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Vectors, EXT2 V1 2019 SPEC2 4

The base of a pyramid is the parallelogram    with vertices at points    and  . The apex (top) of the pyramid is located at  .

  1. Find the values of    and  .  (2 marks)

  2. Find the cosine of the angle between the vectors    and  .  (2 marks)

  3. Find the area of the base of the pyramid.  (2 marks)

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  3. ²
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i.   
   

 

 

ii.   

 
 

 

iii.   
 

  

  

  ²

Statistics, SPEC1-NHT 2019 VCAA 3

The number of cars per day making a U-turn at a particular location is known to be normally distributed with a standard deviation of 17.5. In a sample of 25 randomly selected days, a total of 1450 cars were observed making the U-turn.

  1. Based on this sample, calculate an approximate 95% confidence interval for the number of cars making the U-turn each day. Use an integer multiple of the standard deviation in your calculations.   (3 marks)

  2. The average number of U-turns made at the location is actually 60 per day.

     

    Find an approximation, correct to two decimal places, for the probability that on 25 randomly selected days the average number of U-turns is less than 53.   (1 mark)

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a.      

σ

 

 

b.     μσ
 
         
 
 
         
 

 
 
 

 

Mechanics, SPEC1-NHT 2019 VCAA 1

A 10 kg mass is placed on a rough plane that inclined at 30° to the horizontal, as shown in the diagram below. A force of 40 N is applied to the mass up the slope and parallel to the slope. There is also a frictional resistance force of magnitude that opposes  the motion of the mass.
 


 

  1. Find the magnitude of the frictional resistance force, in newtons, acting up the slope if the force is just sufficient to stop the mass from sliding down the slope.  (2 marks)
     
  2. An additional force of magnitude newtons is applied to the mass  up the slope and parallel to the slope. The sum of the additional force and the frictional resistance force of magnitude that now acts down the slope is such that it is just sufficient to stop the mass from sliding up the slope.  (2 marks)
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a.       

 

 

b.   

 
 

Trigonometry, 2ADV* T1 2011 HSC 24c

A ship sails 6 km from to on a bearing of 121°. It then sails 9 km to .  The
size of angle is 114°.
 

2UG 2011 24c

Copy the diagram into your writing booklet and show all the information on it.

  1. What is the bearing of from ?   (1 mark)

  2. Find the distance . Give your answer correct to the nearest kilometre.   (2 marks)

  3. What is the bearing of from ? Give your answer correct to the nearest degree.   (3 marks)

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  1.  
  2.  
  3.  
Show Worked Solution
STRATEGY: This deserves repeating again: Draw North-South parallel lines through major points to make the angle calculations easier!
i.     2011 HSC 24c

 

 
 

   

 

ii.   

 
 
 

 

iii.   

MARKER’S COMMENT: The best responses showed clear working on the diagram.
 
 

 

 
 
 

Trigonometry, 2ADV* T1 2005 HSC 27c

2UG-2005-27c
 

The bearing of from is 250° and the distance of from is 36 km.

  1. Explain why    is  .   (1 mark)

  2. If    is 15 km due north of  , calculate the distance of    from  , correct to the nearest kilometre.   (3 marks)

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i. 

 

ii.   
 
 
 

Trigonometry, 2ADV* T1 2007 26a

The diagram shows information about the locations of towns  ,    and  .
 

 
 

  1. It takes Elina 2 hours and 48 minutes to walk directly from Town    to Town  .

     

    Calculate her walking speed correct to the nearest km/h.    (1 mark)

  2. Elina decides, instead, to walk to Town    from Town    and then to Town  .

     

    Find the distance from Town    to Town  . Give your answer to the nearest km.   (2 marks)

  3. Calculate the bearing of Town    from Town  .   (1 mark)

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i. 

 

 

 
 

 

ii.  

 
 

 

 

iii.  

 

Vectors, EXT2 V1 SM-Bank 22

are the vertices of a rectangular prism.
  


 

  1. Show that the internal diagonals of the prism, and , intersect.  (2 marks)

  2. Calculate the acute angle, , between the diagonals, to the nearest minute.  (2 marks)

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i.     
   

 


 

 

ii.   
 

 

 

Vectors, EXT2 V1 SM-Bank 15

Consider the two vector line equations

  1. Show that    and    intersect and determine the point of intersection . (2 marks)

  2. What is the acute angle between the vector lines, to the nearest minute.  (2 marks)

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i.   

 


  

 

ii.   

 
 

 

 

Vectors, EXT2 V1 SM-Bank 2

  1. Find values of  , ,   and    such that    is a vector equation of a line that passes through    and  .  (2 marks)

  2. Determine whether    is perpendicular to  .  (1 mark)

  3. Express    in Cartessian form.  (1 mark)

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  1. , or

     

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i.   

 
 

 

 
 

 

 

 
 

 

  


 

ii.   


 

iii.   

Statistics, STD2 S1 EQ-Bank 4

A high school conducted a survey asking students what their favourite Summer sport was.

The Pareto chart shows the data collected.
 


 

  1. What percentage of students chose Hockey as their favourite Summer sport?  (1 mark)

  2. What percentage of students chose Touch Football as their favourite Summer sport?  (1 mark)

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i.   

 


 

 
   

 

ii.  

 

GRAPHS, FUR2 2019 VCAA 3

Members of the association will travel to a conference in cars and minibuses:

  • Let be the number of cars used for travel.
  • Let be the number of minibuses used for travel.
  • A maximum of eight cars and minibuses in total can be used.
  • At least three cars must be used.
  • At least two minibuses must be used.

The constraints above can be represented by the following three inequalities.
 


 

  1. Each car can carry a total of five people and each minibus can carry a total of 10 people.

      

    A maximum of 60 people can attend the conference.

      

    Use this information to write Inequality 4.   (1 mark)

The graph below shows the four lines representing Inequalities 1 to 4.

Also shown on this graph are four of the integer points that satisfy Inequalities 1 to 4. Each of these integer points is marked with a cross (✖).
 


 

  1. On the graph above, mark clearly, with a circle (o), the remaining integer points that satisfy Inequalities 1 to 4.  (1 mark)

Each car will cost $70 to hire and each minibus will cost $100 to hire.

  1. What is the cost for 60 members to travel to the conference?  (1 mark)
  2. What is the minimum cost for 55 members to travel to the conference?  (1 mark)
  3. Just before the cars were booked, the cost of hiring each car increased.

      

    The cost of hiring each minibus remained $100.

      

    All original constraints apply.

      

    If the increase in the cost of hiring each car is more than dollars, then the maximum cost of transporting members to this conference can only occur when using six cars and two minibuses.

      

    Determine the value of  .  (1 mark)

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a. 

 

b.    

 

c.   

 

d.   

 

e.   

   

GRAPHS, FUR2 2019 VCAA 2

Each branch within the association pays an annual fee based on the number of members it has.

To encourage each branch to find new members, two new annual fee systems have been proposed.

Proposal 1 is shown in the graph below, where the proposed annual fee per member, in dollars, is displayed for branches with up to 25 members.
 


 

  1. What is the smallest number of members that a branch may have?  (1 mark)
  2. The incomplete inequality below shows the number of members required for an annual fee per member of $10.

      

    Complete the inequality by writing the appropriate symbol and number in the box provided.   (1 mark)
     

3 ≤ number of members  
 

 

Proposal 2 is modelled by the following equation.

annual fee per member = – 0.25 × number of members + 12.25

  1. Sketch this equation on the graph for Proposal 1, shown below.  (1 mark)

 

 

  1. Proposal 1 and Proposal 2 have the same annual fee per member for some values of the number of members.

      

    Write down all values of the number of members for which this is the case.  (1 mark)

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a. 
 

b. 
 

c.  

 

d.   

GRAPHS, FUR2 2019 VCAA 1

The graph below shows the membership numbers of the Wombatong Rural Women’s Association each year for the years 2008–2018.
 


 

  1. How many members were there in 2009?  (1 mark)
    1. Show that the average rate of change of membership numbers from 2013 to 2018 was − 6 members per year.  (1 mark)
    2. If the change in membership numbers continues at this rate, how many members will there be in 2021?  (1 mark)
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a. 

 

b.i.  
 
 
   

 

b.ii.  
   

GEOMETRY, FUR2 2019 VCAA 3

The following diagram shows a crane that is used to transfer shipping containers between the port and the cargo ship.
 


 

The length of the boom, , is 25 m. The length of the hoist, , is 15 m.

    1. Write a calculation to show that the distance is 20 m.  (1 mark)
    2. Find the angle .

        

      Round your answer to the nearest degree.  (1 mark)

  1. The diagram below shows a cargo ship next to a port. The base of a crane is shown at point .
     

      

      


     

      

    The base of the crane () is 20 m from a shipping container at point . The shipping container will be moved to point , 38 m from . The crane rotates 120° as it moves the shipping container anticlockwise from to .

      

    What is the distance , in metres?

      

    Round your answer to the nearest metre.  (1 mark)

  2. A shipping container is a rectangular prism.

      

    Four chains connect the shipping container to a hoist at point , as shown in the diagram below.
     

      

     
     

      

    The shipping container has a height of 2.6 m, a width of 2.4 m and a length of 6 m.

     

    Each chain on the hoist is 4.4 m in length.

      

    What is the vertical distance, in metres, between point and the top of the shipping container?

      

    Round your answer to the nearest metre.  (2 marks)

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a.i.  
   
   
   

 

a.ii.  
 
   
   

 

b. 

 
 

 

c.  


 


 

 

 

 
 

GEOMETRY, FUR2 2019 VCAA 2

A cargo ship travels from Magadan (60° N, 151° E) to Sydney (34° S, 151° E).

  1. Explain, with reference to the information provided, how we know that Sydney is closer to the equator than Magadan.   (1 mark)
  2. Assume that the radius of Earth is 6400 km.

      

    Find the shortest great circle distance between Magadan and Sydney.

      

    Round your answer to the nearest kilometre.  (1 mark)

  3. The cargo ship left Sydney (34° S, 151° E) at 6 am on 1 June and arrived in Perth (32° S, 116° E) at 10 am on 11 June.

      

    There is a two-hour time difference between Sydney and Perth at that time of year.

      

    How many hours did it take the cargo ship to travel from Sydney to Perth?  (1 mark)

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a.  

 

b.  
   

 

c.   

 
 
 

GEOMETRY, FUR2 2019 VCAA 1

The following diagram shows a cargo ship viewed from above.
 

 
The shaded region illustrates the part of the deck on which shipping containers are stored.

  1. What is the area, in square metres, of the shaded region?  (1 mark)

Each shipping container is in the shape of a rectangular prism.

Each shipping container has a height of 2.6 m, a width of 2.4 m and a length of 6 m, as shown in the diagram below.
 

  1. What is the volume, in cubic metres, of one shipping container?  (1 mark)
  2. What is the total surface area, in square metres, of the outside of one shipping container?  (1 mark)
  3. One shipping container is used to carry barrels. Each barrel is in the shape of a cylinder.

      

    Each barrel is 1.25 m high and has a diameter of 0.73 m, as shown in the diagram below.

      

    Each barrel must remain upright in the shipping container
     
     

      


     
    What is the maximum number of barrels that can fit in one shipping container?  (1 mark)

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  1. ²
  2. ³
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a.  
    ²

 

b.  
    ³

 

c.  
    ²

 

d.  
 
 

 

 

Networks, STD2 N3 2019 FUR2 3

Fencedale High School is planning to renovate its gymnasium.

This project involves 12 activities, to .

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


 

The minimum completion time for the project is 35 weeks.

  1. Identify the critical path and state how many activities are on it?  (2 marks)

  2. Determine the latest start time of activity .  (1 mark)

  3. Which activity has the longest float time?  (1 mark)

It is possible to reduce the completion time for activities and by employing more workers.

  1.  The completion time for each of these five activities can be reduced by a maximum of two weeks.

      

    What is the minimum time, in weeks, that the renovation project could take?  (1 mark)

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a. 
 

 

 


 

b. 
 

c.   


 

d.   


 

 

NETWORKS, FUR2 2019 VCAA 1

Fencedale High School has six buildings. The network below shows these buildings represented by vertices. The edges of the network represent the paths between the buildings.
 


 

  1. Which building in the school can be reached directly from all other buildings?   (1 mark)

  2. A school tour is to start and finish at the office, visiting each building only once.
     i.     What is the mathematical term for this route?   (1 mark)

  3. ii. Draw in a possible route for this school tour on the diagram below.   (1 mark)

 

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  2.  i. 
    ii.  
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a. 
 

b.i.   

 
 

b.ii.   
 

MATRICES, FUR2 2019 VCAA 3

On Sunday, matrix is used when calculating the expected number of visitors at each location every hour after 10 am. It is assumed that the park will be at its capacity of 2000 visitors for all of Sunday.

Let be the state matrix that shows the number of visitors at each location at 10 am on Sunday.

The number of visitors expected at each location at 11 am on Sunday can be determined by the matrix product

 

 

  1. Safety restrictions require that all four locations have a maximum of 600 visitors.
  2. Which location is expected to have more than 600 visitors at 11 am on Sunday?   (1 mark) 
  3. Whenever more than 600 visitors are expected to be at a location on Sunday, the first 600 visitors can stay at that location and all others will be moved directly to Ground World .
  4. State matrix contains the number of visitors at each location hours after 10 am on Sunday, after the safety restrictions have been enforced.
  5. Matrix can be determined from the matrix recurrence relation
  7. where matrix shows the required movement of visitors at 11 am.
    1. Determine the matrix .   (1 mark)

    2. State matrix can be determined from the new matrix rule
    4. where matrix shows the required movement of visitors at 12 noon.
    5. Determine the state matrix .   (1 mark)

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  2.  i.
    ii.
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a.  
 
 
   

 

 

b.i.  
     
 
   
 

 

b.ii.  
   

MATRICES, FUR2 2019 VCAA 2

The theme park has four locations, Air World , Food World , Ground World and Water World .

The number of visitors at each of the four locations is counted every hour.

By 10 am on Saturday the park had reached its capacity of 2000 visitors and could take no more visitors.

The park stayed at capacity until the end of the day

The state matrix, , below, shows the number of visitors at each location at 10 am on Saturday.
 


 

  1. What percentage of the park’s visitors were at Water World at 10 am on Saturday?   (1 mark)

Let be the state matrix that shows the number of visitors expected at each location hours after 10 am on Saturday.

The number of visitors expected at each location hours after 10 am on Saturday can be determined by the matrix recurrence relation below.
 


 

  1. Complete the state matrix, , below to show the number of visitors expected at each location at 11 am on Saturday.   (1 mark)

 

 

  1. Of the 300 visitors expected at Ground World at 11 am, what percentage was at either Air World or Food World at 10 am?   (1 mark)

  2. The proportion of visitors moving from one location to another each hour on Sunday is different from Saturday.

     

    Matrix , below, shows the proportion of visitors moving from one location to another each hour after 10 am on Sunday.

     

     

     
    Matrix is similar to matrix but has the first two rows of matrix interchanged.

     

  3. The matrix product that will generate matrix from matrix is
  5. where matrix is a binary matrix.
  6. Write down matrix .   (1 mark)
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a.   

 

b.  
   

 

c. 


 

d.  
 

 

Calculus, EXT2 C1 2002 HSC 2b

For  ...

let  .

  1.  Show that  .   (1 mark)

  2.  Show that, for  ,
     
         .   (3 marks)

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i.         

 
 
 
 

 

ii.
   
   

 

 
   

 

 
 

 

Calculus, EXT2 C1 2008 HSC 3c

For  , let  .   

  1. Show that for ,
     
         .  (2 marks)

  2. Hence, or otherwise, calculate  .   (2 marks)

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i.
   
   
   

 

   
 

 

 
 
 

 

ii.       

 
 
 
 

MATRICES, FUR2 2019 VCAA 1

The car park at a theme park has three areas, and .

The number of empty and full parking spaces in each of the three areas at 1 pm on Friday are shown in matrix   below.
 


 

  1. What is the order of matrix ?   (1 mark)

  2. Write down a calculation to show that 110 parking spaces are full at 1 pm.   (1 mark)

Drivers must pay a parking fee for each hour of parking.

Matrix , below, shows the hourly fee, in dollars, for a car parked in each of the three areas.
 


 

  1. The total parking fee, in dollars, collected from these 110 parked cars if they were parked for one hour is calculated as follows.  

     

     

     

    where matrix    is a    matrix.

     

    Write down matrix  .   (1 mark)

The number of whole hours that each of the 110 cars had been parked was recorded at 1 pm. Matrix , below, shows the number of cars parked for one, two, three or four hours in each of the areas and .


 

  1. Matrix    is the transpose of matrix  .

      

    Complete the matrix    below.   (1 mark)

      


     

  2. Explain what the element in row 3, column 2 of matrix    represents.   (1 mark)

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a. 
 

b. 
 

c. 
 

d. 
 

e.  
   

CORE, FUR2 2019 VCAA 8

Phil invests $200 000 in an annuity from which he receives a regular monthly payment.

The balance of the annuity, in dollars, after months, , can be modelled by the recurrence relation

  1. What monthly payment does Phil receive?   (1 mark)

  2. Show that the annual percentage compound interest rate for this annuity is 4.2%.   (1 mark)

At some point in the future, the annuity will have a balance that is lower than the monthly payment amount.

  1. What is the balance of the annuity when it first falls below the monthly payment amount?

     

    Round your answer to the nearest cent.   (1 mark)

  2. If the payment received each month by Phil had been a different amount, the investment would act as a simple perpetuity.

     

    What monthly payment could Phil have received from this perpetuity?   (1 mark)

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a. 

b.  
 

  
c. 

 

 

  

d. 

 

Proof, EXT2 P1 SM-Bank 5 MC

Four cards are placed on a table with a letter on one face and a shape on the other.
 

     

 
You are given the rule: "if N is on a card then a circle is on the other side."

Which cards need to be turned over to check if this rule holds?

  1. N and G
  2. G and triangle
  3. circle and N
  4. N and triangle
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Show Worked Solution

 
 

 


 

Financial Maths, 2ADV M1 SM-Bank 15

Phil is a builder who has purchased a large set of tools.

The value of Phil’s tools is depreciated using the reducing balance method.

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


 

  1. Use recursion to show that the value of the tools after two years.   (1 mark)

  2. Phil plans to replace these tools when their value first falls below $20 000.

     

    After how many years will Phil replace these tools?  (1 mark)

  3. Phil has another option for depreciation. He depreciates the value of the tools by a flat rate of 8% of the purchase price per annum.

     

    Let be the value of the tools after years, in dollars.

     

    Write down a recurrence relation, in terms of and , that could be used to model the value of the tools using this flat rate depreciation.  (1 mark)

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a.  
 
 

 

b. 

COMMENT: Dividing by    is dividing by a negative number!

 
 
 
   

 

 

c.