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Calculus, MET1-NHT 2019 VCAA 4

A function has rule  .

  1. State the maximal domain of and the range of over its maximal domain.   (2 marks)

  2. i.  Find the equation of the tangent to the graph of at .   (2 marks)

     

    ii. On the axes below, sketch the graph of the function , labelling any asymptote with its equation.

  3.     Also draw the tangent to the graph of at  .   (4 marks)
      

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  1.  

  2. i. 
    ii. 
     
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a.  
 

 

b.i.  
 
 

 

 

b.ii.  

Calculus, MET1-NHT 2019 VCAA 3

  1. Evaluate    and  .   (2 marks)

  2. Show that  .   (1 mark)

  3. Use your answers to part a. and part b. to evaluate    in the form  , where  and are positive integers.   (1 mark)

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a.  
   
   

 

 
 

 

b.  
   
   

 

c.  
   
   
   
   

Statistics, EXT1 S1 EQ-Bank 23

A light manufacturer knows that 6% of the light bulbs it produces are defective.

Light bulbs are supplied in boxes of 20 bulbs. Boxes are supplied in pallets of 120 boxes.

Calculate the probability that

  1. A box of light bulbs contains exactly 3 defective bulbs.  (1 mark)

  2. A box of light bulbs contains at least 1 defective bulb.  (1 mark)

  3. A pallet contains between 90 and 95 (inclusive) boxes with at least 1 defective bulb (use the probability table attached).  (3 marks)

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i.   

 

 

ii.   
   
   

 

iii.   


 


 


 

 
 

Statistics, EXT1 S1 EQ-Bank 20

Netball Australia records show that 10% of all registered players are over the age of 25.

  1. A random survey of 100 netball players was carried out to find out how many were over 25 years of age.

     

    Assuming the sample proportion is normally distributed, calculate the expected mean and standard deviation of this group.  (2 marks)

  2. Using the probability table attached, estimate the probability that at least 15 players surveyed will be over 25 years of age.  (2 marks)

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i.  
 
   
   
 
   

 

ii. 

 
 
 

Statistics, SPEC2-NHT 2019 VCAA 6

A paint company claims that the mean time taken for its paint to dry when motor vehicles are repaired is 3.55 hours, with a standard deviation of 0.66 hours.

Assume that the drying time for the paint follows a normal distribution and that the claimed standard deviation value is accurate.

  1. Let the random variable    represent the mean time taken for the paint to dry for a random sample of 36 motor vehicles.

     

    Write down the mean and standard deviation of  .   (2 marks)

At a car crash repair centre, it was found that the mean time taken for the paint company's paint to dry on randomly selected vehicles was 3.85 hours. The management of this crash repair centre was not happy and believed that the claim regarding the mean time taken for the paint to dry was too low. To test the paint company's claim, a statistical test was carried out.

  1. Write down suitable null and alternative hypotheses and respectively to test whether the mean time taken for the paint to dry is longer than claimed.   (1 mark)

  2. Write down an expression for the    value of the statistical test and evaluate it correct to three decimal places.   (2 marks)

  3. Using a 1% level of significance, state with a reason whether the crash repair centre is justified in believing that the paint company's claim of mean time taken for its paint to dry of 3.55 hours is too low.  (1 mark)

  4. At the 1% level of significance, find the set of sample mean values that would support the conclusion that the mean time taken for the paint to dry exceeded 3.55 hours. Give your answer in hours, correct to three decimal places.  (2 marks)

  5. If the true time taken for the paint to dry is 3.83 hours, find the probability that the paint company's claim is not rejected at the 1% level of significance, assuming the standard deviation for the paint to dry is still 0.66 hours. Give your answer correct to two decimal places.  (1 mark)

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a.   
 
 

 

b.   

 

c.   
 
 
 

 

d.   

 

e.   

 

f.   
 
 

Calculus, SPEC2-NHT 2019 VCAA 3


 

The vertical cross-section of a barrel is shown above. The radius of the circular base (along the -axis) is 30 cm and the radius of the circular top is 70 cm. The curved sides of the cross-section shown are parts of the parabola with rule  . The height of the barrel is 50 cm.
 
a.  i.   Show that the volume of the barrel is given by  .  (1 marks)
     ii.  Find the volume of the barrel in cubic centimetres.  (1 marks)    

The barrel is initially full of water. Water begins to leak from the bottom of the barrel such that    cubic centimetres per second, where after    seconds the depth of the water is    centimetres, the volume of water remaining in the barrel is    cubic centimetres and the uppermost surface area of the water is    square centimetres.
 
b.     Show that  ?  (2 marks)
c.      Find    in terms of  . Express your answer in the form  , where    and    are positive integers.  (3 marks)
d.     Using a definite integral in terms of  , find the time, in hours, correct to one decimal place, taken for the barrel to empty.  (2 marks)

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a.  i.

      ii. 

b.   

c.   

d.   

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a.i.
 
 
     
 

 

a.ii.
   
   

 

b.   

 
 

 

c.   

 
 
 

  
 
d.   

 
 

Calculus, SPEC2-NHT 2019 VCAA 2

Consider the function with rule  .

  1. State the equations of the asymptotes of the graph of .  (2 marks)

  2. State the coordinates of the stationary points and the point of inflection. Give your answers correct to two decimal places.  (2 marks)

  3. Sketch the graph of from    to    (endpoint coordinates are not required) on the set of axes below, labeling the turning points and the point of inflection with their coordinates correct to two decimal places. Label the asymptotes with their equations.  (3 marks)

Consider the function with rule    where  .

  1. For what values of will have no stationary points?  (2 marks)

  2. For what value of will the graph of have a point of  inflection located on the -axis?  (1 marks)

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i.     

 ii.   
iii.

iv. 

v. 

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i.   


  

ii.   


 


 

iii.

 

iv.   

 

 

v.   

Complex Numbers, SPEC2-NHT 2019 VCAA 1

In the complex plane, is the with equation .

  1.  Verify that the point (0, 0) lies on .   (1 marks)

  2.  The line    can also be expressed in the form  , where  .

     

     Find  in cartesian form.   (2 marks)

  3.  Find, in cartesian form, the points(s) of intersection of    and the graph of  .   (2 marks)

  4.  Sketch    and the graph of    on the Argand diagram below.   (2 marks)

     
     
         
     

  5.  Find the area of the sector defined by the part of    where  , the graph of    where  , and imaginary axis where  .   (1 marks)

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  5.  

     
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a.   

 

b. 
 
 
 
 
 
 


c.


 

 

d.   

 

e.


 

f.    

 

Vectors, EXT1 V1 EQ-Bank 28

A projectile is fired from a canon at ground level with initial velocity ms−1 at an angle of 30° to the horizontal.

The equations of motion are    and  .

  1. Show that  .  (1 mark)

  2. Show that  .  (2 marks)
  3. Hence find the Cartesian equation for the trajectory of the projectile.  (1 mark)

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i.   

 

ii.   


 

 

iii.   

 
 

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

Vectors, EXT2 V1 SM-Bank 21

  1. Find the equation of the vector line    that passes through    and  .  (1 mark)

  2. A sphere has centre    at    and a radius of    units.
    Find the points where the vector line    meets the sphere.  (3 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 6

  1. What vector line equation, , corresponds to the Cartesian equation
  2.   (1 mark)

  3. Express     in Cartesian form where,
  4.   (1 mark)

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  1.  
  2.  
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i.   

COMMENT: Ensure you know the format for conversion from Cartesian to vector form (part 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.   

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 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
  3. ²
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a.  
    ²

 

b.  
    ³

 

c.  
    ²

 

d.  
 
 

 

 

NETWORKS, FUR2 2019 VCAA 2

Fencedale High School offers students a choice of four sports, football, tennis, athletics and basketball.

The bipartite graph below illustrates the sports that each student can play.
  
 


  

Each student will be allocated to only one sport.

  1. Complete the table below by allocating the appropriate sport to each student.   (1 mark)

 

         Student                                 Sport                         
    Blake  
    Charli  
    Huan  
    Marco  

 

  1. The school medley relay team consists of four students, Anita, Imani, Jordan and Lola.

      

    The medley relay race is a combination of four different sprinting distances: 100 m, 200 m, 300 m and 400 m, run in that order.

      

    The following table shows the best time, in seconds, for each student for each sprinting distance.
     

      Best time for each sprinting distance (seconds)
         Student             100 m               200 m               300 m               400 m       
      Anita 13.3 29.6 61.8 87.1
      Imani 14.5 29.6 63.5 88.9
      Jordan 13.3 29.3 63.6 89.1
      Lola 15.2 29.2 61.6 87.9

      
     
    The school will allocate each student to one sprinting distance in order to minimise the total time taken to complete the race.

      

    To which distance should each student be allocated?

      

    Write your answers in the table below.  (2 marks)

     

           Student                                Sprinting distance (m)                         
      Anita  
      Imani  
      Jordan  
      Lola  
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a.           Student                                 Sport                         
    Blake   Tennis
    Charli   Football
    Huan   Basketball
    Marco   Athletics

 

b.           Student                Sprinting distance (m)         
    Anita 400
    Imani 200
    Jordan 100
    Lola 300
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a.   


 

         Student                                 Sport                         
    Blake   Tennis
    Charli   Football
    Huan   Basketball
    Marco   Athletics

 

b.   


 

     Student        100 m         200 m         300 m         400 m    
    Anita 0 2.3 2.1 1.1
    Imani 0 1.1 2.6 1.7
    Jordan 0 2 3.9 3.1
    Lola 0 0 0 0

 


 

     Student        100 m         200 m         300 m         400 m    
    Anita 0 1.2 1 0
    Imani 0 0 1.5 0.6
    Jordan 0 0.9 2.8 2
    Lola 1.1 0 0 0

 

         Student                Sprinting distance (m)         
    Anita 400
    Imani 200
    Jordan 100
    Lola 300

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 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.  
 

 

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. 

 

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. 

CORE, FUR2 2019 VCAA 7

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, , is $48 600.   (1 mark)

  2. What is the annual percentage rate of depreciation used by Phil?   (1 mark)

  3. 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)

  4. 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. 

 
c. 

 
d. 

CORE, FUR2 2019 VCAA 5

The scatterplot below shows the atmospheric pressure, in hectopascals (hPa), at 3 pm (pressure 3 pm) plotted against the atmospheric pressure, in hectopascals, at 9 am (pressure 9 am) for 23 days in November 2017 at a particular weather station.
 

A least squares line has been fitted to the scatterplot as shown.

The equation of this line is

pressure 3 pm = 111.4 + 0.8894 × pressure 9 am

  1. Interpret the slope of this least squares line in terms of the atmospheric pressure at this weather station at 9 am and at 3 pm.   (1 mark)

  2. Use the equation of the least squares line to predict the atmospheric pressure at 3 pm when the atmospheric pressure at 9 am is 1025 hPa.
  3. Round your answer to the nearest whole number.   (1 mark)

  4. Is the prediction made in part b. an example of extrapolation or interpolation?   (1 mark)

  5. Determine the residual when the atmospheric pressure at 9 am is 1013 hPa.
  6. Round your answer to the nearest whole number.   (1 mark)

  7. The mean and the standard deviation of pressure 9 am and pressure 3 pm for these 23 days are shown in Table 4 below.

    1. Use the equation of the least squares line and the information in Table 4 to show that the correlation coefficient for this data, rounded to three decimal places, is  = 0.966   (1 mark)

    2. What percentage of the variation in pressure 3 pm is explained by the variation in pressure 9 am?
    3. Round your answer to one decimal place.   (1 mark)

  1. The residual plot associated with the least squares line is shown below.
     

    1. The residual plot above can be used to test one of the assumptions about the nature of the association between the atmospheric pressure at 3 pm and the atmospheric pressure at 9 am.
    2. What is this assumption?   (1 mark)

    3. The residual plot above does not support this assumption.
    4. Explain why.   (1 mark)

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a.   

 

b.  
   

 

c. 

 

d.  
   
   
   
   

 

e.i.   

   
   
   

 

e.ii.  
 
   

 

f.i.  
 

 

f.ii.