Section A Multiple Choice Questions 20 marks

Instructions for Section A Choose the correct answer by circling the corresponding letter. A correct answer scores 1; an incorrect answer scores 0. Marks will not be deducted for incorrect answers. No marks will be given if more than one answer is completed for any question. Unless otherwise indicated, the diagrams in this book are not drawn to scale.

1. The graph of 13 6xy += + has domain and range respectively as A. { : 1},{ : 6}x x y y> − > B. { : 1},{ : }x x y y R> ∈ C. { : },{ : 6}x x R y y∈ > − D. { : },{ : 6}x x R y y∈ > E. { : },{ : }x x R y y R∈ ∈

2. If log 27qp = , which of the following statements is always true? A. 27p q= B. 27qp = C. 27pq = D. 27q p= E. 27 p q=

3. The function 4: ( , ) , ( ) log (2 1) 3g a R g x x∞ → = + + has a maximal domain and a is a constant. The value of a is A. B. −2 C. −1

D. 12

−

E. 13

−

4. The rule for the inverse function of ( ) 10axf x b= + is given by

A. 1 1( )10axf x

b− =

+

B. 1( ) 10 axf x b− −= − C. 1

10( ) log ( )f x a x b− = −

D. 110

1( ) log ( )f x x ba

− = +

E. 110

1( ) log ( )f x x ba

− = −

5. The graph whose equation is 2 ,xy A B= × + where A and B are constants, is shown below.

The values of A and B are A. 3, 2A B= − = B. 1, 2A B= − = C. 5, 2A B= − = D. 5, 3A B= − = E. 5, 1A B= = −

6. 108° in radians is

A. 35π

B. 0.6

C. 10π

D. 3

5π

E. 53π

7. The period and range respectively of the graph of 2sin(3 ) 1y x= − are A. ,π [−2, 2]

B. 23

,π [−1, 1]

C. ,π [−3, 2]

D. 23

,π [−3, 1]

E. 23

,π [−2, 2]

y

x

2

–3

O

8. If 2sin( )5

θ = , 2π θ π≤ ≤ , the value of cos( )θ is

A. 15

−

B. 35

C. 215

−

D. 215

E. 35

−

9. The graph of 3 tan(2 )y x= for 0 x π≤ ≤ has asymptotes at

A. x =2π only

B. x =4π only

C. x =2π and x =

23π

D. x =4π and x =

43π

E. x = π and x = 2π

10. The solutions to the equation 2cos(3 ) 3x = are x equals

A. (36 1) (36 1), ,18 18n n n Zπ π+ −

∈

B. 11,18 18π π

C. (6 1) ,9

n n Zπ±∈

D. 5 7 11 13 17, , , , ,

9 9 9 9 9 9π π π π π π

E. (12 1) ,

18n n Zπ±

∈

11. The position of an ant, x metres, from its nest after time t minutes is given by 2( ) 2 3 2x t t t= − + . The average rate of change of the ant’s position over the first 10 minutes in metres per minute is A. 10 B. 17 C. 19 D. 37 E. 172

12. If 2( ) , and 0,f x a x a x a R a= − ∈ > has a stationary point at x = 1, the value of a is

A. 13

B. 12

C. 12 2

D. 14

E. 0

13. f is a function such that (3) 0 and ( ) 0f ' f ' x= < when x < 3 and x > 3. When x = 3 the graph of f has a A. local minimum B. local maximum C. stationary point of inflection D. gradient of 3 E. vertical asymptote at x = 3

14. The graph below shows the function g which has a tangent drawn to it. The tangent touches the graph of g at the point (3, 2). The y-intercept of the tangent is 1.

It is true to say that A. '(0) 1g = B. (1) 3g = C. 0)1(' =g

D. 31)3(' =g

E. (3) 3g =

x1 3

2(3,2)

(0,1)

y

)(xgy =

15. The position of a particle (in metres) from a fixed point O after t seconds is given by 3 2( ) 8 20 4.x t t t t= − + − The distance from the origin when the particle first comes to rest is:

A. 2 m B. 12 m C. 0 m D. 16 m E. 4 m

16. If A and B are independent events and are such that Pr( ) 0.4 and Pr( ) 0.3,A B= = then Pr( ')A B∪ is equal to A. 0.42 B. 0.82 C. 0.36 D. 0.92 E. 0.98

17. For the two events A and B with probabilities shown in the two-way table below, Pr( | )B A is equal to

A. 12

A A '

B 1

2

B ' 1

3

1

3

1

B. 13

C. 16

D. 32

E. 56

18. A four-digit lock code consists of digits ranging from 0 to 8. If the first digit must be odd, the last digit must be even and no digit is repeated, the number of possible lock codes is A. 756 B. 840 C. 1120 D. 1440 E. 3024

19. In a batch of 500 screws, 5% are rejected for being undersized and 10% are rejected for being oversized. The probability that a rejected screw is undersized is

A. 12

B. 13

C. 15

D. 120

E. 1

200

20. The probability that the 8:25 am train arrives on time is 0.35. The probability that the train is on time at least once during a working week (Monday to Friday) is A. 51 (0.65)− B. 5(0.35) C. 51 (0.35)− D. 1 45 (0.35) (0.65)× E. 5(0.65)

PLEASE TURN OVER

Section B Extended Response Questions 60 marks

Instructions for Section B Answer all questions in the spaces provided. In all questions where a numerical answer is required, an exact value must be given unless otherwise specified. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise indicated, the diagrams in this book are not drawn to scale.

Question 1 (12 marks) The percentage of the moon visible at night from Earth, P%, t days from the start of the year, when plotted on axes and joined in a smooth curve can be approximated by the rule:

( ) 50 50cos15

tP t .π = −

(a) State the period and amplitude of P.

2 marks (b) What percentage of the moon is visible on the night of January 25 (t = 1 is the night of January 1) according to this model?

2 marks

(c) Sketch a graph of P against t for 0 90t≤ ≤ on the axes below. Label axis intercepts and turning points with their coordinates.

3 marks

t

P

There are 31 days in January. (d) Give the dates in January and February when this model predicts there to be a full moon (100% visible).

2 marks (e) For what percentage of the nights in January is 75% or more of the moon predicted to be visible? Round to the nearest percent.

3 marks

EXAM CONTINUES ON THE NEXT PAGE

Question 2 (15 marks) Inside the Royal Botanical Gardens Laboratory, a team of botanists has just finished a long-term experiment on plants grown from genetically modified seeds. In their research it was found that the average height h, in centimetres, of a batch of genetically modified seedlings t days after they were planted was given by:

0.013( ) 2.8 5 4, [0,120].th t t= × + ∈ (a) For how many days were the plants observed?

1 mark (b) Determine the average height of the seedlings when they were planted.

1 mark (c) On the set of axes below, sketch the graph of the function h. Indicate clearly on the graph the coordinates of any endpoints (correct to one decimal place where necessary).

3 marks

(d) Find the number of days it took for the seedlings to reach an average height of 9 cm after having been planted. Express your answer to the nearest whole day.

2 marks

30 60 90 120

20

40

30

10

t (days)

h (cm)

(e) Find the average rate of change of h(t) between day 10 and day 60, in cm/day, correct to two decimal places.

2 marks

A batch of the same type of plants, but which had grown from non-genetically modified seed, was also observed. The average height n, in centimetres, of these non-genetically modified plants is given by:

( ) 2.8 5 4, [0,120].ktn t t= × + ∈

where t represents the number of days since the seedlings were planted and k is a positive, real constant. The average height of these plants was 9 centimetres, thirty days after the seedlings had been planted.

(f) Show that the value of k is 51 25log30 14

.

3 marks (g) Assume that the suitable height for harvesting is considered to be the average height of the genetically modified seedlings, h(t), 100 days after they had been planted. i. Find the suitable height for harvesting of the genetically modified seedlings, correct to three decimal places.

1 mark This is considered the suitable height for both the plants grown from genetically modified and non-genetically modified seeds. ii. Use the rule for n(t) with k = 0.012 to find how many days sooner the genetically modified plants reach a suitable height for harvesting compared to the batch of non-genetically modified plants. Express your answer to the nearest whole day.

2 marks

Question 3 (13 marks)

Volume of a cylinder: 2r hπ Surface area of a cylinder: 22 2r rhπ π+ Consider the manufacture of 1 litre capacity cylindrical tin cans where the cost of the metal used to manufacture them is to be minimised. Note: 1 litre = 1000 cm3. (a) Use the rule for volume to find an expression for h in terms of r.

2 marks

(b) Hence, show that the total surface area A cm2, is given by 2 20002A rr

π= + .

2 marks (c) Use technology to help you sketch the graph of A against r. Label any significant points with coordinates correct to one decimal place and any straight line asymptotes with their equation.

3 marks

r

A

(d) Using calculus,

i. show that the exact value of r that minimises A is 3500rπ

=

3 marks ii. verify that this value of r gives a minimum stationary point.

1 mark (e) What is the minimum surface area of the tin can if (0,3]r∈ ? Answer correct to one decimal place.

2 marks

EXAM CONTINUES ON THE NEXT PAGE

Question 4 (12 marks) A wedding can have many variable factors that affect the day. (a) A photographer at a wedding randomly selects three people from a group of seven adults and five children standing nearby to hold some equipment for him. i. In how many ways can his selection be made if there are no restrictions?

1 mark ii. Determine the probability that his selection contains only children.

2 marks iii. In how many ways can his selection be made if there must be more adults than children?

2 marks The photographer has learnt that if rain is forecast then there is a 40% chance that the bridal photo can be taken outdoors and that if rain is not forecast then there is a 90% chance that the bridal photos can be taken outdoors. Rain is forecast 30% of the time. (b) Find the probability that at a particular wedding the photographer worked at, rain was forecast given that the photos were not taken outdoors. Space for diagram:

3 marks

(c) The DJ has not arrived for the wedding and so music is being played by shuffling songs from a phone. From the song collection there is a 0.3 chance of a love song being played. Songs on the phone can be repeated. i. In the next 5 songs, determine the probability of hearing at least 2 love songs. Round to four decimal places.

2 marks ii. What is the least number of songs that would need to be played for the probability of at least 1 love song being played to be more than 0.95?

2 marks

EXAM CONTINUES ON THE NEXT PAGE

Question 5 (8 marks)

Part of the graph of a function 29: , ( )

3xh R R h x −

→ = is shown below.

Points A(3, 0) and B(0, 3) are the positive x-intercept and y-intercept of the graph of h, respectively. (a) Determine the rule for ( ).h' x

1 mark The tangent to the graph of h at point C is parallel to the line segment AB. (b) Find the x coordinate of the point on h whose tangent line is parallel to the line segment AB.

2 marks (c) Hence, find the equation of the tangent to the graph of h at the point C.

2 marks

x

y

A

B

O

C

All quadratic graphs have exactly one stationary point. A cubic graph can have 0, 1 or 2 stationary points. (d) Determine the values of n, where n R,∈ for which the function 3 2( ) 2f x x nx x= + + + has two stationary points.

3 marks

END OF EXAMINATION