200. (2 marks)

1200=500×(1.2x )x = log1.2 500x =4.8 months


5

Question 5

The water level (in metres) at a harbour dock on a particular day is modelled by the equation:

h(t) = 1.5cos 4π t25

+ 4

(Time is measured from 12 midnight.)

a) Calculate the time period between successive high tides. (2 marks)

T = 2π

k

T =2π

4π25

=252

= 12.5h

b) State the minimum and maximum heights that the water level reaches. (1 mark)

Minimum: 2.5 m Maximum: 5.5 m

c) Calculate the time of the first low tide. (2 marks)

2.5 = 1.5cos 4πt

25+ 4 ,

−1 = cos 4πt

25

π =

4πt25

, t = 25π

4π= 6.25 h = 6.15 am

d) Boats can only leave the harbour if the water level is above 3.0 m. Use a CAS calculator or another method to find the time periods when boats are stuck in the harbour. (2 marks)

3 = 1.5cos 4πt

25+ 4 gives the times when the water level is at 3.0 m.

There are four solutions:

t=4.58 or t=7.92 or t=17.08 or t=20.42

4.35 am - 7.55 am and 5.05 pm - 8.25 pm


6

Question 6

A stuntman is preparing for a stunt jump in a car. He has used a quadratic equation to model the path he will take, where h(x) is the height above the ramp and x is the horizontal distance covered.(Both distances are measured in metres.)

a) Find the horizontal distance (a) covered by the jump. (1 mark)

h(x) = x − x 2

40= x(1 − x

40) x = 0m and x = 40m

b) Find the derivative dhdx

. (1 mark)

dhdx

= 1 − x20


7

c) Use calculus to show that the maximum height (b) occurs at x = 20m. (2 marks)

0 = 1 − x

20 ,

1 = x

20 , x = 20m

d) Calculate the maximum height reached during the jump. (2 marks)

h(20) = 20 − 202

40= 10m

e) Find the gradient of the launch ramp (when x = 0). (2 marks)

dhdx

= 1 − 020

= 1

f) Calculate the angle of the launch ramp (c) above the horizontal. (1 mark)

θ = tan−1(1) = 45°


8

Question 7

The graph below shows the area under the curve y = x(4 − x) .

a) Find the gradient of the curve at the point x = 1. (1 mark)

y =−x(x −4)

y =4x −x 2

dydx

=4−2x =4−2(1)

dydx

=2

b) Find the equation of the tangent at x = 1. (1 mark)

y −3=2(x −1)

y −3=2x −2

y =2x +1

c) Draw this tangent line on the graph. (1 mark)


9

Question 8

A function f (x) = cos(x) has a number of transformations applied to it.

a) Write the equations of the transformed functions. (4 marks)

i) f (−x )= cos(−x ) iii) 3f (x )=3cos(x )

ii) f (2x )= cos(2x ) iv) f (x −π )+1= cos(x −π )+1

b) The function g(x) = x2 is transformed into the function h(x) by :

dilating by a factor of 3 from the x-axis, then translated 2 units to the left and 4 down . What is the equation of h(x) ? (3 marks)

h(x )=3(x +2)2 −4

Question 9

Two cards are dealt (without replacement) from a standard deck of 52.a) What is the probability that the first card dealt is a king? (1 mark)

452

= 113

b) If the first card dealt is a king, what that is the chance that the second card is also a king? (1 mark)

351

= 117

c) What is the chance that of the two cards dealt, exactly one is a king? (2 marks)

Pr (First card only is a king) = 4

52× 48

51= 192

2652= 16

221

Pr (Second card only is a king) = 48

52× 4

51= 16

221

Pr (Only one king) = 16221

+ 16221

= 32221

d) How many different combinations of cards can be made by dealing five cards? (2 marks)

52C

5= 52!

47!5!= 2,598,960


10

Question 10

2. There are 6 Yr 11 English classes and 5 at Yr 12. A particular teacher (Mr X) has two Yr 11 classes and one Yr 12 class. Students are randomly assigned to classes.

a) Draw a tree diagram showing the possible outcomes and probabilities of having the teacher Mr X.(5 marks)

b) What is the chance that a student has Mr X in Yr 11? (1 mark)

Pr(11)= 1

3

c) What is the chance that a student has Mr X in both Yr 11 and Yr 12? (1 mark)

Pr(11∩12)= 1

15

d) What is the chance that a student has Mr X for only one of the two years? (2 marks)

Pr(11∩12')+Pr(11'∩12)= 6

15= 2

5

e) What is the chance that a student does not ever have Mr X in the two years? (1 mark)

Pr(11'∩12')= 8

15


11

2017 maths methods end of year exam 2 version 4 · unit 1 & 2 maths methods (cas) exam 2 2017...

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