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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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Show Worked Solution

a.  

 

b.  
   
   

 

c.  

♦ Mean mark 35%.

,

 

d.   

♦♦♦ Mean mark 5%.

vcaa-2011-meth-10ai

 
 
 

Calculus, MET1 2011 VCAA 9

Parts of the graphs of the functions

are shown in the diagram below.

The graphs intersect when    and when  

vcaa-2011-meth-9a

The area of the shaded region is 64.

Find the value of and the value of   (4 marks)

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Show Worked Solution

♦ Mean mark 41%.
MARKER’S COMMENT: Few students correctly used the intersection to achieve the final answer.

 

 

 

Probability, MET1 2011 VCAA 7

A biased coin tossed three times. The probability of a head from a toss of this coin is

  1. Find, in terms of , the probability of obtaining

    1. three heads from the three tosses  (1 mark)

    2. two heads and a tail from the three tosses.  (1 mark)

  2. If the probability of obtaining three heads equals the probability of obtaining two heads and a tail, find .  (2 marks)

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Show Worked Solution

a.i.   
   

 

  ii.  

♦ Part (a)(ii) mean mark 41%.

 

b.  

♦ Mean mark 48%.
MARKER’S COMMENT: Many students incorrectly assumed could not be zero.

Algebra, MET1 2011 VCAA 6

Consider the simultaneous linear equations

where is a real constant.

  1. Find the value of for which there are infinitely many solutions.  (3 marks)
  2. Find the values of for which there is a unique solution.  (1 mark)
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Show Worked Solution

a.  

♦ Mean mark 39%.
MARKER’S COMMENT: A number of solutions are possible here, including using the determinant.

 

   

 

 
 
 
 
   

 

 

b.  

♦♦ Mean mark 33%.

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.
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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  .

 

 

 

Functions, MET1 2012 VCAA 6

The graphs of  ,  where is a real constant, have a point of intersection at  

  1. Find the value of .  (2 marks)

  2. If  , find the -coordinate of the other point of intersection of the two graphs.  (1 mark)

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a.  

 

b.  
 
 

Probability, MET1 2012 VCAA 4

On any given day, the number of telephone calls that Daniel receives is a random variable with probability distribution given by

vcaa-2012-meth-4

  1. Find the mean of .   (2 marks)

  2. What is the probability that Daniel receives only one telephone call on each of three consecutive days?   (1 mark)

  3. Daniel receives telephone calls on both Monday and Tuesday.
  4. What is the probability that Daniel receives a total of four calls over these two days?   (3 marks)

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Show Worked Solution

a.   
   
   

 

b.  

MARKER’S COMMENT: Many students understood that the calculation of 0.2³ was required but were not able evaluate correctly.

 

c.  

♦ Mean mark 36%.

Calculus, MET1 2013 VCAA 10

Let  

A right-angled triangle has vertex at the origin, vertex on the -axis and vertex on the graph of , as shown. The coordinates of are
 

 vcaa-2013-meth-10

  1. Find the area, , of the triangle in terms of .   (1 mark)

  2. Find the maximum area of triangle and the value of for which the maximum occurs.   (3 marks)

  3. Let be the point on the graph of on the -axis and let be the point on the graph of with the -coordinate .

     

    Find the area of the region bounded by the graph of and the line segment .   (3 marks)

     

    vcaa-2013-meth-10i

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  2. ²
  3. ²
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a.  
   
 

 

b.  

♦ Mean mark 35%.
 

²

 

c.  

♦♦ Mean mark 32%.


 

 
 
 

 

vcaa-2013-meth-10ii

 
 
 
  ²

Functions, MET1 2013 VCAA 9

The graph of  , is shown below. The graph intersects the -axis where  

vcaa-2013-meth-9

Find the value of   (1 mark)

Note: other parts of this question are out of the syllabus and not included.

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Show Worked Solution

♦ Mean mark 50%.

Probability, MET1 2013 VCAA 7

The probability distribution of a discrete random variable, , is given by the table below

vcaa-2013-meth-7

  1. Show that    or  .   (3 marks)

  2. Let  .

    1. Calculate .   (2 marks)

    2. Find  .   (1 mark)

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a.   

 

b.i.  
   
   
   

 

♦♦ Part (b)(ii) mean mark 32%.

  ii.  
   
   

Calculus, MET1 2013 VCAA 6

Let  ,  where is a real constant.

The average value of on the interval    is  

Find all possible values of    (3 marks)

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Show Worked Solution

♦ Mean mark 39%.
MARKER’S COMMENT: “Far too many” students misunderstood “average value” in this context.

 

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
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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

Probability, MET1 2014 VCAA 9

Sally aims to walk her dog, Mack, most mornings. If the weather is pleasant, the probability that she will walk Mack is , and if the weather is unpleasant, the probability that she will walk Mack is .

Assume that pleasant weather on any morning is independent of pleasant weather on any other morning.

  1. In a particular week, the weather was pleasant on Monday morning and unpleasant on Tuesday morning.
  2. Find the probability that Sally walked Mack on at least one of these two mornings.  (2 marks)

  3. In the month of April, the probability of pleasant weather in the morning was .
  4.  i. Find the probability that on a particular morning in April, Sally walked Mack.  (2 marks)

  5. ii. Using your answer from part b.i., or otherwise, find the probability that on a particular morning in April, the weather was pleasant, given that Sally walked Mack that morning.  (2 marks)

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a.   
   
   

 

b.i.  
 

met1-2014-vcaa-q9-answer1 
 

 

 

♦ Part (b)(ii) mean mark 38%.

b.ii.   
   
   
   

Probability, MET1 2014 VCAA 8

A continuous random variable, , has a probability density function given by
 


 

The median of is  .

  1. Determine the value of  .  (2 marks)
  2. The value of is a number greater than 1.

     

    Find .  (2 marks)

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a.   
 
 
 
 
 

 

 

b.  

♦ Part (b) mean mark 38%.
MARKER’S COMMENT: A common error was assuming obtained in part (a) was equivalent to .
 
 
 
 

Calculus, MET1 2014 VCAA 5

Consider the function  ,  .

  1. Find the coordinates of the stationary points of the function.   (2 marks)

  2. On the axes  below, sketch the graph of .

     

    Label any end points with their coordinates.   (2 marks)

     

     
        met1-2014-vcaa-q5

     

  3. Find the area enclosed by the graph of the function and the horizontal line given by  .   (3 marks)

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  2.  
    met1-2014-vcaa-q5-answer3
  3. ²
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a.   
   
 


 

 

b.    met1-2014-vcaa-q5-answer3

 

♦ Mean mark (c) 48%.
c.    met1-2014-vcaa-q5-answer4

 
 
 
   
²

 

 
 
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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
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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.   

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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.   
 
 
 
 

Probability, MET1 2015 VCAA 8

For events  and  from a sample space,  and  .

  1. Calculate  .   (1 mark)

  2. Calculate  , where denotes the complement of .   (1 mark)

  3. If events and are independent, calculate  .   (1 mark)

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a.  

 

b.    met1-2015-vcaa-q8-answer
 
 

 

c.  

♦♦ Mean mark 28%.
MARKER’S COMMENT: A lack of understanding of independent events was clearly evident.

 

 

Probability, MET1 2015 VCAA 6

Let the random variable be normally distributed with mean 2.5 and standard deviation 0.3

Let be the standard normal random variable, such that  .

  1. Find such that  .  (1 mark)

  2. Using the fact that, correct to two decimal places,  , find  .
  3. Write the answer correct to two decimal places.  (2 marks) 

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Show Worked Solution
♦ Part (a) mean mark 50%.

a.   
   
   

 

♦ Part (b) mean mark 48%.
MARKER’S COMMENT: Students who drew a diagram of a “normal” curve with relevent shaded areas were more successful.

b.    met1-2015-vcaa-q6-answer

Calculus, MET1 2015 VCAA 4

Consider the function  .

  1. Find the coordinates of the stationary points of the function.   (2 marks)

The rule for   can also be expressed as  .

  1. On the axes below, sketch the graph of  , clearly indicating axis intercepts and turning points.

     

    Label the end points with their coordinates.   (2 marks)


      

    met1-2015-vcaa-q4
     

  2. Find the average value of   over the interval    (2 marks)

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  2.  

     

    met1-2015-vcaa-q4-answer

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a.  

 

 

 

b.    met1-2015-vcaa-q4-answer
♦ Part (c) mean mark 50%.
MARKER’S COMMENT: Most students recalled the average value definition but then did not integrate correctly.

 

c.   
   
   
   
   

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 2006 VCAA 3

The company prepares for this expenditure by establishing three different investments.

  1. $7000 is invested at a simple interest rate of 6.25% per annum for eight years.
  2. Determine the total value of this investment at the end of eight years.   (2 marks)
  3. $10 000 is invested at an interest rate of 6% per annum compounding quarterly for eight years.
  4. Determine the total value of this investment at the end of eight years.
  5. Write your answer correct to the nearest dollar.   (1 mark)
  6. $500 is deposited into an account with an interest rate of 6.5% per annum compounding monthly.
  7. Deposits of $200 are made to this account on the last day of each month after interest has been paid.
  8. Determine the total value of this investment at the end of eight years.
  9. Write your answer correct to the nearest dollar.   (1 mark) 

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a.   
   
   

  


    

b.  

 

  


  

c.   

   

CORE*, FUR2 2006 VCAA 1

A company purchased a machine for $60 000.

For taxation purposes the machine is depreciated over time.

Two methods of depreciation are considered.

  1. Flat rate depreciation

    The machine is depreciated at a flat rate of 10% of the purchase price each year.

    i.      By how many dollars will the machine depreciate annually?   (1 mark)

    ii.  Calculate the value of the machine after three years.   (1 mark)

  2. iii. After how many years will the machine be $12 000 in value?   (1 mark)

  3. Reducing balance depreciation

    The value, , of the machine after years is given by the formula .

    i.      By what percentage will the machine depreciate annually?   (1 mark)

    ii.  Calculate the value of the machine after three years.   (1 mark)

  4. iii. At the end of which year will the machine's value first fall below $12 000?   (1 mark)

  1. At the end of which year will the value of the machine first be less using flat rate depreciation than it will be using reducing balance depreciation?  (2 marks)

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  1. i.   
    ii. 
    iii.
  2. i.    
    ii. 
    iii.
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a.i.   
   

a.ii.  

 

a.iii.  

 

 

b.i.   
   


  

b.ii.  

 

 

b.iii.  

  


  

c.   

BUSINESS, FUR2 2006 VCAA 1 Answer

CORE, FUR2 2007 VCAA 4

The books in Khan's office are valued at $10 000.

  1. Calculate the value of these books after five years if they are depreciated by 12% per annum using the reducing balance method. Write your answer correct to the nearest dollar.   (1 mark)

Khan believes his books should be valued at $4000 after five years.

  1. Determine the annual reducing balance depreciation rate that will produce this value. Write your answer as a percentage correct to one decimal place.   (2 marks) 
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Show Worked Solution

a.  

   
b.  

 
 

  

 

CORE*, FUR2 2007 VCAA 3

Khan paid $900 for a fax machine.

This price includes 10% GST (goods and services tax).

  1. Determine the price of the fax machine before GST was added. Write your answer correct to the nearest cent.  (1 mark)

  2. Khan will depreciate his $900 fax machine for taxation purposes.
  3. He considers two methods of depreciation.
  4. Flat rate depreciation
  5. Under flat rate depreciation the fax machine will be valued at $300 after five years.
    1. Calculate the annual depreciation in dollars.   (1 mark)
    2. ii. Determine the value of the fax machine after five years.   (1 mark)

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  2. i.
    ii.  

Show Worked Solution

a.  

MARKER’S COMMENT: Reverse GST questions continue to cause problems for many students.

 
 

  
b.i.  

 

b.ii.  

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.

 

GRAPHS, FUR2 2007 VCAA 2

The Goldsmiths’ car can use either petrol or gas. 

The following equation models the fuel usage of petrol, , in litres per 100 km (L/100 km) when the car is travelling at an average speed of km/h.

The line    is drawn on the graph below for average speeds up to 110 km/h.

GRAPHS, FUR2 2007 VCAA 2

  1. Determine how many litres of petrol the car will use to travel 100 km at an average speed of 60 km/h.

     

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

The following equation models the fuel usage of gas, , in litres per 100 km (L/100 km) when the car is travelling at an average speed of km/h.

  1. On the axes above, draw the line    for average speeds up to 110 km/h.  (1 mark)
  2. Determine the average speeds for which fuel usage of gas will be less than fuel usage of petrol.  (1 mark)

The Goldsmiths'’ car travels at an average speed of 85 km/h. It is using gas.

Gas costs 80 cents per litre.

  1. Determine the cost of the gas used to travel 100 km.

     

    Write your answer in dollars and cents.  (2 marks)

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  2.  
    GRAPHS, FUR2 2007 VCAA 2 Answer
Show Worked Solution

a.  

 

 

b.  

GRAPHS, FUR2 2007 VCAA 2 Answer

 

c.   

MARKER’S COMMENT: Since fuel usage is less for gas, a speed of 75 km/hr was incorrect, as it was equal.

 

 

d.  

 

 

Probability, MET2 2010 VCAA 2

Shoddy Ltd produces statues that are classified as Superior or Regular and are entirely made by machines, on a construction line. The quality of any one of Shoddy’s statues is independent of the quality of any of the others on its construction line. The probability that any one of Shoddy’s statues is Regular is 0.8.

Shoddy Ltd wants to ensure that the probability that it produces at least two Superior statues in a day’s production run is at least 0.9.

Calculate the minimum number of statues that Shoddy would need to produce in a day to achieve this aim.  (3 marks)

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Show Worked Solution

♦♦ Mean mark 27%.

 

Calculus, MET2 2010 VCAA 1

  1. Part of the graph of the function    is shown on the axes below

     

         

    1. Find the rule and domain of  , the inverse function of  .   (3 marks)

    2. On the set of axes above sketch the graph of  . Label the axes intercepts with their exact values.   (3 marks)

    3. Find the values of , correct to three decimal places, for which  .   (2 marks)

    4. Calculate the area enclosed by the graphs of    and  . Give your answer correct to two decimal places.   (2 marks)

  2. The diagram below shows part of the graph of the function with rule
  3.         , where , and are real constants.
     

    • The graph has a vertical asymptote with equation  .
    • The graph has a y-axis intercept at 1.
    • The point on the graph has coordinates  , where is another real constant.
       

      VCAA 2010 1b

    1. State the value of .   (1 mark)

    2. Find the value of .   (1 mark)

    3. Show that  .   (2 marks)

    4. Show that the gradient of the tangent to the graph of at the point is  .   (1 mark)

    5. If the point    lies on the tangent referred to in part b.iv., find the exact value of .   (2 marks)

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  1.   i.
  2.  ii. 
     
  3. iii.
  4. iv. ²
  5.   i.
  6.  ii.
  7. iii.
  8.  iv.
  9.   v.
Show Worked Solution

a.i.  

 

ii.  

 

 iii. 

 

 

  iv.  
    ²

 

b.i.  

 

  ii.  

 

iii.  

 

  iv.  
 
   
   

 

  v.  

♦♦ Mean mark 33%.
MARKER’S COMMENT: Many students worked out the equation of the tangent which was unnecessary and time consuming.

 

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.  

 

   

  

CORE*, FUR2 2008 VCAA 4

Michelle intends to keep a car purchased for $17 000 for 15 years. At the end of this time its value will be $3500.

  1. By what amount, in dollars, would the car’s value depreciate annually if Michelle used the flat rate method of depreciation?   (1 mark)

  2. Determine the annual flat rate of depreciation correct to one decimal place.   (1 mark) 

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Show Worked Solution
a.   
   

 

b.  

CORE*, FUR2 2008 VCAA 3

Michelle purchased a $17 000 car. The car’s value depreciates at the rate of 10% per annum using the reducing balance method.

  1. By what amount, in dollars, does the car’s value depreciate during Michelle’s third year of ownership?   (2 marks)

  2. After how many years of ownership will the car’s value first be below $7000?   (1 mark) 
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Show Worked Solution

a.  

  


  

b.  

  

CORE*, FUR2 2008 VCAA 2

Michelle decided to invest some of her money at a higher interest rate. She deposited $3000 in an account paying 8.2% per annum, compounding half yearly.

  1. Write down an expression involving the compound interest formula that can be used to find the value of Michelle’s $3000 investment at the end of two years. Find this value correct to the nearest cent.   (2 marks)

  2. How much interest will the $3000 investment earn over a four-year period?

     

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

Show Answers Only
Show Worked Solution

a.  

 
 
 

 

b.  

MARKER’S COMMENT: A TVM calculator could also be used to solve this question.
 

 

 
 

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)

Show Answers Only
Show Worked Solution

a.  

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

  

  
b.  


 

  
c.  

 

CORE*, FUR2 2009 VCAA 4

The golf club management purchased new lawn mowers for $22 000.

  1. Use the flat rate depreciation method with a depreciation rate of 12% per annum to find the depreciated value of the lawn mowers after four years.   (2 marks)

  2. Use the reducing balance depreciation method with a depreciation rate of 16% per annum to calculate the depreciated value of the lawn mowers after four years. Write your answer in dollars correct to the nearest cent.   (1 mark)

  3. After 4 years, which method, flat rate depreciation or reducing balance depreciation, will give the greater depreciation? Write down the greater depreciation amount in dollars correct to the nearest cent.   (1 mark)

Show Answers Only
Show Worked Solution

a.  

 

 


 

b.   

 
 
 

 

c.  

CORE*, FUR2 2009 VCAA 3

The golf club’s social committee has $3400 invested in an account which pays interest at the rate of 4.4% per annum compounding quarterly.

  1. Show that the interest rate per quarter is 1.1%.   (1 mark)

  2. Determine the value of the $3400 investment after three years.

     

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

  3. Calculate the interest the $3400 investment will earn over six years.

     

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

Show Answers Only
Show Worked Solution

a.  


 

b.   

 
 
 

 

c.  

 

 

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)

Show Answers Only

  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)

Show Answers Only


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.   
 
 
   

CORE*, FUR2 2010 VCAA 2

$360 000 is invested in a perpetuity at an interest rate of 5.2% per annum.

  1. Find the monthly payment that the perpetuity provides.   (1 mark)

  2. After six years of monthly payments, how much money remains invested in the perpetuity?   (1 mark)

Show Answers Only
Show Worked Solution

a.  

♦♦ Part (b) was poorly answered.
MARKER’S COMMENT: Many students did not understand the concept of a perpetuity.
  

b.   

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) 
Show Answers Only

  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 3

A concrete square pyramid with volume 1.8 m³ sits on the flat top of the hill.

The length of the square base of the pyramid is metres. The height of the pyramid, , is 2.5 metres.

GEOMETRY, FUR2 2010 VCAA 3

Find the value of , correct to two decimal places.  (2 marks)

 

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Show Worked Solution
♦ Mean mark 41%.
 
 

 

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)

Show Answers Only

  1. i. 
  2. ii. 

Show Worked Solution

a.i.     


  

a.ii.   

 


 

b.  

 

CORE*, FUR2 2011 VCAA 3

Tania purchased a house for $300 000.

She will have to pay stamp duty based on this purchase price.

Stamp duty rates are listed in the table below. 

     BUSINESS, FUR2 2011 VCAA 3
 

  1. Calculate the amount of stamp duty that Tania will have to pay.   (1 mark)

  2. Assuming that her house will increase in value at a rate of 3.17% per annum, what will the value of Tania's house be after 5 years?

     

    Write your answer to the nearest thousand dollars.   (1 mark)

Tania bought her house at the start of 2011.

  1. If the rate of increase in value remains at 3.17% per annum, at the start of which year will the value of Tania's house first exceed $450 000?   (2 marks)

Show Answers Only
Show Worked Solution
a.   
   
♦♦ Part (a) was “poorly answered” although exact data is unavailable.
MARKER’S COMMENT: A majority of students did not understand how to use the table.

 

b.  

 
 

 

c.   

 

MARKER’S COMMENT: Many students who correctly found lost a mark by failing to identify the exact year. 

CORE*, FUR2 2011 VCAA 2

Tom and Patty both decided to invest some money from their savings.

Each chose a different investment strategy.

Tom's investment strategy

•  Deposit $5600 into an account with an interest rate of 7.2% per annum, compounding monthly.

•  Immediately after interest is paid into his investment account on the last day of each month, deposit a further $200 into the account.

  1. Determine the total amount in Tom's investment account at the end of the first month.   (1 mark)

Patty's investment strategy

•  Invest $8000 at the start of the year at an interest rate of 7.2% per annum, compounding annually.

  1. The following expression can be used to determine the value of Patty's investment at the end of the first year. Complete the expression by filling in the box.  (1 mark)

BUSINESS, FUR2 2011 VCAA 2

At the end of twelve months, Patty has more money in her investment account than Tom.

  1. How much more does she have?
  2. Write your answer to the nearest cent.  (2 marks)

  3. What annual compounding rate of interest would Patty need in order to earn $1000 interest in one year on her $8000 investment?
  4. Write your answer correct to one decimal place.  (1 mark)

Show Answers Only

Show Worked Solution

a.  


  

b.  


 

c.   

 

 

MARKER’S COMMENT: Many students valued Patty’s investment correctly but then deducted Tom’s after 1 month (from part a), instead of 12.


 

 

d.   
 
 
   

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)

Show Answers Only
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)

Show Answers Only

  3. i. 
  4. ii. 

Show Worked Solution

a.   

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


  

b.  

  


  

c.i.  


  

c.ii.  

CORE*, FUR2 2012 VCAA 2

The value of the equipment will be depreciated using the unit cost method. 

The initial value of the equipment is $8360. It will depreciate by 22 cents per hour of use. 

On average, the equipment will be used for 3800 hours each year.

  1. Calculate the depreciated value of the equipment after three years.   (1 mark)

  2. Show that, in any one year, the flat rate method of depreciation with a depreciation rate of 10% per annum will give the same annual depreciation as the unit cost method.   (1 mark)

  3. After how many years will equipment be written off with a depreciated value of $0?   (1 mark)

  4. Suppose the reducing balance method is used to depreciate the equipment instead of the unit cost method.

     

    The initial value of the equipment is $8360. It will depreciate at a rate of 14% per annum of the reducing balance.

     

    Find, correct to the nearest dollar, the depreciated value of the equipment after ten years.   (1 mark)

Show Answers Only
Show Worked Solution

a.  


  

b.  


  

c.  

  
d.  

 

  

 
 

NETWORKS, FUR1 2008 VCAA 8-9 MC

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

 

networks-fur1-2008-vcaa-8-mc

Part 1

The earliest start time for Activity , in days, is

A.     7

B.   15

C.   16

D.   19

E.   20

 

Part 2

This project currently has one critical path.

A second critical path, in addition to the first, would be created by

A.   increasing the completion time of by 7 days.

B.   increasing the completion time of by 1 day.

C.   increasing the completion time of by 2 days.

D.   decreasing the completion time of by 1 day.

E.   decreasing the completion time of by 2 days.

Show Answers Only

Show Worked Solution

 

♦♦ Mean mark of Part 2 was 35%.

NETWORKS, FUR1 2008 VCAA 6 MC

networks-fur1-2008-vcaa-6-mc

 
For the graph above, the capacity of the cut shown is

A.   

B.   

C.   

D.   

E.   

Show Answers Only

Show Worked Solution

♦♦ Mean mark 35%.
MARKER’S COMMENT: For an individual flow to contribute to the “cut”, it must flow from the source to the sink.

 

NETWORKS, FUR1 2008 VCAA 5 MC

A connected planar graph has five vertices, , , , and .

The degree of each vertex is given in the following table.
 

networks-fur1-2008-vcaa-5-mc
 

Which one of the following statements regarding this planar graph is true?

A.   The sum of degrees of the vertices equals 15.

B.   It contains more than one Eulerian path.

C.   It contains an Eulerian circuit.

D.   Euler’s formula    could not be used.

E.   The addition of one further edge could create an Eulerian path.

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Show Worked Solution

♦ Mean mark 41%.