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Calculus, SPEC1 2018 VCAA 9

A curve is specified parametrically by  .

  1.  Show that the cartesian equation of the curve is  .   (2 marks)

  2.  Find the -coordinates of the points of intersection of the curve    and the line  .   (1 mark)

  3.  Find the volume of the solid of revolution formed when the region bounded by the curve and the line is rotated about the -axis.   (2 marks)

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a.     

 

b.   
 
 
 
 
 

 

♦♦ Mean mark 30%.

 

c.  
 
 
 

 

 
 
 
 
 
 
 

Calculus, SPEC1 2018 VCAA 8

A tank initially holds 16 L of water in which 0.5 kg of salt has been dissolved. Pure water then flows into the tank at a rate of 5 L per minute. The mixture is stirred continuously and flows out of the tank at a rate of 3 L per minute.

  1.  Show that the differential equation for , the number of kilograms of salt in the tank after minutes, is given by
  2.   (1 mark)

  3. Solve the differential equation given in part a. to find as a function of .
  4. Express your answer in the form  , where and are positive integers.  (3 marks)

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  1.  
  2.  
Show Worked Solution

a.

♦ Net mean mark of both parts 44%.


 

MARKER’S COMMENT: Many students took the common factor of 2 from  . This wasn’t necessary and complicated the arithmetic in part b.

b.   

 

 
 

Calculus, MET1 2018 VCAA 8

Let  , where is a positive real constant.

  1.  Show that  .   (1 mark)

  2. Find the value of for which the graphs of    and    have exactly one point of intersection.   (3 marks)

Let  . The diagram below shows sections of the graphs of  and for  .
 


 

Let be the area of the region bounded by the curves  and the line  .

  1. Write down a definite integral that gives the value of .   (1 mark)

  2. Using your result from part a., or otherwise, find the value of such that  .   (3 marks)

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a.   

 

 

b.  

♦♦♦ Mean mark 7%.
MARKER’S COMMENT: Most students found the correct quadratic equation but solved for .

 

 

 

c.   
   
   

♦♦ Mean mark 29%.

 

d.   
 
 
 
 
 
 
 

Calculus, MET1 2018 VCAA 7

Let be a point on the straight line    such that the length of , the line segment from the origin to , is a minimum.

  1.  Find the coordinates of .   (3 marks)

  2.  Find the distance . Express your answer in the form , where and are positive integers.   (2 marks)

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a.   

♦ Mean mark 42%.

 
 
   

 

 

 

 

 

b. 

♦ Mean mark 42%.

 
 
 
 

Calculus, MET2 2018 VCAA 19 MC

The graphs    and  are shown in the diagram below.
 


 

An integral expression that gives the total area of the shaded regions is

A.  

B.  

C.  

               

D.  

E.  

               

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Show Worked Solution

♦ Mean mark 41%.
COMMENT: In each shaded area, it is critical to note which graph is “higher” and which side of the x-axis the shading is.
 

         

 

Algebra, STD2 A2 2007 HSC 27b*

A clubhouse uses four long-life light globes for five hours every night of the year. The purchase price of each light globe is $6.00 and they each cost    per hour to run.

  1. Write an equation for the total cost () of purchasing and running these four light globes for one year in terms of  .    (2 marks)

  2. Find the value of    (correct to three decimal places) if the total cost of running these four light globes for one year is $250.   (1 mark)

  3. If the use of the light globes increases to ten hours per night every night of the year, does the total cost double? Justify your answer with appropriate calculations.   (1 mark)

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i. 

 
 
 

 

ii. 

 
 

 

iii. 

 
   

 

Statistics, STD2 S1 SM-Bank 2 MC

The dot plots show the height of students in Year 9 and Year 12 in a school. They are drawn on the same scale.
 


 

Which statement about the change in heights when comparing Y9 to Y12 is correct?

  1. The mean increased and the standard deviation decreased.
  2. The mean decreased and the standard deviation decreased.
  3. The mean increased and the standard deviation increased.
  4. The mean decreased and the standard deviation increased.
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Financial Maths, STD2 F5 2013 23 MC

Zina opened an account to save for a new car. Six months after opening the account, she made first deposit of $1200 and continued depositing $1200 at the end of each six month period. Interest was paid at 3% per annum, compounded half-yearly.

How much was in Zina's account two years after first opening it?

  1. $4909.08
  2. $4982.72
  3. $5018.16
  4. $5094.55
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Show Worked Solution

♦ Mean mark 41%.


 

Statistics, STD2 S3 2017 HSC 29d*

All the students in a class of 30 did a test.

The marks, out of 10, are shown in the dot plot.
 

  1. Find the median test mark.  (1 mark)

  2. The mean test mark is 5.4. The standard deviation of the test marks is 4.22.
  3. Using the dot plot, calculate the percentage of the marks which lie within one standard deviation of the mean.  (2 marks)
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i.   


 

♦ Mean mark 50%.

ii.   

 
   
   
♦♦ Mean mark 34%.

Financial Maths, STD2 2014 HSC 27a

Alex is buying a used car which has a sale price of  $13 380. In addition to the sale price there are the following costs:

2014 27a1

  1. Stamp Duty for this car is calculated at $3 for every $100, or part thereof, of the sale price.  
  2. Calculate the Stamp Duty payable.   (1 mark)
  3.  
  4. Alex wishes to take out comprehensive insurance for the car for 12 months. The cost of comprehensive insurance is calculated using the following: 
    1.  
    2. 2014 27a2
  5. Find the total amount that Alex will need to pay for comprehensive insurance.   (3 marks)
  6.  
  7. Alex has decided he will take out the comprehensive car insurance rather than the less expensive non-compulsory third-party car insurance.
  8. What extra cover is provided by the comprehensive car insurance?   (1 mark)

 

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  5.  
Show Worked Solution
♦♦♦ Mean mark 12%
IMPORTANT: “or part thereof ..” in the question requires students to round up to 134 to get the right multiple of $3 for their calculation.
(i)   
 

 

(ii)  


 

 
 

 

 
 

 

 

 

♦ Mean mark 34%.
(iii)  
 

Measurement, STD2 M2 2011 HSC 27b

Pontianak has a longitude of 109°E, and Jarvis Island has a longitude of 160°W.

Both places lie on the Equator. 

  1. Calculate the shortest distance between these two places (), to the nearest kilometre, using

     

        where    and  km   (1 mark)

  2. The position of Rabaul is 4° to the south and 48° to the west of Jarvis Island. What is the latitude and longitude of Rabaul?    (2 marks)

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i. 
 
 
 

 

♦♦ Mean mark 33%
ii.
 
 
 
   
 
 
 
 
 

 

Algebra, STD2 A2 SM-Bank 3

The average height, , in centimetres, of a girl between the ages of 6 years and 11 years can be represented by a line with equation

where is the age in years. For this line, the gradient is 6.

  1. What does this indicate about the heights of girls aged 6 to 11?   (1 mark)

  2. Give ONE reason why this equation is not suitable for predicting heights of girls older than 12.   (1 mark)

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i. 


 

ii. 

Networks, STD2 N3 2018 FUR2 1

The graph below shows the possible number of postal deliveries each day between the Central Mail Depot and the Zenith Post Office.

The unmarked vertices represent other depots in the region.

The weighting of each edge represents the maximum number of deliveries that can be made each day.
 


 

  1.  Cut A, shown on the graph, has a capacity of 10.

     

     Two other cuts are labelled as Cut B and Cut C.

    1.  Write down the capacity of Cut B.  (1 mark)

    2.  Write down the capacity of Cut C. (1 mark)

  2.  Determine the maximum number of deliveries that can be made each day from the Central Mail  Depot to the Zenith Post Office.  (1 mark)

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  2.  
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a.i.   
   

 

a.ii.   
   

♦ Mean mark part (b) 32%.
COMMENT: Review carefully! Most common incorrect answer was 9.

 

b.  

Probability, MET1 2018 VCAA 4

Let be a normally distributed random variable with a mean of 6 and a variance of 4. Let be a random variable with the standard normal distribution.

  1.  Find  .  (1 mark)

  2.  Find    such that  .  (1 mark)

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a.   


 

♦ Mean mark part (b) 41%.

b.   
   
   

 

Networks, STD2 N3 2018 FUR1 5 MC

The directed network below shows the sequence of 11 activities that are needed to complete a project.

The time, in weeks, that it takes to complete each activity is also shown.
 


 

How many of these activities could be delayed without affecting the minimum completion time of the project?

  1. 3
  2. 4
  3. 5
  4. 6
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Show Worked Solution

♦ Mean mark 43%.

Graphs, MET2 2018 VCAA 4 MC

The point    lies on the graph of the function  . A transformation maps the graph of  to the graph of ,

where  . The same transformation maps the point to the point .

The coordinates of the point are

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Show Worked Solution

♦ Mean mark 48%.


 


 

GEOMETRY, FUR2 2018 VCAA 2

Frank travelled from Melbourne (38° S, 145° E) to a tennis tournament in Ho Chi Minh City, Vietnam, (11° N, 107° E).

Frank departed Melbourne at 10.30 pm on Monday, 5 February 2018.

Frank arrived in Ho Chi Minh City at 8.00 am on Tuesday, 6 February 2019.

The time difference between Melbourne and Ho Chi Minh City is four hours.

  1.  How long did it take Frank to travel from Melbourne to Ho Chi Minh City?

     

     Give your answer in hours and minutes.   (1 mark)

Ho Chi Minh City is located at latitude 11° N and longitude 107° E.

Assume that the radius of Earth is 6400 km.

  1. i. Write a calculation that shows that the radius of the small circle of Earth at latitude 11° N is 6282 km, rounded to the nearest kilometre.  (1 mark)
  2. ii. Iloilo City, in the Philippines, is located at latitude 11° N and longitude 123° E.

     

        Find the shortest small circle distance between Ho Chi Minh City and Iloilo City.

     

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

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    2. i.
    3. ii.
Show Worked Solution

a.  

♦♦ Mean mark 29%.
COMMENT: Review carefully!

 

 

b.i. 

Mean mark part (b)(i) 52%.

 


 

 

b.ii. 

♦♦ Mean mark part (b)(ii) 33%.
MARKER’S COMMENT: Many students incorrectly used 6400 as the radius in part (b)(ii). Understand why    is the correct radius here!

 
 


 

GRAPHS, FUR1 2018 VCAA 08 MC

A ride-share company has a fee that includes a fixed cost and a cost that depends on both the time spent travelling, in minutes, and the distance travelled, in kilometres.

The fixed cost of a ride is $2.55

Judy’s ride cost $16.75 and took eight minutes. The distance travelled was 10 km.

Pat’s ride cost $30.35 and took 20 minutes. The distance travelled was 18 km.

Roy’s ride took 10 minutes. The distance travelled was 15 km.

The cost of Roy’s ride was

  1. $17.00
  2. $19.55
  3. $20.50
  4. $23.05
  5. $25.60
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Show Worked Solution

♦ Mean mark 48%.

 

GRAPHS, FUR1 2018 VCAA 07 MC

In the diagram below, the shaded region (with boundaries included) represents the feasible region for a linear programming problem.
 


 

The objective function, , has minimum values at both point and point .

Which one of the following could be the objective function?

  1.  
  2.  
  3.  
  4.  
  5.  
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♦ Mean mark 40%.

 

GRAPHS, FUR2 2018 VCAA 4

This year Robert is planning a camping trip for the members of his gold prospecting club.

The club has chosen two camp sites, Bushman’s Track and Lower Creek.

  • Let be the number of members staying at Bushman’s Track.
  • Let be the number of members staying at Lower Creek.
  • A maximum of 10 members can stay at Bushman’s Track.
  • A maximum of 15 members can stay at Lower Creek.
  • At least 20 members in total are attending the camping trip.

The club has decided that at least twice as many members must stay at Lower Creek than at Bushman’s Track.

These constraints can be represented by the following four inequalities.
 


 

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

  1. On the graph above, mark with a cross (×) the five integer points that satisfy Inequalities 1 to 4.   (1 mark)

     

    (answer on the graph above.)

The cost for one member to stay at Bushman’s Track is $130. The cost for one member to stay at Lower Creek is $110.

For budgeting purposes, Robert needs to know the maximum cost of accommodation for both camp sites given Inequalities 1 to 4.

  1. Find the total maximum cost of accommodation.  (1 mark)
  2. When Robert finally made the booking, he was informed that, due to recent renovations, there were two changes to the accommodation at Lower Creek:
  • A maximum of 22 members can now stay at Lower Creek.
  • The cost for one member to stay at Lower Creek is now $140.

Twenty members will be attending the camping trip.

Find the total minimum cost of accommodation for these 20 members.  (1 mark)

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a.  

♦♦ Mean mark part (a) 45%, part (b) 22%.
MARKER’S COMMENT: A poor understanding of this question type notable. Review carefully.

b.    

 

 

c.    

♦♦ Mean mark part (c) 22%.


 

 

GRAPHS, FUR2 2018 VCAA 3

Robert wants to hire a geologist to help him find potential gold locations.

One geologist, Jennifer, charges a flat fee of $600 plus 25% commission on the value of gold found.

The following graph displays Jennifer’s total fee in dollars.
 


 

Another geologist, Kevin, charges a total fee of $3400 for the same task.

  1. Draw a graph of the line representing Kevin’s fee on the axes above.  (1 mark)

     

    (answer on the axes above.)

  2. For what value of gold found will Kevin and Jennifer charge the same amount for their work?  (1 mark)
  3. A third geologist, Bella, has offered to assist Robert.
  4. Below is the relation that describes Bella’s fee, in dollars, for the value of gold found.


  5. The step graph below representing this relation is incomplete.

     

    Complete the step graph by sketching the missing information.  (2 marks)
     

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Show Worked Solution
a.  

 

b.    

♦ Mean mark 41%.

 

 

c.  

GEOMETRY, FUR1 2018 VCAA 6 MC

Aaliyah is bushwalking.

She walks 5.4 km from a starting point on a bearing of 045° until she reaches a hut. From this hut, she walks 2.8 km on a bearing of 300° until she reaches a river.

From the river, she turns and walks back directly to the starting point.

The total distance that she walks, in kilometres, is closest to

  1. 8.2
  2. 13.2
  3. 13.6
  4. 14.1
  5. 14.9
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Show Worked Solution

 

♦ Mean mark 41%.

 

 

 

GEOMETRY, FUR1 2018 VCAA 04 MC

The course for a yacht race is triangular in shape and is marked by three buoys, , and .
 


 

Starting from buoy , the yachts sail on a bearing of 030° to buoy .

From buoy the yachts sail to buoy and then to buoy .

The angle is 69° and the angle is 47°.

The bearing of buoy from buoy is

  1. 034°
  2. 047°
  3. 133°
  4. 279°
  5. 326°
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Show Worked Solution

♦ Mean mark 42%.


 

Networks, STD2 N3 SM-Bank 49

A project requires nine activities (A–I) to be completed. The duration, in hours, and the immediate predecessor(s) of each activity are shown in the table below.


 

  1. Sketch the network diagram. (3 marks)

  2. Find the minimum completion time for this project, in hours.   (1 mark)

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  2. 20
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i.   
 

 
ii. 


 

GEOMETRY, FUR1 2018 VCAA 3 MC

The city of Karachi in Pakistan has latitude 25° N and longitude 67° E.

Assume that the radius of Earth is 6400 km.

The shortest distance along the surface of Earth between Karachi and the North Pole, in kilometres, can be found by evaluating which one of the following products?

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♦ Mean mark 43%.


 

GEOMETRY, FUR2 2018 VCAA 3

Frank owns a tennis court.

A diagram of his tennis court is shown below

Assume that all intersecting lines meet at right angles.

Frank stands at point . Another point on the court is labelled point .
 

  1. What is the straight-line distance, in metres, between point and point ?

     

    Round your answer to one decimal place.   (1 mark)

  2. Frank hits a ball when it is at a height of 2.5 m directly above point .

     

    Assume that the ball travels in a straight line to the ground at point .

     

    What is the straight-line distance, in metres, that the ball travels?

     

    Round your answer to the nearest whole number.   (1 mark)

Frank hits two balls from point .

For Frank’s first hit, the ball strikes the ground at point , 20.7 m from point .

For Frank’s second hit, the ball strikes the ground at point .

Point is metres from point .

Point is 10.4 m from point .

The angle, , formed is 23.5°.
 


 

    1. Determine two possible values for angle .

       

      Round your answers to one decimal place.   (1 mark)

    2. If point is within the boundary of the court, what is the value of ?

       

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

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a.   
 
   
   

 

b.
  


 

 
 

 

c.i.  

♦♦ Mean mark 25%.

 

 

♦♦♦ Mean mark 18%.

c.ii.   
 
   
 
 
 
   
   

NETWORKS, FUR1 2018 VCAA 5 MC

The directed network below shows the sequence of 11 activities that are needed to complete a project.

The time, in weeks, that it takes to complete each activity is also shown.
 


 

How many of these activities could be delayed without affecting the minimum completion time of the project?

  1. 3
  2. 4
  3. 5
  4. 6
  5. 7
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Show Worked Solution

♦ Mean mark 43%.

NETWORKS, FUR2 2018 VCAA 4

Parcel deliveries are made between five nearby towns, to .

The roads connecting these five towns are shown on the graph below. The distances, in kilometres, are also shown.
 

A road inspector will leave from town to check all the roads and return to town when the inspection is complete. He will travel the minimum distance possible.

  1.  How many roads will the inspector have to travel on more than once?   (1 mark)

  2.  Determine the minimum distance, in kilometres, that the inspector will travel.   (1 mark)

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a.  

♦♦ Mean mark 31%.

 

b.  

♦♦♦ Mean mark 14%.


 

NETWORKS, FUR2 2018 VCAA 3

At the Zenith Post Office all computer systems are to be upgraded.

This project involves 10 activities, to .

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

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

  2. The minimum completion time for the project is 15 hours.

     

    Write down the critical path.   (1 mark)

  3. Two of the activities have a float time of two hours.

     

    Write down these two activities.   (1 mark)

  4. For the next upgrade, the same project will be repeated but one extra activity will be added.
    This activity has a duration of one hour, an earliest starting time of five hours and a latest starting time of 12 hours.

     

    Complete the following sentence by filling in the boxes provided.   (1 mark)

     

    The extra activity could be represented on the network above by a directed edge from the

   end of activity   
 
   to the start of activity   
 
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  3.  
Show Worked Solution

a.  

 

 
b. 
 

c.  

♦ Mean mark 45%.

 


 


 

d.  

♦♦ Mean mark 25%.
 

NETWORKS, FUR2 2018 VCAA 1

The graph below shows the possible number of postal deliveries each day between the Central Mail Depot and the Zenith Post Office.

The unmarked vertices represent other depots in the region.

The weighting of each edge represents the maximum number of deliveries that can be made each day.
 


 

  1.  Cut A, shown on the graph, has a capacity of 10.

     

     Two other cuts are labelled as Cut B and Cut C.
     i.  Write down the capacity of Cut B.   (1 mark)

    ii.  Write down the capacity of Cut C.   (1 mark)

  1. Determine the maximum number of deliveries that can be made each day from the Central Mail   Depot to the Zenith Post Office.   (1 mark)

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  1.  i. 
    ii.  
  2.  
Show Worked Solution
a.i.   
   

 

a.ii.   
   

♦ Mean mark part (b) 32%.
COMMENT: Review carefully! Most common incorrect answer was 9.

 

b.  

MATRICES, FUR1 2018 VCAA 8 MC

A public library organised 500 of its members into five categories according to the number of books each member borrows each month.

These categories are

J = no books borrowed per month
K = one book borrowed per month
L = two books borrowed per month
M = three books borrowed per month
N = four or more books borrowed per month

The transition matrix, , below shows how the number of books borrowed per month by the members is expected to change from month to month.
 

 
In the long term, which category is expected to have approximately 96 members each month?

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Show Worked Solution

♦ Mean mark 46%.


 


 

MATRICES, FUR1 2018 VCAA 7 MC

A study of the antelope population in a wildlife park has shown that antelope regularly move between three locations, east (), north () and west ().

Let  be the state matrix that shows the population of antelope in each location months after the study began.

The expected population of antelope in each location can be determined by the matrix recurrence rule
 


 

where

and


 

The state matrix, , below shows the population of antelope three months after the study began.
 


 

The number of antelope in the west () location two months after the study began, as found in the state matrix , is closest to

  1. 2060
  2. 2130
  3. 2200
  4. 2240
  5. 2270
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Show Worked Solution

♦♦ Mean mark 28%.

 

MATRICES, FUR2 2018 VCAA 4

Beginning in the year 2021, a new company takes over the maintenance of the 2700 km highway with a new contract.

Each year sections of highway must be graded , resurfaced or sealed .

The remaining highway will need no maintenance that year.

Let  be the state matrix that shows the highway maintenance schedule of this company for the th year after 2020.

The maintenance schedule for 2020 is shown in matrix  below.

For these 2700 km of highway, the matrix recurrence relation shown below can be used to determine the number of kilometres of this highway that will require each type of maintenance from year to year.
 

where

   

 

  1. Write down the value of in matrix .   (1 mark)

  2. How many kilometres of highway are expected to be graded in the year 2022?   (1 mark)

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Show Worked Solution

a.    

♦ Mean mark 39%.

 

♦♦ Mean mark 23%.

b.   
     
 

 

MATRICES, FUR2 2018 VCAA 3

The Hiroads company has a contract to maintain and improve 2700 km of highway.

Each year sections of highway must be graded , resurfaced or sealed .

The remaining highway will need no maintenance that year.

Let be the state matrix that shows the highway maintenance schedule for the th year after 2018.

The maintenance schedule for 2018 is shown in matrix below.

 

 

The type of maintenance in sections of highway varies from year to year, as shown in the transition matrix , below.
 


 

  1. Of the length of highway that was graded in 2018, how many kilometres are expected to be resurfaced the following year?   (1 mark)

  2. Show that the length of highway that is to be graded in 2019 is 460 km by writing the appropriate numbers in the boxes below.   (1 mark)

     

 
 
 
 

 

The state matrix describing the highway maintenance schedule for the nth year after 2018 is given by


 

  1. Complete the state matrix, , below for the highway maintenance schedule for 2019 (one year after 2018).   (1 mark)

     



     

  2. In 2020, 1536 km of highway is expected to require no maintenance
  3. Of these kilometres, what percentage is expected to have had no maintenance in 2019?
  4. Round your answer to one decimal place.   (1 mark)

  5. In the long term, what percentage of highway each year is expected to have no maintenance ?
  6. Round your answer to one decimal place.   (1 mark)

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Show Worked Solution
a.   
   

♦ Mean mark part (a) 48%.

 

b.   
   

 

c.   
   

 

d.   

 
 

 

e. 

 

MATRICES, FUR2 2018 VCAA 2

The Westhorn Council must prepare roads for expected population changes in each of three locations: main town , villages and rural areas .

The population of each of these locations in 2018 is shown in matrix    below.

The expected annual change in population in each location is shown in the table below.
       

  1. Write down matrix  , which shows the expected population in each location in 2019.   (1 mark)

  2. The expected population in each of the three locations in 2019 can be determined from the matrix product.
  3. where is a diagonal matrix.
  4. Write down matrix  .   (1 mark)

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Show Worked Solution

a.   

♦ Mean mark part (b) 40%.
COMMENT: Many students included 0.04, -0.01 and -0.02 in this matrix. Know why this is incorrect!

 
b.   

CORE, FUR2 2018 VCAA 6

 

Julie has retired from work and has received a superannuation payment of $492 800.

She has two options for investing her money.

Option 1

Julie could invest the $492 800 in a perpetuity. She would then receive $887.04 each fortnight for the rest of her life.

  1. At what annual percentage rate is interest earned by this perpetuity?  (1 mark)

Option 2

Julie could invest the $492 800 in an annuity, instead of a perpetuity.

The annuity earns interest at the rate of 4.32% per annum, compounding monthly.

The balance of Julie’s annuity at the end of the first year of investment would be $480 242.25

    1. What monthly payment, in dollars, would Julie receive?   (1 mark)
    2. How much interest would Julie’s annuity earn in the second year of investment?
    3. Round your answer to the nearest cent.    (1 mark)

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Show Worked Solution

a.   
   

♦ Mean mark 41%.
 

 

 

b.i.   

♦ Mean mark 48%.

 

 

♦♦♦ Mean mark 16%.

b.ii.  
 
 
   

 

 

CORE, FUR2 2018 VCAA 5

After three years, Julie withdraws $14 000 from her account to purchase a car for her business.

For tax purposes, she plans to depreciate the value of her car using the reducing balance method.

The value of Julie’s car, in dollars, after years, , can be modelled by the recurrence relation shown below

  1. For each of the first three years of reducing balance depreciation, the value of is 0.85

     

    What is the annual rate of depreciation in the value of the car during these three years?  (1 mark)

  2. For the next five years of reducing balance depreciation, the annual rate of depreciation in the value of the car is changed to 8.6%.

     

    What is the value of the car eight years after it was purchased?
    Round your answer to the nearest cent.  (2 marks)

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a.  

COMMENT: Note almost half of students answered incorrectly here!

     

  
b.   

♦ Mean mark 42%.

 

 

 
 

MATRICES, FUR1 2018 VCAA 5 MC

Liam cycles, runs, swims and walks for exercise several times a month.

Each time he cycles, Liam covers a distance of kilometres.

Each time he runs, Liam covers a distance of kilometres.

Each time he swims, Liam covers a distance of kilometres.

Each time he walks, Liam covers a distance of kilometres.

The number of times that Liam cycled, ran, swam and walked each month over a four-month period, and the total distance that Liam travelled in each of those months, are shown in the table below.
 

 
The matrix that contains the distance each time Liam cycled, ran, swam and walked, , is

A. B. C.
           
D. E.    
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♦ Mean mark 42%.


 


 

CORE, FUR2 2018 VCAA 3

Table 3 shows the yearly average traffic congestion levels in two cities, Melbourne and Sydney, during the period 2008 to 2016. Also shown is a time series plot of the same data.

The time series plot for Melbourne is incomplete.

  1. Use the data in Table 3 to complete the time series plot above for Melbourne.   (1 mark)

  2. A least squares line is used to model the trend in the time series plot for Sydney. The equation is

       

  1.   i. Draw this least squares line on the time series plot.   (1 mark)

  2.  ii. Use the equation of the least squares line to determine the average rate of increase in percentage congestion level for the period 2008 to 2016 in Sydney.   (1 mark)

    iii. Use the least squares line to predict when the percentage congestion level in Sydney will be 43%.   (1 mark)

The yearly average traffic congestion level data for Melbourne is repeated in Table 4 below.

  1. When a least squares line is used to model the trend in the data for Melbourne, the intercept of this line is approximately –1514.75556
  2. Round this value to four significant figures.   (1 mark)

  3. Use the data in Table 4 to determine the equation of the least squares line that can be used to model the trend in the data for Melbourne. The variable year is the explanatory variable.
  4. Write the values of the intercept and the slope of this least squares line in the appropriate boxes provided below.
  5. Round both values to four significant figures.   (2 marks)

congestion level = 
 
  + 
 
  × year
  1. Since 2008, the equations of the least squares lines for Sydney and Melbourne have predicted that future traffic congestion levels in Sydney will always exceed future traffic congestion levels in Melbourne.

     

    Explain why, quoting the values of appropriate statistics.   (2 marks)

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a.   

 

b.i.   

 

b.ii.  

♦ Mean mark (b)(ii) 36%.
COMMENT: Major problems caused by part (b)(ii). Review!

 

 

b.iii.   
 
 
   

 

c. 

 

d.   

 

e.   

♦♦♦ Mean mark 18%.

 

CORE, FUR1 2018 VCAA 24 MC

Mariska plans to retire from work 10 years from now.

Her retirement goal is to have a balance of $600 000 in an annuity investment at that time.

The present value of this annuity investment is $265 298.48, on which she earns interest at the rate of 3.24% per annum, compounding monthly.

To make this investment grow faster, Mariska will add a $1000 payment at the end of every month.

Two years from now, she expects the interest rate of this investment to fall to 3.20% per annum, compounding monthly. It is expected to remain at this rate until Mariska retires.

When the interest rate drops, she must increase her monthly payment if she is to reach her retirement goal.

The value of this new monthly payment will be closest to

  1. $1234
  2. $1250
  3. $1649
  4. $1839
  5. $1854
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♦ Mean mark 45%.

 

 

 

CORE, FUR1 2018 VCAA 23 MC

Five lines of an amortisation table for a reducing balance loan with monthly repayments are shown below.

The interest rate for this loan changed immediately before repayment number 28.

This change in interest rate is best described as

  1. an increase of 0.24% per annum.
  2. a decrease of 0.024% per annum.
  3. an increase of 0.024% per annum.
  4. a decrease of 0.0024% per annum.
  5. an increase of 0.00024% per annum.
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♦♦ Mean mark 29%.


 


 

 

CORE, FUR1 2018 VCAA 22 MC

Adam has a home loan with a present value of $175 260.56

The interest rate for Adam’s loan is 3.72% per annum, compounding monthly.

His monthly repayment is $3200.

The loan is to be fully repaid after five years.

Adam knows that the loan cannot be exactly repaid with 60 repayments of $3200.

To solve this problem, Adam will make 59 repayments of $3200. He will then adjust the value of the final repayment so that the loan is fully repaid with the 60th repayment.

The value of the 60th repayment will be closest to

  1.   $368.12
  2. $2831.88
  3. $3200.56
  4. $3557.09
  5. $3568.12
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♦♦ Mean mark 29%.

 

 

 

CORE, FUR1 2018 VCAA 20 MC

The graph below shows the value, , of an asset as it depreciates over a period of five months.

Which one of the following depreciation situations does this graph best represent?

  1. flat rate depreciation with a decrease in depreciation rate after two months
  2. flat rate depreciation with an increase in depreciation rate after two months
  3. unit cost depreciation with a decrease in units used per month after two months
  4. reducing balance depreciation with an increase in the rate of depreciation after two months
  5. reducing balance depreciation with a decrease in the rate of depreciation after two months
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♦ Mean mark 39%.

CORE, FUR1 2018 VCAA 14 MC

A least squares line is fitted to a set of bivariate data.

Another least squares line is fitted with response and explanatory variables reversed.

Which one of the following statistics will not change in value?

  1. the residual values
  2. the predicted values
  3. the correlation coefficient
  4. the slope of the least squares line
  5. the intercept of the least squares line
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♦ Mean mark 42%.


 

CORE, FUR1 2018 VCAA 13 MC

The statistical analysis of a set of bivariate data involving variables and resulted in the information displayed in the table below.

Using this information, the value of the correlation coefficient for this set of bivariate data is closest to

  1. 0.88
  2. 0.89
  3. 0.92
  4. 0.94
  5. 0.97
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♦ Mean mark 49%.

 

CORE, FUR1 2018 VCAA 7-9 MC

The scatterplot below displays the resting pulse rate, in beats per minute, and the time spent exercising, in hours per week, of 16 students. A least squares line has been fitted to the data.
 

 
Part 1

Using this least squares line to model the association between resting pulse rate and time spent exercising, the residual for the student who spent four hours per week exercising is closest to

  1. –2.0 beats per minute.
  2. –1.0 beats per minute.
  3. –0.3 beats per minute.
  4.   1.0 beats per minute.
  5.   2.0 beats per minute.

 
Part 2

The equation of this least squares line is closest to

  1. resting pulse rate = 67.2 – 0.91 × time spent exercising
  2. resting pulse rate = 67.2 – 1.10 × time spent exercising
  3. resting pulse rate = 68.3 – 0.91 × time spent exercising
  4. resting pulse rate = 68.3 – 1.10 × time spent exercising
  5. resting pulse rate = 67.2 + 1.10 × time spent exercising

 
Part 3

The coefficient of determination is 0.8339

The correlation coefficient is closest to

  1. –0.913
  2. –0.834
  3. –0.695
  4.   0.834
  5.   0.913
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♦ Mean mark 46%.


 

 

♦ Mean mark 45%.

 

Networks, STD2 N3 SM-Bank 29

The construction of The Royal Easter show involves activities to . The diagram shows these activities and their completion times in days.
 

 
The company contracted to construct it are given a completion deadline of 31 days.

Calculate the float time of Activity .  (3 marks)

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