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Statistics, STD2 S5 2018 HSC 27e

Joanna sits a Physics test and a Biology test.

  1. Joanna’s mark in the Physics test is 70. The mean mark for this test is 58 and the standard deviation is 8.

     

    Calculate the -score for Joanna’s mark in this test.  (1 mark)

  2. In the Biology test, the mean mark is 64 and the standard deviation is 10.

     

    Joanna’s -score is the same in both the Physics test and the Biology test.

     

    What is her mark in the Biology test?  (2 marks)

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i.   

 
 

 

ii.   
 
 

Algebra, STD2 A4 2018 HSC 27d

The graph displays the cost () charged by two companies for the hire of a minibus for hours.
 


  

Both companies charge $360 for the hire of a minibus for 3 hours.

  1. What is the hourly rate charged by Company A?  (1 mark)

  2. Company B charges an initial booking fee of $75.

     

    Write a formula, in the form of  , for the cost of hiring a minibus from Company B for hours.  (2 marks)

  3. A minibus is hired for 5 hours from Company B.

     

    Calculate how much cheaper this is than hiring from Company A.  (2 marks)

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i.   
   

 

ii.   

 

 

iii.   
 

 

Measurement, STD2 M1 2018 HSC 27c

A shade shelter is to be constructed in the shape of half a cylinder with open ends. The diameter is 3.8 m and the length is 10 m.
 

 
The curved roof is to be made of plastic sheeting.

What area of plastic sheeting is required, to the nearest m²?  (2 marks)

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²²

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  ²²

Measurement, STD2 M7 2018 HSC 26g

A field diagram of a block of land has been drawn to scale. The shaded region is covered in grass.
 

 
The actual length of is 24 m.

  1. If the length of on the field diagram is 8 cm, what is the scale of the diagram?  (1 mark)

  2. How much fertiliser would be needed to fertilise the grassed area    at the rate of 26.5 g /m²?  (3 marks)

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♦♦ Mean mark 32%.

i.   
 
 

 

ii.   

²
 

²
 

 

Probability, STD2 S2 2018 HSC 20 MC

During a year, the maximum temperature each day was recorded. The results are shown in the table.
  


  

From the days with a maximum temperature less than 25°C, one day is selected at random.

What is the probability, to the nearest percentage, that the selected day occurred during winter?

  1. 19%
  2. 25%
  3. 32%
  4. 77%
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Statistics, STD2 S1 2018 HSC 11 MC

A set of data is summarised in this frequency distribution table.
 

 
Which of the following is true about the data?

  1. Mode = 7, median = 5.5
  2. Mode = 7, median = 6
  3. Mode = 9, median = 5.5
  4. Mode = 9, median = 6
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Statistics, STD2 S1 2018 HSC 3 MC

A survey asked the following question.

'How many brothers do you have?'

How would the responses be classified?

  1. Categorical, ordinal
  2. Categorical, nominal
  3. Numerical, discrete
  4. Numerical, continuous
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Mechanics, EXT2* M1 2018 HSC 13c

An object is projected from the origin with an initial velocity of  at an angle    to the horizontal. The equations of motion of the object are

  (Do NOT prove this.)

 

  1. Show that when the object is projected at an angle  , the horizontal range is

     

           (2 marks)

  2. Show that when the object is projected at an angle  , the horizontal range is also 

     

         .  (1 mark)

  3. The object is projected with initial velocity to reach a horizontal distance , which is less than the maximum possible horizontal range. There are two angles at which the object can be projected in order to travel that horizontal distance before landing.

     

    Let these angles be   and  , where 

     

    Let    be the maximum height reached by the object when projected at the angle to the horizontal.

     

    Let    be the maximum height reached by the object when projected at the angle to the horizontal.
     
         
     
    Show that the average of the two heights, , depends only on and .  (3 marks)

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i.   

 

 
 
 

 

ii. 

 
 

 

iii. 


  

 
 
 

 

♦ Mean mark 44%.
 

 
 
 
 

 

Quadratic, EXT1 2018 HSC 12e

The points    and    lie on the parabola  . The focus of the parabola is    and the tangents at    and    intersect at  . (Do NOT prove this.)

The tangents at    and    meet the -axis at    and    respectively, as shown.
 

 

  1. Show that  .  (2 marks)
  2. Explain why    are concyclic points.  (1 mark)
  3. Show that the diameter of the circle through    and    has length
     
    .  (2 marks)

 

 

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(i)   

 
 

 


 

 
 

  

 

(ii)   

♦ Mean mark 37%.

 


 


 

   

 

(iii) 

♦ Mean mark 44%.

 
 
 
 
 
 

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.   

 
 

 

 
   

 

 
 

Polynomials, EXT1 2018 HSC 6 MC

The diagram shows the graph of  . The equation    has a solution at  .
 


 

Newton’s method can be used to give an approximation close to the solution  .

Which initial approximation, , will give the second approximation that is closest to the solution  ?

A.     

B.     

C.     

D.     

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Networks, STD2 N3 2007 FUR2 4

A community centre is to be built on the new housing estate.

Nine activities have been identified for this building project.

The directed network below shows the activities and their completion times in weeks.

 

  1. Determine the minimum time, in weeks, to complete this project.  (1 mark)

  2. Determine the float time, in weeks, for activity .  (2 marks)

The builders of the community centre are able to speed up the project.

Some of the activities can be reduced in time at an additional cost.

The activities that can be reduced in time are , , , and .

  1. Which of these activities, if reduced in time individually, would not result in an earlier completion of the project?  (1 mark)

The owner of the estate is prepared to pay the additional cost to achieve early completion.

The cost of reducing the time of each activity is $5000 per week.

The maximum reduction in time for each one of the five activities, , , , , , is weeks.

  1. Determine the minimum time, in weeks, for the project to be completed now that certain activities can be reduced in time.  (1 mark)

  2. Determine the minimum additional cost of completing the project in this reduced time.  (1 mark)

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a.  
 

 

♦ Mean mark of all parts (combined) 40%.
 

 

b.   
 
 

 

c.  


 

d.  

 

e.   
   

Networks, STD2 N3 2006 FUR2 3

The five musicians are to record an album. This will involve nine activities.

The activities and their immediate predecessors are shown in the following table.

The duration of each activity is not yet known.
 

NETWORKS, FUR2 2006 VCAA 31
 

  1. Use the information in the table above to complete the network below by including activities ,  and .  (2 marks)
     

NETWORKS, FUR2 2006 VCAA 32
 

There is only one critical path for this project.

  1. How many non-critical activities are there?  (1 mark)

The following table gives the earliest start times (EST) and latest start times (LST) for three of the activities only. All times are in hours.


Networks, FUR2 2006 VCAA 3_3

  1. Write down the critical path for this project.  (1 mark)

The minimum time required for this project to be completed is 19 hours.

  1. What is the duration of activity ?  (1 mark)

The duration of activity is 3 hours.

  1. Determine the maximum combined duration of activities and .  (1 mark) 
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  1.  
    networks-fur2-2006-vcaa-3-answer
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a.    networks-fur2-2006-vcaa-3-answer

 

b.  


 

 

c.   

♦ Mean mark of parts (c)-(e) (combined) was 36%.

 

d.   
   

 

e.  

♦♦ MARKER’S COMMENT: Many students incorrectly answered 9 hours in part (e).

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 2009 FUR2 4

A walkway is to be built across the lake.

Eleven activities must be completed for this building project.

The directed network below shows the activities and their completion times in weeks.
 

NETWORKS, FUR2 2009 VCAA 4
 

  1. What is the earliest start time for activity E?  (1 mark)

  2. Write down the critical path for this project.  (1 mark)

  3. The project supervisor correctly writes down the float time for each activity that can be delayed and makes a list of these times.

     

    Determine the longest float time, in weeks, on the supervisor’s list.  (1 mark)

A twelfth activity, L, with duration three weeks, is to be added without altering the critical path.

Activity L has an earliest start time of four weeks and a latest start time of five weeks.
 

NETWORKS, FUR2 2009 VCAA 4
 

  1. Draw in activity L on the network diagram above.  (1 mark)
  2. Activity L starts, but then takes four weeks longer than originally planned.

     

    Determine the total overall time, in weeks, for the completion of this building project.  (1 mark)

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  4.  
    NETWORKS, FUR2 2009 VCAA 4 Answer
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a.  

♦ Mean mark of all parts (combined): 44%.

 

b. 
 

 

 

c.   


 

d.    NETWORKS, FUR2 2009 VCAA 4 Answer

 

e.  

Networks, STD2 N2 SM-Bank 12

The following table shows the travelling time, in minutes, between towns which are directly connected by roads.

A dash indicates that towns are not directly connected.
 


 

  1. Draw a network diagram showing the information in this table.  (2 marks) 

  2. What is the shortest travelling time between A and E?  (1 mark)

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  1.  
         
     
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i.  

 

ii.  
    

 

Networks, STD2 N3 SM-Bank 49

The directed graph below shows the sequence of activities required to complete a project.

The time to complete each activity, in hours, is also shown.
 


 

  1. Find the earliest starting time, in hours, for activity .   (2 marks)

To complete the project in minimum time, some activities cannot be delayed.

  1. Calculate the number of activities that cannot be delayed.  (1 mark)
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i.   
 


 

 
 

 

ii.   

Networks, STD2 N3 SM-Bank 17

The network diagram represents a system of roads connecting a shopping centre to the motorway.

Two routes from the shopping centre connect to A and one route connects D to F.

The number on the edge of each road indicates the number of vehicles that can travel on it per hour.
 


 

Draw additional road(s) on the diagram to maximise the capacity. Include the number of vehicles that can travel on each road.  (2 marks)

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Networks, STD2 N3 SM-Bank 16 MC

The network diagram represents a system of roads connecting a shopping centre to the motorway.

Two routes from the shopping centre connect to A and one route connects to D to F.

The number on the edge of each road indicates the number of vehicles that can travel on it per hour.

At present, the capacity of the network from the shopping centre to the motorway is not maximised.

Which additional road(s) would increase the network capacity to its maximum?

A. A road from A to F with a capacity of 20 vehicles per hour
B. A road from B to E with a capacity of 30 vehicles per hour
C. A road from C to F with a capacity of 30 vehicles per hour and a road from E to F with a capacity of 60 vehicles per hour
D. A road from B to F with a capacity of 30 vehicles per hour and a road from D to F with a capacity of 30 vehicles per hour
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Networks, STD2 N3 SM-Bank 47

The directed graph below shows the sequence of activities required to complete a project.

All times are in hours.
 


 

  1.  Find the number of activities that have exactly two immediate predecessors. (1 mark)

  2. Identify the critical path for this project. (3 marks)
  3. If Activity E is reduced by one hour, identify the two new critical paths. (1 mark)

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i.    

 

ii.   
 


 

 

iii.   

Networks, STD2 N3 SM-Bank 48

The network shows the activities that are needed to complete a particular project.
 

networks-fur1-2009-vcaa-5-6-mc
 

  1. Find the total number of activities that need to be completed before activity L can begin.   (1 mark)

The duration of every activity is initially 5 hours. 

  1. If the completion times of both activity F and activity K are reduced to 3 hours each, calculate the effect on the completion time for the project.   (2 marks)

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i.   

 

ii.   
 


 

Networks, STD2 N3 2008 FUR1 8-9 MC

The network below shows the activities that are needed to finish a particular project and their completion times (in days).
 

networks-fur1-2008-vcaa-8-mc

 
Part 1

The earliest start time for Activity K, in days, is

A.     

B.   

C.   

D.   

 

Part 2

This project currently has one critical path.

A second critical path, in addition to the first, would be created by

A.   increasing the completion time of D by 7 days.

B.   increasing the completion time of G by 1 day.

C.   increasing the completion time of I by 2 days.

D.   decreasing the completion time of C by 1 day.

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♦♦ Mean mark of Part 2 was 35%.


 

Networks, STD2 N3 2015 FUR1 9 MC

The table below shows, in minutes, the duration, the earliest starting time (EST) and the latest starting time (LST) of eight activities needed to complete a project.
 

NETWORKS, FUR1 2015 VCAA 9 MC

 
Which one of the following directed graphs shows the sequence of these activities?

A.
B.
C.
D.
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Networks, FUR2 2013 VCE 3

The rangers at the wildlife park restrict access to the walking tracks through areas where the animals breed.

The edges on the directed network diagram below represent one-way tracks through the breeding areas. The direction of travel on each track is shown by an arrow. The numbers on the edges indicate the maximum number of people who are permitted to walk along each track each day.
 

NETWORKS, FUR2 2013 VCAA 31
 

  1. Starting at , how many people, in total, are permitted to walk to each day?  (1 mark)

One day, all the available walking tracks will be used by students on a school excursion.

The students will start at and walk in four separate groups to .

Students must remain in the same groups throughout the walk.

    1. Group 1 will have 17 students. This is the maximum group size that can walk together from to .

      Write down the path that group 1 will take.  (1 mark)

    2. Groups 2, 3 and 4 will each take different paths from to .

      Complete the six missing entries shaded in the table below.  (2 marks)

       


      NETWORKS, FUR2 2013 VCAA 32

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    2.  
      Networks, FUR2 2013 VCAA 3_2 Answer1
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a.   
   
   
♦ Mean mark of all parts (combined) was 41%.

 

 

b.i.  

 

b.ii.   

 

 
  

Networks, FUR2 2013 VCAA 3_2 Answer1

Networks, STD2 N3 SM-Bank 34

The arrows on the diagram below show the direction of the flow of waste through a series of pipelines from a factory to a waste dump.

The numbers along the edges show the number of megalitres of waste per week that can flow through each section of pipeline.
 

NETWORKS, FUR1 2015 VCAA 4 MC

 
The minimum cut is shown as a dotted line.

Calculate the capacity of this cut, in megalitres of waste per week.  (2 marks)

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Networks, STD2 N3 2013 FUR1 9 MC

Alana, Ben, Ebony, Daniel and Caleb are friends. Each friend has a different age.

The arrows in the graph below show the relative ages of some, but not all, of the friends. For example, the arrow in the graph from Alana to Caleb shows that Alana is older than Caleb.
 

 
Using the information in the graph, it can be deduced that the second-oldest person in this group of friends is

A.   Alana

B.   Ben

C.   Caleb

D.   Ebony

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vcaa-networks-fur1-2013-9i

 

Networks, STD2 N2 SM-Bank 5

A network of roads is pictured below, with the distances of each road represented, in kilometres, on each edge.
 

  
A driver wants to travel from A to H in the shortest distance possible.

Describe the possible paths she can take, and the total distance she must travel.  (2 marks)

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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.