Name:

Teacher:

Unit 2 Maths Methods (CAS) Exam 1 2014Monday November 17 (9.00 - 10.45am)

Reading time: 15 Minutes Writing time: 90 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

/20 /40 /60

1

Section A – Multiple choice questions (20 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 3The 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)

Question 4In 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) log2 1.25 = x

b) log2 0.25 = x

c) log1.25 2 = xd) log0.25 2 = x

e) log2.25 1.25 = x

Question 5

Which of the following is equal to x−32 ?

a) −x32

b) x23

c) x3

d)1x3

e)1x−3

3

Question 6Which 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,∞)

Question 7The 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 8Which 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 − 25d) y = 5 − x and y = x − 5

e) y = 25 − x2 and y = 25 + x2

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

4

Question 9

The angle θ is in the second quadrant (π2<θ < π ) .

Which of the following statements about θ is incorrect?

a) sinθ > 0b) tanθ > 0

c) sin θ +π2

⎛⎝⎜

⎞⎠⎟< 0

d) tan θ + π2

⎛⎝⎜

⎞⎠⎟ > 0

e) sin π −θ( ) > 0

Question 10The graph shown here could be described by the equation:a) y = −3cos4x

b) y = −3cos4x −1

c) y = −3sin 4x −1d) y = − cos4x −1

e) y = 3cos4x −1

Question 11An angle θ has a value sinθ = 0.9 . Which of the following is the value of cosθ ?

a) cosθ = 0.1b) cosθ = ±0.19

c) cosθ = ± 0.19

d) cosθ = ± 0.9e) cosθ = 0.9

5

Question 12The graph here shows the price of a particular share over 8 months from the start of January to the end of August. The average change in the price over the time period was:

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

Question 13

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

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

Question 14

The cubic function y =x3

3−3x2

2+1 has 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

⎛⎝⎜

⎞⎠⎟

6

Question 15Which of the following graphs correctly shows the derivative of the function shown here?

a) b)

c) d)

e)

7

Question 16For the matrix multiplication shown below, what is the value of x?

1 4 −2⎡⎣ ⎤⎦7x1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= 17[ ]

a) 0b) 2.5c) 3d) 3.5e) 17

Question 17Which one of the following operations can be performed for the matrices shown below?

A = 123

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

and B = 1 2 3−1 −2 −3

⎡

⎣⎢

⎤

⎦⎥

a) B-1

b) A2

c) A+Bd) ABe) BA

Question 18

What is the determinant of the matrix 4 86 −3

⎡

⎣⎢

⎤

⎦⎥ ?

a) Undefinedb) 60c) -60d) 36e) 0

Question 19The chance of tossing 10 heads from 10 coin tosses is closest to:a) 0.1%b) 1%c) 10%d) 90%e) 99.9%

8

Question 20The 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.

9

Section B – Short answer questions (40 marks)Full workings must be shown.

Question 1

Simplify the following expressions. (4 marks)

a)55 × 252 × a3

(52a)2

b)13log2 27( )− 1

2log2 36( )

10

Question 2

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)

11

Question 3

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)

12

Question 4

Two matrix transformations as shown are applied to a point (x,y). 5 00 1

⎡

⎣⎢

⎤

⎦⎥

0 11 0

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

a) Find the single (2 x 2) matrix that can be used to describe the combined transformations. (2 marks)

b) Find the new co-ordinates if the point (3,6) undergoes this transformation. (1 mark)

13

Question 5

The graph below shows the area under the curve y = −x2 + 6x − 5 between the two x intercepts

(x=1 and x=5).

Use integration to calculate the area between the curve and between the two x intercepts. (3 marks)

14

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

15

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)

16

Question 7

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)

17

Question 8

The probability of a particular football team winning its next game is 75% if it won the previous game and 60% if it lost the previous game.

a) Draw the transition matrix for these outcomes. (1 mark)

b) If the team was successful in the opening game of the season, calculate the probability that it will win the third game of the season. (2 marks)

c) What percentage of games would the team be expected to win over the whole season? (1 mark)

18

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

19

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

Unit 2 Maths Methods (CAS) Exam 2014 Solutions

1

Section B – Short answer questions (40 marks)Full workings must be shown.

Question 1

Simplify the following expressions. (4 marks)

a)

55 ×252 ×a3

(52 a)2 = 55 ×54 ×a3

54 a2 =55 a or a=3125a

b)

13

log2 27( )− 12

log2 36( )

= log2 2713⎛

⎝⎜⎞⎠⎟− log2 36

12⎛

⎝⎜⎞⎠⎟

= log2 3( )− log2 6( )= log236

⎛⎝

⎞⎠

= log212

⎛⎝

⎞⎠ = log2 2−1( )

=−1

Question 2

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

Unit 2 Maths Methods (CAS) Exam 2014 Solutions

2

Question 3

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

h(t) = 1.5cos 4πt25

+ 4

(Time is 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

Unit 2 Maths Methods (CAS) Exam 2014 Solutions

3

Question 4

Two matrix transformations as shown are applied to a point (x,y). 5 00 1

⎡

⎣⎢

⎤

⎦⎥

0 11 0

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

a) Find the single (2 x 2) matrix that can be used to describe the combined transformations. (2 marks)

5 00 1

⎡

⎣⎢

⎤

⎦⎥

0 11 0

⎡

⎣⎢

⎤

⎦⎥=

0 51 0

⎡

⎣⎢

⎤

⎦⎥

b) Find the new co-ordinates if the point (3,6) undergoes this transformation. (1 mark)

0 51 0

⎡

⎣⎢

⎤

⎦⎥

36

⎡

⎣⎢

⎤

⎦⎥=

303

⎡

⎣⎢

⎤

⎦⎥

Question 5

The graph below shows the area under the curve y = −x2 + 6x − 5 between the two x

intercepts.Use integration to calculate the area between the curve and between the two x intercepts (x=1 and x=5). (3 marks)

−x 2 +6x −51

5

∫

= −13

x 3 +3x 2 −5x⎡⎣⎢

⎤⎦⎥1

5

= −13×53 +3×52 −5×5⎡

⎣⎢⎤⎦⎥− −1

3×13 +3×12 −5×1⎡

⎣⎢⎤⎦⎥

= 323

Unit 2 Maths Methods (CAS) Exam 2014 Solutions

4

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

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°

Unit 2 Maths Methods (CAS) Exam 2014 Solutions

5

Question 7

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) = 452

× 4851

= 1922652

= 16221

Pr (Second card only is a king) = 4852

× 451

= 16221

Pr (Only one king) = 16221

+ 16221

= 32221

Question 8

The probability of a particular football team winning its next game is 75% if it won the previous game and 60% if it lost the previous game.

a) Draw the transition matrix for these outcomes. (1 mark)

75% 60%25% 40%

⎡

⎣⎢

⎤

⎦⎥

b) If the team was successful in the opening game of the season, calculate the probability that it will win the third game of the season. (2 marks)

75% 60%25% 40%

⎡

⎣⎢

⎤

⎦⎥

210

⎡

⎣⎢

⎤

⎦⎥=71.25%

c) What percentage of games would the team be expected to win over the whole season? (1 mark)

60%85%

=70.6%

Unit 2 Maths Methods (CAS) Exam 2014 Solutions

6