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Quadratic, EXT1 2016 HSC 14c

The point   lies on the parabola with the equation  .

The tangent to the parabola at meets the directrix at .

The normal to the parabola at meets the vertical line through at the point , as shown in the diagram.

 

ext1-2016-hsc-q14

  1. Show that the point has coordinates .  (1 mark)
  2. Show that the locus of lies on another parabola .  (3 marks)
  3. State the focal length of the parabola .  (1 mark)

It can be shown that the minimum distance between and occurs when the normal to at is also the normal to at .  (Do NOT prove this.)

  1. Find the values of so that the distance between and is a minimum.  (2 marks)
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Show Worked Solution

i.    

 

ii.   

♦♦ Mean mark 32%.
MARKER’S COMMENT: Normal equation is found in the Reference Sheet. Save time by using it!

 

 

 
 
 
 

 

 

iii.  

♦♦♦ Mean mark 15%.
 

 

 

iv.  

♦♦♦ Mean mark 11%.

 

 

Binomial, EXT1 2016 HSC 14b

Consider the expansion of  , where is a positive integer.

  1. Show that  .  (1 mark)
  2. Show that  .  (1 mark)
  3. Hence, or otherwise, show that  .  (2 marks)
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Show Worked Solution

i.    

 

ii.  

 

iii. 

♦♦♦ Mean mark 14%.
 

 

 

 

Plane Geometry, EXT1 2016 HSC 13c

The circle centred at has a diameter . From the point outside the circle the line segments and are drawn meeting the circle at and respectively, as shown in the diagram. The chords and meet at . The line segment produced meets the diameter at .

ext1-2016-hsc-q13_1

Copy or trace the diagram into your writing booklet.

  1. Show that is a cyclic quadrilateral.  (2 marks)
  2. Hence, or otherwise, prove that is perpendicular to .  (2 marks)
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Show Worked Solution
i.  

 


 

ii.  

♦♦♦ Mean mark 14%.

Calculus, 2ADV C3 2016 HSC 16b

Some yabbies are introduced into a small dam. The size of the population, , of yabbies can be modelled by the function

where is the time in months after the yabbies are introduced into the dam.

  1. Show that the rate of growth of the size of the population is
  2. .  (2 marks)

  3. Find the range of the function , justifying your answer.  (2 marks)

  4. Show that the rate of growth of the size of the population can be written as
  5. .  (1 mark)

  6. Hence, find the size of the population when it is growing at its fastest rate.  (2 marks)

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Show Worked Solution
i.  
 
   
   

 

ii.  

♦♦♦ Mean mark (ii) 21%.

 

iii. 

♦♦♦ Mean mark (iii) 18%.

 
 

 

iv.  

♦♦♦ Mean mark (iv) 14%.

 

hsc-2016-16bi

Proof, EXT2 P2 2016 HSC 16c

In a group of people, each has one hat, giving a total of different hats. They place their hats on a table. Later, each person picks up a hat, not necessarily their own.

A situation in which none of the people picks up their own hat is called a derangement.

Let    be the number of possible derangements.

  1. Tom is one of the people. In some derangements Tom finds that he and one other person have each other's hat.

     

    Show that, for  , the number of such derangements is    (1 mark)

  2. By also considering the remaining possible derangements, show that, for  

      (2 marks)

  3. Hence, show that  ,  for    (1 mark)

  4. Given    and  ,  deduce that  ,  for    (1 mark)

  5. Prove by mathematical induction, or otherwise, that for all integers    (2 marks)

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Show Worked Solution

i.   

♦♦♦ Mean mark 22%.

 

ii.   

♦♦♦ Mean mark 6%.

 

 

 

♦♦ Mean mark part (iii) 37%.
iii.   
   

 

 

iv.  

♦♦♦ Mean mark 1%!

 

 

v.  

♦♦♦ Mean mark 13%.

 

 

Complex Numbers, EXT2 N2 2016 HSC 16b

  1. The complex numbers    and    form the vertices of an equilateral triangle in the Argand diagram.

     

    Show that    (2 marks)

  2. Give an example of non-zero complex numbers    and  , so that    and    form the vertices of an equilateral triangle in the Argand diagram.  (1 mark)

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Show Worked Solution

i.  

♦♦ Mean mark 23%.

 

ext2-2016-hsc-16b-answer

 

 

ii.   

♦♦ Mean mark 33%.
 
 

Complex Numbers, EXT2 N2 2016 HSC 16a

  1. The complex numbers    and  , where    and  , satisfy

       

    By considering the real and imaginary parts of      , or otherwise, show that    and   form the vertices of an equilateral triangle in the Argand diagram.  (3 marks)

  2. Hence, or otherwise, show that if the three non-zero complex numbers    and    satisfy

     

     
          AND 

    then they form the vertices of an equilateral triangle in the Argand diagram.      (2 marks)

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Show Worked Solution
i.   
 

 

♦♦♦ Mean mark 25%.
STRATEGY: A clear graphical image simplifies both parts of this question significantly.

 

 
ext2-2016-hsc-16a-answer2 

 

 

ii.  

 

 

♦♦♦ Mean mark 5%.

 

 
ext2-2016-hsc-16a-answer3 
 

Polynomials, EXT2 2016 HSC 15c

  1. Use partial fractions to show that

      (2 marks)

  2. Suppose that for a positive integer

     

    Show that    (3 marks)

  3. Hence, or otherwise, find the limiting sum of

     

      (2 marks)

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i.   

 
 

 

 

ii.  

 

 
   
 
 
 
 

 

iii.   

 

 

 

Algebra, STD2 A4 2016 HSC 29b

The mass kg of a baby pig at age days is given by    where is a constant. The graph of this equation is shown.
 

2ug-2016-hsc-q29_1

  1. What is the value of ?   (1 mark)

  2. What is the daily growth rate of the pig’s mass? Write your answer as a percentage.   (1 mark)

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Show Worked Solution

i.   

♦ Mean mark (i) 48%.
♦♦♦ Mean mark part (ii) 6%!

 
ii.   

Probability, STD2 S2 2016 HSC 23 MC

A group of 485 people was surveyed. The people were asked whether or not they smoke. The results are recorded in the table.
 

A person is selected at random from the group.

What is the approximate probability that the person selected is a smoker OR is male?

  1. 33%
  2. 18%
  3. 68%
  4. 87%
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Show Worked Solution


 

♦♦ Mean mark 34%.

Algebra, 2UG 2016 HSC 18 MC

The value of varies directly with the square of .

It is known that    when  .

What is the value of when  ?

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Show Worked Solution

♦♦♦ Mean mark 22%.
STRATEGY: A well structured answer is important in solving this question type. See solution.

 

 

 

Statistics, STD2 S1 2016 HSC 7 MC

Which set of data is classified as categorical and nominal?

  1. blue, green, yellow
  2. small, medium, large
  3. 5.2 cm, 6 cm, 7.21 cm
  4. 4 people, 5 people, 9 people 
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Show Worked Solution

♦♦ Mean mark 26%.

Probability, MET1 2007 VCAA 6

Two events, and , from a given event space, are such that    and  .

  1. Calculate    when  .  (1 mark)

  2. Calculate    when and are mutually exclusive events.  (1 mark)

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Show Worked Solution

a.   

♦♦ Mean mark (a) 31%.
MARKER’S COMMENTS: Students who drew a Venn diagram or Karnaugh map were the most successful.

met1-2007-vcaa-q6-answer3

 
 

 

♦♦ Mean mark (b) 23%.

b.    met1-2007-vcaa-q6-answer4

Functions, MET1 2008 VCAA 10

Let  .

  1. Find the rule and domain of the inverse function  .   (2 marks)

  2. On the axes provided, sketch the graph of    for its maximal domain.   (1 mark)


     

    met1-2008-vcaa-q2
     

  3. Find    in the form    where , and are real constants.   (2 marks)

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  2.  
    met1-2008-vcaa-q10-answer
Show Worked Solution

a.  

 

 


  

b.  

♦ Mean mark part (b) 19%.
MARKER’S COMMENT: Few students were aware that a function of its own inverse function is the line    over the appropriate domain.

met1-2008-vcaa-q10-answer

 

♦ Mean mark 34%.
c.   
 
   
   
   

Calculus, MET1 2011 VCAA 10

The figure shown represents a wire frame where is a convex quadrilateral. The point is on line segment with   and  , where is a positive constant.

Let    where  

 vcaa-2011-meth-10a

  1. Find and in terms of and .   (2 marks)

  2. Find the length, cm, of the wire in the frame, including length , in terms of and .   (1 mark)

  3. Find  , and hence show that    when  .   (2 marks)

  4. Find the maximum value of if  .   (1 mark)

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a.  

 

b.  
   
   

 

c.  

♦ Mean mark 35%.

,

 

d.   

♦♦♦ Mean mark 5%.

vcaa-2011-meth-10ai

 
 
 

Functions, MET1 2011 VCAA 4

If the function has the rule    and the function has the rule  

  1.  find integers and such that    (2 marks)

  2.  state the maximal domain for which  is defined.   (2 marks)

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a.  
   
   

 
 

 

b.   

♦♦♦ Mean mark 13%.
MARKER’S COMMENT: “Very poorly answered” with a common response of that ignored the information from part (a).


 

 vcaa-2011-meth-4ii

Calculus, MET1 2012 VCAA 10

Let  ,  where is a positive rational number.

  1.  i. Find, in terms of , the -coordinate of the stationary point of the graph of     (2 marks)

  2. ii. State the values of such that the -coordinate of this stationary point is a positive number.   (1 mark)

  3. For a particular value of , the tangent to the graph of    at    passes through the origin.
  4. Find this value of .   (3 marks)

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  1.  i.
  2. ii.
Show Worked Solution

a.i.  

 

 

♦♦♦ Part (a.ii.) mean mark 18%.
  ii.   
 
 
 

 

b.  

♦♦ Mean mark (b) 33%.
MARKER’S COMMENT: Many confused with the more common use of for gradient in  .

 

 

 

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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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. 

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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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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
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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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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. 

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