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Calculus, MET1 2014 VCAA 10

A line intersects the coordinate axes at the points and with coordinates and , respectively, where and are positive real numbers and  .

  1. When , the line is a tangent to the graph of    at the point with coordinates , as shown.

     


    met1-2014-vcaa-q10
     

     

    If and are non-zero real numbers, find the values of and .   (3 marks)

  2. The rectangle has a vertex at on the line. The coordinates of are , as shown.

     
    met1-2014-vcaa-q10_1
     

    1. Find an expression for in terms of .   (1 mark)

    2. Find the minimum total shaded area and the value of for which the area is a minimum.   (2 marks)

    3. Find the maximum total shaded area and the value of for which the area is a maximum.   (1 mark)

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    2. ²
    3. ²
Show Worked Solution

a.  

♦ Mean mark 48%.
MARKER’S COMMENT: The most common error incorrectly stated that (6,0) lay on the parabola.

 

 

 

 

b.i.  

met1-2014-vcaa-q10-answer

♦♦ Mean mark (b.i.) 32%.

 

 
 

 

♦♦♦ Mean mark (b.ii.) 18%.
b.ii.   
   
   

 

 

  ²

 

²
 

b.iii.  

♦♦♦ Mean mark (b.iii.) 8%.

²

Calculus, MET1 2015 VCAA 10

The diagram below shows a point, , on a circle. The circle has radius 2 and centre at the point  with coordinates . The angle is , where  .
  

met1-2015-vcaa-q10
  

The diagram also shows the tangent to the circle at . This tangent is perpendicular to and intersects the -axis at point and the -axis at point .

  1. Find the coordinates of in terms of .   (1 mark)

  2. Find the gradient of the tangent to the circle at in terms of .   (1 mark)

  3. The equation of the tangent to the circle at can be expressed as
  5.  i. Point , with coordinates , is on the line segment .
  6.     Find in terms of .   (1 mark)

  7. ii. Point , with coordinates , is on the line segment .
  8.     Find in terms of .   (1 mark)

  9. Consider the trapezium with parallel sides of length and .
  10. Find the value of for which the area of the trapezium is a minimum. Also find the minimum value of the area.   (3 marks)

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  3.  i.
  4. ii.
  6. ²
Show Worked Solution
a.   
   
 
♦♦♦ Part (a) mean mark 20%.
MARKER’S COMMENT: Many students did not include the “+2” in the -coordinate.

 

 

♦♦♦ Part (b) mean mark 16%.
b.   
   
   
     
 

 

c.i.  

 

c.ii.   

♦ Part (c)(ii) mean mark 47%.
 

 

♦♦♦ Part (d) mean mark 19%.
d.   
   
   

 

 

 
 

 

²

Calculus, MET2 2010 VCAA 4

Consider the function  

  1. Find the -coordinate of each of the stationary points of   and state the nature of each of these stationary points.   (4 marks)

In the following, is the function   where and are real constants.

  1. Write down, in terms of and , the possible values of for which is a stationary point of .   (3 marks)

  2. For what value of does have no stationary points?   (1 mark)

  3. Find in terms of if has one stationary point.   (2 marks)

  4. What is the maximum number of stationary points that can have?  (1 mark)

  5. Assume that there is a stationary point at and another stationary point  where  .
  6. Find the value of .   (3 marks)

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Show Worked Solution

a.  

 


 

 

vcaa-graphs-fur2-2010-4ai


 

b.  

♦ Mean mark (b) 46%.
MARKER’S COMMENT: Issues with use of CAS caused significant difficulties in this question.

 


 

c.  

♦ Mean mark part (c) 47%.


 

d.  

♦♦♦ Mean mark (d) 16%.

 

♦ Mean mark (e) 37%.

e.   

 

f.    


 

♦♦♦ Mean mark (f) 9%.

 

   


 

 

Calculus, MET2 2010 VCAA 3

An ancient civilisation buried its kings and queens in tombs in the shape of a square-based pyramid,

The kings and queens were each buried in a pyramid with  

Each of the isosceles triangle faces is congruent to each of the other triangular faces.

The base angle of each of these triangles is , where  

Pyramid and a face of the pyramid, , are shown here.
 

VCAA 2010 3a

is the midpoint of

  1. i. Find in terms of .   (1 mark)

  2. ii. Find in terms of .   (1 mark)

  3. Show that the total surface area (including the base), , of the pyramid, , is given by 
  4.      .   (2 marks)

  5. Find , the height of the pyramid , in terms of .    (2 marks)

  6. The volume of any pyramid is given by the formula  .
  7. Show that the volume, ³, of the pyramid  is  .   (1 mark)

Queen Hepzabah’s pyramid was designed so that it had the maximum possible volume.

  1. Find    and hence find the exact volume of Queen Hepzabah’s pyramid and the corresponding value of .   (4 marks)

Queen Hepzabah’s daughter, Queen Jepzibah, was also buried in a pyramid. It also had

The volume of Jepzibah’s pyramid is exactly one half of the volume of Queen Hepzabah’s pyramid. The volume of Queen Jepzibah’s pyramid is also given by the formula for obtained in part d.

  1. Find the possible values of , for Jepzibah’s pyramid, correct to two decimal places.   (2 marks)

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a.i.

a.ii.

b.   

c.   

d.   

e.   

f.   

Show Worked Solution
a.i.  
 

 

  ii.  
 

 

b.  
   
 
   
♦ Mean mark 47%.

 

 

 

c.  

 vcaa-graphs-fur2-2010-ci

 
♦♦♦ Mean mark part (d) 22%.

 

d.  
   
   

 

 

 

e.  

♦ Mean mark part (e) 45%.

 

  ³

♦♦♦ Mean mark part (f) 16%.

 

f.  

Probability, MET1 2015 VCAA 9

An egg marketing company buys its eggs from farm A and farm B. Let be the proportion of eggs that the company buys from farm A. The rest of the company’s eggs come from farm B. Each day, the eggs from both farms are taken to the company’s warehouse.

Assume that  of all eggs from farm A have white eggshells and of all eggs from farm B have white eggshells.

  1. An egg is selected at random from the set of all eggs at the warehouse.
  2. Find, in terms of , the probability that the egg has a white eggshell.  (1 mark)

  3. Another egg is selected at random from the set of all eggs at the warehouse.

     

    1. Given that the egg has a white eggshell, find, in terms of , the probability that it came from farm B.  (2 marks)

    2. If the probability that this egg came from farm B is 0.3, find the value of .  (1 mark)

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Show Worked Solution

a.  

met1-2015-vcaa-q9-answer1

 

 
 

 

♦ Part (b) mean mark 41%.
MARKER’S COMMENT: Algebraic fractions “were not handled well”!

b.i.   
   
   
   

 

♦♦♦ Part (c) mean mark 19%.
STRATEGY: Previous parts of a question are gold dust for directing your strategy in many harder questions.

b.ii.   
 
 
 
 

CORE*, FUR2 2006 VCAA 4

A company anticipates that it will need to borrow $20 000 to pay for a new machine.

It expects to take out a reducing balance loan with interest calculated monthly at a rate of 10% per annum.

The loan will be fully repaid with 24 equal monthly instalments.

Determine the total amount of interest that will be paid on this loan.

Write your answer to the nearest dollar.   (2 marks) 

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Show Worked Solution

♦♦ Mean mark 21%.

 

 


 

CORE*, FUR2 2007 VCAA 2

Khan decides to extend his home office and borrows $30 000 for building costs. Interest is charged on the loan at a rate of 9% per annum compounding monthly.

Assume Khan will pay only the interest on the loan at the end of each month.

  1. Calculate the amount of interest he will pay each month.   (1 mark)

Suppose the interest rate remains at 9% per annum compounding monthly and Khan pays $400 each month for five years.

  1. Determine the amount of the loan that is outstanding at the end of five years.

     

    Write your answer correct to the nearest dollar.   (1 mark)

Khan decides to repay the $30 000 loan fully in equal monthly instalments over five years.

The interest rate is 9% per annum compounding monthly.

  1. Determine the amount of each monthly instalment. Write your answer correct to the nearest cent.   (1 mark) 
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Show Worked Solution

a.  

♦ Mean mark of all parts (combined) was 37%.
MARKER’S COMMENT: Many students were confused by the fact that part (a) did not have a given loan period.


  

b.  


  

c.  

GRAPHS, FUR2 2007 VCAA 3

Gas is generally cheaper than petrol. 

A car must run on petrol for some of the driving time.

Let    be the number of hours driving using gas

   be the number of hours driving using petrol

Inequalities 1 to 5 below represent the constraints on driving a car over a 24-hour period.

Explanations are given for Inequalities 3 and 4.

Inequality 1:  

Inequality 2:  

Inequality 3:       The number of hours driving using petrol must not exceed half the number of hours driving using gas.
Inequality 4:       The number of hours driving using petrol must be at least one third the number of hours driving using gas.

Inequality 5:  

 

  1. Explain the meaning of Inequality 5 in terms of the context of this problem.  (1 mark)

The lines    and    are drawn on the graph below.

GRAPHS, FUR2 2007 VCAA 3

  1. On the graph above

     

    1. draw the line    (1 mark)
    2. clearly shade the feasible region represented by Inequalities 1 to 5.  (1 mark)

On a particular day, the Goldsmiths plan to drive for 15 hours. They will use gas for 10 of these hours.

  1. Will the Goldsmiths comply with all constraints? Justify your answer.  (1 mark)

On another day, the Goldsmiths plan to drive for 24 hours.

Their car carries enough fuel to drive for 20 hours using gas and 7 hours using petrol.

  1. Determine the maximum and minimum number of hours they can drive using gas while satisfying all constraints.  (2 marks)

     

     

    Maximum = ___________ hours

     

     

    Minimum = ___________ hours

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  2. i. & ii.
    GRAPHS, FUR2 2007 VCAA 3 Answer




Show Worked Solution

a.  

♦♦ Mean mark of parts (b)-(d) (combined) was 33%.

 

b.i. & ii.

GRAPHS, FUR2 2007 VCAA 3 Answer

 

c.   

MARKER’S COMMENT: A mark was only awarded if a reference was made to the 5 hours driving.

 

d.  

♦♦♦ “Very few” obtained both answers here.
MARKER’S COMMENT: Many ignored that the Goldsmiths planned to drive for 24 hours.

 

CORE*, FUR2 2008 VCAA 5

Michelle took a reducing balance loan for $15 000 to purchase her car. Interest is calculated monthly at a rate of 9.4% per annum.

In order to repay the loan Michelle will make a number of equal monthly payments of $350.

The final repayment will be less than $350.

  1. How many equal monthly payments of $350 will Michelle need to make?   (1 mark)

  2. How much of the principal does Michelle have left to pay immediately after she makes her final $350 payment? Find this amount correct to the nearest dollar.   (1 mark)

Exactly one year after Michelle established her loan the interest rate increased to 9.7% per annum. Michelle decided to increase her monthly payment so that the loan would be fully paid in three years (exactly four years from the date the loan was established).

  1. What is the new monthly payment Michelle will make? Write your answer correct to the nearest cent.   (2 marks)

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Show Worked Solution

a.  

♦♦ Mean mark of all parts (combined) was 23%.
MARKER’S COMMENT: This part was not well done. An area where students consistently struggle each year.

  


  

b.  

  


  

c.  

 

   

  

GRAPHS, FUR2 2008 VCAA 3

An event involves running for 10 km and cycling for 30 km.

Let    be the time taken (in minutes) to run 10 km

   be the time taken (in minutes) to cycle 30 km

Event organisers set constraints on the time taken, in minutes, to run and cycle during the event.

Inequalities 1 to 6 below represent all time constraints on the event.

Inequality 1:   Inequality 4:  
Inequality 2:   Inequality 5:  
Inequality 3:   Inequality 6:  

 

  1. Explain the meaning of Inequality 3 in terms of the context of this problem.  (1 mark)

 

The lines    and    are drawn on the graph below.

GRAPHS, FUR2 2008 VCAA 3

  1. On the graph above

     

    1. draw and label the lines    and    (2 marks)
    2. clearly shade the feasible region represented by Inequalities 1 to 6.  (1 mark)

One competitor, Jenny, took 100 minutes to complete the run.

  1. Between what times, in minutes, can she complete the cycling and remain within the constraints set for the event?  (1 mark)
  2. Competitors who complete the event in 90 minutes or less qualify for a prize. 

     

    Tiffany qualified for a prize.

     

    1. Determine the maximum number of minutes for which Tiffany could have cycled.  (1 mark)
    2. Determine the maximum number of minutes for which Tiffany could have run.  (1 mark)
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  2.  i. & ii.
    GRAPHS, FUR2 2008 VCAA 3 Answer
Show Worked Solution

a.  

 

b.i. & ii.

GRAPHS, FUR2 2008 VCAA 3 Answer

 

c.   

♦♦ Mean mark of parts (c)-(d) (combined) was 19%.

 

d.i.   

 

 

 

d.ii.   

 

CORE*, FUR2 2009 VCAA 5

In order to drought-proof the course, the golf club will borrow $200 000 to develop a water treatment facility. 

The club will establish a reducing balance loan and pay interest monthly at the rate of 4.65% per annum.

  1. $1500 per month will be paid on this loan.

     

    How much of the principal will be left to pay after five years?

     

    Write your answer in dollars correct to the nearest cent.   (1 mark)

  2. Determine the total interest paid on the loan over the five-year period.

     

    Write your answer in dollars correct to the nearest cent.   (1 mark)

  3. When the amount outstanding on the loan has reduced to $95 200, the interest rate increases to 5.65% per annum.

     

    Calculate the new monthly repayment that will fully repay this amount in 60 equal instalments.

     

    Write your answer in dollars correct to the nearest cent.   (1 mark)

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Show Worked Solution

a.  

♦♦ Mean mark of all parts (combined) was 28%.

  

  
b.  


 

  
c.  

 

GRAPHS, FUR2 2009 VCAA 4

Cheapstar Airlines wishes to find the optimum number of flights per day on two of its most popular routes: Alberton to Bisley and Alberton to Crofton.

Let    be the number of flights per day from Alberton to Bisley

   be the number of flights per day from Alberton to Crofton

Table 4 shows the constraints on the number of flights per day and the number of crew per flight.

GRAPHS, FUR2 2009 VCAA 41

The lines    and    are graphed below.

GRAPHS, FUR2 2009 VCAA 42

A profit of $1300 is made on each flight from Alberton to Bisley and a profit of $2100 is made on each flight from Alberton to Crofton.

Determine the maximum total profit that Cheapstar Airlines can make per day from these flights.  (2 marks)

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Show Worked Solution

GRAPHS, FUR2 2009 VCAA 4 Answer

♦♦♦ Mean mark 9%.
MARKER’S COMMENT: Many students incorrectly substituted the point of intersection (4.5,5.5) into the equation. Integer values within the feasible region were required.

 

CORE*, FUR2 2010 VCAA 4

A home buyer takes out a reducing balance loan of $250 000 to purchase an apartment.

Interest on the loan will be calculated and paid monthly at the rate of 6.25% per annum.

  1. The loan will be fully repaid in equal monthly instalments over 20 years.
    1. Find the monthly repayment, in dollars, correct to the nearest cent.   (1 mark)
    2. Calculate the total interest that will be paid over the 20 year term of the loan.
    3. Write your answer correct to the nearest dollar.   (2 marks)
  2. After 60 monthly repayments have been made, what will be the outstanding principal on the loan?
  3. Write your answer correct to the nearest dollar.   (1 mark)

By making a lump sum payment after nine years, the home buyer is able to reduce the principal on his loan to $100 000. At this time, his monthly repayment changes to $1250. The interest rate remains at 6.25% per annum, compounding monthly.

  1. With these changes, how many months, in total, will it take the home buyer to fully repay the $250 000 loan?   (1 mark)

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  1. i. 
  2. ii. 

Show Worked Solution

a.i.  

♦♦ This question (all parts) were generally poorly answered, although exact data unavailable.

  

 

a.ii.  

  
b.  

  

 

c.  

  

MARKER’S COMMENT: A common error was to forget to add the initial 108 months.

CORE*, FUR2 2010 VCAA 3

Simple Saver is a simple interest investment in which interest is paid annually.

Growth Plus is a compound interest investment in which interest is paid annually.

Initially, $8000 is invested with both Simple Saver and Growth Plus.

The graph below shows the total value (principal and all interest earned) of each of these investments over a 15 year period.

The increase in the value of each investment over time is due to interest
 

BUSINESS, FUR2 2010 VCAA 3
 

  1. Which investment pays the highest annual interest rate, Growth Plus or Simple Saver?

     

    Give a reason to justify your answer.   (1 mark)

  2. After 15 years, the total value (principal and all interest earned) of the Simple Saver investment is $21 800.

     

    Find the amount of interest paid annually.   (1 mark)

  3. After 15 years, the total value (principal and all interest earned) of the Growth Plus investment is $24 000.

     

    1. Write down an equation that can be used to find the annual compound interest rate, .   (1 mark)

    2. Determine the annual compound interest rate.

       

      Write your answer as a percentage correct to one decimal place.   (1 mark)

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Show Worked Solution

a.  

♦♦♦ Part (a) was “very” poorly answered although exact data unavailable.
MARKER’S COMMENT: Most students ignored the word “rate” and instead referred to the eventual return of each investment.


  

b.  


  

c.i.  


  

c.ii.   
 
 
   

GRAPHS, FUR2 2010 VCAA 3

Let be the number of Softsleep pillows that are sold each week and be the number of Resteasy pillows that are sold each week.

A constraint on the number of pillows that can be sold each week is given by

Inequality 1:  

  1. Explain the meaning of Inequality 1 in terms of the context of this problem.  (1 mark)

Each week, Anne sells at least 30 Softsleep pillows and at least Resteasy pillows.

These constraints may be written as

Inequality 2:  

 

Inequality 3:  

The graphs of    and    are shown below.

GRAPHS, FUR2 2010 VCAA 3

  1. State the value of .  (1 mark)
  2. On the axes above

     

    1. draw the graph of    (1 mark)
    2. shade the region that satisfies Inequalities 1, 2 and 3.  (1 mark)
  3. Softsleep pillows sell for $65 each and Resteasy pillows sell for $50 each.

     

    What is the maximum possible weekly revenue that Anne can obtain?  (2 marks)

Anne decides to sell a third type of pillow, the Snorestop.

She sells two Snorestop pillows for each Softsleep pillow sold. She cannot sell more than 150 pillows in total each week.

  1. Show that a new inequality for the number of pillows sold each week is given by

     

          Inequality 4:  

     

     

    where      is the number of Softsleep pillows that are sold each week

     

        and      is the number of Resteasy pillows that are sold each week.  (1 mark)

Softsleep pillows sell for $65 each.

Resteasy pillows sell for $50 each.

Snorestop pillows sell for $55 each.

  1. Write an equation for the revenue, dollars, from the sale of all three types of pillows, in terms of the variables and .  (1 mark)
  2. Use Inequalities 2, 3 and 4 to calculate the maximum possible weekly revenue from the sale of all three types of pillow.  (2 marks) 
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  3. i. & ii.
    GRAPHS, FUR2 2010 VCAA 3 Answer
Show Worked Solution

a.   

 

b.  

 

c.i. & ii.   

GRAPHS, FUR2 2010 VCAA 3 Answer

 

d.  

 

e.  

♦♦ Mean mark of parts (e)-(g) (combined) was 24%.

 

f.   
   
   

 

g.   

GRAPHS, FUR2 2010 VCAA 3 Answer1

GEOMETRY, FUR2 2010 VCAA 4

A flying fox suspension wire begins at , 12.5 metres above as shown in the diagram below. It ends at , 4.5 metres above .
 

GEOMETRY, FUR2 2010 VCAA 4
 

At , the flying fox wire passes over .

The horizontal distances and are 95 metres and 65 metres respectively.

Calculate the vertical distance, , in metres.  (2 marks)

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Show Worked Solution

GEOMETRY, FUR2 2010 VCAA 4 Answer 
 

♦♦♦ Mean mark 11%.
 

 

 
 

 

 

 

 

CORE*, FUR2 2011 VCAA 4

Tania takes out a reducing balance loan of $265 000 to pay for her house.

Her monthly repayments will be $1980.

Interest on the loan will be calculated and paid monthly at the rate of 7.62% per annum.

    1. How many monthly repayments are required to repay the loan? Write your answer to the nearest month.   (1 mark)
    2. Determine the amount that is paid off the principal of this loan in the first year. Write your answer to the nearest cent.   (1 mark)

Immediately after Tania made her twelfth payment, the interest rate on her loan increased to 8.2% per annum, compounding monthly.

Tania decided to increase her monthly repayment so that the loan would be repaid in a further nineteen years.

  1. Determine the new monthly repayment.
  2. Write your answer to the nearest cent.   (1 mark)

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  1. i. 
  2. ii. 

Show Worked Solution

a.i.     


  

a.ii.   

 


 

b.  

 

CORE*, FUR2 2012 VCAA 4

Arthur invested $80 000 in a perpetuity that returns $1260 per quarter. Interest is calculated quarterly.

  1. Calculate the annual interest rate of Arthur’s investment.   (1 mark)

  2. After Arthur has received 20 quarterly payments, how much money remains invested in the perpetuity?   (1 mark)

  3. Arthur’s wife, Martha, invested a sum of money at an interest rate of 9.4% per annum, compounding quarterly.

     

     

    She will be paid $1260 per quarter from her investment.

     

     

    After ten years, the balance of Martha’s investment will have reduced to $7000.

     

     

    Determine the initial sum of money Martha invested.

     

     

    Write your answer, correct to the nearest dollar.   (1 mark)

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Show Worked Solution

a.  

♦♦ Mean mark of all parts (combined) was 26%.
 
   

 

b.   

 

c.   

 

CORE*, FUR2 2012 VCAA 3

An area of a club needs to be refurbished.

$40 000 is borrowed at an interest rate of 7.8% per annum. 

Interest on the unpaid balance is charged to the loan account monthly. 

Suppose the $40 000 loan is to be fully repaid in equal monthly instalments over five years.

  1. Determine the monthly payment, correct to the nearest cent.   (1 mark)
  2. If, instead, the monthly payment was $1000, how many months will it take to fully repay the $40 000?   (1 mark)
  3. Suppose no payments are made on the loan in the first 12 months.
    1. Write down a calculation that shows that the balance of the loan account after the first 12 months will be $43 234 correct to the nearest dollar.   (1 mark)
    2. After the first 12 months, only the interest on the loan is paid each month.
    3. Determine the monthly interest payment, correct to the nearest cent.   (1 mark)

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  3. i. 
  4. ii. 

Show Worked Solution

a.   

♦♦ Mean mark of all parts (combined) 28%.


  

b.  

  


  

c.i.  


  

c.ii.  

NETWORKS, FUR1 2006 VCAA 8 MC

Euler’s formula, relating vertices, faces and edges, does not apply to which one of the following graphs?

networks-fur1-2006-vcaa-8-mc-ab

networks-fur1-2006-vcaa-8-mc-cd

networks-fur1-2006-vcaa-8-mc-e

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Show Worked Solution

♦♦ Mean mark 28%.
MARKER’S COMMENT: Over a third of students incorrectly chose option , unaware that any graph with 4 or less vertices must be planar.

networks-fur1-2006-vcaa-8-mc-answer

NETWORKS, FUR1 2010 VCAA 9 MC

The table below shows the time (in minutes) that each of four people, Aiden, Bing, Callum and Dee, would take to complete each of the tasks and
 

     vcaa-networks-fur1-2010-9
 

If each person is allocated one task only, the minimum total time for this group of people to complete all four tasks is

A.   

B.   

C.   

D.   

E.   

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Show Worked Solution

vcaa-networks-fur1-2010-9i

 

NETWORKS, FUR1 2006 VCAA 9 MC

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


 

The project is to be crashed by reducing the completion time of one activity only.

This will reduce the completion time of the project by a maximum of

A.   1 hour 

B.   2 hours

C.   3 hours

D.   4 hours

E.   5 hours

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Show Worked Solution

♦♦♦ Mean mark 17%.
MARKER’S COMMENT: When choosing an activity to crash, take care that a new critical path is not created.

 

MATRICES*, FUR1 2009 VCAA 9 MC

Five soccer teams played each other once in a tournament. In each game there was a winner and a loser.

A table of one-step and two-step dominances was prepared to summarise the results.
 

     networks-fur1-2009-vcaa-9-mc
 

One result in the tournament that must have occurred is that

A.   Elephants defeated Bears.

B.   Elephants defeated Aardvarks.

C.   Aardvarks defeated Donkeys.

D.   Donkeys defeated Bears.

E.   Bears defeated Chimps. 

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Show Worked Solution

♦♦♦ Mean mark 18%.
MARKER’S COMMENT: The critical step here is to realise that if Team A beats Team B, then Team B’s one-step dominances become Team A’s two-step dominances.

 

NETWORKS, FUR1 2009 VCAA 8 MC

An undirected connected graph has five vertices.

Three of these vertices are of even degree and two of these vertices are of odd degree.

One extra edge is added. It joins two of the existing vertices.

In the resulting graph, it is not possible to have five vertices that are

A.   all of even degree.

B.   all of equal degree.

C.   one of even degree and four of odd degree.

D.   three of even degree and two of odd degree.

E.   four of even degree and one of odd degree. 

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Show Worked Solution

♦♦♦ Mean mark 25%.


 

matrices-fur1-2009-vcaa-8-mc-answer
 

NETWORKS, FUR1 2011 VCAA 9 MC

An Euler path through a network commences at vertex  and ends at vertex .

Consider the following five statements about this Euler path and network.

•  In the network, there could be three vertices with degree equal to one.

•  The path could have passed through an isolated vertex.

•  The path could have included vertex more than once.

•  The sum of the degrees of vertices and could equal seven.

•  The sum of the degrees of all vertices in the network could equal seven.

How many of these statements are true?

A.   

B.   

C.   

D.   

E.   

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Show Worked Solution

♦♦ Mean mark 30%.

NETWORKS, FUR1 2011 VCAA 7 MC

Andy, Brian and Caleb must complete three activities in total (K, L and M)

The table shows the person selected to complete each activity, the time it will take to complete the activity in minutes and the immediate predecessor for each activity.
 

   

 
All three activities must be completed in a total of 40 minutes.

The instant that Andy starts his activity, Caleb gets a telephone call.

The maximum time, in minutes, that Caleb can speak on the telephone before he must start his allocated activity is

A.    5

B.   13

C.   18

D.   24

E.   34

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Show Worked Solution

♦♦ Mean mark 33%.
MARKER’S COMMENT: Many students incorrectly answered the earliest starting time, C.

NETWORKS, FUR1 2012 VCAA 9 MC

John, Ken and Lisa must work together to complete eight activities, and , in minimum time.

The directed network below shows the activities, their completion times in days, and the order in which they must be completed.

net9

Several activities need special skills. Each of these activities may be completed only by a specified person.

  • Activities and may only be completed by John.
  • Activities and may only be completed by Ken.
  • Activities and may only be completed by Lisa.
  • Activities and may be completed by any one of John, Ken or Lisa.

With these conditions, the minimum number of days required to complete these eight activities is

A.  14

B.  17

C.  20

D.  21

E.  24

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♦♦♦ Mean mark 18%.

 

NETWORKS, FUR1 2012 VCAA 8 MC

networks-fur1-2012-vcaa-8-mc 
 

Eight activities, , must be completed for a project.

The graph above shows these activities and their usual duration in hours.

The duration of each activity can be reduced by one hour.

To complete this project in 16 hours, the minimum number of activities that must be reduced by one hour each is

A.   1

B.   2

C.   3

D.   4

E.   5

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Show Worked Solution

♦♦ Mean mark 30%.

 

NETWORKS, FUR1 2012 VCAA 7 MC

Vehicles from a town can drive onto a freeway along a network of one-way and two-way roads, as shown in the network diagram below.

The numbers indicate the maximum number of vehicles per hour that can travel along each road in this network. The arrows represent the permitted direction of travel.

One of the four dotted lines shown on the diagram is the minimum cut for this network.
  

networks-fur1-2012-vcaa-7-mc-1

 
The maximum number of vehicles per hour that can travel through this network from the town onto the freeway is

A.  

B. 

C. 

D. 

E. 

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Show Worked Solution

♦♦♦ Mean mark 23%.

NETWORKS, FUR1 2013 VCAA 8 MC

   
 

The graph above shows five activities, , that must be completed to finish a project.

The number next to each letter shows the completion time, in hours, for the activity.

Each of the five activities can have its completion time reduced by a maximum of one hour at a cost of $100 per hour.

The least cost to achieve the greatest reduction in the time taken to finish the project is

A.  $100

B.  $200

C.  $300

D.  $400

E.  $500

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Show Worked Solution

♦♦♦ Mean mark 23%.

 

NETWORKS, FUR2 2007 VCAA 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.
 

NETWORKS, FUR2 2007 VCAA 4
 

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

  2. Determine the slack 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 2 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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Show Worked Solution

a.  

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

 

b.   
 
 

  
c.   


  

d.   

 

e.   
   

NETWORKS, FUR2 2007 VCAA 3

As an attraction for young children, a miniature railway runs throughout the new housing estate.

The trains travel through stations that are represented by nodes on the directed network diagram below.

The number of seats available for children, between each pair of stations, is indicated beside the corresponding edge.
 

NETWORKS, FUR2 2007 VCAA 3

 
Cut 1, through the network, is shown in the diagram above.

  1. Determine the capacity of Cut 1.   (1 mark)

  2. Determine the maximum number of seats available for children for a journey that begins at the West Terminal and ends at the East Terminal.   (1 mark)

On one particular train, 10 children set out from the West Terminal.

No new passengers board the train on the journey to the East Terminal.

  1. Determine the maximum number of children who can arrive at the East Terminal on this train.   (1 mark)

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Show Worked Solution

a.  

♦♦ Mean mark for all parts (combined) was 33%.
MARKER’S COMMENT: A common error was counting the edge with “10” in the reverse direction (it should be ignored).

 

b.    networks-fur2-2007-vcaa-3-answer
 
 

 

c.  


  

NETWORKS, FUR1 2014 VCAA 9 MC

A network of tracks connects two car parks in a festival venue to the exit, as shown in the directed graph below.
 

 
The arrows show the direction that cars can travel along each of the tracks and the numbers show each track’s capacity in cars per minute.

Five cuts are drawn on the diagram.

The maximum number of cars per minute that will reach the exit is given by the capacity of

A.  Cut A

B.  Cut B

C.  Cut C

D.  Cut D

E.  Cut E

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Show Worked Solution

♦♦♦ Mean mark 24%.
COMMENT: Note that the “4” is not included in Cut D as it is flowing in the opposite direction.

NETWORKS, FUR1 2014 VCAA 6-7 MC

Consider the following four graphs.
 


 

Part 1

How many of these four graphs have an Eulerian circuit?

A. 

B. 

C. 

D. 

E. 

 

Part 2

How many of these four graphs are planar?

A. 

B. 

C. 

D. 

E. 

Show Answers Only

Show Worked Solution

 

♦♦♦ Mean mark part 2: 7%!
MARKER’S COMMENT: A majority of students chose option A, not understanding that a graph with intersecting edges can still be planar.

 

 

NETWORKS, FUR2 2013 VCAA 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. i. 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. ii. 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
Show Worked Solution
a.   
   
   
♦ Mean mark of all parts (combined) was 41%.

 


  

b.i.   
  

b.ii.   

 

 

Networks, FUR2 2013 VCAA 3_2 Answer1

NETWORKS, FUR2 2009 VCAA 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 ?   (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, , with duration three weeks, is to be added without altering the critical path.

Activity 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 on the network diagram above.   (1 mark)

  2. Activity 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
Show Worked Solution

a.  

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

b.  
  

c.   


  

d.    NETWORKS, FUR2 2009 VCAA 4 Answer

  
e.  

NETWORKS, FUR2 2011 VCAA 4

Stormwater enters a network of pipes at either Dunlop North (Source 1) or Dunlop South (Source 2) and flows into the ocean at either Outlet 1 or Outlet 2.

On the network diagram below, the pipes are represented by straight lines with arrows that indicate the direction of the flow of water. Water cannot flow through a pipe in the opposite direction.

The numbers next to the arrows represent the maximum rate, in kilolitres per minute, at which stormwater can flow through each pipe.

 

NETWORKS, FUR2 2011 VCAA 4_1
 

  1. Complete the following sentence for this network of pipes by writing either the number 1 or 2 in each box.  (1 mark)

NETWORKS, FUR2 2011 VCAA 4_2

  1. Determine the maximum rate, in kilolitres per minute, that water can flow from these pipes into the ocean at Outlet 1 and Outlet 2.  (2 marks)

A length of pipe, show in bold on the network diagram below, has been damaged and will be replaced with a larger pipe.

 

NETWORKS, FUR2 2011 VCAA 4_3
 

  1. The new pipe must enable the greatest possible rate of flow of stormwater into the ocean from Outlet 2.
  2. What minimum rate of flow through the pipe, in kilolitres per minute, will achieve this?  (1 mark)

Show Answers Only


Show Worked Solution

a.  

♦ Mean mark of all parts (combined) was 35%.

 

b.    NETWORKS, FUR2 2011 VCAA 4 Answer

 

 

c.   

NETWORKS, FUR2 2014 VCAA 4

To restore a vintage train, 13 activities need to be completed.

The network below shows these 13 activities and their completion times in hours.
 

NETWORKS, FUR2 2014 VCAA 4
 

  1. Determine the earliest starting time of activity .   (1 mark)

The minimum time in which all 13 activities can be completed is 21 hours.

  1. What is the latest starting time of activity ?   (1 mark)

  2. What is the float time of activity ?   (1 mark)

Just before they started restoring the train, the members of the club needed to add another activity, , to the project.

Activity will take seven hours to complete.

Activity has no predecessors, but must be completed before activity starts.

  1. What is the latest starting time of activity if it is not to increase the minimum completion time of the project?   (1 mark)

Activity  can be crashed by up to four hours at an additional cost of $90 per our.

This may reduce the minimum completion time for the project, including activity .

  1. Determine the least cost of crashing activity to give the greatest reduction in the minimum completion time of the project.   (1 mark)

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Show Worked Solution

a.   

♦ Mean mark for all parts (combined) was 42%.

  
b.  


  

c.  


  

d.   


  


  

e.   

 


 

 

MATRICES, FUR2 2006 VCAA 3

Market researchers claim that the ideal number of bookshops (), sports shoe shops () and music stores () for a shopping centre can be determined by solving the equations

  1. Write the equations in matrix form using the following template.   (1 mark)

     

     

     

     

  2. Do the equations have a unique solution? Provide an explanation to justify your response.   (1 mark)

  3. Write down an inverse matrix that can be used to solve these equations.   (1 mark)

  4. Solve the equations and hence write down the estimated ideal number of bookshops, sports shoe shops and music stores for a shopping centre.   (1 mark)

Show Answers Only
  1.  
  2.  
  3.  
Show Worked Solution
a.   
♦ Mean mark 35% for all parts (combined).

 

b.   

 

 

c.   

 

d. 

MATRICES, FUR2 2007 VCAA 2

To study the life-and-death cycle of an insect population, a number of insect eggs (), juvenile insects () and adult insects () are placed in a closed environment.

The initial state of this population can be described by the column matrix

A row has been included in the state matrix to allow for insects and eggs that die ().

  1. What is the total number of insects in the population (including eggs) at the beginning of the study?   (1 mark)

In this population

    • eggs may die, or they may live and grow into juveniles
    • juveniles may die, or they may live and grow into adults
    • adults will live a period of time but they will eventually die.

In this population, the adult insects have been sterilised so that no new eggs are produced. In these circumstances, the life-and-death cycle of the insects can be modelled by the transition matrix
 


 

  1. What proportion of eggs turn into juveniles each week?   (1 mark)

    1. Evaluate the matrix product     (1 mark)

    2. Write down the number of live juveniles in the population after one week.   (1 mark)

    3. Determine the number of live juveniles in the population after four weeks. Write your answer correct to the nearest whole number.   (1 mark)

    4. After a number of weeks there will be no live eggs (less than one) left in the population.
    5. When does this first occur?   (1 mark)

    6. Write down the exact steady-state matrix for this population.  (1 mark)

  3. If the study is repeated with unsterilised adult insects, eggs will be laid and potentially grow into adults.
  4. Assuming 30% of adults lay eggs each week, the population matrix after one week, , is now given by
  6. where     and  
     

    1. Determine  (1 mark)

    2. This pattern continues. The population matrix after weeks, , is given by
    4. Determine the number of live eggs in this insect population after two weeks.  (1 mark)

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  3.  
Show Worked Solution

a.  
 

b.   
 

c.i.   
   
   

 
c.ii.   
 

c.iii.   
   

 

 

c.iv. 

 

 

c.v.  

 

d.i.  
   

 

♦♦ Mean mark for part (d) was 30%.
d.ii.  

 

MATRICES, FUR2 2011 VCAA 3

A breeding program is started in the wetlands. It is aimed at establishing a colony of native ducks.

The matrix displays the number of juvenile female ducks () and the number of adult female ducks () that are introduced to the wetlands at the start of the breeding program.

  1. In total, how many female ducks are introduced to the wetlands at the start of the breeding program?   (1 mark)

The number of juvenile female ducks () and the number of adult female ducks () in the colony at the end of Year 1 of the breeding program is determined using the matrix equation

In this equation, is the breeding matrix

  1. Determine   (1 mark)

The number of juvenile female ducks () and the number of adult female ducks () in the colony at the end of Year of the breeding program is determined using the matrix equation

The graph below is incomplete because the points for the end of Year 3 of the breeding program are missing.

 

MATRICES, FUR2 2011 VCAA 3

    1. Use the matrices to calculate the number of juvenile and the number of adult female ducks expected in the colony at the end of Year 3 of the breeding program.
    2. Plot the corresponding points on the graph.   (2 marks)

    3. Use matrices to determine the expected total number of female ducks in the colony in the long term.
    4. Write your answer correct to the nearest whole number.   (1 mark)

The breeding matrix assumes that, on average, each adult female duck lays and hatches two female eggs for each year of the breeding program.

If each adult female duck lays and hatches only one female egg each year, it is expected that the duck colony in the wetland will not be self-sustaining and will, in the long run, die out.

The matrix equation

with a different breeding matrix

and the initial state matrix

models this situation.

  1. During which year of the breeding program will the number of female ducks in the colony halve?   (1 mark)

Changing the number of juvenile and adult female ducks at the start of the breeding program will also change the expected size of the colony.

  1. Assuming the same breeding matrix, , determine the number of juvenile ducks and the number of adult ducks that should be introduced into the program at the beginning so that, at the end of Year 2, there are 100 juvenile female ducks and 50 adult female ducks.   (2 marks)

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  3.  
    1.  
      MATRICES, FUR2 2011 VCAA 3 Answer

Show Worked Solution

a.  


 

b.   
   

 

c.i.    MATRICES, FUR2 2011 VCAA 3 Answer


 

c.ii.  

 


 

d.  

♦♦♦ Mean mark of parts (d)-(e) was 20%.


 

e.   
   

 

MATRICES, FUR2 2013 VCAA 2

10 000 trout eggs, 1000 baby trout and 800 adult trout are placed in a pond to establish a trout population.

In establishing this population

    • eggs () may die () or they may live and eventually become baby trout ()
    • baby trout () may die () or they may live and eventually become adult trout ()
    • adult trout () may die () or they may live for a period of time but will eventually die.

From year to year, this situation can be represented by the transition matrix , where
 


 

  1. Use the information in the transition matrix to
    1. determine the number of eggs in this population that die in the first year.   (1 mark)

    2. complete the transition diagram below, showing the relevant percentages.   (2 marks)

       

          Matrices, FUR2 2013 VCAA 2_a

The initial state matrix for this trout population, , can be written as
 


 

Let  represent the state matrix describing the trout population after years.

  1. Using the rule  , determine each of the following.

     

    1.    (1 mark)

    2. the number of adult trout predicted to be in the population after four years   (1 mark)

  2. The transition matrix predicts that, in the long term, all of the eggs, baby trout and adult trout will die.
    1. How many years will it take for all of the adult trout to die (that is, when the number of adult trout in the population is first predicted to be less than one)?   (1 mark)

    2. What is the largest number of adult trout that is predicted to be in the pond in any one year?   (1 mark)

  3. Determine the number of eggs, baby trout and adult trout that, if added to or removed from the pond at the end of each year, will ensure that the number of eggs, baby trout and adult trout in the population remains constant from year to year.   (2 marks)

The rule    that was used to describe the development of the trout in this pond does not take into account new eggs added to the population when the adult trout begin to breed.

  1. To take breeding into account, assume that 50% of the adult trout lay 500 eggs each year.
  2. The matrix describing the population after one year, , is now given by the new rule
  4. where      
    1. Use this new rule to determine .   (1 mark)

  5. This pattern continues so that the matrix describing the population after years, , is given by the rule
  6.      
     

    1. Use this rule to determine the number of eggs in the population after two years   (2 marks)

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a.i.   

a.ii. 

b.i.   

b.ii.

c.i.   

c.ii.  

d.   

e.i.   

e.ii.   

Show Worked Solution

a.i.   


 

MARKER’S COMMENT: A 100% cycle drawn at was a common omission. Do not draw loops and edges of 0%!
a.ii.   

Matrices-FUR2-2013-VCAA-2_a Answer
b.i.   
   

 

b.ii.   
   

 

 

c.i.   


 


 

c.ii.   


 


 

d.   

 

 

e.i.   
   
   
   

 

e.ii.   
   

       
   

MATRICES, FUR1 2008 VCAA 7-9 MC

A large population of mutton birds migrates each year to a remote island to nest and breed. There are four nesting sites on the island, A, B, C and D.

Researchers suggest that the following transition matrix can be used to predict the number of mutton birds nesting at each of the four sites in subsequent years. An equivalent transition diagram is also given.
 

     VCAA MATRICES FUR2 2008 7i

Part 1

Two thousand eight hundred mutton birds nest at site C in 2008.

Of these 2800 mutton birds, the number that nest at site A in 2009 is predicted to be

A.   

B.   

C. 

D. 

E. 

 

Part 2

This transition matrix predicts that, in the long term, the mutton birds will

A.  nest only at site A.

B.  nest only at site B.

C.  nest only at site A and C.

D.  nest only at site B and D.

E.  continue to nest at all four sites.

 

Part 3

Six thousand mutton birds nest at site B in 2008.

Assume that an equal number of mutton birds nested at each of the four sites in 2007. The same transition matrix applies.

The total number of mutton birds that nested on the island in 2007 was

A. 

B. 

C. 

D. 

E. 

Show Answers Only

Show Worked Solution

 

 

♦♦♦ Mean mark 25%.

 
 

 

MATRICES, FUR1 2010 VCAA 9 MC

Robbie completed a test of four multiple-choice questions.

Each question had four alternatives, A, B, C or D.

Robbie randomly guessed the answer to the first question.

He then determined his answers to the remaining three questions by following the transition matrix
 


 

Which of the following statements is true?

A.  It is impossible for Robbie to give the same answer to all four questions.

B.  Robbie would always give the same answer to the first and fourth questions

C.  Robbie would always give the same answer to the second and third questions.

D.  If Robbie answered A for question one, he would have answered B for question two

E.  It is possible that Robbie gave the same answer to exactly three of the four questions.

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Show Worked Solution

♦♦ Mean mark 29%.

 

 

 

MATRICES, FUR1 2010 VCAA 6 MC

Vince, Nev and Rani all service office equipment.

The matrix shows the time that it takes (in minutes) for each of Vince (V), Nev (N) and Rani (R) to service a photocopier (P) a fax machine (F) and a scanner (S).
 


 

The matrix  below displays the number of photocopiers, fax machines and scanners to be serviced in three schools, Alton (A), Borton (B) and Carlon (C).
 


 

A matrix that displays the time that it would take each of Vince, Nev and Rani, working alone, to service the photocopiers, fax machines and scanners in each of the three schools is

A.  B. 
       
C.  D. 
       
E.     

 

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Show Worked Solution

♦♦♦ Mean mark 19%.
MARKER’S COMMENT: Most students incorrectly calculated matrix . Make sure you know why this is incorrect!
 

 

MATRICES, FUR1 2011 VCAA 9 MC

Matrix  is a 3 x 3  matrix. Seven of the elements in matrix  are zero.

Matrix  contains six elements, none of which are zero.

Assuming the matrix product    is defined, the minimum number of zero elements in the product matrix    is

A.   

B.   

C.   

D.   

E.   

Show Answers Only

Show Worked Solution


 


 

 


 

 

MATRICES, FUR2 2015 VCAA 3

A new model for the number of students in the school after each assessment takes into account the number of students who are expected to leave the school after each assessment.

After each assessment, students are classified as beginner (), intermediate (), advanced () or left the school ().

Let matrix be the transition matrix for this new model.

Matrix , shown below, contains the percentages of students who are expected to change their ability level or leave the school after each assessment.
  

  1. An incomplete transition diagram for matrix is shown below.
  2. Complete the transition diagram by adding the missing information.   (2 marks)


     

    Matrices, FUR2 2015 VCAA 3

The number of students at each level, immediately before the first assessment of the year, is shown in matrix below.

Matrix , repeated below, contains the percentages of students who are expected to change their ability level or leave the school after each assessment.

  1. What percentage of students is expected to leave the school after the first assessment?   (1 mark)

  2. How many advanced-level students are expected to be in the school after two assessments.
  3. Write your answer correct to the nearest whole number.   (1 mark)

  4. After how many assessments is the number of students in the school, correct to the nearest whole number, first expected to drop below 50?   (1 mark)

Another model for the number of students in the school after each assessment takes into account the number of students who are expected to join the school after each assessment.

Let be the state matrix that contains the number of students in the school immediately after assessments.

Let be the matrix that contains the number of students who join the school after each assessment. 

Matrix is shown below.

The expected number of students in the school after assessments can be determined using the matrix equation

where


 

  1. Consider the intermediate-level students expected to be in the school after three assessments.
  2. How many are expected to become advanced-level students after the next assessment?
  3. Write your answer correct to the nearest whole number.   (2 marks)

Show Answers Only
  1.  
Show Worked Solution
a.   

 

b.  

 

♦♦ Mean mark of parts (b)-(e) was 32%.


 

c.  

 

 

d.  


 

MARKER’S COMMENT: Understand why the common error is incorrect!
e.  
   
 
   

 

MATRICES, FUR2 2012 VCAA 3

When a new industrial site was established at the beginning of 2011, there were 350 staff at the site. 

The staff comprised 100 apprentices (), 200 operators () and 50 professionals (). 

At the beginning of each year, staff can choose to stay in the same job, move to a different job at the site or leave the site ().

The number of staff in each category at the beginning of 2011 is given in the matrix

The transition diagram below shows the way in which staff are expected to change their jobs at the site each year.

Matrices, FUR2 2012 VCAA 3

  1. How many staff at the site are expected to be working in their same jobs after one year?   (1 mark)

The information in the transition diagram has been used to write the transition matrix .

  1. Explain the meaning of the entry in the fourth row and fourth column of transition matrix .   (1 mark)

If staff at the site continue to change their jobs in this way, the matrix  will contain the number of apprentices (), operators (), professionals () and staff who leave the site () at the beginning of the th year.

  1. Use the rule    to find

     

    1.    (1 mark)

    2. the expected number of operators at the site at the beginning of 2013   (1 mark)

    3. the beginning of which year the number of operators at the site first drops below 30   (1 mark)

    4. the total number of staff at the site in the longer term.   (1 mark)

Suppose the manager decides to bring 30 new apprentices, 20 new operators and 10 new professionals to the site at the beginning of each year.

The matrix will then be given by

  where     and  

  1. Find the expected number of operators at the site at the beginning of 2013.   (2 marks) 

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a. 

♦ Mean mark for parts (a)-(d) (combined) was 37%.


 

b.   

MARKER’S COMMENT: Most students couldn’t explain the meaning of the 1.0 figure in this context.

 

c.i.  
   
   

 

c.ii.  
   

 

 

c.iii.  
   
 
   

 

MARKER’S COMMENT: “In the 10th year” recieved no marks – make sure you answer with specific years when actual years are used.

 
c.iv.  


 

d.  
   
     
 
   
MARKER’S COMMENT: A common error was  . Know why this is not correct!

 

GRAPHS, FUR2 2012 VCAA 3

A company repairs phones and laptops.

Let   be the number of phones repaired each day

  be the number of laptops repaired each day.

It takes 35 minutes to repair a phone and 50 minutes to repair a laptop. 

The constraints on the company are as follows.

Constraint 1    

Constraint 2    

Constraint 3    

Constraint 4    

  1. Explain the meaning of Constraint 3 in terms of the time available to repair phones and laptops.  (1 mark)
  2. Constraint 4 describes the maximum number of phones that may be repaired relative to the number of laptops repaired.

     

    Use this constraint to complete the following sentence.

     

    For every ten phones repaired, at most _______ laptops may be repaired.  (1 mark)

The line  is drawn on the graph below.

GRAPHS, FUR2 2012 VCAA 3

  1. Draw the line    on the graph.  (1 mark)
  2. Within Constraints 1 to 4, what is the maximum number of laptops that can be repaired each day?  (1 mark)
  3. On a day in which exactly nine laptops are repaired, what is the maximum number of phones that can be repaired?  (1 mark)

The profit from repairing one phone is $60 and the profit from repairing one laptop is $100.

    1. Determine the number of phones and the number of laptops that should be repaired each day in order to maximise the total profit.  (2 marks)
    2. What is the maximum total profit per day that the company can obtain from repairing phones and laptops?  (1 mark) 
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a.  

 

b.  

 

 

 

c.    GRAPHS, FUR2 2012 VCAA 3 Answer

 

d.  

♦♦ Mean mark of parts (c) – (f) combined was 32%.

 

e.  

 

 

f.i.   

 

 

 

f.ii.  

CORE*, FUR2 2013 VCAA 4

Hugo took out a reducing balance loan of $25 000 to compete in road races overseas.

Interest was charged at a rate of 12% per annum compounding quarterly.

His loan is to be repaid fully in four years with equal quarterly payments.

After two years, how much of the $25 000 will Hugo have repaid?

Write your answer, correct to the nearest dollar.   (2 marks)

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♦♦ Mean mark 25%.
MARKER’S COMMENT: Very few students showed any TVM input or other working in this question. Labelling steps to organise thoughts is a feature of the most successful answers.

 

 

 

CORE*, FUR2 2013 VCAA 3

Hugo paid $7500 for a second bike under a hire-purchase agreement.

A flat interest rate of 8% per annum was charged.

He will fully repay the principal and the interest in 24 equal monthly instalments.

  1. Determine the monthly instalment that Hugo will pay.

     

    Write your answer in dollars, correct to the nearest cent.   (2 marks)

  2. Find the effective rate of interest per annum charged on this hire-purchase agreement.

     

    Write your answer as a percentage, correct to two decimal places.   (1 mark)

  3. Explain why the effective interest rate per annum is higher than the flat interest rate per annum.   (1 mark)

  4. The value of his second bike, purchased for $7500, will be depreciated each year using the reducing balance method of depreciation.

     

    One year after it was purchased, this bike was valued at $6375.

     

    Determine the value of the bike five years after it was purchased. 

     

    Write your answer, correct to the nearest dollar.   (2 marks)

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a.   
   
   
 
 

 

♦ Mean mark of all parts combined was 37%.
b.   
   
   

  
c.   


  

d.   
   
   

GRAPHS, FUR2 2013 VCAA 4

The school group may hire two types of camp sites: powered sites and unpowered sites.

Let   be the number of powered camp sites hired

  be the number of unpowered camp sites hired.

Inequality 1 and inequality 2 give some restrictions on and .

inequality 1   

inequality 2   

There are 48 students to accommodate in total.

A powered camp site can accommodate up to six students and an unpowered camp site can accommodate up to four students.

Inequality 3 gives the restrictions on and based on the maximum number of students who can be accommodated at each type of camp site.

inequality 3  

  1. Write down the values of and in inequality 3.  (1 mark)

     


          Graphs and relations, FUR2 2013 VCAA 4

School groups must hire at least two unpowered camp sites for every powered camp site they hire.

  1. Write this restriction in terms of and as inequality 4.  (1 mark)

     

The graph below shows the three lines that represent the boundaries of inequalities 1, 3 and 4.

GRAPHS, FUR2 2013 VCAA 4

  1. On the graph above, show the points that satisfy inequalities 1, 2, 3 and 4.  (1 mark)
  2. Determine the minimum number of camp sites that the school would need to hire.  (1 mark)
  3. The cost of each powered camp site is $60 per day and the cost of each unpowered camp site is $30 per day.

     

    1. Find the minimum cost per day, in total, of accommodating 48 students.  (1 mark)

School regulations require boys and girls to be accommodated separately.

The girls must all use one type of camp site and the boys must all use the other type of camp site.

  1. Determine the minimum cost per day, in total, of accommodating the 48 students if there is an equal number of boys and girls.  (1 mark)
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  3.  
    GRAPHS, FUR2 2013 VCAA 4 Answer
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a.   

 

b.  

 

c.    GRAPHS, FUR2 2013 VCAA 4 Answer

 

d.  

♦♦ Mean mark of parts (c)-(e) combined was 22%.

 

 

e.i.  

 

 

e.ii.   

 

GEOMETRY, FUR2 2013 VCAA 5

Daniel threw a javelin a distance of 68.32 m on a bearing of 057° on his first throw.

On his second throw from the same point, he threw the javelin a distance of 72.51 m.

The second throw landed at a point on a bearing of 125°, measured from the point where the first throw landed.

Determine the distance between the point where Daniel’s first throw landed and the point where his second throw landed.

Write your answer in metres, correct to one decimal place.  (2 marks) 

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GEOMETRY, FUR2 2013 VCAA 5 Answer

 
 

 
 

GEOMETRY, FUR2 2014 VCAA 4

A chicken escapes into a neighbouring field through an open gate.

The chicken’s owner is 50 m due north of the gate, searching for the chicken.

The chicken is 40 m from the gate on a bearing of 295°.

What is the bearing of the chicken from its owner?

Write your answer, correct to the nearest degree.  (2 marks)

 

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♦♦ Mean mark 28%.
 

 

 
 

 
 

 

GEOMETRY, FUR2 2013 VCAA 4

Competitors in the intermediate division of the discus use a smaller discus than the one used in the senior division, but of a similar shape. The total surface area of each discus is given below.
 

GEOMETRY, FUR2 2013 VCAA 4
 

By what value can the volume of the intermediate discus be multiplied to give the volume of the senior discus?  (2 marks)

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♦♦ Mean mark well below 50% (exact data unavailable.
MARKER’S COMMENT: A common error was to use a ratio where which was obviously wrong since the volume needed to be scaled up.