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 distance between the points with coordinates (−1, 3) and (2, −2) is A. 6 B. 34 C. 4 2 D. 2 2 E. 2

2. The simultaneous linear equations 5 3mx y+ = and 5x my m+ = have a unique solution only for A. m = 5 or m = −5 B. 5m ≠ C. \{ 5,5}m R∈ − D. m = 3 or m = 5 E. \{3,5}m R∈

3. 40 cm of wire is used to form a rectangle. A rule for the area, A cm2, of the rectangle could be given by A. 220A x x= − B. (20 2 )A x x= − C. 210A x=

D. 40

2xA −

=

E. (40 )A x x= −

4. The range of the graph of 2( ) ( 1) 5, [0,4]f x x x= − − ∈ is A. [ 4, 4]− B. [ 5, 4]− C. [0, 4] D. [ 5, )− ∞ E. [ 6, 4]−

5. For which values of m is 2 2 0mx x c+ + ≥ given c > 0?

A. 1mc

≤

B. 16mc

≤

C. 1mc

≥

D. 4m c≥ E. 0 8m c≤ ≤

6. Which of the following is a polynomial of degree 4? A. 5 4 2( ) 5P x x x x= + − + B. 4 2( ) 3 1P x x x= − + C. 2( ) 4( 1) 2P x x= − + D. 3 2( ) 4 2P x x x x= + − E. 4( ) 2 3P x x x= − +

7. The remainder when 3 2( ) 3 2 4P x x x x= + + − is divided by 2x + is A. 30 B. −22 C. 26 D. −16 E. 24

8. If ( ) 3 5f x x= − , then ( )2 1f x + is equal to A. 6 8x − B. 6 5x − C. 3 5x − D. 3 4x − E. 6 2x −

9. The graph of a cubic function is shown below. A possible rule for the graph is A. 3( 2)y x= −

B. 3( 2)y x= + C. 3( 2) 1y x= − + − D. 3( 2)y x= − + E. 3( 2) 1y x= − − −

y

x-2-1

-1

10. The solutions to the cubic inequality 2( ) ( ) 0x a x b− + > , where a and b are both positive constants, are given by A. ( , )x a b∈ − B. ( , )x b∈ − ∞ C. ( , )x a∈ ∞ D. ( , ) ( , )x b a a∈ − ∪ ∞ E. ( , ) ( , )x a b∈ −∞ − ∪ ∞

11. The values on the number line shown below can be represented by A. ( , 2) [1, )x∈ −∞ − ∪ ∞ B. ( , 2] (1, )x∈ −∞ − ∩ ∞ C. ( , ) \ ( 2,1)x∈ −∞ ∞ − D. ( , 2] (1, )x∈ −∞ − ∪ ∞ E. ( , 2) [1, )x∈ −∞ − ∩ ∞

12. Which one of the graphs below represents a one-to-one function? A.

B.

C.

D.

E.

13. The inverse function of has range A. [2, )∞ B. [ 2,1]− C. R D. ( ,1]−∞ E. [1, )∞

14. A circle has centre coordinates (3, −2) and radius 4. The equation of the circle is A. 2 2( 3) ( 2) 2x y− + − = B. 2 2( 3) ( 2) 4x y− + + = C. 2 2( 3) ( 2) 16x y+ + − = D. 2 2( 3) ( 2) 16x y− + + = E. 2 2( 3) ( 2) 4x y+ + − =

15. For the function 24 , 1

( )2 , 1

x xf x

x x − ≤ −

= > −

, ( 1)f − is equal to

A. −2 B. 3 C. 5 D. −1 E. 2

16. The equations of the asymptotes of the graph of 2

2 3( 1)

yx

= −+

are

A. 1x = and 3y = B. 3y = − and 2x = C. 1y = and 3x = − D. 1x = − and 3y = − E. 1x = and 3y = −

17. The implied (maximal) domain of 1 3y x= − is A. 0x ≥

B. 13

x ≥

C. 13

x ≤

D. 3x ≤ E. 3x ≥

18. For the function 2( ) 2 8 1f x x x= − + , '( ) 0f x > for A. 2x >

B. 12

x >

C. 1x > D. 0 2x< < E. 2x < −

19. A container is being filled at a constant rate, and the height of the water (h cm) in the container after t seconds is given by the function ( ) 4h t t= for t ≥ 0. The container’s shape is most similar to

A.

B.

C.

D.

E. It could be any one of A – D; we need more information.

20. The curve with equation ( )y f x= passes through the point (1, 2) and 2'( ) 3 4 5f x x x= − + . The rule for ( )f x is A. 3 2( ) 2 5 2f x x x x= − + + B. 3 2( ) 2 5 2f x x x x= − + − C. ( ) 6 4f x x= − D. 3 2( ) 2 5f x x x x= − +

E. 3

23( ) 2 5 22xf x x x= − + +

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 (13 marks) In 2012, American Nik Wallenda made a successful tight-rope crossing on a wire over Niagara Falls, from the United States side of the Falls to the Canadian side.

Up-and-coming stunt artist, Heroic Harry, improves on Wallenda’s achievement by beginning on the American side and walking towards the Canadian side. After reaching Canada, he turns around and walks back.

His position, d metres, from the edge of the cliff on the American side, t minutes after he starts is given by the function:

222:[0,50] , ( ) 44 .25

d R d t t t→ = − +

(a) Sketch a graph of ( )d t over the specified domain, labelling key features with coordinates.

3 marks (b) State both the domain and range of ( )d t .

2 marks

t

d

(c) Hence, what was the total distance covered by Heroic Harry?

1 mark (d) Find the average rate of change of position of Heroic Harry, in m/min, between t = 5 and t = 40.

2 marks (e) Find '( )d t and calculate the rate of change of position at t = 20, in m/min.

2 marks (f) Find the time when the rate of change of position is 12 m/min. Answer correct to the nearest second.

3 marks

Question 2 (14 marks) Initial designs for the Olympic torch for the Tokyo 2020 Games involved the cone shape shown (drawn in two dimensions). The height of the cone is h cm, the radius is r cm and the slant height is 80 cm. (a) Find an expression for r in terms of h.

2 marks

(b) Given that the volume of a cone is given by 213

V r hπ= , show that the volume of the Olympic torch,

in cm3, can be expressed as 36400

3 3h hV π π

= − .

2 marks (c) State the possible values of h, giving a reason.

2 marks (d) Sketch a graph of V against h over the domain specified in part (c). Label endpoints with coordinates.

2 marks

h cm 80 cm

r cm

h

V

(e) The volume of the cone must be at least 150 000 cm3 to fit enough lighter fluid to keep the flame burning. Find the possible values of h that satisfy this condition, answering correct to one decimal place.

3 marks (f) Find the maximum possible volume of the cone (to the nearest cm3) and the corresponding values of h and r correct to one decimal place.

3 marks

Question 3 (13 marks) A waterslide is to be built modelled from where it starts to where it enters the water by the cubic function

2:[0, 4] , ( ) ( 2) ( 4)h R h x x x→ = − − − where h is the height of the waterslide above the ground in metres and x the horizontal distance of the slide in metres.

(a) What is the initial height of the water slide in metres?

1 mark (b) Give the exact coordinates of the mini-bump, A.

2 marks (c) The ‘thrillseeker’ rating for waterslides is based on their gradient function, '( )h x .

i. Use CAS to find the gradient function, '( )h x , for this waterslide.

ii. Give the exact coordinates of the point on the slide where it is ‘maximum thrillseeker’ (steepest gradient).

1 mark 2 marks

A ladder is also to be built going from the ground at point B with coordinates ( 2,0)− , to the start of the waterslide (when x = 0) as shown in the diagram.

(d) Determine the gradient of the line that represents the ladder.

1 mark (e) Write the rule for the line that represents the ladder, ( )f x , in formal function notation.

2 marks (f) To be legally safe, the ladder must have an angle between it and the ground of no more than 60º. i. Determine the angle the current ladder makes with the ground (the positive direction of the x-axis). Give your answer to the nearest whole degree. Is it legally safe?

2 marks ii. Find the rule for the line that represents the steepest legal ladder, given it must connect to the start of the waterslide.

2 marks

Question 4 (9 marks)

Consider the curve given by ( ) af xx

= where .a Z∈

(a) The point 54,4

lies on the curve ( ) af xx

= . Show that a = 5.

1 mark

Consider the following graph of ( ) 5: 0, , ( )f R f xx

∞ → = .

(b) Find the midpoint of the line segment that joins A to B.

1 mark (c) Find the equation of the line perpendicular to AB that passes through the midpoint of line segment AB.

3 marks

(d) Find '( ).f x

1 mark (e) Hence, find the coordinates of the point P on the curve, at which the tangent line is parallel to the line segment AB.

3 marks

Question 5 (11 marks) One of the most recognisable symbols in the world is the McDonald’s logo. Over the years, it has undergone some small changes. One store is looking to build the ‘golden arches’ out the front of its store (where the original design idea came from). On a set of axes, the first arch of the logo is modelled by the rule 2( ) 2 8 , [0,4]g x x x x= − + ∈ . (a) Show that in turning point form this rule can be expressed as 2( ) 2( 2) 8.g x x= − − +

3 marks (b) Hence, describe the sequence of transformations that maps 2( )f x x= to 2( ) 2 8 .g x x x= − +

3 marks (c) The second arch is formed by translating ( )g x 4 units in the positive direction of the x-axis. Give the rule for the second arch.

1 mark (d) Hence, specify the rule for the entire logo as a hybrid (piecewise) function, including domains.

1 mark

The final design that is chosen is to make the arches appear narrower to more accurately resemble the logo. The designer alters the transformations applied to the graph of 2( )f x x= . (e) Give the new rule for the first arch by applying the following sequence of transformations to

2( )f x x= .

• Dilate by factor 12

from the y-axis

• Reflect in the x-axis • Translate 1 unit in the positive-direction of the x-axis and 4 units in the positive direction of the y-

axis.

3 marks

END OF EXAMINATION