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Calculus, EXT1 C1 2018 HSC 12b

A ferris wheel has a radius of 20 metres and is rotating at a rate of 1.5 radians per minute. The top of a carriage is metres above the horizontal diameter of the ferris wheel. The angle of elevation of the top of the carriage from the centre of the ferris wheel is .
 


 

  1. Show that  .  (1 mark)

  2. At what speed is the top of the carriage rising when it is 15 metres higher than the horizontal diameter of the ferris wheel? Give your answer correct to one decimal place.  (2 marks)

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i.   

 

ii.   

 
 

 

 
   

 

 
 

Functions, EXT1 F2 2018 HSC 11a

Consider the polynomial  .

  1. Show that    is a zero of  .  (1 mark)

  2. Find the other zeros.  (2 marks)

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i.   

 

ii.   


 

 

 

Networks, STD2 N3 2013 FUR2 2

A project will be undertaken in the wildlife park. This project involves the 13 activities shown in the table below. The duration, in hours, and predecessor(s) of each activity are also included in the table.


NETWORKS, FUR2 2013 VCAA 21
 

Activity is missing from the network diagram for this project, which is shown below.

 
NETWORKS, FUR2 2013 VCAA 22
 

  1. Complete the network diagram above by inserting activity .  (1 mark)
  2. Determine the earliest starting time of activity .  (1 mark)

  3. Given that activity  is not on the critical path

     

    1. write down the activities that are on the critical path in the order that they are completed  (1 mark)

    2. find the latest starting time for activity .  (1 mark)

  4. Consider the following statement.

     

    ‘If the time to complete just one of the activities in this project is reduced by one hour, then the minimum time to complete the entire project will be reduced by one hour.’

    Explain the circumstances under which this statement will be true for this project.  (1 mark)

  5. Assume activity  is reduced by two hours.
    What will be the minimum completion time for the project?  (1 mark)

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a.

networks-fur2-2013-vcaa-2-answer

b. 

c.i.  

c.ii. 

d. 
     

e. 

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a.    networks-fur2-2013-vcaa-2-answer

 

b.  


 

c.i.   

♦♦ Mean mark of parts (c)-(e) (combined) was 40%.
 

c.ii.  

 
 

d.  

MARKER’S COMMENT: Most students struggled with part (d).


 

e.  

Networks, STD2 N3 2012 FUR2 2

Thirteen activities must be completed before the produce grown on a farm can be harvested. 

The directed network below shows these activities and their completion times in days.
 


 

  1. Determine the earliest starting time, in days, for activity .  (1 mark)

  2. An activity with zero duration starts at the end of activity .

     

    Explain why this activity is used on the network diagram.  (1 mark)

  3. Determine the earliest starting time, in days, for activity .  (1 mark)

  4. In order, list the activities on the critical path.  (1 mark)

  5. Determine the latest starting time, in days, for activity .  (1 mark)

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a.   
   
♦ Mean mark of all parts (combined) 47%.

 

b.  


 

♦♦ “Few students” were able to correctly deal with the zero duration activity in part (c).

c.  
 

  

 

d.  
 


 

e.  

MARKER’S COMMENT: A correct calculation based on an incorrect critical path in part (d) gained a consequential mark here. Show your working!.


 

 

Networks, STD2 N3 2006 FUR1 5 MC

For a particular project there are ten activities that must be completed.

These activities and their immediate predecessors are given in the following table.
 

networks-fur1-2006-vcaa-5-mc

 
A directed graph that could represent this project is

 

A.
B.
C.
D.
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Networks, STD2 N2 SM-Bank 13

An estate has large open parklands that contain seven large trees.

The trees are denoted as vertices A to G on the network diagram below.

Walking paths link the trees as shown.

The numbers on the edges represent the lengths of the paths in metres.
 

NETWORKS, FUR2 2007 VCAA 2

 
Jamie is standing at A and Michelle is standing at D.

Write down the shortest route that Jamie can take and the distance travelled to meet Michelle at D.  (1 mark) 

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Networks, STD2 N2 SM-Bank 2

A school is designing a computer network between five key areas within the school.

The cost of connecting the rooms is shown in the diagram below.
  


 

  1. Complete the spanning tree below that creates the school's network at a minimum cost.  (1 mark)
      


     
     

  2. What is the minimum cost of the network?  (1 mark)

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  1.  
         
     
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i.   


  

 

ii.   
   

Networks, STD2 N2 SM-Bank 1 MC

This diagram shows the possible paths (in km) for laying gas pipes between various locations.
 

 
Gas is to be supplied from one location. Any one of the locations can be the source of the supply.

What is the minimum total length of the pipes required to provide gas to all the locations?

A. 32 km
B. 34 km
C. 36 km
D. 38 km
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Networks, STD2 N2 SM-Bank 14

Water will be pumped from a dam to eight locations on a farm.

The pump and the eight locations (including the house) are shown as vertices in the network diagram below.

The numbers on the edges joining the vertices give the shortest distances, in metres, between locations.
 

NETWORKS, FUR2 2012 VCAA 1

 

  1. How many vertices on the network diagram have an odd degree?  (1 mark)

     

    The total length of all edges in the network is 1180 metres.

     

    The total length of pipe that supplies water from the pump to the eight locations on the farm is a minimum.

     

    This minimum length of pipe is laid along some of the edges in the network.

  2. On the diagram below, draw the minimum length of pipe that is needed to supply water to all locations on the farm.  (2 marks)
     
     
    NETWORKS, FUR2 2012 VCAA 1

  3. What is the mathematical term that is used to describe this minimum length of pipe?  (1 mark)

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  2.  
    NETWORKS, FUR2 2012 VCAA 1 Answer

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i.  

MARKER’S COMMENT: Many students, surprisingly, had problems with part (i).
 

ii. 


 

NETWORKS, FUR2 2012 VCAA 1 Answer


iii.   

Networks, FUR2 2015 VCE 2

The factory supplies groceries to stores in five towns, , , , and , represented by vertices on the graph below.

 

Networks, FUR2 2015 VCAA 2

The edges of the graph represent roads that connect the towns and the factory.

The numbers on the edges indicate the distance, in kilometres, along the roads.

Vehicles may only travel along the road between towns and in the direction of the arrow due to temporary roadworks.

Each day, a van must deliver groceries from the factory to the five towns.

The first delivery must be to town , after which the van will continue on to the other four towns before returning to the factory.

Describe the order in which these deliveries would follow to achieve the shortest possible circuit and the length, in kilometres, of the circuit.  (2 marks)

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Networks, STD2 N2 2011 FUR2 1

Aden, Bredon, Carrie, Dunlop, Enwin and Farnham are six towns.

The network shows the road connections and distances between these towns in kilometres.
 

  1. In kilometres, what is the shortest distance between Farnham and Carrie?  (1 mark)

  2. How many different ways are there to travel from Farnham to Carrie without passing through any town more than once?  (1 mark)

An engineer plans to inspect all of the roads in this network.

He will start at Dunlop and inspect each road only once.

  1. At which town will the inspection finish?  (1 mark)

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a.  

 

b.  

 

c.   

Networks, STD2 N2 2010 FUR2 2

The diagram below shows a network of tracks (represented by edges) between checkpoints (represented by vertices) in a short-distance running course. The numbers on the edges indicate the time, in minutes, a team would take to run along each track.

 

Network, FUR2 2011 VCAA 2_1
 

A challenge requires teams to run from checkpoint to checkpoint using these tracks.

What would be the shortest possible time for a team to run from checkpoint to checkpoint ? Mark the shortest route on the diagram below. (2 marks)

 
  Network, FUR2 2011 VCAA 2_2

 
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Networks, STD2 N3 2009 FUR2 2

One of the landmarks in a city is a hedge maze. The maze contains eight statues. The statues are labelled to on the following directed graph. Walkers within the maze are only allowed to move in the directions of the arrows.
 

NETWORKS, FUR2 2009 VCAA 2
 

  1. Write down the two statues that a walker could not reach from statue .  (1 mark)

  2. One way that statue can be reached from statue is along path .

     

    List the three other ways that statue can be reached from statue .  (1 mark)

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a.   

b.  

Networks, STD2 N2 2017 FUR2 1

Bus routes connect six towns.

The towns are Northend (), Opera (), Palmer (), Quigley (), Rosebush () and Seatown ().

The graph below gives the cost, in dollars, of bus travel along these routes.

Bai lives in Northend () and he will travel by bus to take a holiday in Seatown ().
 


 

  1. Bai considers travelling by bus along the route Northend () – Opera () – Seatown ().
    How much would Bai have to pay?  (1 mark)

  2. If Bai takes the cheapest route from Northend () to Seatown (), which other town(s) will he pass through?  (1 mark)

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a.   
   

 

b.   

 

Networks, STD2 N2 2013 FUR2 1

The vertices in the network diagram below show the entrance to a wildlife park and six picnic areas in the park: , , , , and .

The numbers on the edges represent the lengths, in metres, of the roads joining these locations.

 


 

  1. In this graph, what is the degree of the vertex at the entrance to the wildlife park?  (1 mark)

  2. What is the shortest distance, in metres, from the entrance to picnic area ?  (1 mark)

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a.  

b.  
 


 

Networks, FUR2 2012 VCE 1

Water will be pumped from a dam to eight locations on a farm.

The pump and the eight locations (including the house) are shown as vertices in the network diagram below.

The numbers on the edges joining the vertices give the shortest distances, in metres, between locations.
 

NETWORKS, FUR2 2012 VCAA 1
 

a.i.  Determine the shortest distance between the house and the pump.  (1 mark)

a.ii. How many vertices on the network diagram have an odd degree?  (1 mark)

The total length of pipe that supplies water from the pump to the eight locations on the farm is a minimum.

This minimum length of pipe is laid along some of the edges in the network.

b.i.  On the diagram below, draw the minimum length of pipe that is needed to supply water to all locations on the farm.  (1 mark)

 

   
  NETWORKS, FUR2 2012 VCAA 1
 

b.ii. What is the mathematical term that is used to describe this minimum length of pipe in part b.i.?  (1 mark)

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a.i. 

a.ii.

b.i. 

NETWORKS, FUR2 2012 VCAA 1 Answer

b.ii.

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a.i.   

MARKER’S COMMENT: Many students, surprisingly, had problems with part (a)(ii).

 

a.ii.  

 

b.i.    NETWORKS, FUR2 2012 VCAA 1 Answer

 

b.ii.   

Networks, STD2 N2 2015 FUR2 1

A factory requires seven computer servers to communicate with each other through a connected network of cables.

The servers, , , , , , and , are shown as vertices on the graph below.
 

Networks, FUR2 2015 VCAA 11

 
The edges on the graph represent the cables that could connect adjacent computer servers.

The numbers on the edges show the cost, in dollars, of installing each cable.

  1. What is the cost, in dollars, of installing the cable between server and server ?  (1 mark)

  2. What is the cheapest cost, in dollars, of installing cables between server and server ?  (1 mark)

  3. The computer servers will be able to communicate with all the other servers as long as each server is connected by cable to at least one other server.

    1. The cheapest installation that will join the seven computer servers by cable in a connected network follows a minimum spanning tree.

       

      Draw the minimum spanning tree on the plan below.  (1 mark) 
       

      Networks, FUR2 2015 VCAA 12
       

    2. The factory’s manager has decided that only six connected computer servers will be needed, rather than seven.

       

      How much would be saved in installation costs if the factory removed computer server  from its minimum spanning tree network?

       

      A copy of the graph above is provided below to assist with your working.  (1 mark)

      Networks, FUR2 2015 VCAA 12

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    1.  
      Networks, FUR2 2015 VCAA 12 Answer
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a.   

 

b.  


 

MARKER’S COMMENT: Many students had difficulty finding the minimum spanning tree, often incorrectly excluding or .

c.i.  


 

Networks, FUR2 2015 VCAA 12 Answer


c.ii.   

 

Networks, STD2 N2 2011 FUR2 2

At the Farnham showgrounds, eleven locations require access to water. These locations are represented by vertices on the network diagram shown below. The dashed lines on the network diagram represent possible water pipe connections between adjacent locations. The numbers on the dashed lines show the minimum length of pipe required to connect these locations in metres.
 

NETWORKS, FUR2 2011 VCAA 2 

 
All locations are to be connected using the smallest total length of water pipe possible.

  1. On the diagram, show where these water pipes will be placed.  (1 mark)
  2. Calculate the total length, in metres, of water pipe that is required.  ( 1 mark) 
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  1.  
    NETWORKS, FUR2 2011 VCAA 2 Answer
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a.


 

NETWORKS, FUR2 2011 VCAA 2 Answer


b.  

Networks, STD2 N2 2008 FUR2 1

James, Dante, Tahlia and Chanel are four children playing a game.

In this children’s game, seven posts are placed in the ground.

The network below shows distances, in metres, between the seven posts.

The aim of the game is to connect the posts with ribbon using the shortest length of ribbon.

This will be a minimal spanning tree.

 

NETWORKS, FUR2 2008 VCAA 11

  1. Draw in a minimal spanning tree for this network on the diagram below.  (1 mark)


NETWORKS, FUR2 2008 VCAA 12

  1. Determine the length, in metres, of this minimal spanning tree.  (1 mark)

  2. How many different minimal spanning trees can be drawn for this network?  (1 mark)

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  1.  


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a. 

 
 

 

b.   


 

c.   

Networks, STD2 N2 2013 FUR1 3 MC

 
The vertices of the graph above represent nine computers in a building. The computers are to be connected with optical fibre cables, which are represented by edges. The numbers on the edges show the costs, in hundreds of dollars, of linking these computers with optical fibre cables.

Based on the same set of vertices and edges, which one of the following graphs shows the cable layout (in bold) that would link all the computers with optical fibre cables for the minimum cost?
 

 

vcaa-networks-fur1-2013-3ii

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Networks, STD2 N2 2017 FUR1 2 MC

Two graphs, labelled Graph 1 and Graph 2, are shown below.
 

 
The sum of the degrees of the vertices of Graph 1 is

  1. two less than the sum of the degrees of the vertices of Graph 2.
  2. one less than the sum of the degrees of the vertices of Graph 2.
  3. equal to the sum of the degrees of the vertices of Graph 2.
  4. two more than the sum of the degrees of the vertices of Graph 2.
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GRAPHS, FUR2 2017 VCAA 3

Lifeguards are required to ensure the safety of swimmers at the beach.

Let be the number of junior lifeguards required.

Let be the number of senior lifeguards required.

The inequality below represents the constraint on the relationship between the number of senior lifeguards required and the number of junior lifeguards required.

 Constraint 1 
 

  1. If eight junior lifeguards are required, what is the minimum number of senior lifeguards required?  (1 mark)

 
There are three other constraints.

 Constraint 2 

 Constraint 3 

 Constraint 4 

  1. Interpret Constraint 4 in terms of the number of junior lifeguards and senior lifeguards required.  (1 mark)

 
The shaded region of the graph below contains the points that satisfy Constraints 1 to 4.

All lifeguards receive a meal allowance per day.

Junior lifeguards receive $15 per day and senior lifeguards receive $25 per day.

The total meal allowance cost per day, , for the lifeguards is given by



  1. Determine the minimum total meal allowance cost per day for the lifeguards.  (2 marks)
  2. On rainy days there will be no set minimum number of junior lifeguards or senior lifeguards required, therefore:

     

    • Constraint 2    and Constraint 3    are removed

     

    • Constraint 1 and Constraint 4 are to remain.
     

     

              Constraint 1 

     

              Constraint 4 

     


    The total meal allowance cost per day,     , for the lifeguards remains as

     

     


    How many junior lifeguards and senior lifeguards work on a rainy day if the total meal allowance cost     is to be a minimum?

     

    Write your answers in the boxes provided below.  (1 mark)


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a.   

b.   

c.   

d.   
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a.   

 

b.   

 

c.   

 

 

d.   

♦♦ Mean mark 24%.
MARKER’S COMMENT: A common incorrect answer was 10 and 3.

GRAPHS, FUR2 2017 VCAA 1

The graph below shows the depth of water in a sea bath from 6.00 am to 8.00 pm.
 

 

  1. What was the maximum depth, in metres, of water in the sea bath?   (1 mark)

  2. The sea bath was open to the public when the depth of water was above 1.5 m.
  3. Between which times was the sea bath open?   (1 mark)
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a.  

b.   

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a.  

b.   

GEOMETRY, FUR2 2017 VCAA 1

Miki is planning a gap year in Japan.

She will store some of her belongings in a small storage box while she is away.

This small storage box is in the shape of a rectangular prism.

The diagram below shows that the dimensions of the small storage box are 40 cm × 19 cm × 32 cm.
 

The lid of the small storage box is labelled on the diagram above.

    1. What is the surface area of the lid, in square centimetres?  (1 mark)
    2. What is the total outside surface area of this storage box, including the lid and base, in square centimetres?  (1 mark)
  2. Miki has a large storage box that is also a rectangular prism.

      

    The large storage box and the small storage box are similar in shape.

      

    The volume of the large storage box is eight times the volume of the small storage box.

      

    The length of the small storage box is 40 cm.

    What is the length of the large storage box, in centimetres?  (1 mark)

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a.i.   ²

a.ii.  ²

b.     

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a.i.   
    ²

 

a.ii.   
    ²

 

b.   

♦♦ Mean mark 32%.
MARKER’S COMMENT: A lack of understanding between linear and volume scale factors was again notable.

NETWORKS, FUR2 2017 VCAA 1

Bus routes connect six towns.

The towns are Northend (), Opera (), Palmer (), Quigley (), Rosebush () and Seatown ().

The graph below gives the cost, in dollars, of bus travel along these routes.

Bai lives in Northend () and he will travel by bus to take a holiday in Seatown ().
 


 

  1. Bai considers travelling by bus along the route Northend () – Opera () – Seatown ().

     

    How much would Bai have to pay?   (1 mark)

  2. If Bai takes the cheapest route from Northend () to Seatown (), which other town(s) will he pass through?   (1 mark)

  3. Euler’s formula, , holds for this graph.

    Complete the formula by writing the appropriate numbers in the boxes provided below.   (1 mark)

     

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a.   

b.   

c. 

       

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a.   
   

 

b.   

 

c.   

MATRICES, FUR2 2017 VCAA 1

A school canteen sells pies (), rolls () and sandwiches ().

The number of each item sold over three school weeks is shown in matrix .

 

  1. In total, how many sandwiches were sold in these three weeks?   (1 mark)

  2. The element in row and column of matrix is .
  3. What does the element indicate?  (1 mark)

  4. Consider the matrix equation



    where = cost of one pie, = cost of one roll and = cost of one sandwich.
  5.  i. What is the cost of one sandwich?   (1 mark)

The matrix equation below shows that the total value of all rolls and sandwiches sold in these three weeks is $915.60

Matrix in this equation is of order .

  1. ii. Write down matrix .   (1 mark)

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a.   
   

 
b.   


 

c.i.   
   

 

 

c.ii.   

CORE, FUR2 2017 VCAA 5

Alex is a mobile mechanic.

He uses a van to travel to his customers to repair their cars.

The value of Alex’s van is depreciated using the flat rate method of depreciation.

The value of the van, in dollars, after years, , can be modelled by the recurrence relation shown below.

  1. Recursion can be used to calculate the value of the van after two years.

     

    Complete the calculations below by writing the appropriate numbers in the boxes provided.   (2 marks)


    1. By how many dollars is the value of the van depreciated each year?   (1 mark)
    2. Calculate the annual flat rate of depreciation in the value of the van.
    3. Write your answer as a percentage.   (1 mark)
  3. The value of Alex’s van could also be depreciated using the reducing balance method of depreciation.
  4. The value of the van, in dollars, after years, , can be modelled by the recurrence relation shown below.

     

           

    At what annual percentage rate is the value of the van depreciated each year?   (1 mark)

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a.

b.i.  

b.ii.

c.  

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a.   

  
b.i.   
  

b.ii.   
   

 

c.   
   

Algebra, MET2 2017 VCAA 2

Sammy visits a giant Ferris wheel. Sammy enters a capsule on the Ferris wheel from a platform above the ground. The Ferris wheel is rotating anticlockwise. The capsule is attached to the Ferris wheel at point . The height of above the ground, , is modelled by  , where is the time in minutes after Sammy enters the capsule and is measured in metres.

Sammy exits the capsule after one complete rotation of the Ferris wheel.
 


 

  1. State the minimum and maximum heights of above the ground.   (1 mark)

  2. For how much time is Sammy in the capsule?   (1 mark)

  3. Find the rate of change of with respect to and, hence, state the value of at which the rate of change of is at its maximum.   (2 marks)

As the Ferris wheel rotates, a stationary boat at , on a nearby river, first becomes visible at point . is 500 m horizontally from the vertical axis through the centre of the Ferris wheel and angle , as shown below.
 

   
 

  1. Find in degrees, correct to two decimal places.   (1 mark)

Part of the path of is given by  , where and are in metres.

  1. Find .   (1 mark)

As the Ferris wheel continues to rotate, the boat at is no longer visible from the point  onwards. The line through and is tangent to the path of , where angle .
 

   
 

  1. Find the gradient of the line segment in terms of and, hence, find the coordinates of , correct to two decimal places.   (3 marks)

  2. Find in degrees, correct to two decimal places.   (1 mark)

  3. Hence or otherwise, find the length of time, to the nearest minute, during which the boat at is visible.   (2 marks)

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a.   
     

 

b.   

 

c.   

♦ Mean mark 50%.
MARKER’S COMMENT: A number of commons errors here – 2 answers given, calc not in radian mode, etc …

 

   

 

d.   

♦ Mean mark 36%.
MARKER’S COMMENT: Choosing degrees vs radians in the correct context was critical here.

 

 

e.   

 

f.   

 

♦♦♦ Mean mark part (f) 18%.
MARKER’S COMMENT: Many students were unable to use the rise over run information to calculate the second gradient.

 

 

 

 
 

 

 

♦♦♦ Mean mark part (g) 7%.

g.   
   
 

 

h.   

♦♦♦ Mean mark 5%.

 

 
 

Calculus, MET2 2017 VCAA 1

Let  . Part of the graph of is shown below.
 

  1. Find the coordinates of the turning points.   (2 marks)

  2.   and   are two points on the graph of .

     

    1. Find the equation of the straight line through and .   (2 marks)

    2. Find the distance .   (1 mark)

Let  .

  1. Let  and be two points on the graph of .

     

    1. Find the distance in terms of .   (2 marks)

    2. Find the values of such that the distance is equal to  .   (1 mark)

  2. The diagram below shows part of the graphs of and  . These graphs intersect at the points with the coordinates and .
  3.  
       
  4.  
    1. Find the value of in terms of .   (1 mark)

    2. Find the area of the shaded region in terms of .   (2 marks)

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    2. ²
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a.   

 


 

b.i.  

 

MARKER’S COMMENT: Students skilled in the use of technology will be much more efficient and minimise errors here.
b.ii.   
   
   

 

c.i.   
   
   

 

c.ii.   


 

d.i.   


 

d.ii.   
    ²

Probability, MET1 2017 VCAA 5

For Jac to log on to a computer successfully, Jac must type the correct password. Unfortunately, Jac has forgotten the password. If Jac types the wrong password, Jac can make another attempt. The probability of success on any attempt is . Assume that the result of each attempt is independent of the result of any other attempt. A maximum of three attempts can be made.

  1. What is the probability that Jac does not log on to the computer successfully?   (1 mark)

  2. Calculate the probability that Jac logs on to the computer successfully. Express your answer in the form , where and are positive integers.   (1 mark)

  3. Calculate the probability that Jac logs on to the computer successfully on the second or on the third attempt. Express your answer in the form , where and are positive integers.   (2 marks)

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Show Worked Solution

a. 

 

b. 

 

c. 

Calculus, MET1 2017 VCAA 3

Let  

  1. Show that  .   (1 mark)

  2. Sketch the graph of   on the axes below. Label the axis intercepts and any stationary points with their coordinates.   (3 marks)

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  2.  
Show Worked Solution

a. 

 

b.  


 

 

Harder Ext1 Topics, EXT2 2017 HSC 15b

Consider the curve  , for    and  , where is a positive constant.

  1. Show that the equation of the tangent to the curve at the point    is given by  .  (2 marks)
  2. The tangent to the curve at the point meets the and axes at   and  respectively. Show that  , where is the origin.  (3 marks)

 

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(i)   

 

 

 

(ii)